Question

Starting from known thermodynamic potentials, create a new potential that has independent variables of temperature, pressure, and chemical potential (T, P, mu), and create a Maxwell relation from your new potential involving entropy and particle number.

Answer 1

Our discussion of thermodynamic potentials dealt hitherto mostly with the equilibrium state. The second law of thermodynamics,

T dS ≥ δQ = dU + P dV − µdN ,

allows however also to study the impact of irreversible processes, that is the evolution of the system toward the equilibrium. We have used for the differential form and together with the generalized first law

dU = δQ + δW + µdN, δW = −P dV ,

where we have taken the δW appropriate for a gas.

Equilibrium internal energy. The entropy is an exact differential dS = δQ/T at equilibrium. One can then consider situations like

dU = T dS − P dV ⇒ dU = −P dV S = const. (work)

dU = T dS V = const. (heat)