For the entropy expressed as s(T,v), the total differential is:
The coefficient of dT is CV/T, as can be demonstrated from du = Tds - pdv by dividing by dT while holding v constant. The coefficient of dv is eliminated using the Maxwell relation Eq (6.19), leading to the final result:
The total differential of h is:
Starting from s(T,p) yields an alternative entropy differential:
The fundamental differential dh = Tds + vdp gives CP/T as the coefficient of dT and use of the Maxwell relation Eq (6.20) gives the coefficient of dp. The end result is:
When integrated for an ideal gas with constant heat capacity, Eqs (6.23b) and (6.24b) reduce to Eqs (3.9) and (3.10), respectively. The first pair of equations apply to any one-component substance, but to integrate them, a path v = F(T) or p = G(T) must be specified. The resulting change in entropy, however, is independent of the path chosen.
For solids, the coefficients of dv in Eq (6.23b) and dp in Eq (6.24b) are best replaced by a/p and av, respectively.
Equations (6.23b) and (6.24b) are sometimes called the "ds" equations. There is a third variant in which the independent variables are p and v. This equation is derived in problem 6.6.
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