4 -- Formal Structure of Thermodynamics

8/18/2019 4 -- Formal Structure of Thermodynamics

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MSEG 803Equilibria in Material Systems

4: Formal Structure of TD

Prof. ue!un "# $u

%u!ue!un&u'el.e'u


2/29

(st la) an' *n' la) in a sim+le system

(st la):

*n' la):

T%e functions U(S, V, N) an' S(U,V, N) are calle'

fundamental equations of a system. Eac% one of t%em

contains full information about a system.

Generally ener,y re+resentation

entro+y re+resentation

dU Q PdV δ = −

Q TdS δ =

dU TdS PdV = −1 P

dS dU dV T T

= +

i i

i

dU TdS y dx= + ∑1 i

i

i

ydS dU dx

T T = − ∑


3/29

Equations of state

T%e intensi-e -ariables in t%e fun'amental

equations )ritten as functions of t%e etensi-e

-ariables "for fie' mole numbers#:

Generally

dU TdS PdV = −

1 P dS dU dV

T T

= +

⇒

⇒

( , )T T S V = ( , ) P P S V − = −

1 1( , )S V

T T

= ( , ) P P

S V

T T

=

1 2( , ,..., ,...)i i i y y x x x=


4/29

/%emical +otential an' +artial molar quantities

/%emical +otential µ i for t%e com+onent i

uasi1static c%emical )or2

T%e +artial molar quantity x " x is an etensi-e function#

associate' )it% t%e com+onent i ")%en T P are constant#

+artial molar -olume

, ,...

i

i S V

U

N µ

∂= ÷∂ , ,...

i

i S V

S

N T

µ ∂= − ÷∂

c i i

i

W dN δ µ = ∑ i ii

dU TdS PdV dN µ = − + ∑

°

, , ( ) j

i

i T P N j i

x x

N ≠

∂= ÷∂

°

, , ( ) j

i

i T P N j i

V V

N ≠

∂= ÷∂


5/29

Euler relation

U an' S are bot% %omo,eneous first or'er

functions of etensi-e +arameters

λ is a constant

et λ = 1

Sim+le systems

1 2 1 2( , ,..., ,...) ( , ,..., ,...)i iU X X X U X X X λ λ λ λ =

1 2 1 2( , , ..., ,...) ( , ,..., ,...)i iU x x x U x x xλ λ λ λ

λ λ

∂ ∂=∂ ∂

1 2 1 2( , ,..., ,...) ( ) ( , ,..., ,...)

( ) ( )

i i ii

i ii i

U x x x x U x x xU x

x x

λ λ λ λ λ λ λ

λ λ λ

∂ ∂ ∂= × = ×

∂ ∂ ∂∑ ∑

1 2( , ,..., ,...)

( )

ii i i

i ii

U x x xU x y x

x

∂= × = ×

∂∑ ∑

i i

i

U TS PV N µ = − + ∑ 1 i ii

P S U V N

T T T

µ = + − ∑


6/29

Gibbs1Du%em relation

(st la) of TD:

in sim+le systems

5n a sin,le com+onent sim+le system:

i i

i

U TS PV N µ = − + ∑ ( ) ( ) ( )i ii

dU d TS d PV d N µ = − + ∑⇒

i i

i

dU TdS PdV dN µ = − + ∑

( ) ( ) ( )i i i ii i

TdS PdV dN d TS d PV d N µ µ − + = − +∑ ∑

0i i

i

SdT VdP N d µ − + =∑1

( ) ( ) ( ) 0iii

P Ud Vd N d

T T T

µ + − =∑

d sdT vdP µ = − +


7/29

Summary of t%e formal structure of TD

T%e fun'amental equation by itself contains full

information about t%e system

6 of con!u,ate -ariables: N

6 of ,enerali7e' )or2 terms: N - 1 6 of -ariables: 2N

6 of in'e+en'ent -ariables "t%ermo'ynamic 'e,ree of

free'om#: N - 1

6 of equations of state: N n in'i-i'ual equation of state 'oes not com+letely

c%aracteri7e t%e system

ll equations of state to,et%er contain full information about

t%e system "Euler relation#


8/29

Eam+le: i'eal ,as

9ot a fun'amental equation

9ot an eq. of state in t%e ener,y re+resentation

Gibbs1Du%am eq. in t%e

entro+y re+resentation

/ombine all 3 equations of state:

