Question

For a single component system at vapor-liquid equilibrium, the extensive variables A, V, S and N denote the Helmholtz free energy, volume, entropy, and number of moles, respectively, in a given phase. If superscripts \( (\nu) \) and \( (\ell) \) denote the vapor and liquid phase, respectively, the relation that is NOT CORRECT is

• A. \( \left( \frac{\partial A^{(\ell)}}{\partial V^{(\ell)}} \right)_{T,N^{(\ell)}} = \left( \frac{\partial A^{(\nu)}}{\partial V^{(\nu)}} \right)_{T,N^{(\nu)}} \)
• B. \( \left( \frac{\partial A^{(\ell)}}{\partial N^{(\ell)}} \right)_{T,V^{(\ell)}} = \left( \frac{\partial A^{(\nu)}}{\partial N^{(\nu)}} \right)_{T,V^{(\nu)}} \)
• C. \( \left( \frac{A + PV}{N} \right)^{(\ell)} = \left( \frac{A + PV}{N} \right)^{(\nu)} \)
• D. \( \left( \frac{A + TS}{N} \right)^{(\ell)} = \left( \frac{A + TS}{N} \right)^{(\nu)} \)

Hint:

In vapor-liquid equilibrium systems, the properties such as Helmholtz free energy, entropy, and number of moles vary between the liquid and vapor phases. Ensure to check phase-dependent relations before applying thermodynamic identities.

Answer 1

The Correct Option is D

In a vapor-liquid equilibrium system, the Helmholtz free energy \( A \), the volume \( V \), the entropy \( S \), and the number of moles \( N \) are related in both the liquid and vapor phases.

Step 1: Analyze the relations.
The first three options (A), (B), and (C) are correct relations derived from thermodynamic identities. However, option (D) is not correct because the relation \( \left( \frac{A + TS}{N} \right)^{(\ell)} = \left( \frac{A + TS}{N} \right)^{(\nu)} \) is not valid for a general vapor-liquid system, as the specific values of \( A \), \( T \), and \( S \) differ between the liquid and vapor phases.

Step 2: Conclusion.
The relation in option (D) is not correct. Hence, the correct answer is (D).

Final Answer: (D) \( \left( \frac{A + TS}{N} \right)^{(\ell)} = \left( \frac{A + TS}{N} \right)^{(\nu)} \)