Physical Chemistry , 1st ed.

Notice the slight difference in the answers. Such slight differences may be lost

in the significant figures of a calculation (as they would in this case), but in

very precise measurements these differences will be noticed.

There is one further conclusion about internal energy changes. Consider the

change in Ufor the process illustrated in Figure 2.9: an insulated system in

which an ideal gas is in one chamber, and then a valve is opened and the gas

expands into a vacuum. Because this is a free expansion, work is zero. The in-

sulation keeps any heat from being exchanged between the system and the sur-

roundings, so q0 also. This means that U0 for this process. By equa-

tion 2.21, this means that

0 

U

T



V

dT (^) 

U

V



T

dT

Barring the possible coincidence that the two terms might cancel each other

out exactly, the right side of the equation will be zero only if both of the terms

themselves are zero. The derivative in the first term, ( U/ T)V, is notzero be-

cause temperature is a measure of energy of the system. As the temperature

changes, of course the energy changes; this is what a nonzero heat capacity im-

plies. Therefore dT, the change in temperature, must equal zero and the process

is isothermic. Consider the second term, however. We know that dVis nonzero

because the ideal gas expands, and in doing so changes its volume. In order for

the second term to be zero, then, the partial derivative ( U/ V)Tmustbe zero:



U

V



T

 0 for an ideal gas (2.28)

This derivative says that the change in internal energy with respect to volume

changes at constant temperaturemust be zero for an ideal gas. Because we as-

sume that in an ideal gas the individual particles do not interact with each other,

a change in the volume of the ideal gas (which would tend to separate the in-

dividual particles more, on average) does not change the total energy if the tem-

perature remains constant. In fact, equation 2.28 is one of the two criteria for

an ideal gas. An ideal gas is any gas that (a) follows the ideal gas law as an equa-

tion of state, as discussed in Chapter 1, and (b) has an internal energy that does

not change if the temperature of the gas does not change. For real gases, equa-

tion 2.28 does not apply and the total energy willchange with volume. This is

because there are interactions between the atoms and molecules of real gases.

One can do similar things with the infinitesimal for enthalpy,dH. We have al-

ready mentioned that we will use temperature and pressure for enthalpy. Hence,

dH

H

T



p

dT+ 

H

p



T

dp (2.29)

If a change occurs at constant pressure, then dp0 and we have

dH

H

T



p

dT

where we can now define a constant-pressure heat capacity Cpjust as we defined

CV. Only now, we define our heat capacity in terms ofH:

Cp

H

T



p

(2.30)

2.6 Changes in State Functions 41

Vacuum

Insulated system

Closed

Gas

Open

Figure 2.9 An adiabatic, free expansion of an
ideal gas leads to some interesting conclusions
about U. See text for discussion.