Deviation from Ideal Behaviour and the van der Waals Equation

The ideal gas equation, $PV = nRT$, describes real gases only approximately — and the approximation breaks down

noticeably at high pressure and low temperature. This deviation is neatly captured by the

compressibility factor, $Z = \dfrac{PV}{nRT}$. For an ideal gas, $Z = 1$ under all conditions, by definition.

For a real gas, $Z$ deviates from $1$: at very low pressures, $Z \approx 1$ (real gases behave almost ideally when

the molecules are far apart); as pressure rises to moderate values, $Z$ typically dips below $1$, because

intermolecular attractive forces pull the molecules closer together than the ideal-gas model predicts, reducing

the observed volume (or pressure) below the ideal value; at high pressure, $Z$ rises above $1$, because now

the finite volume actually occupied by the molecules themselves becomes a significant fraction of the total gas

volume, and the molecules literally cannot be compressed into an arbitrarily small space. This produces a

characteristic dip-then-rise curve when $Z$ is plotted against $P$ for a real gas, in contrast to the flat,

horizontal line at $Z=1$ for an ideal gas.

The kinetic theory's postulates fail for real gases in exactly two ways, and these are precisely the two

deviations just described: real gas molecules do occupy a small but non-zero volume (postulate 2 fails), and real

gas molecules do exert weak attractive forces on one another (postulate 3 fails). Johannes van der Waals corrected

the ideal gas equation for both effects, producing the van der Waals equation for $n$ moles of a real gas:

The pressure correction term, $\dfrac{an^2}{V^2}$, is added to the measured pressure $P$: because

intermolecular attractions pull molecules near the container wall slightly inward (away from the wall), a real gas

strikes the walls slightly less forcefully than an ideal gas would at the same conditions, so its measured

pressure is somewhat lower than the "true" pressure the molecules would exert without attraction; adding back

$\dfrac{an^2}{V^2}$ compensates for this. The constant $a$ measures the strength of intermolecular attraction —

larger for gases with stronger dispersion forces or permanent dipoles (e.g. $\text{NH}_3$ or $\text{CO}_2$),

smaller for weakly interacting gases (e.g. $\text{H}_2$, $\text{He}$).

The volume correction term, $nb$, is subtracted from the measured volume $V$: since the molecules themselves …