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

  1. $\left ( \dfrac{\partial A^{\left ( l \right )}}{\partial V^{\left ( l \right )}} \right )_{T, N^{\left ( l \right )}} = \left ( \dfrac{\partial A^{\left ( v \right )}}{\partial V^{\left ( v \right )}} \right )_{T, N^{\left ( v \right )}}$
  2. $\left ( \dfrac{\partial A^{\left ( l \right )}}{\partial N^{\left ( l \right )}} \right )_{T, V^{\left ( l \right )}} = \left ( \dfrac{\partial A^{\left ( v \right )}}{\partial N^{\left ( v \right )}} \right )_{T, V^{\left ( v \right )}}$
  1. $\left ( \dfrac{A + PV}{N} \right )^{\left ( l \right )} = \left ( \dfrac{A + PV}{N} \right )^{\left ( v \right )}$
  1. $\left ( \dfrac{A + TS}{N} \right )^{\left ( l \right )} = \left ( \dfrac{A + TS}{N} \right )^{\left ( v \right )}$
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