van der Waals Equation of State

VAN DER WAALS EQUATION OF STATE

The Ideal Gas Law, PV = nRT, can be derived by assuming that the molecules that make up the gas have negligible sizes, that their collision with themselves and the wall are perfectly elastic, and that the molecules have no interactions with each other.
The van der Waal's equation is a second order approximation of the equation of state of a gas that will work even when the density of the gas is not low.

Here a and b are constants particular to a given gas.

Some van der Waals Constants

Substance a
(J^.m³/mole²) b
(m³/mole) P_c
(MPa) T_c
(K)

Air .1358 3.64x10^-5 3.77 133 K

Carbon Dioxide (CO₂) .3643 4.27x10^-5 7.39 304.2 K

Nitrogen (N₂) .1361 3.85x10^-5 3.39 126.2 K

Hydrogen (H₂) .0247 2.65x10^-5 1.30 33.2 K

Water (H₂O) .5507 3.04x10^-5 22.09 647.3 K

Ammonia (NH₃) .4233 3.73x10^-5 11.28 406 K

Helium (He) .00341 2.34x10^-5 0.23 5.2 K

Freon (CCl₂F₂) 1.078 9.98x10^-5 4.12 385 K

The parameter b is related to the size of each molecule. The volume that the molecules have to move around in is not just the volume of the container V, but is reduced to ( V - nb ).
The parameter a is related to intermolecular attractive force between the molecules, and n/V is the density of molecules. The net effect of the intermolecular attractive force is to reduce the pressure for a given volume and temperature.
When the density of the gas is low (i.e., when n/V is small and nb is small compared to V) the van der Waals equation reduces to that of the ideal gas law.
One region where the van der Waals equation works well is for temperatures that are slightly above the critical temperature T_c of a substance

Observe that inert gases like Helium have a low value of a as one would expect since such gases do not interact very strongly, and that large molecules like Freon have large values of b.
There are many more equations of state that are even better approximation of real gases than the van der Wall equation.

Substance	a (J^.m³/mole²)	b (m³/mole)	P_c (MPa)	T_c (K)
Air	.1358	3.64x10^-5	3.77	133 K
Carbon Dioxide (CO₂)	.3643	4.27x10^-5	7.39	304.2 K
Nitrogen (N₂)	.1361	3.85x10^-5	3.39	126.2 K
Hydrogen (H₂)	.0247	2.65x10^-5	1.30	33.2 K
Water (H₂O)	.5507	3.04x10^-5	22.09	647.3 K
Ammonia (NH₃)	.4233	3.73x10^-5	11.28	406 K
Helium (He)	.00341	2.34x10^-5	0.23	5.2 K
Freon (CCl₂F₂)	1.078	9.98x10^-5	4.12	385 K