STATISTICAL MECHANICS NOTES

Padobedaqve

HHEIRMODYNATUIIE sect

A lot can be denied from general properties of the entropy In

fact if The entropy as a function of the so called extensive variables

is known everything else about the Thermodynamic behavior of the

system can be found

Bottyouggnns D

entropy

-DS = KB ln P

s

instead of P ( phaserArrdoue occupied by the mioocauonirl ensemble )iBoltzmann constant

IT s here for historicalThere will be

reasons

If we measure temperature some refinementsin units of energy ( as we should ) Then 193=1 on This definition

Otherwise later

KB x 138times15 Jq

lt 86 X is eV(degK

A good Thing To reweubeu is

10000 degK plusmn ev

We now list a few properties satisfied by The entropy

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S as a function of The extensive variables contains all the-

info you need

y

extensive quantities arte ones That double

if we double The system for instance

gas wl one molecule type E U N

gas wl several kinds of molecules E

U N Nz

magnet E IN ( magnetic moment )

dielectric E P ( electric dipole moment )elastic medium E

Eij ( strain Tensor )rubber band E

L ( length )black body E

V

Temperature pressure external magnetic fields

are

intensity quantities ( two identical

blocks of ice next To each other have

Twice the volume energy of one block 150T

the same Temperature )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is extensive

S ( d E d V

d N ) = d SCEN N )

T follows fromp =p Tz neglecting

surface effects

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 fg 30 ( we ll see That This means TZO )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 SCE V N ) is

concave

SCEHSE VHJV

N tan ) + SCE - DE VIV

N DN )

f

2 SCE V N )

Ss

- SCE )scEtorArr

- - - - -

bullset - iffsees t

- - - - amp

if SCETSE ) + SCE D t )fixE- DE

E

ETDE

E

( we ll see later that this

implies stability )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 5=0 at I =o ( T=o )24oE

( This property depends on

how we normalize the number of microstates T

quantummechanics determines The night

normalization That implies SCTTDTO

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831676983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is maximized ( within the constraints set by the physicalset up ) in equilibrium

( This is a direct consequence thatall microstates are equally probable )

Inverting SCEN N ) we get ECS U micro )

If SN Nare changed slowly ( adiabatic ally

) so the system stays in

equilibrium at all intermediate states

DE = 0yen ds +0yen dv + golf DN

-

tin

-lav

-

ten

T - P micro ( chemical potential

-

-

= energy required

heat put into work done

to add one partThe system yen The energy ofthe

system particle added

similar

=

stet

de

tstvkY+yenoyen

PIT

- MT

we used

I

f y1

8Th micro= of we used

Tolan

Eopoundlsifsplusmnh )

yenhyeneyenyenhu

larrayenyendivideseHrArrtlArr s

Its important to notice that S is defined for equilibriummacroscopic states only

So des or DE refer To changes slow

enough such that the system remains in equilibrium at allintermediate stages

For instance if The value is changed

suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be

larger ) Move insight on this will come later when we

discuss the quantum use and the canonical ensemble

equztiousof The relations

state divide

offensive

F=yenetmicroinedividemicro tP=

3eurofmicro=

PCSIYN)

larr

yen=ofyenmicro= - PENyen

TENN )

micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs

of state

They are frequently accessible to

experiments Notice Thet the same quantities as function of other

variables are nyen eqs of state and may not contain The same

information

As mentioned before all Thermo dynamical problems can be reduced To

before

~ after

extentFEyenttsot

Eicui Ni Eicuiirz

ELVORN

A little thought implies That

wall

allows

energy

exchange IT

= Tzl

wall allows volume exchange V

= vi

wall allow particle exchange micro

= microi

Non - relativistic non

interact

monatomic gas

Finally let us compute S for one example

Acri pi ) = i

yin

( oc

rick)

PCEMN ) = SHEdri dpi SCE - Eakin )

= microtimes ]

sectdegd Pixdpiydpz dpmicrotimesdpmicroydP

zm Scant yenYpi2 )

= VN on 53CR=zmE=

area of a 3N -dimensional

spherical hyper surface wl radius

R=zmEM

N 3Nz= Zm lArr

a yen

S = ka hnp = kBlnV + 3yen bn E

- but ( 3142 ) +312 bytangymicro Stirling approx

Inn

micro Nlnw +

microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]

312= Nka In Vmicroldivide

The expression above violates postulates ( extensively ) and

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising

the behavior of a gas at To is dictated by quantum mechanics

not classical mechanics ( The atom s wavelength is much larger ku

the separation between them )

what is surprising is that

quantum mechanics carrots the calculation above even at target

What is missing from The classical calculation above is that

the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )

