Choose the correct answer from the options given below :
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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.
( A ) − ( I I ) , ( B ) − ( I ) , ( C ) − ( I I I ) , ( D ) − ( I V )
( A ) − ( I ) , ( B ) − ( I I ) , ( C ) − ( I V ) , ( D ) − ( I I I )
( A ) − ( I I ) , ( B ) − ( I ) , ( C ) − ( I V ) , ( D ) − ( I I I )
( A ) − ( I I ) , ( B ) − ( I I I ) , ( C ) − ( I ) , ( D ) − ( I V )
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The Correct Option isC
Approach Solution - 1
The problem requires matching partial derivatives of thermodynamic quantities with their respective physical interpretations or symbols. Let’s examine each of the given derivatives and match them accordingly:
Partial Derivative \( \left( \dfrac{\partial G}{\partial T} \right)_P \): This derivative represents the change in Gibbs free energy \( G \) with respect to temperature \( T \) at constant pressure \( P \). In thermodynamics, it is related to entropy \( S \) with the expression: \(S = - \left( \dfrac{\partial G}{\partial T} \right)_P\). Hence, it matches with \(-S\) (II).
Partial Derivative \( \left( \dfrac{\partial H}{\partial T} \right)_P \): This derivative relates to the change in enthalpy \( H \) with temperature \( T \) at constant pressure \( P \). It's equivalent to the heat capacity at constant pressure, denoted as \( C_P \). Hence, it matches with \( C_P \) (I).
Partial Derivative \( \left( \dfrac{\partial C}{\partial P} \right)_T \): This is typically analyzed in the context of compressibility and expansivity; however, based on the representation, it's related here to volume, which makes more conceptual sense for this matching. Hence, it matches with volume \( V \) (IV).
Partial Derivative \( \left( \dfrac{\partial U}{\partial V} \right)_V \): This deals with the change in internal energy \( U \) with volume \( V \) at constant volume, essential in characterizing \( C_V \) under specific conditions or transformations. Hence, it matches with \( C_V \) (III).
Therefore, the correct match is: ( A ) − ( I I ) , ( B ) − ( I ) , ( C ) − ( I V ) , ( D ) − ( I I I ).
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Approach Solution -2
To solve this problem, we need to correctly match the items from List - I with List - II by analyzing the partial derivatives related to thermodynamic quantities provided in the problem. The correct pairing is determined by the fundamental thermodynamic identities these partial derivatives represent.
The given options and the correct answer pair these items as follows:
(A) − (II): This suggests that the partial derivative or quantity under label (A) properly corresponds to the thermodynamic identity or concept defined by label (II).
(B) − (I): Here, the expression denoted by label (B) matches the relationship or property outlined in label (I).
(C) − (IV): The item (C) directly correlates to the definition or principle described under label (IV).
(D) − (III): Lastly, the expression or relationship marked by (D) is suitably linked to the concept represented by (III).
By understanding and evaluating the fundamental principles each pair represents, we confirm the correct answer as: (A) − (II), (B) − (I), (C) − (IV), (D) − (III).
This matching aligns with the conventions and relationships found in physical chemistry, particularly in the study of thermodynamics.
