The equation of state for a real gas

The most common form and the most accurate simple equation for the behaviour of a real gas is that proposed by Johannes van der Waals in 1872.

He modified the ideal gas equations to allow for the fact that two of the assumptions made in their derivation may not be correct, that is:
(a) the volume of the molecules may not be negligible when compared with the volume of the gas, and
(b) the forces between the molecules may not be negligible.

Clearly both these effects become much more noticeable at high pressures and small volumes when the molecules are packed tightly together.

Consider first the volume of the molecules. The actual volume given in the equation must be reduced because the number of collisions will be greater. The equation becomes:

P(V - b) = RT

where b is the effective volume of the molecules.
(It has been found that b is about 4.2 x the volume of the molecules.)

Considering now the attractive forces between the gas molecules, you can see that the pressure in the body of the gas is higher than that at the edges since molecules are pulled back into the centre by other molecules. We assume the attraction to be proportional to the number of molecules per unit volume (that is, to N/V).

The number of impacts per second is also affected and both these numbers are proportional to the density. So

pressure reduction = a/V²

where a is a constant. Taking both these corrections into consideration, van der Waals' equation for one mole of a gas thus becomes:

Van der Waals' equation: (P + a/V²)(V - b) = RT

where the observed pressure and volume are P and V.

Van der Waals' equation fits the isothermals of actual gases above the critical temperature, but below this the equation must be modified considerably.