Physical Chemistry , 1st ed.

to get H:

HG TS

If we know how Gbehaves with respect to its natural variables, we know

( G/ T)p. This partial derivative is equal to S, so we can substitute to get

HGT

G

T



p

which gives us H.

It is worth stating again how useful the natural variable equations are. If we
know how any one of the energies varies in terms of its natural variables, we
can use the various definitions and equations from the laws of thermodynam-
ics to construct expressions for any other energy.The mathematics of thermo-
dynamics is becoming powerful indeed.

4.5 The Maxwell Relationships

The equations involving partial derivatives of the thermodynamic energies can
be taken a step further. However, some definitions are necessary.
We have repeatedly made the point that some thermodynamic functions are
state functions, and that changes in state functions are independent of the ex-
act path taken. In other words, the change in a state function depends only on
the initial and final conditions, not on how the initial conditions became the
final conditions.
Consider this in terms of the natural variable equations. They all have two
terms, a change with respect to one state variable, and a change with respect to
the other state variable. For instance, the natural variable equation for dHis

dH^

H

S



p

dS (^) ^

H

p



S

dp (4.27)

where the overall change in His separated into a change as the entropy Svaries,
and a change as the pressure pvaries. The idea of path-independent changes
in state functions means that it does not matter which change occurs first. It
does not matter in what order the partial derivatives in Hoccur. As long as
both of them change from designated initial values to designated final values,
the overall change in Hhas the same value.
There is a mathematical parallel to this idea. If you have a mathematical
“state function” that depends on two variables F(x,y), then you can determine
the overall change in Fby setting up a “natural variable” equation for the over-
all change in Fas

dF^

F

x



y

dx (^) ^

F

y



x

dy (4.28)

The function F(x,y) changes with respect to xand with respect to y. Suppose
you were interested in determining the simultaneous change ofFwith respect
to xand y; that is, you wanted to know the secondderivative ofFwith respect
to xand y. In what order do you perform the differentiation? Mathematically,
it does not matter. This means that the following equality exists:



x



F

y



xy



y



F

x



yx

(4.29)

4.5 The Maxwell Relationships 99