Heat Capacities

Consider a gas with arbitrary equation of state P = f (τ, V), at a temperature τ, τ < τ_cr, where τ_cr is a critical temperature of this gas.

a) Calculate C_P - C_V for this gas in terms of f. Does C_P - C_V always have the same sign?

b) Using the result of (a), calculate C_P - C_V for one mole of a van der Waals gas.

SOLUTION

From the definition of C_P for a gas,

(1)

Since we are interested in a relation between C_P and C_V it is useful to transform to other variables than in (1), namely τ, V instead of τ, P. We will use the Jacobian transformation:

(2)

A useful identity is obtained from

(3)

(4)

So

(5)

Since

and

b) Let us write the van der Waals equation for one mole of the gas in the form

(6)

from which we obtain

(7)

Substituting for P in (5) yields

We can see that C_P – C_V = N_A (in regular units C_P – C_V = R) for an ideal gas where a = b = 0.