Return of Heat Capacities
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 26
3-9-2016
1311
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 cp - cv 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 cp or cv, but we can find cp (P) and cv (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 cp - cv from (a), we infer that f1 = f2 ≡ f, and finally
(10)
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