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GATE 2021 ME (ME1) – Question 39

Thermodynamics · Thermodynamic relations and property charts · 2 marks · Multiple choice

The fundamental thermodynamic relation for a rubber band is given by $dU=T\,dS+\tau\,dL$, where $T$ is the absolute temperature, $S$ is the entropy, $\tau$ is the tension in the rubber band, and $L$ is the length of the rubber band. Which one of the following relations is CORRECT:

  1. $\tau=(\partial U/\partial S)_L$
  2. $(\partial T/\partial L)_S=(\partial\tau/\partial S)_L$
  3. $(\partial T/\partial S)_L=(\partial\tau/\partial L)_S$
  4. $T=(\partial U/\partial S)_\tau$

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Correct answer: (B) $(\partial T/\partial L)_S=(\partial\tau/\partial S)_L$

Explanation

From U(S,L), $T=(\partial U/\partial S)_L$ and $\tau=(\partial U/\partial L)_S$. Equality of mixed derivatives gives $\partial^2U/\partial L\partial S=\partial^2U/\partial S\partial L$.

Hence $(\partial T/\partial L)_S=(\partial\tau/\partial S)_L$, **B**.