Physical Chemistry , 1st ed.

from equation 4.29. Similarly, for the other energies:



V



U

S



VS



S



U

V



SV

(4.31)



V



A

T



VT



T



V

A



TV

(4.32)



T



G

p



Tp



p



G

T



pT

(4.33)

For each of these relationships, we know the inside partial derivative on both

sides of the equations: they are given in equations 4.18–4.25. Substituting for

the inside partial derivatives from equation 4.30, we get



p

T

S



S

V

p

or rather,



T

p



S



V

S



p

(4.34)

This is an extremely useful relationship, as we no longer need to measure the

change in volume with respect to entropy at constant pressure: it equals the

isentropic change in temperature with respect to pressure. Notice that we have

lost any direct relationship to any energy.

Using equations 4.31–4.33, we can also derive the following expressions:



V

T



S



p

S



V

(4.35)



V

S



T



T

p



V

(4.36)



p

S



T



V

T



p

(4.37)

Equations 4.34–4.37 are called Maxwell relationshipsor Maxwell relations,af-

ter the Scottish mathematician and physicist James Clerk Maxwell (Figure 4.3),

who first presented them in 1870. (Although the derivation of equations

4.34–4.37 may seem straightforward now, it wasn’t until that time that the ba-

sics of thermodynamics were understood well-enough for someone like

Maxwell to derive these expressions.)

The Maxwell relationships are extremely useful for two reasons. First, all of

them are generally applicable. They are not restricted to ideal gases, or even

just gases. They apply to solid and liquid systems as well. Second, they express

certain relationships in terms of variables that are easier to measure. For ex-

ample, it might be difficult to measure entropy directly and determine how

entropy varies with respect to volume at constant temperature. The Maxwell

relationship in equation 4.36 shows that we don’t have to measure it directly.

If we measure the change in pressure with respect to temperature at constant

volume, ( p/ T)V, we know ( S/ V)T. They are equal. The Maxwell relation-

ships are also useful in deriving new equations that we can apply to thermo-

dynamic changes in systems, or in determining the values of changes in state

functions that might be difficult to measure directly by experiment. The fol-

lowing examples use the same Maxwell relationship in two different ways.

4.5 The Maxwell Relationships 101

Figure 4.3 James Clerk Maxwell (1831–1879),

Scottish mathematician. Maxwell made many

important contributions before his untimely

death just before his 48th birthday. Among them

is the Maxwell theory of electromagnetism, which

even today forms the basis of electrical and mag-

netic behavior. He also contributed to the kinetic

theory of gases and the development of the sec-

ond law of thermodynamics. He was one of the

few people to understand Gibbs’s work.

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