Van der Waal was the first to introduce systematically the correction terms due to the above two invalid assumptions in the ideal gas equation `PV = nRT` . His corrections are given below.

`V` in the ideal gas equation represents the volume where the molecules can move freely. In real gases, a part of the total volume is, however, occupied by the molecules of the gas . If `b` represents the effective volume occupied by the molecules of `1` mole of a gas, then for the amount `n` moles of the gas `V`; is given by `V = V_(text(container)) -nb`

where `b` is called the excluded volume or co-volume. The numerical value of `b` is four times the actual volume occupied by the gas molecules. This can be shown as follows. If we consider only bimolecular collisions, then the volume occupied by the sphere of radius `2r` represents the excluded volume per pair of molecules as shown in Fig

Thus excluded volume per pair of molecules

`= 4/3 pi (2r)^3=8(4/3 pi r^3)`

Excluded volume per molecule

`=1/2[8(4/3 pi r^3)] -4(4/3 pi r^3) = 4xx` (volume occupied by a molecule)

Since `b` represents excluded volume per mole of the gas, it is obvious that `b= N_A[4(4/3 pi r^3)]`

Consider a molecule `A` in the bulk of a vessel as shown in fig. This molecule is surrounded by other molecules in symmetrical manner, with the result that this molecule on the whole experiences no net force of attraction. Now, consider a molecule `B` near the side of the vessel, which is about to strike one of its sides, thus contributing towards the total pressure of the gas. There are gas molecules only on one side of the vessel, i.e. towards its centre, with the result that this molecule experiences a net force of attraction towards the centre of the vessel. This results in decreasing the velocity of the molecule, and hence its momentum. Thus, the molecule does not contribute as much force as it would have, had there been no force of attraction. Thus, the pressure of a real gas would be smaller than the corresponding pressure or an ideal gas, i.e. `p_i = P_r + text(correction term)`

This correction term depends upon two factors :

When these expressions are substituted in the ideal gas equation `p_i V_i = nRT`, we get

`(p+ (n^2 a)/V^2)(V - nb) = nRT`

This equation is applicable to real gases and is known as the Van der Waals equation.

Van der Waals constant for attraction (`a`) and excluded volume (`b`) are characteristic for a given gas. Some salient features of `a` & `b` are :

i) For a given gas Van der Waal's constant of attraction `'a'` is always greater than Van der Waals constant of excluded volume `b`.

ii) The gas having higher value of `'a'` can be liquefied easily and tl1erefore `H_2` & `He` are not liquefied easily.

iii) The units of `a = text(litre)^2 atm mol^(-2)` & that of `b = text(litre) mol^(-1)`

iv) The numerical values of `a` & `b` are in the order of `10^(-1)` to `10^(-2)` & `10^(-2)` to `10^(-4)` respectively.

v) Volume correction factor, depends on molecuJar si.ze and larger molecule will have larger `b`.

For example, size of `He`, `CH_4`, `CF_4`, `C_4H_10` are in order of `He < CH_4 < CF_4 < C_4H_10` and same will be the order of `b`.

vi) Pressure correction factor `((n^2 a)/V^2)` depends on intermolecular force of attraction. Hence, larger the intermolecular force of attraction larger the value of `'a'`, for same `n` and `V`. For example, intermolecular force of attraction among the molecules `H_2`, `CO_2`, `NH_3` are in order of `H_2 < CO_2 < NH_3` (`H`-bonding) thus same is the order of `a`.