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The ideal gas law assumes that gas molecules do not have volume. This approximation is useful at low pressures, as the volume of the container is much larger than the volume taken up by the molecules themselves. At high pressures, however, the volume of the molecules becomes significant, and the ideal gas equation will predict a larger volume than is actually there. At this point, the volume term in the Ideal Gas Equation must be corrected to account for the space taken up by the gas molecules:
The (bn) correction factor is simply the number of moles of gas multiplied by a constant that is related to the molecular volume. Thus, the corrected volume term in the "modified" ideal gas equation indicates the free volume available to the gas molecules (as it should).
The second assumption required by the ideal gas equation is that gases do not interact with each other. Gas pressure is a measure of the force of the gas molecules on the walls of the container. If the gas molecules are attracting each other, they will more likely collide with each other than the container. This interaction will make the measured pressure appear to be less than the pressure predicted from the ideal gas law. As with the "volume effect" described above, this effect is also observed only at high pressures. The appropriate correction for this effect is given below:
The correction for the pressure term is not as bad as it looks. The term (n/V) is the concentration of the gas. This value is squared to indicate that two molecules participate in any intermolecular force. The "a" value is just a proportionality constant that is determined experimentally.
The above equation is called the van der Waals equation, and it is mostly applied in high-pressure or low-temperature situations.