Question

Determine the following when a gas obeys Van der Waals’ equation,

\(\left(p+\cfrac{a}{v^2}\right)\) (v – b) = RT

(i) Change in internal energy ; 

(ii) Change in enthalpy ; 

(iii) Change in entropy.

Answer 1

(i) Change in internal energy : 

The change in internal energy is given by

(ii) Change in enthalpy : 

The change in enthalpy is given by

Let us consider p = f(v, T)

From equation (1),

Substituting the value of (dp)_T from eqn. (2), we get

Using the cyclic relation for p, v, T which is

Substituting this value in eqn. (3), we get

For Van der Waals equation

Substituting the values of eqns. (5) and (6) in equation (1), we get

(iii) Change in entropy : 

The change in entropy is given by

ds = cp \(\cfrac{dT}T\) = \(\left(\cfrac{\partial p}{\partial T}\right)_v\). dv

For Van der Waals equation,

\(\left(\cfrac{\partial p}{\partial T}\right)_v\) = \(\cfrac{R}{v-b}\) ...as per eqn. (6)