Page 156 - Physical Chemistry

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U, S, and T. The reciprocal of this relation, (
S/
U) 1/T, 4.20 A reversible adiabatic process is an isentropic (constant-
V
shows that entropy always increases when internal energy in- entropy) process. (a) Let a V 1 (
V/
T) . Use the first
S
S
creases at constant volume. Use the Gibbs equation for dU to Maxwell equation in (4.44) and Eqs. (1.32), (1.35), and (4.31)
show that (
S/
V) P/T. to show that a C k/TVa. (b) Evaluate a for a perfect
S
V
S
U
gas. Integrate the result, assuming that C is constant, and ver-
V
4.5 Verify the Maxwell relations (4.44) and (4.45).
ify that you obtain Eq. (2.76) for a reversible adiabatic process
4.6 For water at 30°C and 1 atm, use data preceding Eq. (4.54) in a perfect gas. (c) The adiabatic compressibility is k
S
1
to find (a) (
U/
V) ; (b) m . V (
V/
P) . Starting from (
V/
P) (
V/
T ) (
T/
P) ,
S
S
S
S
JT
T
prove that k C k/C .
S
P
V
3
4.7 Given that, for CHCl at 25°C and 1 atm, r 1.49 g/cm ,
3
1
C 116 J/(mol K), a 1.33 10 3 K , and k 9.8 4.21 Since all ideal gases are perfect (Sec. 4.4) and since for a
P,m
1
10 5 atm , find C for CHCl at 25°C and 1 atm. perfect gas (
H/
P) 0 [Eq. (2.70)], it follows that (
H/
P) T
T
V,m 3
0 for an ideal gas. Verify this directly from (4.48).
1
4.8 For a liquid with the typical values a 10 3 K , k
3
1
10 4 atm , V 50 cm /mol, C P,m 150 J/mol-K, calculate 4.22 This problem finds an approximate expression for
m
at 25°C and 1 atm (a) (
H /
T ) ; (b) (
H /
P) ; (c) (
U/
V) ; U intermol , the contribution of intermolecular interactions to U. As
T
T
m
m
P
(d) (
S /
T ) ; (e) (
S /
P) ; ( f ) C V,m ; (g) (
A/
V) . the volume V changes at constant T, the average distance be-
m
P
T
T
m
tween molecules changes and so the intermolecular interaction
4.9 Show that (
U/
P) TVa PVk (a) by starting from
T energy changes. The translational, rotational, vibrational, and
the Gibbs equation for dU; (b) by starting from (4.47) for electronic contributions to U depend on T but not on V
(
U/
V) . (Sec. 2.11). Infinite volume corresponds to infinite average
T
4.10 Show that (
U/
T ) C PVa (a) by starting from distance between molecules and hence to U intermol 0.
P
P
dU TdS PdV; (b) by substituting (4.26) into (4.30). Therefore U(T, V) U(T,
) U intermol (T, V). (a) Verify that
U intermol (T, V ) V
(
U/
V) dV, where the integration is at
T
4.11 Starting from dH TdS VdP, show that (
H/
V) constant T, and V is some particular volume. (b) Use (4.57) to
T
aT/k 1/k. show that for a van der Waals gas U intermol,m a/V . (This is
m
only a rough approximation since it omits the effects of inter-
4.12 Consider solids, liquids, and gases not at high pressure.
molecular repulsions, which become important at high densi-
For which of these is C C usually largest? Smallest?
P,m V,m
ties.) (c) For small to medium-size molecules, the van der Waals
2
4.13 Verify that [
(G/T)/
T] P H/T . This is the a values are typically 10 to 10 cm atm mol 2 (Sec. 8.4).
6
7
6
Gibbs–Helmholtz equation. Calculate the typical range of U intermol,m in a gas at 25°C and 1
atm. Repeat for 25°C and 40 atm.
4.14 Derive the equations in (4.31) for (
S/
T ) and (
S/
T )
P V
from the Gibbs equations (4.33) and (4.34) for dU and dH. 4.23 (a) For liquids at 1 atm, the attractive intermolecular
forces make the main contribution to U . Use the van der
1
4.15 Show that m (P aTk )/C , where m is the Joule intermol
J
J
V
coefficient. Waals expression in Prob. 4.22b and the van der Waals a value
6
6
of 1.34 10 cm atm mol 2 for Ar to show that for liquid or
4.16 A certain gas obeys the equation of state PV m gaseous Ar,
RT(1 bP), where b is a constant. Prove that for this gas 5 3 2
2
2
(a) (
U/
V) bP ; (b) C P,m C V,m R(1 bP) ; (c) m 0. U m 11.36 10 J cm >mol 2>V m
T
JT
112.5 J>mol-K2T const.
4.17 Use Eqs. (4.30), (4.42), and (4.48) to show that
2
2
(
C /
P) T (
V/
T ) . The volumes of substances in- (b) Calculate the translational and intermolecular energies in liq-
P T P
2
2
crease approximately linearly with T, so
V/
T is usually uid and in gaseous Ar at 1 atm and 87.3 K (the normal boiling
3
point). The liquid’s density is 1.38 g/cm at 87 K. (c) Estimate
quite small. Consequently, the pressure dependence of C can
P
usually be neglected unless one is dealing with high pressures. U for the vaporization of Ar at its normal boiling point and
m
compare the result with the experimental value 5.8 kJ/mol.
4.18 The volume of Hg in the temperature range 0°C to 100°C
2
at 1 atm is given by V V (1 at bt ), where a 0.18182 Section 4.5
0
2
1
10 3 °C , b 0.78 10 8 °C , and where V is the vol- 4.24 True or false? (a) G is undefined for a process in which
0
ume at 0°C and t is the Celsius temperature. The density of T changes. (b) G 0 for a reversible phase change at constant
3
mercury at 1 atm and 0°C is 13.595 g/cm . (a) Use the result of T and P.
Prob. 4.17 to calculate (
C P,m /
P) for Hg at 25°C and 1 atm.
T
(b) Given that C P,m 6.66 cal mol 1 K 1 for Hg at 1 atm and 4.25 Calculate G and A when 2.50 mol of a perfect gas with
4
25°C, estimate C P,m of Hg at 25°C and 10 atm. C V,m 1.5R goes from 28.5 L and 400 K to 42.0 L and 400 K.
4.26 For the processes of Probs. 2.45a, b, d, e, and f, state
4.19 For a liquid obeying the equation of state V c c T
m 1 2 whether each of A and G is positive, zero, or negative.
2
c T c P c PT [Eq. (1.40)], find expressions for each of
4
5
3
the following properties in terms of the c’s, C , P, T, and V: 4.27 Calculate A and G when a mole of water vapor ini-
P
(a) C C ; (b) (
U/
V) ; (c) (
S/
P) ; (d) m ; (e) (
S/
T ) ; tially at 200°C and 1 bar undergoes a cyclic process for which
P
T
P
T
JT
V
(f) (
G/
P) . w 145 J.
T