So The correct way of

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

HHEIRMODYNATUIIE sect

A lot can be denied from general properties of the entropy In

fact if The entropy as a function of the so called extensive variables

is known everything else about the Thermodynamic behavior of the

system can be found

Bottyouggnns D

entropy

-DS = KB ln P

s

instead of P ( phaserArrdoue occupied by the mioocauonirl ensemble )iBoltzmann constant

IT s here for historicalThere will be

reasons

If we measure temperature some refinementsin units of energy ( as we should ) Then 193=1 on This definition

Otherwise later

KB x 138times15 Jq

lt 86 X is eV(degK

A good Thing To reweubeu is

10000 degK plusmn ev

We now list a few properties satisfied by The entropy

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S as a function of The extensive variables contains all the-

info you need

y

extensive quantities arte ones That double

if we double The system for instance

gas wl one molecule type E U N

gas wl several kinds of molecules E

U N Nz

magnet E IN ( magnetic moment )

dielectric E P ( electric dipole moment )elastic medium E

Eij ( strain Tensor )rubber band E

L ( length )black body E

V

Temperature pressure external magnetic fields

are

intensity quantities ( two identical

blocks of ice next To each other have

Twice the volume energy of one block 150T

the same Temperature )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is extensive

S ( d E d V

d N ) = d SCEN N )

T follows fromp =p Tz neglecting

surface effects

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 fg 30 ( we ll see That This means TZO )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 SCE V N ) is

concave

SCEHSE VHJV

N tan ) + SCE - DE VIV

N DN )

f

2 SCE V N )

Ss

- SCE )scEtorArr

- - - - -

bullset - iffsees t

- - - - amp

if SCETSE ) + SCE D t )fixE- DE

E

ETDE

E

( we ll see later that this

implies stability )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 5=0 at I =o ( T=o )24oE

( This property depends on

how we normalize the number of microstates T

quantummechanics determines The night

normalization That implies SCTTDTO

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831676983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is maximized ( within the constraints set by the physicalset up ) in equilibrium

( This is a direct consequence thatall microstates are equally probable )

Inverting SCEN N ) we get ECS U micro )

If SN Nare changed slowly ( adiabatic ally

) so the system stays in

equilibrium at all intermediate states

DE = 0yen ds +0yen dv + golf DN

-

tin

-lav

-

ten

T - P micro ( chemical potential

-

-

= energy required

heat put into work done

to add one partThe system yen The energy ofthe

system particle added

similar

=

stet

de

tstvkY+yenoyen

PIT

- MT

we used

I

f y1

8Th micro= of we used

Tolan

Eopoundlsifsplusmnh )

yenhyeneyenyenhu

larrayenyendivideseHrArrtlArr s

Its important to notice that S is defined for equilibriummacroscopic states only

So des or DE refer To changes slow

enough such that the system remains in equilibrium at allintermediate stages

For instance if The value is changed

suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be

larger ) Move insight on this will come later when we

discuss the quantum use and the canonical ensemble

equztiousof The relations

state divide

offensive

F=yenetmicroinedividemicro tP=

3eurofmicro=

PCSIYN)

larr

yen=ofyenmicro= - PENyen

TENN )

micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs

of state

They are frequently accessible to

experiments Notice Thet the same quantities as function of other

variables are nyen eqs of state and may not contain The same

information

As mentioned before all Thermo dynamical problems can be reduced To

before

~ after

extentFEyenttsot

Eicui Ni Eicuiirz

ELVORN

A little thought implies That

wall

allows

energy

exchange IT

= Tzl

wall allows volume exchange V

= vi

wall allow particle exchange micro

= microi

Non - relativistic non

interact

monatomic gas

Finally let us compute S for one example

Acri pi ) = i

yin

( oc

rick)

PCEMN ) = SHEdri dpi SCE - Eakin )

= microtimes ]

sectdegd Pixdpiydpz dpmicrotimesdpmicroydP

zm Scant yenYpi2 )

= VN on 53CR=zmE=

area of a 3N -dimensional

spherical hyper surface wl radius

R=zmEM

N 3Nz= Zm lArr

a yen

S = ka hnp = kBlnV + 3yen bn E

- but ( 3142 ) +312 bytangymicro Stirling approx

Inn

micro Nlnw +

microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]

