Van der Waal’s equation of state derivation
Experiments have shown that actual gases do not obey the ideal gas equation PV = RT. Even the so called permanent gases show marked departure when they approach critical conditions. While deriving the ideal gas equation, it has been assumed that the volume occupied by a molecule is negligible when compared to the volume of the gas and that the molecules exert no force on one another. van der Waal, argued that that these assumptions are not true for an actual gas and modified the ideal gas equation by making allowance for (1) the finite site of the molecule and (2) intermolecular force of attraction.
(1) Correction of the finite size of the molecule:
Assuming that each molecule of a gas has finite size, the molecule will occupy some space. Hence the volume actually available for the molecules to move freely will be less than the volume of the container. If V is the volume of the container, the effective volume available for the molecule to move about is (V — b); where b is the correction for the finite size of the molecule.
Assuming that each molecule of a gas has finite size, the molecule will occupy some space. Hence the volume actually available for the molecules to move freely will be less than the volume of the container. If V is the volume of the container, the effective volume available for the molecule to move about is (V — b); where b is the correction for the finite size of the molecule.
(2) Correction for intermolecular force of attraction:
A molecule like A well within the body of the gas is attracted equally in all directions by the surrounding molecules. Hence the resultant force on it is zero. A molecule like B near the wall of the container experiences an inward pull due to attraction of the molecules behind it. Hence the molecules strike the wall with reduced momentum and so the effective pressure of the gas is reduced. The true pressure inside the gas is obtained by adding a correction dp to the observed pressure P. The correction dp depends on (1) the number of molecules striking the wall and (2) the number of molecules attracting them backwards. Each of these factors depends on the number of molecules per unit volume of the gas; i.e., the density ρ of the gas.
A molecule like A well within the body of the gas is attracted equally in all directions by the surrounding molecules. Hence the resultant force on it is zero. A molecule like B near the wall of the container experiences an inward pull due to attraction of the molecules behind it. Hence the molecules strike the wall with reduced momentum and so the effective pressure of the gas is reduced. The true pressure inside the gas is obtained by adding a correction dp to the observed pressure P. The correction dp depends on (1) the number of molecules striking the wall and (2) the number of molecules attracting them backwards. Each of these factors depends on the number of molecules per unit volume of the gas; i.e., the density ρ of the gas.
The correction dp α ρ2 α 1/ V2; where V is the volume of the gas.
Therefore, dp = a/ V2; where a is a constant.
Corrected pressure = P + dp = P + (a/ V2)
Substituting the corrected values of P and V in the gas equation we get,
(P + (a/V2)) (V - b) = RT
This is known as van der Waal's equation of state for real gases. The constants a and b are called van der Waal's constants.