You already know the ideal gas equation PV=nRT. It is clean, simple, and works beautifully for most gases at room temperature and atmospheric pressure. But if you squeeze a gas hard enough or cool it down enough, that neat equation starts to fail. The gas stops behaving "ideally." Why?
The ideal gas law assumes two things that are not true for real molecules. First, it assumes gas molecules are point particles — they take up zero volume. Second, it assumes there are no forces between the molecules — they just bounce off each other like perfect billiard balls. Real molecules have a finite size, and they do attract each other (weakly, but measurably). At high pressure, you push the molecules so close together that their own volume matters. At low temperature, the molecules move slowly enough that the attractive forces between them become significant.
So a real gas deviates from ideal behaviour in two opposite directions, depending on which correction dominates.
The two causes of deviation are:
- Finite molecular volume — molecules occupy space, so the free volume available for motion is less than the container volume.
- Intermolecular attraction — molecules pull on each other, reducing the impact force on the walls, so the measured pressure is less than the ideal pressure.
The van der Waals equation — one correction for each cause
Johannes van der Waals modified the ideal gas equation to account for both effects. The result is the van der Waals equation:
(P+V2an2)(V−nb)=nRT
Here a and b are constants specific to each gas. Let's see what each term does.
Volume correction (V−nb): The term nb is the total volume occupied by the molecules themselves. The free volume available for the gas to move in is not V but V−nb. So you replace V with V−nb. This correction increases the pressure compared to the ideal case — because the molecules are crammed into a smaller effective space.
Pressure correction (P+an2/V2): Molecules in the bulk of the gas are pulled inward by neighbours on all sides. But a molecule about to hit the wall feels a net pull backward into the gas, because there are no molecules outside the wall to pull it forward. This reduces the force of impact. The measured pressure P is therefore less than the ideal pressure. The correction term an2/V2 accounts for this: the ideal pressure would be P+an2/V2.
At high pressure, the volume correction (V−nb) dominates — the gas is harder to compress than an ideal gas. At low temperature, the pressure correction (P+an2/V2) dominates — the gas is easier to compress than an ideal gas. The two effects oppose each other.
The compressibility factor — a single number to measure deviation
Instead of working with the full van der Waals equation every time, we define a dimensionless quantity called the compressibility factor Z:
Z=nRTPV
For an ideal gas, Z=1 always. For a real gas:
- Z>1 means the gas is harder to compress than ideal (repulsive forces / finite volume dominate).
- Z<1 means the gas is easier to compress than ideal (attractive forces dominate). …