5.3 Van der Waals equation 103
useful rule of thumb, the saturation vapor pressure doubles for every 10
◦
C increase
in temperature (at least in the range of interest for atmospheric science). Even so,
at moderate temperatures the saturation vapor pressure is very small compared to
atmospheric pressure near the surface (usually 5 to 30 hPa compared to 1000 hPa).
Does the presence of dry air affect the saturation vapor pressure of water? Perhaps
the added pressure of the air on the liquid surface squeezes more water molecules
into the vapor phase. But on the contrary, some air dissolves in the liquid and thereby
might hinder the flux of molecules out of the liquid surface. Both effects are present
but together their impact is less than 1% of the saturation vapor pressure.
5.3 Van der Waals equation
As we learned earlier, the approximation of an ideal gas works well if we can neglect
the intermolecular forces. This is virtually always the case for the major constituents
of air at Earth-like conditions. But as a gas nears its critical temperature and the
liquid or solid state can coexist with the gas phase, the departure from ideality is
important. As we see from Figure 5.2 the ideal gas equation of state describes the
behavior of real gases in limiting cases of high temperatures and low pressures.
Isotherms for an ideal gas are rectangular hyperbolae (p ∝ 1/V ). A small pressure
decrease leads to a large increase in volume (B to A in Figure 5.2). However, the
ideal gas equation of state is no longer a good approximation when the temperature
of the gas is below its critical point, and the volume is in the range where the
isotherms become horizontal (see the flat segment C to B in Figure 5.2); i.e., there
is a mixture of liquid and gas in equilibrium together.
A very useful equation that describes the behavior of many substances over a
wide range of temperatures and pressures was derived by van der Waals. The van
der Waals equation for 1 mol of gas is:
p +
a
v
2
(
v − b) = R
∗
T [van der Waals equation] (5.2)
where a and b are constants (different for different substances) and
v is the volume
per mole of the gas (i.e., the reduced volume or the specific volume). The term b
in (5.2) is due to the finite size of the molecules, while the term a/
v
2
is due to the
effect of the attractive molecular forces. For a = b = 0, the van der Vaals equation
reduces to the Ideal Gas Law (5.2).
Usually the van der Waals equation is written in the form
p =
R
∗
T
v − b
−
a
v
2
. (5.3)
Figure 5.4 shows an example of isotherms calculated using the van der Waals
equation. If we compare Figures 5.2 and 5.4, we see that van der Waals isotherms