Question

The plot of $1 / x_A$ versus $1 / y_A$ (where $x_A$ and $y_A$ are the mole fractions of $A$ in liquid and vapour phases respectively) is linear whose slope and intercept respectively are given as

(A) $\frac{p_A^{\circ}}{p_B^{\circ}}, \frac{\left(p_A^{\circ}-p_B^{\circ}\right)}{p_B^{\circ}}$

(B) $\frac{p_A^{\circ}}{p_B^{\circ}}, \frac{\left(p_B^{\circ}-p_A^{\circ}\right)}{p_B^{\circ}}$

(C) $\frac{p_B^{\circ}}{p_A^{\circ}}, \frac{\left(p_A^{\circ}-p_B^{\circ}\right)}{p_B^{\circ}}$

(D) $\frac{p_B^{\circ}}{p_A^{\circ}}, \frac{\left(p_B^{\circ}-p_A^{\circ}\right)}{p_B^{\circ}}$

Answer 1

$P_A=x_A p_A^{\circ}, P_B=x_B p_B^{\circ}$

$\begin{array}{l}y_A=\frac{p_A}{p_A+p_B}=\frac{x_A p_A^{\circ}}{x_A p_A^{\circ}+x_B p_B^{\circ}} \\=\frac{x_A p_A^{\circ}}{x_A p_A^{\circ}+\left(1-x_A\right) p_B^{\circ}}=\frac{x_A p_A^{\circ}}{x_A\left(p_A^{\circ}-p_B^{\circ}\right)+p_B^{\circ}}\end{array}$

or, $\frac{1}{y_A}=\frac{x_A\left(p_A^{\circ}-p_B^{\circ}\right)+p_B^{\circ}}{x_A p_A^{\circ}}$

or, $\frac{1}{y_A}=\frac{p_A^{\circ}-p_B^{\circ}}{p_A^{\circ}}+\frac{p_B^{\circ}}{p_A^{\circ}} \frac{1}{x_A}$

or, $\frac{1}{x_A}=\frac{p_A^{\circ}}{p_B^{\circ}} \frac{1}{y_A}+\frac{p_B^{\circ}-p_A^{\circ}}{p_B^{\circ}}$

Hence, plot of $\frac{1}{x_A} v s \frac{1}{y_A}$ will be linear with slope $=\frac{p_A^{\circ}}{p_B^{\circ}}$ and intercept $=\frac{\left(p_B^{\circ}-p_A^{\circ}\right)}{p_B^{\circ}}$.

So, The correct option is (B).