Physical Chemistry Third Edition

164 4 The Thermodynamics of Real Systems

c.For an isothermal process in a closed system,

∆S

∫

c

dS

∫V 2

V 1

(

∂S

∂V

)

rn

dV

∫V 2

V 1

[

R

Vm

+

R

Vm^2

(

B 2 +T

dB 2

dT

)]

dV

n

∫V 2

V 1

R

Vm

dVm+n

(

B 2 +T

dB 2

dT

)∫V 2

V 1

R

Vm^2

dVm

nRln

(

Vm2

Vm1

)

−n

(

B 2 +T

dB 2

dT

)(

1

Vm2

−

1

Vm1

)

d. ∆S(1.000 mol)(8.3145 J K−^1 mol−^1 )ln

(

0. 05000

0. 02500

)

−(1.000 mol)(8.3145 J−^1 mol−^1 )

×

(

− 15. 8 × 10 −^6 m^3 mol−^1

)

+(298.15 K)(0. 20 × 10 −^6 m^3 mol−^1 K−^1 )

×

(

1

0 .05000 m^3 mol−^1

−

1

0 .02500 m^3 mol−^1

)

 5 .763JK−^1 + 0 .00729 J K−^1  5 .770JK−^1

The correction for nonideality, 0.007JK−^1 , is numerically almost insignificant in

this case.

We can also derive an expression for∆Sfor an isothermal pressure change, using

another Maxwell relation:

∆S

∫P 2

P 1

(

∂S

∂P

)

T,n

dP−

∫P 2

P 1

(

∂V

∂T

)

P,n

dP (4.2-24)

EXAMPLE 4.4

a.Find an expression for∆Sfor an isothermal pressure change on a pure liquid assuming

that the volume of the liquid is constant.

b.Find an expression for∆Sfor an isothermal pressure change on a pure liquid, assuming

that the volume of the liquid is given by

V(T,P)V(T 1 ,P 1 )[1+α(T−T 1 )−κT(P−P 1 )]

whereαandκTare equal to constants and whereP 1 andT 1 are a reference pressure and

a reference temperature.

c.Find∆Sfor pressurizing 1.000 mol of liquid water isothermally at 298.15 K from a

pressure of 1.00 atm to a pressure of 100.00 atm.

Solution

a.

∆S

∫P 2

P 1

(

∂S

∂P

)

T,n

dP−

∫P 2

P 1

(

∂V

∂T

)

P,n

dP−

∫P 2

P 1

V αdP 0

∆S0 sinceα0ifVis constant.