Page 177 - Physical Chemistry

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Chapter 5 entropy contribution of each such solid–solid phase transition must be included in
Standard Thermodynamic (5.29) as an additional term H /T , where H is the molar enthalpy change of
Functions of Reaction trs m trs trs m
the phase transition at temperature T .
trs
For a substance that is a gas at 1 bar and T , we include the S of vaporization
2
m
at the boiling point T and the S of heating the gas from T to T .
b
m
2
b
In addition, since the standard state is the ideal gas at 1 bar P°, we include the small
correction for the difference between ideal-gas and real-gas entropies. The quantity
S (T, P°) S (T, P°) is calculated from the hypothetical isothermal three-step process
re
id
(5.13). For step (a) of (5.13), we use (
S/
P) (
V/
T) [Eq. (4.50)] to write S
P
T
a
0 P° (
V/
T) dP P° (
V/
T) dP. For step (b) of (5.13), we use a result of statistical
P
P
0
mechanics that shows that the entropy of a real gas and the entropy of the corresponding
ideal gas (no intermolecular interactions) become equal in the limit of zero density
(see Prob. 21.93). Therefore S 0. For step (c), the use of (
S/
P) (
V/
T) P
T
b
[Eq. (4.50)] and PV nRT gives S (nR/P) dP. The desired S is the sum S
P°
0
a
c
S S ; per mole of gas, we have
b
c
P° 0V m R
S m,id 1T, P°2 S m,re 1T, P°2 ca b d dP (5.30)
0 0T P P
where the integral is evaluated at constant T. Knowledge of the P-V-T behavior of the real
gas allows calculation of the contribution (5.30) to S° , the conventional standard-state
m
molar entropy of the gas. (See Sec. 8.8.) Some values of S m,id S m,re in J/(mol K) at 25°C
and 1 bar are 0.15 for C H (g) and 0.67 for n-C H (g).
10
2
6
4
The first integral in (5.29) presents a problem in that T 0 is unattainable
(Sec. 5.11). Also, it is impractical to measure C° (s) below a few degrees Kelvin.
P,m
Debye’s statistical-mechanical theory of solids (Sec. 23.12) and experimental data
show that specific heats of nonmetallic solids at very low temperatures obey
3
C° P,m C° aT very low T (5.31)
V,m
where a is a constant characteristic of the substance. At the very low temperatures to
2
which (5.31) applies, the difference TVa /k between C and C [Eq. (4.53)] is negli-
V
P
gible, because both T and a vanish (see Prob. 5.58) in the limit of absolute zero. For
metals, a statistical-mechanical treatment (Kestin and Dorfman, sec. 9.5.2) and exper-
imental data show that at very low temperatures
3
C° P,m C° aT bT metal at very low T (5.32)
V,m
where a and b are constants. (The term bT arises from the conduction electrons.) One
uses measured values of C° at very low temperatures to determine the constant(s) in
P,m
(5.31) or (5.32). Then one uses (5.31) or (5.32) to extrapolate C° P,m to T 0 K. Note
that C vanishes as T goes to zero.
P
For example, let C° (T ) be the observed value of C° of a nonconductor at the
P,m
P,m
low
lowest temperature for which C° P,m is conveniently measurable (typically about 10 K).
Provided T low is low enough for (5.31) to apply, we have
aT 3 low C° 1T low 2 (5.33)
P,m
We write the first integral in (5.29) as
T fus C° dT T low C° dT T fus C° P,m dT (5.34)
P,m
P,m
0 T 0 T T low T