Derive the van der Waals equation of state. Explain the importance of van der Waals' gas equation.

Explanation

Vander Waal.s equation of state : Vander Waal.s proposed an approximate equation of state which involves the intermolecular interactions that contribute to the deviations of a gas from perfect gas law. It may be explained as follows. The repulsive interactions between two molecules cannot allow them to come closer than a certain distance. Therefore, for the gas molecules the available volume for free travel is not the 1. volume of the container V but reduced to an extent proportional to the number of molecules present and the volume of each exclude. Therefore, in the perfect gas equation a volume correction is made by changing v to (v-nb). Here, .b. is the proportionality constant between the reduction in volume and the amount of molecules present in the container.
` P=(nRT)/(V-nb)`
If pressure is low, the volume is large compared with the volume excluded by the molecules (V ` gt gt ` nb). The nb can be neglected in the denominator and the equation reduces to the perfect gas equation of state.
The effect of attractive interactions between molecules is to reduce the pressure that the gas exerts. The attraction experienced by a given molecule is proportional to the concentration / of molecules in that container. As the attractions slow down the molecules, the molecules strike the waals less frequently and strike with a weaker impact. Therefore, we can expect the reduction.in pressure to be proportional to the square of the molar-concentration, one factor of W showing the reduction in frequency of collisions and the other factor the reduction in the strength of their impulse.
Reduction in pressure` prop (n/v)^2`
Reduction in pressure = a. `(n/v)^2`
Where a = the proportionality constant: Vander Waals equation is
`(P +(a n^2)/(V^2)) (V- nb ) = nRT`
` therefore P=(nRT) /(V-nb ) - a(n/v)^2`
` therefore (P +(a n^2)/( V^2) ) (v-nb ) = nRT` The equation is called Vander Waals equation of state. The constants .a. and .b. known as Vander Waals parameters (or) empirical parameters. They depend on the nature of the gas independent of temperature.
Significance :-
` to` .a. is the measure of magnitude of inter molecular forces (attractive) with in the gas and is independent of temperature and pressure. If .a. value is high the gas can be easily liquified.
` to ` .b . is the effective volume of the gas molecule. It indicates the effective size of the gas molecules If the value of .b. is constant over a long range of temperature and pressure then the gas cannot be compressed easily.