312= Nka In Vmicroldivide

The expression above violates postulates ( extensively ) and

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising

the behavior of a gas at To is dictated by quantum mechanics

not classical mechanics ( The atom s wavelength is much larger ku

the separation between them )

what is surprising is that

quantum mechanics carrots the calculation above even at target

What is missing from The classical calculation above is that

the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )

So The correct way of

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

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9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

We now list a few properties satisfied by The entropy

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S as a function of The extensive variables contains all the-

info you need

y

extensive quantities arte ones That double

if we double The system for instance

gas wl one molecule type E U N

gas wl several kinds of molecules E

U N Nz

magnet E IN ( magnetic moment )

dielectric E P ( electric dipole moment )elastic medium E

Eij ( strain Tensor )rubber band E

L ( length )black body E

V

Temperature pressure external magnetic fields

are

intensity quantities ( two identical

blocks of ice next To each other have

Twice the volume energy of one block 150T

the same Temperature )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is extensive

S ( d E d V

d N ) = d SCEN N )

T follows fromp =p Tz neglecting

surface effects

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 fg 30 ( we ll see That This means TZO )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 SCE V N ) is

concave

SCEHSE VHJV

N tan ) + SCE - DE VIV

N DN )

f

2 SCE V N )

Ss

- SCE )scEtorArr

- - - - -

bullset - iffsees t

- - - - amp

if SCETSE ) + SCE D t )fixE- DE

E

ETDE

E

( we ll see later that this

implies stability )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 5=0 at I =o ( T=o )24oE

( This property depends on

how we normalize the number of microstates T

quantummechanics determines The night

normalization That implies SCTTDTO

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831676983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is maximized ( within the constraints set by the physicalset up ) in equilibrium

( This is a direct consequence thatall microstates are equally probable )

Inverting SCEN N ) we get ECS U micro )

If SN Nare changed slowly ( adiabatic ally

) so the system stays in

equilibrium at all intermediate states

DE = 0yen ds +0yen dv + golf DN

-

tin

-lav

-

ten

T - P micro ( chemical potential

-

-

= energy required

heat put into work done

to add one partThe system yen The energy ofthe

system particle added

similar

=

stet

de

tstvkY+yenoyen

PIT

- MT

we used

I

f y1

8Th micro= of we used

Tolan

Eopoundlsifsplusmnh )

yenhyeneyenyenhu

larrayenyendivideseHrArrtlArr s

Its important to notice that S is defined for equilibriummacroscopic states only

So des or DE refer To changes slow

enough such that the system remains in equilibrium at allintermediate stages

For instance if The value is changed

suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be

larger ) Move insight on this will come later when we

discuss the quantum use and the canonical ensemble

equztiousof The relations

state divide

offensive

F=yenetmicroinedividemicro tP=

3eurofmicro=

PCSIYN)

larr

yen=ofyenmicro= - PENyen

TENN )

micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs

of state

They are frequently accessible to

experiments Notice Thet the same quantities as function of other

variables are nyen eqs of state and may not contain The same

information

As mentioned before all Thermo dynamical problems can be reduced To

before

~ after

extentFEyenttsot

Eicui Ni Eicuiirz

ELVORN

A little thought implies That

wall

allows

energy

exchange IT

= Tzl

wall allows volume exchange V

= vi

wall allow particle exchange micro

= microi

Non - relativistic non

interact

monatomic gas

Finally let us compute S for one example

Acri pi ) = i

yin

( oc

rick)

PCEMN ) = SHEdri dpi SCE - Eakin )

= microtimes ]

sectdegd Pixdpiydpz dpmicrotimesdpmicroydP

zm Scant yenYpi2 )

= VN on 53CR=zmE=

area of a 3N -dimensional

spherical hyper surface wl radius

R=zmEM

N 3Nz= Zm lArr

a yen

S = ka hnp = kBlnV + 3yen bn E

- but ( 3142 ) +312 bytangymicro Stirling approx

Inn

micro Nlnw +

microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]

312= Nka In Vmicroldivide

The expression above violates postulates ( extensively ) and

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising

the behavior of a gas at To is dictated by quantum mechanics

not classical mechanics ( The atom s wavelength is much larger ku

the separation between them )

what is surprising is that

quantum mechanics carrots the calculation above even at target

What is missing from The classical calculation above is that

the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )

So The correct way of

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 fg 30 ( we ll see That This means TZO )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 SCE V N ) is

concave

SCEHSE VHJV

N tan ) + SCE - DE VIV

N DN )

f

2 SCE V N )

Ss

- SCE )scEtorArr

- - - - -

bullset - iffsees t

- - - - amp

if SCETSE ) + SCE D t )fixE- DE

E

ETDE

E

( we ll see later that this

implies stability )

