Van der Waals Equation

J. D. van der Waals carried out the first serious mathematical treatment of real-gas behaviour, by identifying exactly which two ideal-gas assumptions fail and building a correction term for each into the ideal gas equation $PV=nRT$: a pressure correction and a volume correction.

Pressure correction. Gas pressure arises from molecules striking the container wall. But a molecule heading toward the wall is slowed very slightly by the attractive pull of its neighbours behind it (Figure 6.10) -- a molecule deep inside the gas feels balanced attraction from every direction and is unaffected, but one near the wall, about to collide with it, feels a net inward pull. So it strikes the wall a little less forcefully than it would if there were no attraction at all, and the measured pressure of a real gas is somewhat lower than the true "ideal" pressure it would exert with no attraction.

Van der Waals found this attractive effect near the wall to be proportional to the square of the gas density $\rho=n/V$:

where $a$ is a proportionality constant (the first van der Waals constant) whose value depends on how strongly a particular gas's molecules attract one another. The "ideal" pressure the corrected equation should use is therefore the measured pressure plus this correction:

Volume correction. Because each molecule of a real gas occupies some actual volume, the space genuinely available for molecules to move around in is less than the full container volume V. Van der Waals worked this correction out explicitly by modelling gas molecules as hard spheres of radius r. Two such spheres cannot have their centres closer than $2r$ apart (Figure 6.11), so the "excluded volume" around any one molecule -- the region into which a second molecule's centre simply cannot enter -- is a sphere of radius $2r$:

where $V_m$ is the volume of a single molecule. Since this excluded volume is shared between the two molecules of the colliding pair, the excluded volume attributable to a single molecule is half of that, $4V_m$. For n moles of gas, the total excluded volume is $n(4V_m)=nb$, where $b=4V_m$ is the second van der Waals constant (a molar "co-volume"). So the volume genuinely available to the gas is

$V_{ideal}=V-nb\qquad(6.21)$ …