Indian Forest Service (Mains), Chemistry optional · 2023 · Paper I

Q1(d). Derivatives of thermodynamic potentials

Question

Derive the following thermodynamic relations: (i) S = −(∂A/∂T)_V, (ii) P = −(∂A/∂V)_T, (iii) S = −(∂G/∂T)_P, (iv) V = (∂G/∂P)_T.

For a closed simple compressible system of fixed composition, dA = −S dT − P dV and dG = −S dT + V dP; their coefficients give all four relations.

Combine the first and second laws for reversible changes: dU = T dS − P dV.
Helmholtz energy is A = U − TS. Hence dA = dU − T dS − S dT = −S dT − P dV.
Since A = A(T, V), compare its total differential to obtain (∂A/∂T)_V = −S and (∂A/∂V)_T = −P.
Gibbs energy is G = A + PV. Thus dG = dA + P dV + V dP = −S dT + V dP.
With G = G(T, P), (∂G/∂T)_P = −S and (∂G/∂P)_T = V.

The full model answer

As written for the answer booklet: the diagram carries the working, the lines stay short. Read here only; it is not offered as a download.

CoversHelmholtz energyGibbs energyfundamental equationsthermodynamic relationsMaxwell relations