Page 137 - Physical Chemistry

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Chapter 4 where we defined the functions M and N as
Material Equilibrium
M 10z>0x2 , N 10z>0y2 x (4.41)
y
From Eq. (1.36), the order of partial differentiation does not matter:

0 0z 0 0z
a b a b (4.42)
0y 0x 0x 0y
Hence Eqs. (4.40) to (4.42) give
0M 0N
a b a b if dz M dx N dy (4.43)*
0y x 0x y
Equation (4.43) is the Euler reciprocity relation. Since M appears in front of dx in the
expression for dz, M is (
z/
x) and since we want to equate the mixed second partial
y
derivatives, we take (
M/
y) .
x
The Maxwell Relations
The Gibbs equation (4.33) for dU is

dU T dS P dV M dx N dy where M T, N P, x S, y V

The Euler relation (
M/
y) (
N/
x) gives
y
x
10T>0V2 301 P2>0S4 10P>0S2 V
V
S
Application of the Euler relation to the other three Gibbs equations gives three more
thermodynamic relations. We find (Prob. 4.5)
0T 0P 0T 0V
a b a b , a b a b (4.44)
0V S 0S V 0P S 0S P
0S 0P 0S 0V
a b a b , a b a b (4.45)
0V T 0T V 0P T 0T P
These are the Maxwell relations (after James Clerk Maxwell, one of the greatest of
nineteenth-century physicists). The first two Maxwell relations are little used. The last
two are extremely valuable, since they relate the isothermal pressure and volume vari-
ations of entropy to measurable properties.
The equations in (4.45) are examples of the powerful and remarkable relationships
that thermodynamics gives us. Suppose we want to know the effect of an isothermal
pressure change on the entropy of a system. We cannot check out an entropy meter
from the stockroom to monitor S as P changes. However, the relation (
S/
P)
T
(
V/
T) in (4.45) tells us that all we have to do is measure the rate of change of the
P
system’s volume with temperature at constant P, and this simple measurement enables
us to calculate the rate of change of the system’s entropy with respect to pressure at
constant T.

Dependence of State Functions on T, P, and V
We now find the dependence of U, H, S, and G on the variables of the system. The most
common independent variables are T and P. We shall relate the temperature and pres-
sure variations of H, S, and G to the directly measurable properties C , a, and k. For U,
P
the quantity (
U/
V) occurs more often than (
U/
P) , so we shall find the tempera-
T
T
ture and volume variations of U.