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
29-8-2016
1467
Heat Capacities
Consider a gas with arbitrary equation of state P = f (τ, V), at a temperature τ, τ < τcr, where τcr is a critical temperature of this gas.
a) Calculate CP - CV for this gas in terms of f. Does CP - CV always have the same sign?
b) Using the result of (a), calculate CP - CV for one mole of a van der Waals gas.
SOLUTION
From the definition of CP for a gas,
(1)
Since we are interested in a relation between CP and CV 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 CP – CV = NA (in regular units CP – CV = R) for an ideal gas where a = b = 0.
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