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 5=0 at I =o ( T=o )24oE

( This property depends on

how we normalize the number of microstates T

quantummechanics determines The night

normalization That implies SCTTDTO

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831676983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 S is maximized ( within the constraints set by the physicalset up ) in equilibrium

( This is a direct consequence thatall microstates are equally probable )

Inverting SCEN N ) we get ECS U micro )

If SN Nare changed slowly ( adiabatic ally

) so the system stays in

equilibrium at all intermediate states

DE = 0yen ds +0yen dv + golf DN

-

tin

-lav

-

ten

T - P micro ( chemical potential

-

-

= energy required

heat put into work done

to add one partThe system yen The energy ofthe

system particle added

similar

=

stet

de

tstvkY+yenoyen

PIT

- MT

we used

I

f y1

8Th micro= of we used

Tolan

Eopoundlsifsplusmnh )

yenhyeneyenyenhu

larrayenyendivideseHrArrtlArr s

Its important to notice that S is defined for equilibriummacroscopic states only

So des or DE refer To changes slow

enough such that the system remains in equilibrium at allintermediate stages

For instance if The value is changed

suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be

larger ) Move insight on this will come later when we

discuss the quantum use and the canonical ensemble

equztiousof The relations

state divide

offensive

F=yenetmicroinedividemicro tP=

3eurofmicro=

PCSIYN)

larr

yen=ofyenmicro= - PENyen

TENN )

micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs

of state

They are frequently accessible to

experiments Notice Thet the same quantities as function of other

variables are nyen eqs of state and may not contain The same

information

As mentioned before all Thermo dynamical problems can be reduced To

before

~ after

extentFEyenttsot

Eicui Ni Eicuiirz

ELVORN

A little thought implies That

wall

allows

energy

exchange IT

= Tzl

wall allows volume exchange V

= vi

wall allow particle exchange micro

= microi

Non - relativistic non

interact

monatomic gas

Finally let us compute S for one example

Acri pi ) = i

yin

( oc

rick)

PCEMN ) = SHEdri dpi SCE - Eakin )

= microtimes ]

sectdegd Pixdpiydpz dpmicrotimesdpmicroydP

zm Scant yenYpi2 )

= VN on 53CR=zmE=

area of a 3N -dimensional

spherical hyper surface wl radius

R=zmEM

N 3Nz= Zm lArr

a yen

S = ka hnp = kBlnV + 3yen bn E

- but ( 3142 ) +312 bytangymicro Stirling approx

Inn

micro Nlnw +

microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]

312= Nka In Vmicroldivide

The expression above violates postulates ( extensively ) and

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising

the behavior of a gas at To is dictated by quantum mechanics

not classical mechanics ( The atom s wavelength is much larger ku

the separation between them )

what is surprising is that

quantum mechanics carrots the calculation above even at target

What is missing from The classical calculation above is that

the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )

So The correct way of

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

Inverting SCEN N ) we get ECS U micro )

If SN Nare changed slowly ( adiabatic ally

) so the system stays in

equilibrium at all intermediate states

DE = 0yen ds +0yen dv + golf DN

-

tin

-lav

-

ten

T - P micro ( chemical potential

-

-

= energy required

heat put into work done

to add one partThe system yen The energy ofthe

system particle added

similar

=

stet

de

tstvkY+yenoyen

PIT

- MT

we used

I

f y1

8Th micro= of we used

Tolan

Eopoundlsifsplusmnh )

yenhyeneyenyenhu

larrayenyendivideseHrArrtlArr s

Its important to notice that S is defined for equilibriummacroscopic states only

So des or DE refer To changes slow

enough such that the system remains in equilibrium at allintermediate stages

For instance if The value is changed

suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be

larger ) Move insight on this will come later when we

discuss the quantum use and the canonical ensemble

equztiousof The relations

state divide

offensive

F=yenetmicroinedividemicro tP=

3eurofmicro=

PCSIYN)

larr

yen=ofyenmicro= - PENyen

TENN )

micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs

of state

They are frequently accessible to

experiments Notice Thet the same quantities as function of other

variables are nyen eqs of state and may not contain The same

information

As mentioned before all Thermo dynamical problems can be reduced To

before

~ after

extentFEyenttsot

Eicui Ni Eicuiirz

ELVORN

A little thought implies That

wall

allows

energy

exchange IT

= Tzl

wall allows volume exchange V

= vi

wall allow particle exchange micro

= microi

Non - relativistic non

interact

monatomic gas

Finally let us compute S for one example

Acri pi ) = i

yin

( oc

rick)

