Ratio of Mole Fractions in Ideal Binary Solution

To find the value of the ratio $x_A / x_B$, we apply Raoult's law for an ideal binary solution.

Step 1: Determine the vapour pressure of pure liquid $\mathbf{B}$ ($P_B^\circ$)

According to Raoult's law, the total vapour pressure $P_{\text{total}}$ of an ideal binary solution containing liquids $\mathbf{A}$ and $\mathbf{B}$ is given by: $P_{\text{total}} = x_A P_A^\circ + x_B P_B^\circ$

For an equimolar solution of $\mathbf{A}$ and $\mathbf{B}$, the mole fractions are: $x_A = x_B = 0.5$

Given that $P_{\text{total}} = 45\text{ Torr}$ and the vapour pressure of pure liquid $\mathbf{A}$ is $P_A^\circ = 20\text{ Torr}$, we have: $45 = 0.5 \times 20 + 0.5 \times P_B^\circ$ $45 = 10 + 0.5 P_B^\circ$ $0.5 P_B^\circ = 35$ $P_B^\circ = 70\text{ Torr}$

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Step 2: Find the mole fractions $x_A$ and $x_B$ in the new solution

For the new solution, the vapour pressure is $22.5\text{ Torr}$. Using Raoult's law: $P_{\text{total}} = x_A P_A^\circ + x_B P_B^\circ = 22.5\text{ Torr}$

Since the sum of mole fractions is unity, $x_A + x_B = 1 \implies x_B = 1 - x_A$: $20 x_A + 70(1 - x_A) = 22.5$ $70 - 50 x_A = 22.5$ $50 x_A = 70 - 22.5$ $50 x_A = 47.5$ $x_A = \frac{47.5}{50} = 0.95$

Consequently, the mole fraction of liquid $\mathbf{B}$ is: $x_B = 1 - x_A = 1 - 0.95 = 0.05$

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Step 3: Calculate the ratio $x_A / x_B$

$\frac{x_A}{x_B} = \frac{0.95}{0.05} = 19$