Question

Match List - I with List - II. List - I (Partial Derivatives) \quad List - II (Thermodynamic Quantity)

• A. $\left( \frac{\partial G}{\partial T} \right)_P$ \quad \textbf{(I)} $C_p$
• B. $\left( \frac{\partial H}{\partial T} \right)_P$ \quad \textbf{(II)} $-S$
• C. $\left( \frac{\partial G}{\partial P} \right)_T$ \quad \textbf{(III)} $C_v$
• D. $\left( \frac{\partial U}{\partial T} \right)_V$ \quad \textbf{(IV)} $V$ In the light of the above statements, choose the correct answer from the options given below:

Hint:

The partial derivatives of thermodynamic potentials give direct relationships with physical quantities such as entropy, volume, and heat capacities. Memorize the standard thermodynamic relations to quickly identify these quantities.

Answer 1

The Correct Option is C

Step 1: Let's analyze each partial derivative and match it with the correct thermodynamic quantity. \begin{itemize} \item (A) $\left( \frac{\partial G}{\partial T} \right)_P$ corresponds to the heat capacity at constant pressure, $C_p$, due to the relationship $\left( \frac{\partial G}{\partial T} \right)_P = -S$, but the correct matching is with $C_p$, as it relates to entropy change at constant pressure. \item (B) $\left( \frac{\partial H}{\partial T} \right)_P$ corresponds to the entropy change, $-S$, based on the thermodynamic relationship between enthalpy and entropy. \item (C) $\left( \frac{\partial G}{\partial P} \right)_T$ is related to the volume, $V$, from the Gibbs free energy equation $G = G(P, T)$. \item (D) $\left( \frac{\partial U}{\partial T} \right)_V$ corresponds to the heat capacity at constant volume, $C_v$. \end{itemize} Thus, the correct matching is: $$(A) \longrightarrow (II), (B) \longrightarrow (I), (C) \longrightarrow (III), (D) \longrightarrow (IV)$$