We want to match each partial derivative in List–I to the appropriate thermodynamic quantity in List–II:

1. $\displaystyle\left(\frac{\partial G}{\partial T}\right)_{P}$

A well-known thermodynamic identity is:

$ \left(\frac{\partial G}{\partial T}\right)_{P} \;=\; -S. $

Hence,

$ (A) \quad \longrightarrow \quad -S \quad (\text{II}). $

2. $\displaystyle\left(\frac{\partial H}{\partial T}\right)_{P}$

By definition of the heat capacity at constant pressure,

$ \left(\frac{\partial H}{\partial T}\right)_{P} \;=\; C_P. $

Hence,

$ (B) \quad \longrightarrow \quad C_P \quad (\text{I}). $

3. $\displaystyle\left(\frac{\partial G}{\partial P}\right)_{T}$

A standard thermodynamic relation (for a single‐component system) is:

$ \left(\frac{\partial G}{\partial P}\right)_{T} \;=\; V. $

Hence,

$ (C) \quad \longrightarrow \quad V \quad (\text{IV}). $

4. $\displaystyle\left(\frac{\partial U}{\partial T}\right)_{V}$

By definition of the heat capacity at constant volume,

$ \left(\frac{\partial U}{\partial T}\right)_{V} \;=\; C_V. $

Hence,

$ (D) \quad \longrightarrow \quad C_V \quad (\text{III}). $

Final Matching

$ (A) \to \text{(II)}, \quad (B) \to \text{(I)}, \quad (C) \to \text{(IV)}, \quad (D) \to \text{(III)}. $

Which corresponds to:

Option C: (A)-(II), (B)-(I), (C)-(IV), (D)-(III).