0 0

0 0

( ) ln( ) ln( ) P V u v

s c s c RT u v

µ = − = + +

PV NRT =

V U Nc T =

1 V c

T u=

P R

T v= 1( ) ( ) ( )

P d u d v d

T T T

µ = × + ×

( ) ( )V du dv

c Ru v

= − −

1( ) ( ) ( ) P S U V N T T T

µ = + −


9/29

Ener,y minimum +rinci+le

Entro+y maimum +rinci+le: in an isolate' system

equilibrium is reac%e' if S is maimi7e' %en dU = 0 "isolate' system# S is maimi7e' in equilibrium

Ener,y minimum +rinci+le: for a ,i-en -alue of total

entro+y of a system equilibrium is reac%e' if U isminimi7e' %en dS = 0 U is minimi7e' in equilibrium

0U

S x

∂ = ÷∂

2

2 0

U

S x

∂ ÷

∂


10/29


t state S ta2es t%e maimum -alue if U is ta2en as a constant;

similarly U ta2es t%e minimum -alue if S is ta2en as a constant.


11/29


Start )it%

See /allen section


12/29

T 1 T 2

Eam+le (: equilibrium in an isolate'

system after remo-al of an a'iabatic

+artition "i.e. only allo)s %eat flo)

bet)een sub1systems#

is a constant/onstraint:

t%ermal equilibrium

9o) instea' of t%e enclosure con'ition "dU = 0 # let=s start from

t%e ne) constraint t%at dStot = dS1 + dS2 = 0

t%ermal equilibrium: t%e same equilibrium state results>

1 21 2 1 2 1

1 2 1 2

1 1( ) 0tot

S S dS dS dS dU dU Q

U U T T δ

∂ ∂= + = × + × = × − =

÷ ÷∂ ∂

1 2tot U U U = + ⇒ 2 1Q Qδ δ = −

1 2T T =⇒

1 21 2 1 2 1 1 2

1 2

( ) 0tot U U

dU dU dU dS dS dS T T S S

∂ ∂= + = × + × = × − = ÷ ÷∂ ∂

1 2

T T =⇒


13/29

Sim+le mec%anical systems

Entro+y remains constant in a

+urely mec%anical system

)%ere

k is t%e s+rin, constant

( )U mgx F x dx= − + ∫ F kx=

0dU mgdx Fdx= − + =

mg F kx= =⇒


14/29

e,en're transformations

?ot% S an' U are functions of etensi-e -ariables; %o)e-er in+ractical e+eriments ty+ically t%e controlle' -ariables

"constraints# are t%e intensi-e ones>

e,en're transformations: fun'amental relations e+resse'

as functions of intensi-e -ariables

e,en're transformations +reser-e t%e informational content

e,en're transform of a fun'amental equation is also a

fun'amental equation


15/29

Ent%al+y H(S, P, N) = U + PV = TS + µ N

Partial e,en're transform of U : re+laces t%e etensi-e +arameter V )it% t%e intensi-e +arameter P

For a com+osite system in mec%anical contact )it% a +ressure

reser-oir t%e equilibrium state minimi7es t%e ent%al+y o-er t%e mani1

fol' of states of constant +ressure "equal to t%at of t%e reser-oir#.

Ent%al+y c%an,e in an isobaric +rocess is equal to %eat ta2en in or

,i-en out from t%e sim+le system

Differential: ( )dH d U PV TdS VdP dN µ = + = + + ∑


16/29

Ent%al+y minimi7ation +rinci+le

/onsi'er a com+osite system )%ere all sub1systems

are in contact )it% a common +ressure reser-oir

t%rou,% )alls non1restricti-e )it% res+ect to -olume

++ly U minimum +rinci+le to reser-oir @ system:

T%e system is in mec%anical equilibrium )it% t%e

reser-oir:

0r r r r tot sys sys sys sysdU dU dU dU P dV dU P dV = + = − = + =

r

P P =⇒ ( ) 0tot sys sys sys sys sys sys sysdU dU P dV d U P V dH = + = + = =

2 2 2( , ) ( ) 0r sys sys sys sys sys sysd U S V d U P V d H = + = >


17/29

$elm%olt7 +otential F(T, V, N) = U - TS = - PV + µ N

Partial e,en're transform of U : re+laces t%e etensi-e

+arameter S )it% t%e intensi-e +arameter T

For a com+osite system in t%ermal contact )it% a

t%ermal reser-oir t%e equilibrium state minimi7es t%e

$elm%olt7 +otential o-er t%e manifol' of states ofconstant tem+erature "equal to t%at of t%e reser-oir#.

