Physical Chemistry , 1st ed.

Solution

The partial derivative of interest is



V

p



T, n

which we can evaluate in a fashion similar to the example above, using

pn

V

RT

only this time taking the derivative with respect to Vinstead ofT. Following

the rules of taking derivatives, and treating n,R, and Tas constants, we get



V

p

T, n

n

V

R

2

T

for this change. Notice that although in our earlier example the change did

not depend on T, here the change in pwith respect to Vdepends on the in-

stantaneous value ofV. A plot of pressure versus volume will notbe a straight

line. (Determine the numerical value of this slope for 1 mole of gas having a

volume of 22.4 L at a temperature of 273 K. Are the units correct?)

Substituting values into these expressions for the slope must give units that

are appropriate for the partial derivative. For example, the actual numerical

value of ( p/ T)V, n,for V22.4 L and 1 mole of gas, is 0.00366 atm/K. The

units are consistent with the derivative being a change in pressure (units of

atm) with respect to temperature (units of K). Measurements of gas pressure

versus temperature at a known, constant volume can in fact provide an exper-

imental determination of the ideal gas law constant R. This is one reason why

partial derivatives of this type are useful. They can sometimes provide us with

ways of measuring variables or constants that might be difficult to determine

directly. We will see more examples of that in later chapters, all ultimately de-

riving from partial derivatives of just a few simple equations.

Finally, the derivative in Example 1.3 suggests that any true ideal gas goes to

zero volume at 0 K. This ignores the fact that atoms and molecules themselves

have volume. However, gases do not act very ideally at such low temperatures

anyway.

1.6 Nonideal Gases

Under most conditions, the gases that we deal with in reality deviate from the

ideal gas law. They are real gases, not ideal gases. Figure 1.7 shows the behav-

ior of a real gas compared to an ideal gas. The behavior of real gases can also

be described using equations of state, but as might be expected, they are more

complicated.

Let us first consider 1 mole of gas. If it is an ideal gas, then we can rewrite

the ideal gas law as

p

R

V

T

 1 (1.15)

where Vis the molar volumeof the gas. (Generally, any state variable that is

written with a line over it is considered a molar quantity.) For a nonideal gas,

10 CHAPTER 1 Gases and the Zeroth Law of Thermodynamics