A student starts from and writes down ( S V )_ T = - ( p T )_ V (as claimed)

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 student starts from dF=−S dT−p dV\mathrm{d}F = -S\,\mathrm{d}T - p\,\mathrm{d}V and writes down (∂S∂V)T=−(∂p∂T)V(as claimed),\left(\frac{\partial S}{\partial V}\right)_{T} = -\left(\frac{\partial p}{\partial T}\right)_{V}\qquad\text{(as claimed)}, and concludes that an ideal gas loses entropy when it expands at fixed temperature. Identify the mistake in this step, write the relation with its correct sign, and evaluate (∂S/∂V)T(\partial S/\partial V)_{T} for an ideal gas of N=2.0×1023N = 2.0\times10^{23} molecules held at T=300 KT = 300\,\mathrm{K} in a volume V=5.0×10−3 m3V = 5.0\times10^{-3}\,\mathrm{m^{3}}. Take kB=1.381×10−23 J K−1k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}.

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