The P = P(v, T) equation of state of a van der Waals gas is in which R, a, and b are three…

The P = P(v, T) equation of state of a van der Waals gas is

in which R, a, and b are three constants.

(a) Prove that if cv is also constant the
“caloric” equation of state u = u(T, v) of the same gas is

where u0 = u(T0, v0).

(b) Derive the expression for the entropy function s = s(T,
v), which is valid under the same conditions.

(c) Combining this last result with dh = T ds + v dP, derive
the corresponding expression for enthalpy, h = h(T, v).