Physical Chemistry Third Edition

4.6 Euler’s Theorem and the Gibbs–Duhem Relation 197

d.Find the final volume of 1.000 mol of liquid water if it

is compressed adiabatically from 1.000 bar and

25.00◦C to a pressure of 100.00 bar. Assume that the

adiabatic compressibility is constant.

e.Find the final temperature for the process of part d.

f.Find∆Ufor the process of part d.

4.52 For each of the following proposed processes, say:
(1) whether the process is spontaneous, nonspontaneous, or
reversible; (2) whether∆Gis positive, negative, or equal
to zero; (3) whether∆His positive, negative, or equal to
zero; (4) whether∆Sis positive, negative, or equal to zero;
and (5) whether∆His smaller or larger in magnitude than
T∆S. Assume that each initial state is a metastable state.
a.Liquid water is vaporized at 1.000 atm and 100◦C.
b. Liquid water is vaporized at 1.000 atm and 105◦C.
c.Liquid water is vaporized at 1.000 atm and 95◦C.
d.Solid water melts at 1.000 atm and 5◦C.
e.Solid water melts at 1.000 atm and 0◦C.
f. Water vapor at 25◦C and a partial pressure of 23.756
torr condenses to a liquid.
g.Solid water melts at 10.000 atm and 0◦C.

4.53A nonideal gas is described equally well by either of two
truncated virial equations of state:

PVm

RT

 1 +

B 2

RT

and PVmRT+A 2 P

whereA 2 andB 2 , the second virial coefficients, are

functions ofT, and can be shown to equal each other.

a.Find expressions for the following:

i.Gm(T,P′)−G◦m(T)

ii.Sm(T,P′)−S◦m(T)

iii. Hm(T,P′)−Hm◦(T)

iv.Am(T,P′)−A◦m(T)

v. Um(T,P′)−Um◦(T)

b. Evaluate each of the quantities in part a for carbon

dioxide at 0◦C, using data in Table A.4.

4.54 Show that (∂T /∂V)S,n<0 unless (∂V /∂T)P,n<0. What

is the sign of (∂T /∂V)S,nfor water in the temperature

range between 0.00◦C and 3.98◦C?

4.55 a. Use the fundamental equation of Problem 4.4 to obtain

an expression forS−S 0 as a function ofTandPfor a

monatomic ideal gas. Use the facts thatU 3 nRT / 2

and thatPVnRT.

b. Carry out a line integration ofdSfrom (T 0 ,P 0 )to

(T′,P′) for a closed ideal gas system withnn 0 to

obtain the same expression.

c.Use the fundamental equation of Problem 4.4 to obtain

an expression forS−S 0 as a function ofTandVfor

an ideal gas. Use the facts thatU 3 nRT /2 and that

PVnRT.

d.Carry out a line integration ofdSfrom (T 0 ,V 0 )to

(T′,V′) for a closed ideal gas system withnn 0 to

obtain the same expression.

4.56 Consider the reaction

2H 2 (g)+O 2 (g)→2H 2 O(l)

a. Calculate the value of∆H◦,∆U◦,∆G◦, and∆A◦for

this reaction at 298.15 K. Calculate the value of∆A◦

from the value of∆G◦in the same way that you

calculate the value of∆U◦from the value of∆H◦(see

Chapter 2).

b. If the heat from this reaction is used to power a steam

turbine with an efficiency that is 60.0% as great as that

of a Carnot engine operating between 200. 0 ◦C and

400. 0 ◦C, find the maximum amount of work that

can be done by the combustion of 2.000 mol of

hydrogen gas.

c.This reaction is carried out in fuel cells in spacecraft.

Calculate the maximum amount of net work (work

other thanP–Vwork) that can be done by the reaction

of 2.000 mol of hydrogen gas. Calculate the total

amount of work that can be done. Comment on the

comparison between your results from parts b and c.

4.57 Calculate∆Vand∆Sfor pressurizing 2.000 mol of liquid

benzene from 1.000 atm to 1000.0 atm at 25.00◦C. The

density of benzene at this temperature and 1.000 atm is

0.8765 g cm−^3. Use an average value for the isothermal

compressibility over this range of pressures.