A system has internal energy U(S,V)= V-2/3S2, where λ is constant, V volume and S entropy. Which equation of state follows?

$P/(V^{1/3}T)=\mathrm{constant}$

GATE 2020 PH – Question 48

Thermodynamics and Statistical Mechanics · Laws of thermodynamics · 2 marks · Multiple choice

A system has internal energy $U(S,V)=\lambda V^{-2/3}S^2$, where λ is constant, V volume and S entropy. Which equation of state follows?

$PV^{1/3}/T^2=\mathrm{constant}$
$PV/T^{1/3}=\mathrm{constant}$
$P/(V^{1/3}T)=\mathrm{constant}$
$PV^{2/3}/T=\mathrm{constant}$

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Correct answer: (A) $PV^{1/3}/T^2=\mathrm{constant}$

Explanation

Thermodynamic derivatives give $T=(\partial U/\partial S)_V=2\lambda V^{-2/3}S$ and $P=-(\partial U/\partial V)_S=(2/3)\lambda V^{-5/3}S^2$. Eliminating S using $S=TV^{2/3}/(2\lambda)$ gives $P=T^2V^{-1/3}/(6\lambda)$. Hence $PV^{1/3}/T^2=1/(6\lambda)$, A.

GATE 2020 PH – Question 48

Explanation

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