Physical Chemistry , 1st ed.

4.6 Using Maxwell Relationships

The Maxwell relationships can be extremely useful in deriving other equations
for thermodynamics. For example, since

dHT dS V dp

then if we hold Tconstant and divide everything by dp,we get



d

d

H

p



T

^

H

p



T

T

p

S

T^ V

Measuring the change in entropy with respect to pressure is difficult, but us-
ing a Maxwell relationship we can substitute some other expression. Since
( S/ p)Tequals ( V/ T)p,we get



H

p



T

VT

V

T



p

(4.38)

where we have switched the order of the terms. Why is this equation useful?
Because once we know the equation of state (for example, the ideal gas law),
we know V,T, and how Vvaries with Tat constant pressure—and we can use
that information to calculate how the enthalpy varies with pressure at constant
temperature, all without having to measure the enthalpy.
The enthalpy derivative in equation 4.38 can be used with the Joule-
Thomson coefficient,JT. Recall that by the cyclic rule of partial derivatives,

JT^

T

p



H



H

T

p

H

p



T



C

1

p



H

p

T

We can now substitute for the differential ( H/ p)Tfrom equation 4.38 and get

JT

C

1

p

VT

V

T



p

(4.39)



C

1

p

T^

V

T



p

V

and now we can calculate the Joule-Thomson coefficient of a gas if we know its
equation of state and its heat capacity. Equation 4.39 does not require any knowl-
edge of the enthalpy of the system, beyond its heat capacity at constant pressure.
These are just two examples of how useful the Maxwell relationships are.

Example 4.9

Use equation 4.39 to determine the value ofJTfor an ideal gas. Assume

molar quantities.

Solution

An ideal gas has the ideal gas law as its equation of state:

pVRT

In order to evaluate equation 4.39, we need to determine ( V/ T)p.We

rewrite the ideal gas law as

VR

p

T

4.6 Using Maxwell Relationships 103