Chapter 16, Problem 16.93P

Interpretation Introduction

Interpretation:

The given below expression has to be proved.

  $y_A=\frac{x_A{P}_A^{\circ }}{x_A\left(P_A^{\circ }-P_B^{\circ }\right)+P_B^{\circ }}$

Where, $y_A$ is the mole fraction of component A in the vapor over a two-component ideal solution.

Concept Introduction:

Raoult’s law:

It states that the equilibrium vapor pressure of the solvent over a solution is directly proportional to the mole fraction of the solvent in the solution.

The expression for the vapor pressure lowering is given below.

  $\Delta P={\chi }_2{P}^{\circ }$

Where, $\Delta P$ is vapor pressure lowering, ${\chi }_2$ is the mole fraction of solute and $P^{\circ }$ is vapor pressure of the pure solvent.

Dalton’s law of partial pressure:

According to Dalton’s law of partial pressure, the total pressure of a mixture is the sum of partial pressures each component.

The partial pressure of each component can also be represented as given below.’

  $\begin{array}{l}{P}_1=x_1{P}_1^{\circ }\\ \\ P_2=x_2{P}_2^{\circ }\end{array}$

Therefore, the total pressure of the mixture is given below.

  $\begin{array}{l}{P}_T=P_1+P_2=x_1{P}_1^{\circ }+x_2{P}_2^{\circ }=x_1{P}_1^{\circ }+\left(1-x_1\right)P_2^{\circ }\\ \\ \left[\because x_1+x_2=1\right]\end{array}$