Question

A thermodynamic system is described by the \( P, V, T \) coordinates. Choose the valid expression(s) for the system.

• A. \( \left( \frac{\partial P}{\partial V} \right)_T \left( \frac{\partial V}{\partial T} \right)_P = - \left( \frac{\partial P}{\partial T} \right)_V \)
• B. \( \left( \frac{\partial P}{\partial V} \right)_T \left( \frac{\partial V}{\partial P} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V \)
• C. \( \left( \frac{\partial V}{\partial T} \right)_P \left( \frac{\partial T}{\partial P} \right)_V = \left( \frac{\partial V}{\partial P} \right)_T \)
• D. \( \left( \frac{\partial V}{\partial T} \right)_P \left( \frac{\partial T}{\partial P} \right)_V = \left( \frac{\partial V}{\partial P} \right)_T \)

Hint:

In thermodynamics, Maxwell's relations provide identities between partial derivatives of state variables. Use these relations to derive various thermodynamic properties.

Answer 1

The Correct Option is A, C

Step 1: Understanding thermodynamic identities.
The relations between partial derivatives of state variables in thermodynamics are derived from Maxwell’s relations, which are based on the thermodynamic potentials. For a system with the \( P, V, T \) variables, the correct relations come from the use of the Helmholtz free energy or other thermodynamic potentials.

Step 2: Analyzing the options.
(A) \( \left( \frac{\partial P}{\partial V} \right)_T \left( \frac{\partial V}{\partial T} \right)_P = - \left( \frac{\partial P}{\partial T} \right)_V \): Correct. This is a valid thermodynamic identity derived from Maxwell’s relations.
(B) \( \left( \frac{\partial P}{\partial V} \right)_T \left( \frac{\partial V}{\partial P} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V \): Incorrect. This does not hold as a valid identity.
(C) \( \left( \frac{\partial V}{\partial T} \right)_P \left( \frac{\partial T}{\partial P} \right)_V = \left( \frac{\partial V}{\partial P} \right)_T \): Incorrect. This is not a valid thermodynamic identity.
(D) \( \left( \frac{\partial V}{\partial T} \right)_P \left( \frac{\partial T}{\partial P} \right)_V = \left( \frac{\partial V}{\partial P} \right)_T \): Incorrect. This does not hold as a valid identity.

Step 3: Conclusion.
The correct answer is (A) as it is a valid thermodynamic identity based on Maxwell’s relations.