Behaviour of Real Gases: Deviation from Ideal Gas Behaviour

Where the ideal picture starts to fail

The kinetic-molecular model matches experiment well overall, but trouble appears when we test how precisely $pV=nRT$ reproduces the actual pressure-volume-temperature behaviour of real gases. A convenient test is to plot $pV$ against $p$: for an ideal gas at constant $T$, Boyle's law says $pV$ is constant, so this plot should be a straight horizontal line. Fig. 5.10 shows this plot for several real gases at 273 K, using real data.

The real curves are not flat. Two patterns emerge: for H$_2$ and He, $pV$ rises steadily with increasing pressure — a positive deviation throughout. For gases like CO and CH$_4$, $pV$ first dips below the ideal line (a negative deviation), reaches a minimum, then rises steeply, crosses the ideal line, and continues upward as a positive deviation at higher pressures.

A second view, plotting pressure against volume (Fig. 5.11), shows the same story differently: at low pressure the real-gas and ideal-gas curves nearly coincide, but at high pressure the measured volume exceeds the volume Boyle's law would predict.

Why real gases deviate — the flawed assumptions

Two postulates of kinetic theory turn out not to hold exactly:

• (a) there is no force of attraction between gas molecules, and
• (b) the volume of the molecules themselves is negligible compared to the space they occupy.

If (a) were strictly true, gases would never liquefy — yet they clearly do, on cooling and compression. If (b) were strictly true, the measured and Boyle's-law-predicted $p$–$V$ curves would coincide exactly — they do not.

Attraction correction: at high pressure, molecules are close enough for attractive forces to act; molecules heading for the wall get "pulled back" slightly by their neighbours, so they strike with less than full force. The observed pressure is therefore lower than the ideal pressure by a correction term:

Volume correction: repulsive forces (short-range, significant only when molecules are nearly touching) make the molecules behave like small, incompressible spheres, so the volume actually available for free movement is not $V$ but $(V - nb)$, where $nb$ approximates the volume taken up by the molecules themselves.

Applying both corrections to the ideal gas equation gives the van der Waals equation:

Here $a$ and $b$ are the van der Waals constants, characteristic of each gas — $a$ measures the strength of intermolecular attraction (independent of temperature and pressure). Real gases also deviate more strongly at very low temperature, since slow-moving molecules are more easily captured by attractive forces. Real gases approach ideal behaviour as pressure approaches zero.

The compressibility factor

Deviation from ideality is conveniently measured by the compressibility factor:

For an ideal gas, $Z=1$ at every temperature and pressure, giving a flat horizontal line on a $Z$ vs $p$ plot (Fig. 5.12). Real gases deviate from $Z=1$: at very low pressure nearly all gases have $Z\approx1$ (behaving ideally); at high pressure all gases show $Z>1$ (harder to compress than an ideal gas); at intermediate pressures most gases show $Z<1$. Gases behave most ideally when pressure is low enough that the molecular-volume correction becomes negligible. …