Physical Chemistry Third Edition

174 4 The Thermodynamics of Real Systems

b.Find the change in enthalpy if 1.000 mol of solid water

(ice) is pressurized at 0.00◦C from 1.000 atm to

50.000 atm.

c.From the results of parts a and b, find the value of the

molar enthalpy change of fusion of water at 0.00◦C and

50.00 atm.

4.18 Find the values of the isothermal compressibility and the
adiabatic compressibility of helium gas at 1.000 bar and
298.15 K. Assume the gas to be ideal. Explain in words
why the values are different, and explain why the larger
value is larger.

4.19 Derive the identify

κSκT−

α^2 VT

CP

4.20 Find an expression for

(

∂CV

∂V

)

T,n

for a gas obeying the

van der Waals equation of state.

4.21 Up to a pressure of 100 bar, N 2 gas is adequately described
by the truncated virial equation of state

PVm

RT

 1 +

B 2

Vm

where at 298.15 K

B 2 − 5. 4 × 10 −^5 m^3 mol−^1

dB 2

dT

 0. 020 × 10 −^5 m^3 mol−^1 K−^1

a.Findwfor a reversible isothermal expansion of

1.000 mol of nitrogen gas from a volume of 0.25 L

to a volume of 50.0 L at 298.15 K.

b.Find∆Sfor the same process.

c.Findqand∆Ufor the same process.

d.If the gas expands irreversibly into a vacuum, but still

isothermally and still with the same initial and final

volumes at 298.15 K, findw,q,∆S, and∆U.

4.22 Give a numerical value (in SI units) for each of the
following:

a.

(

∂H

∂S

)

P,n

for 1.000 mol of an ideal gas at 400.0 K

and 1.000 atm.

b.

(

∂S

∂V

)

T,n

for 2.000 mol of an ideal gas at 298.15 K

and 1.000 bar.

c.

(

∂U

∂V

)

S,n

for 3.000 mol of a monatomic ideal gas at

298.15 K and 2.000 atm.

4.23 a.Find an expression forSm−S◦mas a function ofVmfor

a van der Waals gas.

b. Evaluate the expression for carbon dioxide at

T 298 .15 K andVm 10 .00 L mol−^1.

4.24 Xenon gas is adequately described by

PVm

RT

 1 +

B 2

Vm

or

PVmRT+A 2 P

where it can be shown thatA 2 B 2. At 298.15 K,

B 2 − 1. 302 × 10 −^4 m^3 mol−^1 , and

dB 2 /dT 7. 2 × 10 −^7 m^3 mol−^1 K−^1.

a.Find the pressure of a sample of xenon at 298.15 K and

a molar volume of 24.465 L mol−^1.

b.Derive a formula giving

(

∂U

∂V

)

T,n

for xenon gas.

c.Find the value of

(

∂U

∂V

)

T,n

for xenon gas at 298.15 K

and a molar volume of 24.465 L mol−^1.

4.25 The Berthelot equation of state is

P

RT

Vm−b

−

a

TVm^2

a.Find an expression for

(

∂U

∂V

)

T,n

for a gas obeying this

equation of state.

b. Find the value of∆Uif 1.000 mol of argon gas expands

from a volume of 1.000 L to a volume of 10.00 L at a

constant temperature of 298.15 K.

4.26 CalculateCV, mfor benzene at 298.15 K. Assume that the

value ofαin the appendix for 20◦C can be used for 25◦C.

The density of benzene is 0.8765 g cm−^3 at 20◦C. Assume

that this value can be used at 25◦C.

4.27 a.Derive a formula for∆Sfor a pressure change of a

liquid. Assume that the coefficient of thermal expansion

is independent of pressure.

b. Evaluate∆Sfor 1.000 mol of liquid water at 20◦Cif

the pressure is changed from 1.00 atm to 100.00 atm.

α 2. 07 × 10 −^4 K−^1.