Question

For a pure substance, the Maxwell's relation obtained from the fundamental property relation \( dU = T dS - P dV \) is:

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

Hint:

Maxwell's relations connect different thermodynamic variables and are derived from exact differentials of thermodynamic potentials.

Answer 1

The Correct Option is B

Step 1: State the First Law of Thermodynamics.
For an isolated system, the relation is: \[dU = T dS - P dV\]

Step 2: Derive the Maxwell relation.
Applying the properties of exact differentials to the fundamental relation yields: \[\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T\]

Step 3: Final Result.
The derived Maxwell relation corresponds to option (B).