(a) For the form , with and independent variables, and , evaluate and state whether…

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) For the form

M\,\mathrm{d}x+N\,\mathrm{d}y

, with

x

and

y

independent variables,

M=3x^2y

and

N=x^3+2y

, evaluate

(\partial M/\partial y)_x-(\partial N/\partial x)_y

and state whether the form is exact. (b) Write the differential of the Helmholtz free energy

F

in its natural variables and read off

(\partial F/\partial V)_T

. (c) A fixed quantity of

N=3.01\times10^{23}

molecules of an ideal gas sits in a cylinder in thermal contact with a reservoir at

T=320\,\mathrm{K}

, so the temperature stays fixed. The gas expands from

V_1=2.00\,\mathrm{L}

V_2=6.00\,\mathrm{L}

. Find

\Delta F

for the gas and the greatest work it can deliver to its surroundings.

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