Question

For a closed, simple compressible system, which of the following thermodynamic relations is always valid, irrespective of the process, assuming only reversible processes and equilibrium states?

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

Hint:

A helpful mnemonic for Maxwell's relations is the thermodynamic square.
The variables $S$ and $P$ are diagonally opposite to $V$ and $T$, and the negative sign arises when moving between variables of opposite differential signs.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question focuses on Maxwell's thermodynamic relations, which are derived from the definitions of thermodynamic potentials and the exactness of differential properties.

Step 2: Key Formula or Approach:
The four primary Maxwell relations for a simple compressible system are:
1. $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$ (from $dU = TdS - PdV$)
2. $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$ (from $dH = TdS + VdP$)
3. $\left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial S}{\partial V}\right)_T$ (from $dA = -SdT - PdV$)
4. $\left(\frac{\partial V}{\partial T}\right)_P = -\left(\frac{\partial S}{\partial P}\right)_T$ (from $dG = -SdT + VdP$)

Step 3: Detailed Explanation:

• Maxwell relations are based on the mathematical properties of exact differentials of thermodynamic potentials (internal energy, enthalpy, Helmholtz function, and Gibbs free energy).

• Let us consider the Gibbs free energy function, which is defined as:
\[ G = H - TS \]
Its differential form for a simple compressible substance in a reversible process is:
\[ dG = V dP - S dT \]

• Since $G$ is a state function, $dG$ is an exact differential. Therefore, by reciprocity:
\[ \left(\frac{\partial V}{\partial T}\right)_P = -\left(\frac{\partial S}{\partial P}\right)_T \]

• Rearranging this yields:
\[ \left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P \]

• This corresponds exactly to Option C.

Step 4: Final Answer:
The valid relation is $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$.