Differential: ( )dF d U TS SdT PdV dN µ = − = − + ∑


18/29

$elm%olt7 free ener,y

System in t%ermal contact )it% a

%eat reser-oir

T%e )or2 'eli-ere' in a re-ersible

+rocess by a system in contact )it%

a t%ermal reser-oir is equal to t%e

'ecrease in t%e $elm%olt7 +otential

of t%e system

$elm%olt7 Afree ener,yB: a-ailable

)or2 at constant tem+erature

System

δ Q

δ W

State → ?: dF

$eat

reser-oir at T

( )

r r W dU Q dU T dS

d U TS dF

δ δ = − = +

= − =


19/29

or2 +erforme' by a system in contact )it%

%eat reser-oir cylin'er contains an internal +iston on eac% si'e of

)%ic% is one mole of a monatomic i'eal ,as. T%e cylin'er

)alls are 'iat%ermal an' t%e system is immerse' in a

%eat reser-oir at tem+erature 0C/. T%e initial -olumes oft%e t)o ,aseous subsystems "on eit%er si'e of t%e

+iston# are (0 an' ( res+ecti-ely. T%e +iston is no)

mo-e' re-ersibly so t%at t%e final -olumes are an' <

res+ecti-ely. $o) muc% )or2 is 'eli-ere' Solution (: 'irect inte,ration of PdV "isot%ermal +rocess#

Solution *: ∆W = ∆F


20/29

$elm%olt7 +otential minimi7ation +rinci+le

/onsi'er a com+osite system )%ere all sub1systems

are in t%ermal contact )it% a common %eat reser-oir

t%rou,% )alls non1restricti-e )it% res+ect to %eat flo)

++ly U minimum +rinci+le to reser-oir @ system:

T%e system is in t%ermal equilibrium )it% t%e reser-oir:

0r r r r tot sys sys sys sysdU dU dU dU T dS dU T dS = + = + = − =

r T T = ⇒ ( ) 0tot sys sys sys sysdU d U T S dF = − = =2 2 2( ) ( , ) 0 sys sys sys sys sys sys sysd F d U T S d U S V = − = >


21/29

Gibbs +otential G(T, P, N) = U - TS + PV = µ N

e,en're transform of U : re+laces bot% S an' V )it% t%eintensi-e +arameters T an' P

For a com+osite system in contact )it% a t%ermal

reser-oir an' a +ressure reser-oir t%e equilibrium state

minimi7es t%e Gibbs +otential o-er t%e manifol' of statesof constant tem+erature an' +ressure.

Differential: ( )dG d U TS PV SdT VdP dN µ = − + = + + ∑


22/29

Gibbs free ener,y an' c%emical +otential

Sim+le systems:

Sin,le com+onent systems:

Multi1com+onent systems:

molar Gibbs +otential

+artial molarGibbs +otential

/onsi'er a c%emical reaction:

c%emical equilibrium con'ition

i i

i

G U TS PV N µ = − + = ∑G

N µ =

i i

i

G x N

µ = ∑0i i

i

v A∑ ƒ °i

i

dN

const d N dv = = °

0i i i ii i

dG SdT VdP dN v d N µ µ = + + = =∑ ∑⇒0i i

i

v µ =∑


23/29

First or'er +%ase transition

t T m 0 C/ an' ( atm liqui' )ater an' ice can

coeist dGwate-i!e = d(H - TS) = 0 at T m 0 C/ ( atm

∆Swate-i!e = ∆H wate-i!e "T m ∆U wate-i!e "T m at T m 0 C/ (atm

T%e 'iscontinuity of H an' U are c%aracteristic of

first or'er +%ase transition

t T H 0 C/ an' ( atm ice s+ontaneously melts dGwate-i!e = d(H - TS) # 0 at T H 0 C/ ( atm

