Physical Chemistry , 1st ed.

Thus dUhas one term that varies with temperature and one term that varies
with volume. The two partial derivatives represent slopes in the plot ofUver-
sus Tand V, and the total infinitesimal change in U,dU, can be written in
terms of those slopes. Figure 2.8 illustrates a plot ofUand the slopes that are
represented by the partial derivatives.
Recall that there is another definition for dU:dUdq+ dwdqp dV.
If we equate these two definitions of dU:



U

T



V

dT+ 

U

V



T

dVdqp dV

Solving for the change in heat,dq:

dq

U

T



V

dT+ 

U

V



T

dV+ p dV

Grouping the two terms in dVgives

dq

U

T



V

dT+ 

U

V



T

• pdV

If our gaseous system undergoes a change in which the volume does not
change, then dV0 and the above equation simplifies to

dq^

U

T



V

dT (2.22)

We can also rewrite this by dividing both sides of the equation by dT:



d

d

T

q



U

T



V

The change in heat with respect to temperature, which equals the change in
the internal energy with respect to temperature at constant volume, is defined
as the constant volume heat capacityof the system. (Compare this definition
to that of equation 2.9, where we define the heat in terms of the change in
temperature using a constant we called specific heat.) In terms of the partial
derivative above,



U

T



V

CV (2.23)

where we now use the symbol CVfor the constant volume heat capacity.
Equation 2.22 can therefore be written as

dqCVdT (2.24)

2.6 Changes in State Functions 39

U

V

T

U

(V)T

U

(T)V

Figure 2.8 An illustration that the overall change in Ucan be separated into a change with re-
spect to temperature [labeled ( U/ T)V] and a change with respect to volume [labeled ( U/ V)T].