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50 CHAPTER 3 The Importance of State Functions: Internal Energy and Enthalpy

3.2 The Dependence of U on V and T

In this section, the fact that dU is an exact differential is used to establish how U
varies with T and V. For a given amount of a pure substance or a mixture of fixed
composition, U is determined by any two of the three variables P, V, and T. One
could choose other combinations of variables to discuss changes in U. However,
the following discussion will demonstrate that it is particularly convenient to
choose the variables T and V. Because U is a state function, an infinitesimal change
in U can be written as
0U 0U
dU = a b dT + a b dV (3.12)
0T V 0V T
This expression says that if the state variables change from T, V to T + dT, V + dV,
the change in U, dU, can be determined in the following way. We determine the
slopes of U(T,V) with respect to T and V and evaluate them at T, V. Next, these
slopes are multiplied by the increments dT and dV, respectively, and the two terms
are added. As long as dT and dV are infinitesimal quantities, higher order deriva-
tives can be neglected.
How can numerical values for (0U>0T) V and (0U>0V) T be obtained? In the follow-
ing, we only consider P–V work. Combining Equation (3.12) and the differential
expression of the first law,
0U 0U
dq - P external dV = a b dT + a b dV (3.13)
0T V 0V T

The symbol dq is used for an infinitesimal amount of heat as a reminder that heat is not
a state function. We first consider processes at constant volume for which dV = 0 , so
that Equation (3.13) becomes
0U
dq = a b dT (3.14)
V
0T V
Note that in the previous equation, dq V is the product of a state function and an exact
differential. Therefore, dq V behaves like a state function, but only because the path
(constant V) is specified. The quantity dq is not a state function.
Although the quantity (0U>0T) V looks very abstract, it can be readily measured.
For example, imagine immersing a container with rigid diathermal walls in a water
bath, where the contents of the container are the system. A process such as a chem-
ical reaction is carried out in the container and the heat flow to the surroundings is
measured. If heat flow dq V occurs, a temperature increase or decrease dT is
observed in the system and the water bath surroundings. Both of these quantities
can be measured. Their ratio, dq >dT , is a special form of the heat capacity dis-
V
cussed in Section 2.5:
dq V = a 0U b = C (3.15)
dT 0T V V

where dq >dT corresponds to a constant volume path and is called the heat capacity
V
at constant volume.
The quantity C is extensive and depends on the size of the system, whereas C V, m is
V
an intensive quantity. As discussed in Section 2.5, C V, m is different for different sub-
stances under the same conditions. Observations show that C V, m is always positive for a
single-phase, pure substance or for a mixture of fixed composition, as long as no chem-
ical reactions or phase changes take place in the system. For processes subject to these
constraints, U increases monotonically with T.
With the definition of C , we now have a way to experimentally determine changes
V
in U with T at constant V for systems of pure substances or for mixtures of constant
composition in the absence of chemical reactions or phase changes. After C has been
V