van der waals equation

Aug 26, 2022, 16:45 IST

van der waals equation derivation

The ideal gas laws are derived from the kinetic theory of gases which is based on the following two important assumptions:

(i)The volume occupied by the molecules is negligible in comparison to the total volume of the gas.

(ii)The molecules exert no forces of attraction upon one another. 

It is because neither of these assumptions can be regarded as applicable to real gases that the real gases show departure from the ideal behaviour.

Van der waal Equation derivation 

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.

Volume Correction in Van der waal Equation

V in the ideal gas equation represents an 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 Vi is given by  V = V container  nbwhere 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 given in pdf belwo 

Pressure Correction in Van der waal Equation

Consider a molecule A in the bulk of a vessel as shown in fig. This molecule issurrounded 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 of an ideal gas, i.e.     pi = pr + correction term

This correction term depends upon two factors:

(i) The number of molecules per unit volume of the vessel :- larger this number, larger will be the force of attraction with which the molecule B is dragged behind. This results in a greater decrease in the velocity of the molecule B and hence a greater decrease in the rate of change of momentum. Consequently, the correction term also has a large value. If n is the amount of the gas present in the volume V of the container

(ii) The number of molecules striking the side of the vessel per unit time:- Larger this number, larger will be the decrease in the rate of change of momentum. Consequently, the correction term also has a larger value. Now, the number of molecules striking the side of vessel in a unit time also depends upon the number of molecules present in unit volume of the container, and hence in the present case:

• The constants a & b: 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 Vander Waal’s constant of attraction ‘a’ is always greater than Vander Waals  constant of excluded volume (b).

ii) The gas having higher value of ‘a’ can be liquefied easily and therefore H2 & He are not  liquefied easily.

iii) The units of a = litre2 atm mole–2 & that of b = litre mole –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 molecular size and larger molecule will have larger b.