Physical Chemistry , 1st ed.

this quotient may not equal 1. It can also be less than or greater than 1.
Therefore, the above quotient is defined as the compressibility factor Z:

Z

p

R

V

T

 (1.16)

Specific values for compressibility depend on the pressure, volume, and tem-
perature of the real gas, but generally, the farther Zis from 1, the less ideally
the gas behaves. Figure 1.8 shows two plots of compressibility, one with respect
to pressure and another with respect to temperature.
It would be extremely useful to have mathematical expressions that provide
the compressibilities (and therefore an idea of the behavior of the gas toward
changing state variables). These expressions are equations of state for the real
gases. One common form for an equation of state is called a virial equation.
Virialcomes from the Latin word for “force” and implies that gases are non-
ideal because of the forces between the atoms or molecules. A virial equation
is simply a power series in terms of one of the state variables, either por V.
(Expressing a measurable, in this case the compressibility, in terms of a power
series is a common tactic in science.) Virial equations are one way to fit the be-
havior of a real gas to a mathematical equation.
In terms of volume, the compressibility of real gases can be written as

Z

p

R

V

T

 1 

V

B





V

C

 2 V

D

 3  (1.17)

where B,C,D,...are called the virial coefficientsand are dependent on the na-
ture of the gas and the temperature. The constant that would be labeled Ais
simply 1, so the virial coefficients “start” with B.Bis called the secondvirial
coefficient;Cis the third virial coefficient, and so forth. Because the denomi-
nator, the power series in V, gets larger and larger as the exponent increases,
successive coefficients make a smaller and smaller contribution to the com-
pressibility. The largest single correction is due to the Bterm, making it the

1.6 Nonideal Gases 11

14

12

8

6

4

2

10

0

0

Volume (liters)

Ideal gas

Real gas

2 4 6 8 10

Pressure (atm)

Figure 1.7 The pVbehavior of an ideal gas compared to a real gas.