The thermodynamic identity for a closed system of fixed particle number is
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Thermodynamic potentials and the formal structure of thermodynamics · Thermodynamic potentials and the formal structure of thermodynamics
Thermal Physics, Ch Thermodynamic potentials, Legendre transforms and Maxwell relations
Problem
The thermodynamic identity for a closed system of fixed particle number is dU=TdS−pdV. The Gibbs free energy is G=U−TS+pV.
((a)) Show, naming the two Legendre transforms involved (one in S, one in V), that dG=−SdT+Vdp, and read off S and V as derivatives of G.
((b)) Using the equality of the mixed second partial derivatives of G, derive the Maxwell relation connecting (∂S/∂p)T to a derivative of the volume. Evaluate both sides of your relation for one mole of an ideal gas at T=300.0K and p=1.013×105Pa (take NkB=R=8.314JK−1mol−1).
((c)) A sample sits in a bath that holds its temperature at T and its pressure at p, and the bath is large enough that its own T and p do not change. The heat delivered to the sample in a change is Q, so the bath's entropy changes by −Q/T, and at constant pressure Q=ΔH. Show from the entropy increase principle that any change of the sample obeys ΔG≤0.
((d)) Supercooled liquid mercury is held at T=230.0K and p=1.013×105Pa. For the freezing of one mole, the values at the melting point are ΔH=−2.295kJmol−1 and ΔS=−9.794JK−1mol−1; take both as constant over this small temperature range. Compute ΔG per mole, keeping four figures in the two terms, and say whether the mercury freezes.