The Foundations of Chemistry

PmeasuredPideally exerted

or

Pideally exertedPmeasured

In this correction term, large values of aindicate strong attractive forces. When

more molecules are present (greater n) and when the molecules are close together

(smaller V^2 in the denominator), the correction term becomes larger. The correc-

tion term becomes negligibly small, however, when the volume is large.

When we substitute these two expressions for corrections into the ideal gas equation,
we obtain the equation

Pmeasured (Vmeasured
nb)nRT

or

P (V
nb)nRT

This is the van der Waals equation.In this equation, P, V, T,and nrepresent the measured
values of pressure, volume, temperature (expressed on the absolute scale), and number of
moles, respectively, just as in the ideal gas equation. The quantities aand bare experi-
mentally derived constants that differ for different gases (Table 12-5). When aand bare
both zero, the van der Waals equation reduces to the ideal gas equation.
We can understand the relative values of aand bin Table 12-5 in terms of molecular
properties. Note that afor helium is very small. This is the case for all noble gases and
many other nonpolar molecules, because only very weak attractive forces, called disper-
sion forces, exist between them. Dispersion forcesresult from short-lived electrical
dipoles produced by the attraction of one atom’s nucleus for an adjacent atom’s electrons.
These forces exist for all molecules but are especially important for nonpolar molecules,
which would never liquefy if dispersion forces did not exist. Polar molecules such as
ammonia, NH 3 , have permanent charge separations (dipoles), so they exhibit greater forces
of attraction for one another. This explains the high value of afor ammonia. Dispersion
forces and permanent dipole forces of attraction are discussed in more detail in Chapter 13.
Larger molecules have greater values of b. For instance, H 2 , a first-row diatomic mole-
cule, has a greater bvalue than the first-row monatomic He. The bvalue for CO 2 , which
contains three second-row atoms, is greater than that for N 2 , which contains only two
second-row atoms.
The following example illustrates the deviation of methane, CH 4 , from ideal gas
behavior under high pressure.

EXAMPLE 12-23 van der Waals Equation

Calculate the pressure exerted by 1.00 mole of methane, CH 4 , in a 500.-mL vessel at 25.0°C
assuming (a) ideal behavior and (b) nonideal behavior.

n^2 a



V^2

n^2 a



V^2 measured

n^2 a



V^2 measured

n^2 a



V^2 measured

The van der Waals equation, like the

ideal gas equation, is known as an

equation of state,that is, an equation

that describes a state of matter.

12-15 Real Gases: Deviations from Ideality 473

TABLE 12-5 van der Waals

Constants

ab

Gas (L^2 atm/mol^2 ) (L/mol)

H 2 0.244 0.0266

He 0.034 0.0237

N 2 1.39 0.0391

NH 3 4.17 0.0371

CO 2 3.59 0.0427

CH 4 2.25 0.0428