Real gases and the van der Waals equation

The ideal gas law, PV = nRT, rests on two simplifications: molecules are treated as dimensionless points with no volume of their own, and they are assumed never to interact except in perfectly elastic collisions. Real molecules fail both assumptions — they do occupy actual space, and they do attract each other weakly through intermolecular forces — and the van der Waals equation is the ideal gas law with a correction term added for each.

The volume correction subtracts a term accounting for the space molecules themselves take up, since the volume actually available for the gas to move through is less than the container's total volume once real molecular size is counted. The pressure correction adds a term accounting for the weak mutual attraction between molecules, which pulls them slightly toward each other and so reduces the force with which they strike the container walls compared to what non-interacting point particles would produce — real pressure ends up slightly lower than the ideal prediction for the same density, unless corrected for.

Both effects are negligible when a gas is spread out and moving fast: at low pressure, molecules are far apart on average, so their finite size barely matters; at high temperature, they move quickly enough that the weak attractive forces have little time to act before a collision separates them again. It is precisely the opposite regime — high pressure, low temperature — where molecules are forced close together and moving slowly, that both corrections become significant, and not coincidentally, this is also the regime approaching condensation into a liquid. The van der Waals equation is in fact good enough to predict roughly where that transition happens, something the ideal gas law, with no attraction between molecules at all, cannot capture even qualitatively — an ideal gas can never condense, because nothing in its model gives molecules any reason to stick together.