Important Thermodynamic equations

SdTVdpdG

pdVSdTdA

VdpTdSdH

pdVTdSdU

0)(

0

,

TP

surrsys

Gd

dSdS

Conditions for equilibrium between Phases

other.

of expense at the in volume increase would

phase one not, If same. thebe should phases

two theof pressure theT, and Vconstant At

Equil.) l(Mechanica

0

re, temperatuand olumeconstant vAt

0 (2)

another. tophase one from flow

heat will same,not is re temperatu theIf

Equil.) (Thermal

0

0

system, isolatedFor

0 )1(

,

,

pp

dVpdVp

dA

TT

T

dq

T

dq

dSdSdS

dS

TV

VU

other.

tophase one from place takewillcomponent

that of transfer thenot, If phases. in two

same thebe shouldcomponent a of potential

chemical thePressure, and Temp.constant At

Equil.) (Chemical

0

0

Pressure, and emperatureconstant tAt

,,

system,open an For

0 )3(

,

,

ii

iiii

iiPT

ii

PT

dndn

dndG

dnVdpSdTdG

nPTfG

dG

Various Equilibrium conditions

Capacity

Factor

Intensity

Factor

Equilibrium

Condition

S T

At const. U,V

V P

At const. V,T

ni i

At const. P,T

TT

pp

ii

jnVTjnpSjnVSi

ii

jnPTi

ii

n

A

n

H

n

U

dnpdVTdSdU

n

G

dnVdpSdTdG

,,,,,,

,,

Chemical Potential

Intergration of function

ii

ii

ii

nG

TSPVUTSHG

nPVTSU

dnPdVTdSdU

function, extensive are alln and V S, U,Here,

Phase changes of pure componentvapourliquidsolid

Function sGibb'molar

system,component oneFor

1

n

G

nG

nG i

N

ii

lower. is potential chemical its wherephase a

higher to is potential chemical its wherephase

thefromsly spontaneou diffuses speciesA

0

0

0

reaction, sspontaneou aFor

,

ii

ii

iiii

PT

dndn

dG

Physical meaning of Chemical potential

• Tendency of a system to give particles.

(s) (l) (g)

quantity) (negative

mmm

mp

mp

m

p

SSS

ST

ST

G

ST

G

VdpSdTdG

Temperature dependence of phase stability

Ice-skating

• As we glide across the ice, we exert pressure on the thin blade, and are therefore creating a small stream of water in our path by melting that ice. The water between the blade and the ice is what we really glide across.

Response of melting temperature to the applied pressure

m

T

Vp

Relationship between equilibrium temperature and pressure

e.g. Solid-liquid equilibria

VS

dTdP

dPSV

LVdT

SS

LS

dPLVdT

LSdP

SVdT

SS

ldd

lls

ls

l

)()(

s

's

'

(i)-(ii)

(ii) ''

2T tochanges re temperatumelting the,

2P toPressure thechangingon Now

(i) s

phasesboth in same iscomponent of potential chemical the,1

P and 1

Tat So

1T is re temperatumelting the,

1P Pressureat Suppose

Liquid Solid

Relation between dp and dT

equation) (Clapeyron

),(),(

equal. still are

but changes potentials chemical theb,At

),(),(

,,,,

,,,,

V

S

dT

dp

dpSSdTVV

dpVdTSdpVdTS

dd

TpTp

TpTp

trs

trs

mmmm

mmmm

bb

aa

The solid –liquid boundary

curve Steep

large ve,

smallbut ve

ve

VT

H

dT

dp

fus

fus

Liquid –Vapour boundary

small ve,

large ve,

ve

VT

H

dT

dp

vap

vap

Equation)Clapeyron (Clausius

ln

(g)

2RT

H

dT

pd

p

RTT

H

dT

dp

VV

vap

vap

mvap

Solid-Vapour Boundary

vap-liquidan Steeper th

ve

large ve,

ve

VT

H

VT

H

dT

dp

vap

vap

sub

sub

1 bar

Elevation of boiling point

• Boiling point of water is raised if the pressure above water is increased; it is lowered if the pressure is reduced. This explains why water boils at 70 °C up in the Himalaya

Elevation of boiling point

• An application of this effect is the pressure cooker. A pressure cooker traps steam inside it, raising the pressure to about twice the atmospheric pressure. As a result, the water boils at 120 °C. Food cooks quicker at the higher temperature. There is a safety valve on the cover which opens if the pressure is too high.

