Vanderwall’s Equation

The Ideal gas equation, PV = nRT, is not obeyed by the real gas at high pressure & low temperatures. In order to get an equation which is obeyed by the real gases over wide ranges of temperatures and pressure, appropriate corrections have to be applied so as to take into account.

(1). The volume of the gas molecules, and

(2). The forces of attraction between the gas molecules.

A number of attempts have been made to get such equations. However, the earliest and the best known so for is the one given by J. D Vanderwaals in 1873. He applied the corrections as fallows;

Volume Correction

Suppose ‘V’ is the actual volume occupied by one mole of gas molecules. Then since the gas molecules are in motion, it has been found that the effective volume occupied by the gas molecules is four times the actual volume i.e. equal to ‘4V’. Let it be represented by ‘b’. The volume ‘b’ is called Co-Volume or Excluded Volume. Thus the free space available for the movement of the gas molecules or for compression of gas is (v–b). Hence in Ideal gas equation ‘V’ should be replaced by(v–b).

Pressure Correction

A gas molecules lying in the interior of the gas is attracted by all other gas molecules surrounding it. Hence the net force of attraction exerted on such a molecule by the other molecules is zero. However, for a molecule lying near the wall of the container, the molecules lying inside the bulk of gas exert some inward pull. Thus the effect of such an inward pull is “dragging back” of the molecule. Consequently, the pressure with which the gas molecule strikes the wall of the vessel is less than the pressure that would have been exerted if there were no such inward pull. In other words, the observed pressure is less than the ‘Ideal pressure’. Thus Ideal pressure would be greater than observed pressure by a factor p, hence,

Corrected Pressure =P+P

Where ‘p’ is the ‘corrected factor’ due to inward pull

On the basis of calculation’s it has been observed that pressure correction factor (p) is equal to a*(n/V) ². Where ‘n’ is number of moles, v is volume and ‘a’ is constant depends upon the nature of gas.

For 1 mole

P = a/v²

Correct pressure=P + a/v²

Putting the corrected values of volume and pressure in Ideal gas equation PV = nRT for 1 mole of gas, the equation is modified to

(P + (a/v²))(v – b)= RT

This equation is known as Vanderwall’s Equation. The constants ‘a’ and ‘b’ are called Vanderwall’s Constants whose value depends upon nature of gas.

For n moles of gas

(P + (an²/v²)) (v – nb)=nRT.