Page 27 - Instant notes

Non-ideal gases 13

Numerous attempts have been made to modify the perfect gas equation of state in order
to describe real gases. The two most significant equations are the virial equation and the
Van der Waals equation of state.

The virial equation

The virial equation is primarily a mathematical attempt to describe the deviation from
ideality in terms of powers of the molar volume, V m. It takes the form:

The coefficients B, C, D, etc. are the virial coefficients. The expression converges very
rapidly, so that B>C>D, and the expression is usually only quoted with values for B and
C at best. For a gas at low pressure, V m is large, making the second and subsequent terms
very small, reducing the equation to that of the perfect gas equation of state. For an ideal
gas, B, C, D, etc. are equal to zero, and the equation again reduces to that for an ideal gas.
Although the virial equation provides an accurate description of the behavior of a real
gas, the fit is empirical, and the coefficients B, C, D, etc, are not readily related to the
molecular behavior.

The van der Waals equation of state

The van der Waals equation of state attempts to describe the behavior of a non-ideal
gas by accounting for both the attractive and repulsive forces between molecules. The
equation has the form:

The coefficients a and b are the van der Waals parameters, and have values which
convey direct information about the molecular behavior.
The term (V−nb) models the repulsive potential between the molecules. This potential
has the effect of limiting the proximity of molecules, and so reducing the available
volume. The excluded volume is proportional (through the coefficient, b) to the number
2
of moles of gas, n. The term (p+a(n/V) ) reflects the fact that the attractive potential
reduces the pressure. The reduction in pressure is proportional to both the strength and
number of molecular collisions with the wall. Because of the attractive potential, both of
these quantities are reduced in proportion to the density of the particles (n/V), and the
2
overall pressure reduction is therefore a(n/V) where a is a constant of proportionality.
At high temperatures and large molar volumes (and therefore low pressures), the
correction terms become relatively unimportant compared to V and T and the equation
reduces to the perfect gas equation of state.