A closed system contains a single-phase substance that undergoes an infinitesimally small change at constant entropy (S) and pressure (P). The change is such that the entropy inventories of the two...

A closed system contains a single-phase substance that undergoes an infinitesimally small change at constant entropy (S) and pressure (P). The change is such that the entropy inventories of the two halves of the system become, respectively,
 (S + 𝛿S) and
 (S − 𝛿S), where S is the original (constant) entropy inventory of the entire system. Invoking the enthalpy minimum principle for constant S and P, show that the system is in a state of stable equilibrium if the system’s temperature increases during heating at constant pressure, in other words, if

c_P
> 0

Show also that the “positive c_P” requirement is consistent with the criteria for internal thermal and mechanical equilibria, c_v
> 0 and k > 0.