∆Swate-i!e # ∆H wate-i!e "T = ∆Qwate-i!e "T : irre-ersible

+rocess

Th d i E t


24/29

Constraints Thermodynamic

potentialExtremumprinciple

Example

U V constant

dU = 0, dV = 0

S (U, V, N) = U"T + PV"T -

µ N"T

dS = 1"T$dU + P"T$dV - µ "T$dN

S ma

dS = 0, d

2

S % 0

5solate' systems

S V constantdS = 0, dV = 0

U (S, V, N) = TS - PV + µ N

dU = TdS - PdV + µ dN

U mindU = 0, d 2 U # 0

Sim+le mec%anicalsystems consistin, of

ri,i' bo'ies

S P constantdS = 0, dP = 0

H (S, P, N) = TS + µ N

dH = TdS + VdP + µ dN

H mindH = 0, d 2 H # 0

Systems in contact )it%a +ressure reser-oir'urin, a re-ersiblea'iabatic +rocess

T V constantdT = 0, dV = 0

F (S, V, N) = - PV + µ N dF = - SdT - PdV + µ dN

F mindF = 0, d 2 F # 0

Ieactions in a ri,i''iat%ermal container at

room tem+erature

T P constantdT = 0, dP = 0

G (T, P, N) = µ N

dG = - SdT + VdP + µ dN

G mindG = 0, d 2 G # 0

E+eriments +erforme'at room tem+erature an'

atmos+%eric +ressure


25/29

Generali7e' Massieu functions

e,en're transforms of entro+y S

Maimum +rinci+les of Massieu functions a++ly


26/29

General case

e,en're transform re+laces a -ariable )it% its con!u,ate

For a t%ermo'ynamic system its TD function )ill be t%e

e,en're transform )%ere t%e -ariables are constraine'

/ontrolle' -ariables: ε && , σ xx , σ ''

is t%e TD +otential t%at

is minimi7e' in equilibrium

?eam

)all )all

7

y

0, ,

ii iii x y z dU TdS dN V d µ σ ε == + + ∑

0 0

0 0 0

( ) xx xx yy yy

zz zz xx xx yy yy

d d U TS V V

SdT dN V d V d V d

φ σ ε σ ε

µ σ ε ε σ ε σ

= − − −

= − + + − −

( , , , , ) xx yy zz T N φ σ σ ε


27/29

Deri-in, equilibrium con'itions

T, N 1 T ,N 2 P P

Equilibrium in a system surroun'e'by 'iat%ermal im+ermeable )alls in

contact )it% a +ressure reser-oir after

remo-al of an im+ermeable +artition

"i.e. allo)s mass flo) bet)een sub1

systems#.are constants/onstraints:

is a constant

c%emical equilibrium

Gibbs +otential minimi7ation:

,T P

1 2tot N N N = + ⇒ 2 1dN dN = −

1 21 2

1 2

1 1 2( ) 0

tot

G GdG dN dN

N N

dN µ µ

∂ ∂= × + × ÷ ÷∂ ∂

= × − =

⇒ 1 2 µ µ =


28/29

/ou+le' equilibrium@

@

@@

@

@

@

@

V e

bo of an electrically con'ucti-e solutioncontainin, a +ositi-ely c%ar,e' ion "+e#

s+ecies is se+arate' into t)o +arts by an

im+ermeable electrically insulatin, internal

+artition. -olta,e ∆V is a++lie' across

+ -

t%e t)o +arts. 5f t%e +artition becomes +ermeable to t%e ion but still remains

insulatin, )%at is t%e equilibrium con'ition )it% res+ect to t%e ion "assumin,constant tem+erature an' +ressure#

Fun'amental equation:

/onstraints:

dG SdT VdP dN Vd! µ = − + + +

1 1 1 11 2 1 2

1 2 1 2

1 1 2 1 ,1 , 2 1 1 2 1( ) ( ) ( )

tot

" " "

G G G GdG dN dN d! d!

N N ! !

dN d! V V dN d! V µ µ µ µ ∆

∂ ∂ ∂ ∂= × + × + × + ×

÷ ÷ ÷ ÷∂ ∂ ∂ ∂ = × − + × − = × − + ×

1 2dN dN = −

1 2d! d!= − 1 1d! #" dN = ×

1

( ) 0tot "

dG dN #" V ∆µ ∆= × + = ⇒ 1 ,1 2 ,2" " #"V #"V µ µ + = +


29/29

Electroc%emical +otential

Describes t%e equilibrium con'ition of c%ar,e'

c%emical s+ecies "ions electrons#

/%emical +otential: Electroc%emical +otential:

)%ere is t%e -alence numberof t%e ion "'imensionless# e is

t%e elementary c%ar,e an' V

is t%e local electrical +otential

Eam+le: ion 'iffusion across

cell membrane

" #"V µ µ = + µ

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4 -- Formal Structure of Thermodynamics