Page 157 - Physical Chemistry

lev38627_ch04.qxd 2/29/08 3:13 PM Page 138

138
4.28 (a) Find G for the fusion of 50.0 g of ice at 0°C and equal the molar Gibbs energy of solid sucrose at 300 K and
1 atm. (b) Find G for the supercooled-water freezing process 1 bar. (c) The chemical potential of sucrose in a saturated solu-
of Prob. 3.14. tion of sucrose in water at 300 K and 1 bar must equal the molar
Gibbs energy of solid sucrose at 300 K and 1 bar. (d) If phases
4.29 Find A and G when 0.200 mol of He(g) is mixed at a and b are in equilibrium with each other, the chemical poten-
constant T and P with 0.300 mol of O (g) at 27°C. Assume ideal tial of phase a must equal the chemical potential of phase b.
2
gases.
4.39 For each of the following closed systems, write the con-
4.30 Suppose 1.00 mol of water initially at 27°C and 1 atm dition(s) for material equilibrium between phases: (a) ice in
undergoes a process whose final state is 100°C and 50 atm. Use equilibrium with liquid water; (b) solid sucrose in equilibrium
data given preceding Eq. (4.54) and the approximation that the with a saturated aqueous solution of sucrose; (c) a two-phase
temperature and pressure variations of a, k, and C can be system consisting of a saturated solution of ether in water and a
P
neglected to calculate: (a) H; (b) U; (c) S.
saturated solution of water in ether; (d) ice in equilibrium with
4.31 Calculate G for the isothermal compression of 30.0 g an aqueous solution of sucrose. (e) Solid sucrose and solid glu-
of water from 1.0 atm to 100.0 atm at 25°C; neglect the varia- cose in equilibrium with an aqueous solution of these two
tion of V with P. solids.
4.32 A certain gas obeys the equation of state PV m 4.40 For each of the following pairs of substances, state which
2
RT(1 bP cP ), where b and c are constants. Find expres- substance, if either, has the higher chemical potential:
sions for H and S for a change of state of this gas from (a) H O(l) at 25°C and 1 atm vs. H O(g) at 25°C and 1 atm;
2
2
m
m
(P , T ) to (P , T ); neglect the temperature and pressure depen- (b) H O(s) at 0°C and 1 atm vs. H O(l) at 0°C and 1 atm;
2
2
1
1
2
2
dence of C P,m . (c) H O(s) at 5°C and 1 atm vs. supercooled H O(l) at 5°C
2
2
and 1 atm; (d) C H O (s) at 25°C and 1 atm vs. C H O (aq)
6
12
6
12
6
6
4.33 If 1.00 mol of water at 30.00°C is reversibly and adiabat- in an unsaturated aqueous solution at 25°C and 1 atm;
ically compressed from 1.00 to 10.00 atm, calculate the final (e) C H O (s) at 25°C and 1 atm vs. C H O (aq) in a saturated
6
6
6
12
12
6
volume by using expressions from Prob. 4.20 and neglecting the solution at 25°C and 1 atm; ( f) C H O (s) at 25°C and
6
12
6
temperature and pressure variation in k . Next calculate the 1 atm vs. C H O (aq) in a supersaturated solution at 25°C and
S
6
6
12
final temperature. Then use the first law and the (
V/
P) ex- 1 atm. (g) Which substance in (a) has the higher G ?
S
m
pression in Prob. 4.20 to calculate U; compare the result with
the approximate answer of Prob. 2.47. See Eq. (4.54) and data 4.41 Show that for ice in equilibrium with liquid water at 0°C
preceding it. and 1 atm the condition of equality of chemical potentials is
equivalent to G 0 for H O(s) → H O(l).
2
2
4.34 Use a result of the example after Eq. (4.55) to derive an
expression for U for a gas obeying the van der Waals equation
and undergoing a change of state. Section 4.8
4.42 Give the value of the stoichiometric number n for each
Section 4.6 species in the reaction C H (g) 5O (g) → 3CO (g) 4H O(l).
3
8
2
2
2
4.35 True or false? (a) The chemical potential m is a state
i 4.43 Write the reaction equilibrium condition for N 3H ∆
function. (b) m is an intensive property. (c) m in a phase must 2 2
i i 2NH in a closed system.
remain constant if T, P, and x remain constant in the phase. 3
i
(d) The SI units of m are J/mol. (e) The definition of m for a 4.44 Suppose that in the reaction 2O → 3O , a closed system
3
2
i i
. ( f ) The chemical initially contains 5.80 mol O and 6.20 mol O . At some later
3
2
i
single-phase system is m 10G i >0n i 2 T,P,n j i
potential of pure liquid acetone at 300 K and 1 bar equals G of time, 7.10 mol of O is present. What is j at this time?
3
m
liquid acetone at 300 K and 1 bar. (g) The chemical potential of
benzene in a solution of benzene and toluene at 300 K and 1 bar
must be equal to G of pure benzene at 300 K and 1 bar. General
m
4.45 For H O(s) at 0°C and 1 atm and H O(l) at 0°C and
2
2
4.36 Show that m i 10U>0n i 2 S,V,n j i 10H>0n i 2 S,P,n j i 1 atm, which of the following quantities must be equal for the
. two phases? (a) S ; (b) U ; (c) H ; (d) G ; (e) m; ( f ) V .
m m m m m
10A>0n i 2 T,V,n j i
4.37 Use Eq. (4.75) to show that dq TdS m dn for a
i i i 4.46 Consider a two-phase system that consists of liquid
one-phase closed system with P-V work only in mechanical and water in equilibrium with water vapor; the system is kept in a
thermal equilibrium. This expression gives dq during a chemi- constant-temperature bath. (a) Suppose we reversibly increase
cal reaction. Since the reaction is irreversible, dq T dS. the system’s volume, holding T and P constant, causing some of
the liquid to vaporize. State whether each of H, S, S univ , and
Section 4.7 G is positive, zero, or negative. (b) Suppose we suddenly re-
4.38 True or false? (a) The chemical potential of benzene in a move some of the water vapor, holding T and V constant. This
solution of benzene and toluene must equal the chemical poten- reduces the pressure below the equilibrium vapor pressure of
tial of toluene in that solution. (b) The chemical potential of su- water, and liquid water will evaporate at constant T and V until
crose in a solution of sucrose in water at 300 K and 1 bar must the equilibrium vapor pressure is restored. For this evaporation