Page 139 - Physical Chemistry

lev38627_ch04.qxd 2/29/08 3:13 PM Page 120

120
Chapter 4 Pressure Dependence of S
Material Equilibrium The Euler reciprocity relation applied to the Gibbs equation dG SdT VdP gives

0S 0V
a b a b aV (4.50)
0P T 0T P
as already noted in Eq. (4.45).

Temperature and Pressure Dependences of G
In dG SdT VdP, we set dP 0 to get (
G/
T ) S. In dG SdT
P
VdP, we set dT 0 to get (
G/
P) V. Thus [Eq. (4.38)]
T
0G 0G
a b S, a b V (4.51)
0T P 0P T

Summary on Finding T, P, and V Dependences of State Functions
To find (
/
P) , (
/
V) , (
/
T) , or (
/
T) of U, H, A, or G, one starts with the Gibbs
P
V
T
T
equation for dU, dH, dA, or dG [Eqs. (4.33) to (4.36)], imposes the condition of con-
stant T, V, or P, divides by dP , dV , dT , or dT , and, if necessary, uses one of the
T
V
T
P
Maxwell relations (4.45) or the heat-capacity relations (4.31) to eliminate (
S/
V) ,
T
(
S/
P) , (
S/
T) , or (
S/
T ) . To find (
U/
T) and (
H/
T) , it is faster to simply
T
V
P
V
P
write down the C and C equations (4.29) and (4.30).
V
P
In deriving thermodynamic identities, it is helpful to remember that the tempera-
ture dependences of S [the derivatives (
S/
T ) and (
S/
T) ] are related to C and C V
V
P
P
[Eq. (4.31)] and the volume and pressure dependences of S [the derivatives (
S/
P) T
and (
S/
V) ] are given by the Maxwell relations (4.45). Equation (4.45) need not be
T
memorized, since it can quickly be found from the Gibbs equations for dA and dG by
using the Euler reciprocity relation.
As a reminder, the equations of this section apply to a closed system of fixed com-
position and also to closed systems where the composition changes reversibly.
Magnitudes of T, P, and V Dependences of U, H, S, and G
We have (
U /
T) C V,m and (
H /
T) C P,m . The heat capacities C P,m and C V,m
P
m
V
m
are always positive and usually are not small. Therefore U and H increase rapidly
m
m
with increasing T (see Fig. 5.11). An exception is at very low T, since C P,m and C V,m go
to zero as T goes to absolute zero (Secs. 2.11 and 5.7).
Using (4.47) and experimental data, one finds (as discussed later in this section)
that (
U/
V) (which is a measure of the strength of intermolecular forces) is zero for
T
ideal gases, is small for real gases at low and moderate pressures, is substantial for
gases at high pressures, and is very large for liquids and solids.
Using (4.48) and typical experimental data (Prob. 4.8), one finds that (
H /
P) is
m
T
rather small for solids and liquids. It takes very high pressures to produce substantial
changes in the internal energy and enthalpy of a solid or liquid. For ideal gases
(
H /
P) 0 (Prob. 4.21), and for real gases (
H /
P) is generally small.
m
T
m
T
From (
S/
T) C /T, it follows that the entropy S increases rapidly as T in-
P
P
creases (see Fig. 5.11).
We have (
S /
P) aV . As noted in Sec. 1.7, a is somewhat larger for gases
m
T
m
than for condensed phases. Moreover, V at usual temperatures and pressures is about
m
3
10 times as great for gases as for liquids and solids. Thus, the variation in entropy with
pressure is small for liquids and solids but is substantial for gases. Since a is positive
for gases, the entropy of a gas decreases rapidly as the pressure increases (and the vol-
ume decreases); recall Eq. (3.30) for ideal gases.
For G, we have (
G /
P) V . For solids and liquids, the molar volume is rela-
m
m
T
tively small, so G for condensed phases is rather insensitive to moderate changes in
m