139.

Derive an expression for the Vander Waal's equation of state.

Vander Waal in 1873, modified the general gas equation (PV = nRT) by introducing two correction terms:

(i) Volume correction. He pointed out that for real gases the volume of the gas molecules is not to be neglected in comparison to the total volume of the gas. This means that the molecules are not free to move in the whole of volume V but the free volume is less than the observed volume. In other words, the ideal volume of the gas is less than the observed volume. Vander Waal suggested that
V_ideal = (V – b) for one mole
where b is the correction term known as co-volume or excluded volume. It is related (not equal) to the actual volume of gas molecules. It has been found that b is about four times the actual volume of the gas molecules.
For n moles of a gas,
V_ideal = (V – nb)

(ii) Pressure correction. There is an attractive force of attraction between molecules of a gas.
A molecule (M) in the interior of the container is surrounded uniformly on all sides by other molecules, so there is no net force of attraction on it.
But when a molecule (N) is about to strike the wall of the container, the molecules are present only on one side of it will attract the striking molecule and pull it back. Thus, the molecule strikes the wall with a lesser force than it would have done if there were no attractive forces. Therefore, the observed pressure is less than the ideal pressure. Consequently, some correction factor must, therefore, be added to the observed pressure in order to calculate ideal pressure.

β is the correction term. The necessary correction term (β) will depend on upon the number of molecules in the surface layer and also the number of molecules in the body of the gas. Both these are directly proportional to the pressure and inversely proportional to the volume.
For 1 mole of a gas, Vander Waal suggested the factor 



 

Substituting the corrected values in PV = nRT, we have,



The constants a and b are called Vander Waal's constants.