Kinetic Theory : Molecular Attraction

We have so far assumed that the forces exerted on one another by gas molecules are negligibly small. This is only approximately true. Gases are usually more compressible than according to Boyle's law, and this may be explained by assuming that the molecules attract one another, the attraction becoming greater the closer the molecules come together. When the gas is liquefied the molecular attraction is sufficient to prevent the molecules flying off into space, as they do in an open vessel. A liquid is much less compressible than a gas, and the compressibility of a gas also falls off considerably at high pressures. This effect is assumed to be due to the space occupied by the molecules, x; if this is comparable with the total space, v, only the intermolecular space (v - x) is available for compression.

These two factors are taken into account by the equation of Van der Waals, which replaces the ideal gas equation pv = RT by:

where a and b are constants. The term a/v² is the molecular attraction correction, which is inversely proportional to the square of the volume; it adds itself to the external pressure: b is the correction for the space occupied by the molecules. According to Van der Waals, b is equal to four times the total volume of the molecules, but it appears to be 4√2 times the latter. This equation gives good results with some gases (e.g., ethylene), but the attraction term depends on the temperature, hence D. Berthelot has used the equation:

with remarkably good results at moderate pressures, and it has been given a theoretical foundation by Keesom (1912). The constants in Van der Waals's equation are related to the critical constants as follows:

If we assume that the molecules are spherical and of radius r, we have

For carbon dioxide, b = 42.8 cm³ per mol, hence:

or d = 3.23 x 10^-3 cm., which is smaller than the value (3.39 x 10^-8) calculated from the viscosity.

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