Physical Chemistry , 1st ed.

constant V, which is what we are trying to substitute for; we substitute and

rearrange as follows:

(V)

V

1





1



T

p



V

where we have brought the partial derivative we need to substitute for to the

other side of the equation. In doing so, we get the partial derivative ofpres-

surewith respect to temperature.On the left side, the volumes cancel, and the

negative signs on both sides cancel. We gather everything together to get



T

p



V









Substituting into our equation for ( U/ V)T:



U

V



T

T





p

where we now have what is required: an equation for ( U/ V)Tin terms of

parameters easily measured experimentally: the temperature T, the pressure

p, and the coefficients and .

Example 4.10 above actually has an important lesson. The ability to math-
ematically derive expressions like this—which provide us with quantities in
terms of experimentally determined values—is a major talent of the mathe-
matics of thermodynamics. The mathematics of thermodynamics is a useful
tool. Yes, it can get complicated. But there is a lot we can know and say about
a system using these tools, and ultimately that is part of what physical chem-
istry is all about.

4.7 Focus on G

We have found how U,H, and Svary with temperature. For the two energies,
the changes with respect to temperature are called heat capacities, and we de-
rived several equations for the change in Swith respect to temperature (like
equation 3.18,Sn C ln(Tf/Ti), or the integral form previous to equa-
tion 3.18 for a nonconstant heat capacity). Since we are making the point that
Gis the most useful energy state function, how does Gvary with temperature?
From the natural variable equation for dG, we found one relationship be-
tween Gand T:



G

T



p

S (4.40)

As temperature changes, the change in Gis equal to the negative of the entropy
of the system. Notice the negative sign on the entropy in this equation: it im-
plies that as temperature goes up, the free energy goes down, and vice versa.
This might seem intuitively wrong at first glance: an energy goes downas the
temperature increases? But recall the original definition of the Gibbs free en-
ergy:GHTS. The negative sign in front of the term that includes tem-
perature does indeed imply that as Tincreases,Gwill be lower.
There is another expression that relates the temperature-dependence ofG,
but in a slightly different fashion. If we start with the definition ofG:

GHTS

4.7 Focus on G 105