PCEMN ) = SHEdri dpi SCE - Eakin )

= microtimes ]

sectdegd Pixdpiydpz dpmicrotimesdpmicroydP

zm Scant yenYpi2 )

= VN on 53CR=zmE=

area of a 3N -dimensional

spherical hyper surface wl radius

R=zmEM

N 3Nz= Zm lArr

a yen

S = ka hnp = kBlnV + 3yen bn E

- but ( 3142 ) +312 bytangymicro Stirling approx

Inn

micro Nlnw +

microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]

312= Nka In Vmicroldivide

The expression above violates postulates ( extensively ) and

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising

the behavior of a gas at To is dictated by quantum mechanics

not classical mechanics ( The atom s wavelength is much larger ku

the separation between them )

what is surprising is that

quantum mechanics carrots the calculation above even at target

What is missing from The classical calculation above is that

the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )

So The correct way of

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

Its important to notice that S is defined for equilibriummacroscopic states only

So des or DE refer To changes slow

enough such that the system remains in equilibrium at allintermediate stages

For instance if The value is changed

suddenly and then the system is left To relax To equilibriumthe change in entropy is not given by lzdv ( will be

larger ) Move insight on this will come later when we

discuss the quantum use and the canonical ensemble

equztiousof The relations

state divide

offensive

F=yenetmicroinedividemicro tP=

3eurofmicro=

PCSIYN)

larr

yen=ofyenmicro= - PENyen

TENN )

micro=oyenw Msn yen=yenkw=Iyenedivideiare called eqs

of state

They are frequently accessible to

experiments Notice Thet the same quantities as function of other

variables are nyen eqs of state and may not contain The same

information

As mentioned before all Thermo dynamical problems can be reduced To

before

~ after

extentFEyenttsot

Eicui Ni Eicuiirz

ELVORN

A little thought implies That

wall

allows

energy

exchange IT

= Tzl

wall allows volume exchange V

= vi

wall allow particle exchange micro

= microi

Non - relativistic non

interact

monatomic gas

Finally let us compute S for one example

Acri pi ) = i

yin

( oc

rick)

PCEMN ) = SHEdri dpi SCE - Eakin )

= microtimes ]

sectdegd Pixdpiydpz dpmicrotimesdpmicroydP

zm Scant yenYpi2 )

= VN on 53CR=zmE=

area of a 3N -dimensional

spherical hyper surface wl radius

R=zmEM

N 3Nz= Zm lArr

a yen

S = ka hnp = kBlnV + 3yen bn E

- but ( 3142 ) +312 bytangymicro Stirling approx

Inn

micro Nlnw +

microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]

312= Nka In Vmicroldivide

The expression above violates postulates ( extensively ) and

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising

the behavior of a gas at To is dictated by quantum mechanics

not classical mechanics ( The atom s wavelength is much larger ku

the separation between them )

what is surprising is that

quantum mechanics carrots the calculation above even at target

What is missing from The classical calculation above is that

the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )

So The correct way of

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

As mentioned before all Thermo dynamical problems can be reduced To

before

~ after

extentFEyenttsot

Eicui Ni Eicuiirz

ELVORN

A little thought implies That

wall

allows

energy

exchange IT

= Tzl

wall allows volume exchange V

= vi

wall allow particle exchange micro

= microi

Non - relativistic non

interact

monatomic gas

Finally let us compute S for one example

Acri pi ) = i

yin

( oc

rick)

PCEMN ) = SHEdri dpi SCE - Eakin )

= microtimes ]

sectdegd Pixdpiydpz dpmicrotimesdpmicroydP

zm Scant yenYpi2 )

= VN on 53CR=zmE=

area of a 3N -dimensional

spherical hyper surface wl radius

R=zmEM

N 3Nz= Zm lArr

a yen

S = ka hnp = kBlnV + 3yen bn E

- but ( 3142 ) +312 bytangymicro Stirling approx

Inn

micro Nlnw +

microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]

312= Nka In Vmicroldivide

The expression above violates postulates ( extensively ) and

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising

the behavior of a gas at To is dictated by quantum mechanics

not classical mechanics ( The atom s wavelength is much larger ku

the separation between them )

what is surprising is that

quantum mechanics carrots the calculation above even at target

What is missing from The classical calculation above is that

the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )

