Physical Chemistry , 1st ed.

But from the natural variable equation, we know that

dUT dSp dV

If we compare these two equations, the terms multiplying the dSmust be
equal, as must the terms multiplying the dV. That is,



U

S



V

dST dS



U

V



S

dVp dV

We therefore have the following expressions:



U

S



V

T (4.18)



U

V



S

p (4.19)

Equation 4.18 states that the change in internal energy as the entropy changes at
constant volume equals the temperature of the system. Equation 4.19 shows that
the change in internal energy as the volume changes at constant entropy equals
the negative of the pressure. What fascinating relationships! It means that we do
not have to actually measure the change in internal energy versus volume at con-
stant entropy—if we know the pressure of the system, the negative value of it
equals that change. Since these changes represent slopes of plots of internal en-
ergy versus entropy or volume, we know what those slopes are for our system. So,
if we know how Uvaries with Sand V, we also know Tand pfor our system.
Furthermore, many such partial derivatives can be constructed that cannot
be determined experimentally. (Example: Can you construct an experiment in
which the entropy remains constant? That can sometimes be extremely diffi-
cult to guarantee.) Equations like 4.18 and 4.19 eliminate the need to do that:
they tell us mathematically that the change in internal energy with respect to
volume at constant entropy equals the negative of the pressure, for example.
There is no need to measure internal energy versus volume. All we need to
measure is the pressure.
Finally, in many derivations, partial derivatives like these will show up.
Equations like 4.18 and 4.19 allow us to substitute simple state variables for
more complicated partial derivatives. This will be extremely useful in our fur-
ther development of thermodynamics and accounts partially for its real power.

Example 4.4

Show that the expression on the left-hand side of Equation 4.18 yields units

of temperature.

Solution

The units ofUare J/mol, and the units of entropy are J/mol K. Changes in

Uand Sare also described using those units. Therefore, the units on the de-

rivative (which is a change in Udivided by a change in S) are



J/

J

m

/m

o

o

l

l

K



1/

1

K

K

which is a unit of temperature.

4.4 Natural Variable Equations and Partial Derivatives 97