(a) The curve is taken for , where it is convex. Its Legendre transform is the…

PHYSICS · Subjective · 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

(a) The curve y=x4y = x^4 is taken for x>0x > 0, where it is convex. Its Legendre transform is the intercept ϕ(s)=y−sx\phi(s) = y - sx of the tangent whose slope is s=dy/dxs = \mathrm{d}y/\mathrm{d}x. Show that ϕ(s)=−3(s/4)4/3\phi(s) = -3(s/4)^{4/3}, and show that x=−dϕ/dsx = -\mathrm{d}\phi/\mathrm{d}s recovers the original variable. (b) Starting from dF=−S dT−p dV\mathrm{d}F = -S\,\mathrm{d}T - p\,\mathrm{d}V and the equality of mixed partials, derive the Maxwell relation connecting (∂S/∂V)T(\partial S/\partial V)_T and (∂p/∂T)V(\partial p/\partial T)_V. Verify it for an ideal gas of NN molecules, using pV=NkBTpV = Nk_BT and the result that an isothermal change of volume alters its entropy by ΔS=NkBln⁡(V2/V1)\Delta S = Nk_B\ln(V_2/V_1). (c) A system is in thermal contact with a reservoir at fixed temperature TT and its volume is held fixed, so no work is done on it. In some change it receives heat QQ, the reservoir's entropy changes by −Q/T-Q/T, and the entropy of system plus reservoir cannot decrease. Show that the system's Helmholtz free energy F=U−TSF = U - TS obeys ΔF≤0\Delta F \le 0.

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