So The correct way of

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

= VN on 53CR=zmE=

area of a 3N -dimensional

spherical hyper surface wl radius

R=zmEM

N 3Nz= Zm lArr

a yen

S = ka hnp = kBlnV + 3yen bn E

- but ( 3142 ) +312 bytangymicro Stirling approx

Inn

micro Nlnw +

microrarrorarro KB [ N KV +3yen In E - 3yen KN + IN hit ]

312= Nka In Vmicroldivide

The expression above violates postulates ( extensively ) and

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 ( Nevis postulate ) The violation of 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831675983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 is not surprising

the behavior of a gas at To is dictated by quantum mechanics

not classical mechanics ( The atom s wavelength is much larger ku

the separation between them )

what is surprising is that

quantum mechanics carrots the calculation above even at target

What is missing from The classical calculation above is that

the gas is composed of identical particles ( bosons or fermions=it doesnt matter at high T )

So The correct way of

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

counting states is

S = Kalp = ka fETyd4divideg SCE - EE

piyzm )

N

D

avoids double

euro This factor wont mattercounting of microstates now but we ll see its The

with the same particle correct one when we consider

Positions

the classical limit of quantum gases

32=D KB N h

micro - DOO NFZ

From this entropy we can find the eqs of state

t =

feurotfmicro=3yenDyen or E

= 3=143 NT

- Tf

- sectmicro

= -

kBdivide or PV = KBN T

et = If

micro

= shumicroyenh - Eke

Notice That we found The Boyle - Marriott law wo even

discussing Fans colliding against walls or other Kinetic toyconsiderations

Theres a lot behind The assumption of evgodicity

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

Euler relation SC dE dv t N ) = d SCE

v N )

yen he

E

Ee + v 0pound +nosectmicro = s

in

in

-

YT - PIT divide

E

= TS - PV + p N

or

S

- Ee - Ny tiff

Recovering The entropy

-energy ) from edivideIf 3 e as are available

TCSNIN ) = 3yen|vmicro- P ( Siu N ) - zEugmicro yenf

EGYM = TQYMS - PCS YN)Vtmicro( gu micro ) N

micro ( SNID

kTulsmicro

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

If only 2 eds

are available ECSN N ) = N

E ( Shu YN i )

k extensity

T TTL microi TCSN M= TCYN YN i )

f-no N

T

and P are intensive

r-

Pizeurofgmicro - PCSW N)=tPC9microMmicro I )

sw

E(9microYmicro )

= fd(4micro ) TCSINMMI ) + EC 04N i )

deg

T Ebslvav

IN

= sectd(4micro ) Tcsln YN ) -

sectDCYN ) PC9Ymicro i ) teaECSINYIN l )

can be found up to one

constant independent ofSN or N

-

If only 1 e as

cant be done

i available

Example non - rd

monoztomic non

interacting gasTKBN

s= ken hVEplusmn E

= ( duke 4=13 eyenaNNSTZ 613

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

+ = yenhmicroidividendivideeyenN- P -

fuelsmicro

= yen5 e 243

Ec sauna n=

Esau) divideydivideP

eyenplusmn- Plan

tea at

= plusmn

euro53 rArrdivide[eyenn - if - dividetrArrI3kt 83g

e

yenn -

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

Mining EnergyPrinciple

divide s start with an analogy

A circle is the figure with the largestarea for a given perimeter

Of course its also The one with

the smallest perimeter for a given area Even if a given

circle was built by someone with a fixed length string tryingto enclose the largest area it will still be True that The

final circle minimizes The perimeter for a given area

The same Thing is True for The maximum entropy principleAfter some constraints are changed the final state will be The

one with The largest entropy for a given +0yen energy C the

constraint That the Total energy is unchanged is always there )

But The final equilibrium state can also be characterized as

the state with the smallest energy for a fixed entropy

S

maximum S

Sdeg

ysorfiteaesqImTnYe3EsfixedetgpKedstqpnE E

E

E

Convince yourself that 5 as a function of total E and

subsystem energy E

has the general shape depicted above

The minimum energy principle follows from staring at the pictureabove

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

Let us see how

the minimum energy principle works in Our

archetypical example

diathermy wall

( energy transfer allowed )actuality

yen pretend

-

-

=

- -

lArrrArr M lArr

E=EtEz fixed

rArr- Sitszfixed

S changes

- E changes

-

+- t

E = Ee ( Si v micro ) +Ez(

IUyNr )

ss

minimum E

forfixeds fldivide=o if oldivide

- Itf =o rate as expected )