Equation)Clapeyron (Clausius

11ln

ln

211

2

2

TTR

H

p

p

RT

H

dT

pd

vap

vap

Autoclave

• Sterilization of both materials and contaminated media is an essential process in a laboratory for in vitro cultures.

• This sterilization is usually done in a device called an autoclave.

• Essentially, an autoclave is a container that exposes the material to be sterilized to temperatures above the boiling point of water, which is achieved by increasing pressure.

Q. It has been suggested that the surface melting of ice plays a role in enabling speed skaters to achieve peak performance. Carry out the following calculation to test this hypothesis. At 1 atm pressure, ice melts at 273.15 K, ΔHf =6010 J mol-1, the density of ice is 920 Kg m-3, and the density of liquid water is 997 kg m-3.

(a) What pressure is required to lower the melting temperature by 5.0 ºC ?

(b) Assume the width of the skate in contact with ice has been reduced by sharpening to 25*10-3 cm, and that the length of the contact area is 15 cm. If the skater of mass 85 kg is balanced on one skate, what pressure is exerted at the interface of the skate and the ice ?

©What is the melting point of ice under this pressure?

(d) If the temperature of ice is -5 ºC, do you expect melting of ice at ice-skate interface to occur ?

6010 fusH 6010 fusH 6010 fusH

H of vaporization of water is 539.4 cal/g at the normal boiling point. Many bacteria can survive at 100 C by forming spores. Hence autoclaves used to sterlize medical and laboratory instruments are pressurized to raise the boiling point of water to 120 C. What will be the pressure required to increase the boiling point of water to 120 ºC ?

Q. Ar has normal melting and boiling points of 83.8 and 87.3 K; its triple point is at 83.8 K and 0.7 atm, and its critical temperature and pressure are 151 K and 48 atm. State whether Ar is a solid, liquid, or gas under each of the following conditions (a) 0.9 atm and 90 K (b) 0.7 atm and 80 K (c) 0.8 atm and 88 K (d) 0.8 atm and 84 K (e) 1.2 atm and 83.5 K.

Q. Dry ice is frozen carbondioxide. A block of dry ice has a surface temperature of -78.5 C. If you want to send something frozen across the country, you can pack it in dry ice. It will be frozen when it reaches its destination, and there will be no messy liquid left over like you would have with normal ice. Explain this phenomenon with the help of a phase diagram?

The vapor pressure of zinc varies with temperature as log p (mmHg)= -6850/T – 0.755log T + 11.24

and that of liquid Zn as log p (mmHg) = -6620/T -1.255 log T + 12.34

calculate (a) boiling pt of Zn.(b) the triple point.© the heat of evaporation at the boiling point.(d) heat of fusion.(e) the difference in Cps of solid and liquid Zn.

The effect of applied pressure on vapor pressure

rises. pressure vapour its phase,

condensed a toapplied is pressureWhen

ln (l)

(l)

(v) (l)

equili. preserves that changeany For

)()(

m,equilibriuAt

1

2

p

pRTPV

dpp

RTdPV

dd

gl

m

m

Phase Rule

F=C-P+2

C = number of component

Degree of Freedom

• Number of independent intensive variables needed to specify its intensive state.

T, P and mole fraction in each phase.

Total number of intensive variable = pc +2

Mole fraction of components in each phase must be one

Number of relation =number of phases= p

For chemical equilibrium,

Chemical potential of individual component will be same in each of the phases.

In each of the component

(p-1) relation for each of the component

Total number of relation for chemical equilibrium = c(p-1)

Degree of Freedom = pc + 2 – p - c(p-1)

= c – p + 2

.......1111

What happens to boiling point or freezing point of a liquid in the presence of a solute?

What happens to chemical potential of a liquid in the presence of solute?

solute. volatile-non ofaddition on

decreases liquid a of potential Chemical

) law s(Raoult' ln

ln

ln

(g) (l)

solute a of presenceIn

ln

(g) (l)

solvent pureFor

*

AAAA

AAA

AAA

AA

AAA

AA

xppxRT

p

pRT

pRT

pRT

There is elevation of boiling point upon addition of a solute.

There is depression of freezing point upon addition of a solute.