Return of Heat Capacities

In a certain range of temperature τ and pressure p, the specific volume v of a substance is described by the equation

where v₁, τ₁, P₁ are positive constants. From this information, determine (insofar as possible) as a function of temperature and pressure the following quantities:

SOLUTION

a) We will again use the Jacobian transformation to find c_p - c_v as a function of τ, P.

(1)

where we used

So, we obtain

(2)

Substituting v(τ, P) into (2) yields

(3)

b,c) We cannot determine the temperature dependence of c_p or c_v, but we can find c_p (P) and c_v (v), as follows:

(4)

Similarly,

(5)

where F is the Helmholtz free energy, and we used

From (4) and the equation of state, we have

(6)

and from (5),

(7)

(since v = const implies τ/τ₁ = P/P₁). Integrating (6) and (7), we obtain

(8)

and

(9)

where f₁ and f₂ are some functions of temperature. Since we know c_p - c_v from (a), we infer that f₁ = f₂ ≡ f, and finally

(10)