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

Efficiency of engines

An engine is a machine That Takes energy from some hot

Source ( a furnace say ) uses some of This energy to do work

and eject The remaining energy if any To a colder sink ( usuallyThe atmosphere

We would like To do This over and over again so

The machine should go to a closed cycle That is end up in

the same macroscopic state as it started

For now we will assume

that all The changes occur slowly ( adiabatic ally ) so at every pointin The cycle The machine is in equilibrium and its macroscopic

state is specified by EN N or SN N

IT turns out That its

better To use instead The variables ITS N

Keep N fixed by

having impermeable walls separating The machine from The rest ofthe Universe

A cycle can Then be represented by a closed loopin the its

stages aD

s

Let us stat by considering one specific cycle The G=T cycle

T

t

deg 0yen

- -

ideal gas

toLItaE inside

Ds

Md LEE

hot

coldsource

sdividethe piston is exposed to

five or ice and its motionmoves a wheel

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831671983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas in contact w hot source

absorbs heat

To and expands keeping T=Th= cast

In - 9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831672983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gas isolated from siuklsoovcl expands

+ 40euro40 wlo absorbing heat cools down and does work

c

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831673983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167 gzs in contact w| cold sink Shrinks

rArr z

5 and Transfers heat tote cold sink

9830409830419830429830439830449830459830469830479830489830499830509830519830529830539830549830559830569830579830589830599830609830619830629830639830649830659830669830679830689830699830709830719830729830739830749830759830769830779830789830799830809830819830829830839830849830859830869830879830889830899830909830919830929830939830949830959830969830979830989830999831009831019831029831039831049831059831069831079831089831099831109831119831129831139831149831159831169831179831189831199831209831219831229831239831249831259831269831279831289831299831309831319831329831339831349831359831369831379831389831399831409831419831429831439831449831459831469831479831489831499831509831519831529831539831549831559831569831579831589831599831609831619831629831639831649831659831669831674983040983041983042983043983044983045983046983047983048983049983050983051983052983053983054983055983056983057983058983059983060983061983062983063983064983065983066983067983068983069983070983071983072983073983074983075983076983077983078983079983080983081983082983083983084983085983086983087983088983089983090983091983092983093983094983095983096983097983098983099983100983101983102983103983104983105983106983107983108983109983110983111983112983113983114983115983116983117983118983119983120983121983122983123983124983125983126983127983128983129983130983131983132983133983134983135983136983137983138983139983140983141983142983143983144983145983146983147983148983149983150983151983152983153983154983155983156983157983158983159983160983161983162983163983164983165983166983167

gas isolated from sink ) source contracts

and hzs work dove on it

Forfeited N DE - T ds - PDU so

Sz

DQ

= STDS =

Th ( Sz - S )

T

DQ

si

divide n n

DQ ftds

= - I

Csz - S )~gt

T

sectDDW1

Szt-

J

D

S

DQZ Sz

S

Since after a full cycle The energy of the system goes back

to what it was in The beginning

DQI + DQZ = DW

= area inside the cycle

T heat work done C The Txs plane)

T

absorbed yen The system

A - _

Dwi t Dwz

t tc

-

i

S

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

For practical reasons wed like To

get as much work as possible

from a given amount of heat absorbed ( $15 ) The efficiency

of The engine is defined as

e = I

= Ethos - l -

yen ( l

DQ In (52-9)

The efficiency E is smaller Than one because some of the energyis wasted on the cold sink Can we change The cycle to obtain a

larger

efficiency

For That we would want To maximize The area

inside of The cycle ( DW in red )

and minimize TheTp area below The top come ( DQ

blue

dashedTh

- -

yendivideg - area on the figure ) Its clear

for instance

Tc - -

yenyenpound

-

that the cycle on the left can have its

13

efficiency improved by

changing it

as

shown below

We end op then

sw and DQ wl the Cannot cycle which

increases by the same is the best it can be done

p ( amount E goes up

Thus we have an universal result

In

plusmnlArrdivideg - -

+ yenyenyenFyen euro divideyenasyentFyen_

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

Notice that we only used ( implicitly actually ) The fact That

an ideal gas was used inside the engine to describe the differentphases of the cycle in words ( in Q the gas expands

)

The

main result E

( l - Tyth does not depend on that and is valid

universally

Finally let us revisit

The assumption of adiabatic it namely

that all phases of The cycle were infinitely slow

In each phase

of the cycle the machine goes from one initial state A to a

final state B exchanging a certain amount of heat DQ

( with a source at Temperature T ) and

performing

some work

DW ( positive or negative )

