Physical Chemistry , 1st ed.

dww

dqq (2.13)

but dUU

The same relationship exists for most of the other state functions as well.
(There is one exception, which we will see in the next chapter.) The differen-
tials dwand dqare called inexact differentials,meaning that their integrated
values wand qare path-dependent. By contrast,dUis an exact differential,
meaning that its integrated value Uis path-independent. All changes in state
functions are exact differentials.
Another way to illustrate equations 2.13 is to note that
UUfUi
but: w wfwi and q qfqi

The equations 2.13 imply that, for an infinitesimal change in a system,

dUdq+ dw

which is the infinitesimal form of the first law. But when we integrate this
equation, we get

Uq+ w

The difference in the treatment ofqand wversus Uis because Uis a state func-
tion. We can know qand wabsolutely, but they are dependent on the path that
the system takes from initial to final conditions.Udoes not, although we can-
not know the absolute value ofUfor the initial and final states of a system.
Although these definitions might not seem useful, consider that any ran-
dom change of any gaseous system might not be simply described as isother-
mal, adiabatic, and so on. However, in many cases, we can go from initial con-
ditions to final conditions in small, ideal steps, and the overall change in a state
function will be the combination of all of the steps. Since the change in a state
function is path-independent, the change in the state function calculated in
steps is the same as the change in the state function for a one-step process. We
will see examples of this idea shortly.
If no work†is performed during the course of a process, then dUdqand
Uq. There are two common conditions where work equals zero. The first
is for a free expansion. The second is when the system does not change volume
for a process. Since dV0, any expression that gives the work of the process
will also be exactly zero. The relationship with heat under these conditions is
sometimes written as

dUdqV (2.14)

UqV (2.15)

where the subscript Von qimplies that the volume of the system during the
change remains constant. Equation 2.15 is important because we can measure
qvalues directly for many processes. If these processes occur at constant vol-
ume, we therefore know U.

2.4 State Functions 35

†Although we focus initially on pressure-volume work, there are other types of work, like
electrical or gravitational work. Here we are assuming that none of these other types of work
are performed on or by the system.