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Ratio of isothermal and isentropic compressibilities
Finite whole-system compressibility and heat-capacity ratio on a positive-temperature C2 simple-compressible chart at fixed composition with valid isothermal and isentropic response inverses, positive isothermal compressibility, and positive constant-volume heat capacity.
Statement
On an open C2 Helmholtz chart at fixed composition , assume , , , and valid local response inverses for both the isothermal and isentropic compressibilities. Then , , and
The heat capacities are whole-system values; is defined in Isentropic compressibility on a valid local chart. The hypotheses exclude zero-capacity and singular-chart limits.
Facts & Assumptions
Given: The valid fixed-composition charts and positivity assumptions in the Statement.
and are explicit response hypotheses on this chart; they are not inferred from notation alone.
Thermodynamic response functions on valid local charts: , , , and on the valid response chart; these definitions assert no identities between response functions.
Isentropic compressibility on a valid local chart: on a valid isentropic inverse chart.
Differentials of the thermodynamic potentials: and on the named valid charts.
Heat-capacity relation on a stable response chart: on the stated stable response chart.
The chain rule for total derivatives: : Differentiating a composition gives the usual total-derivative chain rule.
Proof
By [F1], . On the fixed-volume path in the Helmholtz chart, [F3] gives ; composing with the chart map and applying [F6] yields . Likewise, by [F1], , and on the fixed-pressure path in the Gibbs chart [F3] gives ; the chain rule yields . Thus at fixed , . Along an isentrope , the valid chart gives .
Differentiate along the same isentrope using [F6]: .
On the valid isothermal response inverse, . Differentiating this identity in at fixed gives ; by [F1] and , . Differentiating it in at fixed gives , so [F1] yields . Thus [F5] is equivalently . Substitution into step 2.1 yields . Since and , [F5] also gives , hence .
On the valid isentropic inverse , the identity gives by [F6]. Hence the definitions [F1]–[F2] give and . By step 3.1, . All denominators are nonzero by the valid-chart and positivity hypotheses.