Ill assume The work is done on

a reversible work source namely a system which can exchange work

but not heat ( a paperweight on top of a piston is an example )

Similarlyheat is exchanged with a reversible heat source

- - - -

-

euro3

universe

I

i

Dw gto

5yen

DQ gt0

1- - - -

- - --

iFor given initial and final states A and 13 ( and consequently

given changes in energy Eg E and entropy ) The maximum amount ofwork will be delivered if

DQ is a minimum

as Ed - Ea = - COWTDQI )

The process with the smallest DQ will correspond To The smallest

possible increase of The entropy of the universe ( system + heat

source The work source was no entropy )

How much is the minimum

amount of heat To be Transferred To the heat Source As

The Total entropy of The universe must increase ( otherwise

The system would stay at A ) we have

SB - SAIIBQ 30 temperatureoftteThat

source

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

That means That The minimum amount of heat wasted on the

heat source happens when the Total entropy of the universe is

unchanged as is

given Q

=

SAI-

r

If more heat tan Syen is delivered to the heat source the universe s

entropy will increase and less work will be delivered to the work source

A similar analysis applies when The system is absorbing heat or

work The main point is That reify Those where

the entropy of The universe does not change lead To the maximum

efficiency Thus our bound on the efficiency of an engine

obtained assuming adiabatic ity is not violated by non adiabatic

processes Notice That The adiabaticity condition means that the

engine works really slowly Practical machines which need

to do Their job fast are not adiabatic and much less

efficient Than The theoretical limit

E f 1- Tatty

would suggest ( roughly a factor 3 less efficient )

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

Thermodynamic stability-

We determined

The equilibrium condition between two

subsystemsTiitz

R=Pz micro

=

microz7

to

7if

datherurl if

movable if porouswall

wall wall

using the maximum entropy principle But we only imposed That

Stsitsz was a critical point it could have been a maximum

a minimum or a saddle point

we ll now see what The

consequences are of demanding that The final equilibrium state

is indeed a maximum

Consider first The case of Two subsystems separated by a diathermyo

well

The condition That 5=4+52 is a maximum implies

allowsenergy

exchange

IT tyenoIdivide=Idivide

divide vv Ez

vuNz fyen lt 0 rArru

DI +325yen lt 0

in

E-Et The quantity 5 sloe has a physical interpretation

final stateof

equilibrium

Hel micro=poundChjyenz yenhmicro

= - Foetal microplusmnn Tspecific heat

D

at constant volk

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

The stability condition is Then

- taut

- yen at

lt o or 1- it gto

ya Mci nai

T =Tz

in equilibrium

By Taking one subsystem to be much longer we conclude that

each one of Ci and Gi should be positive This result is physically

obvious

Suppose Cv lt 0

Then itd get cdyen by absorbing

heat therefore absorbing even move heat

This is an unstable

situation

Let us consider now The ease where both E and U are

exchanged between The Subsystems we have

deg= yen

- 3yen

- 01

OEZM

o= divide = st

- IIdivide iv

N Ei

Vi Nz

in ETOgtd2s=(aeiduiy yeniyenyenui ) (fffinal state 01

plusmn

of oeiavi

ovizequilibrium

+ 1 larr

we need This To be

negative for any dehdu

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

For The second variation Is to be negative for any E

U

its

necessary That The eigeuvcwes of01

EeiyenpFoe

be negative

Let us wore out a general condition for this to

be

tv j

o

d

- catch tae 5 o

teddies

t= at

plusmn(adivideg2-ceaT

co

the eigewawesd ate lt o and ac -5 )

0

irArr a lt 0 and

c - I ( 0

a

torus at Yah micro

= EecrArrhmicro

= - ttz seekmicro=dividezmicroodividedivide

me condition

2

Cr ) 0 as

o ) c - plusmn

= Is - (zEsDyeny

beforeOV

ENotthis can be written in

a pretty way wl a TransparentIt CyenlEyenfeh)

physical meaning

-- - -

IeuroQplusmnejhiyenyen

aspyent=crArr

DV

=xtyenhyeneyenwtimestyenh

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one

z

2

= Buy + t Ile 1-

I

2

oeov

+zzeu one

so osmyent=divide stdivide[dividesdividetI=dividedivide

isothermal

compressibility

The stability condition implies That the isothermal compressibilityis positive

This is not so obvious A negative compressibility

means That the system expands

as The pressure increases

That would

increase The pressure of the other system leading to a further expansionand so on

What I dont find obvious is that The isotevwd

compressibility ( as opposed To say adiabatic compressibility ks=rArrzfs

)is the relevant one