Question

Consider a binary solution of two volatile liquid components 1 and 2. \(x_1\) and \(y_1\) are the mole fractions of component 1 in the liquid and vapor phase, respectively. The slope and intercept of the linear plot of \( \frac{1}{x_1} \) vs \( \frac{1}{y_1} \) are given respectively as:

• A. \( \frac{p_1^0}{p_2^0} - \frac{p_1^0}{p_2^0} \)
• B. \( \frac{p_2^0}{p_1^0} - \frac{p_1^0}{p_2^0} \)
• C. \( \frac{p_1^0}{p_2^0} - \frac{p_2^0}{p_1^0} \)
• D. \( \frac{p_2^0}{p_1^0} - \frac{p_2^0}{p_1^0} \)

Hint:

For binary solutions, Raoult's Law governs the relationship between mole fractions in the liquid and vapor phases. The slope of the plot of \( \frac{1}{x_1} \) vs \( \frac{1}{y_1} \) depends on the vapor pressures of the components.

Answer 1

The Correct Option is A

To address this problem, we must examine the relationship between mole fractions in the liquid and vapor phases for a binary ideal solution containing volatile components 1 and 2. This relationship is established using Raoult's Law and Dalton's Law for ideal solutions and vapors.

1. Let \(x_1\) represent the mole fraction of component 1 in the liquid phase and \(y_1\) represent it in the vapor phase.
2. Raoult's Law states that the partial pressure of component 1, denoted as \(p_1\), is calculated as:

\[p_1 = x_1 \cdot p_1^0\]

1. Here, \(p_1^0\) signifies the vapor pressure of pure component 1.
2. The same principle applies to component 2:

\[p_2 = x_2 \cdot p_2^0\]

1. where \(p_2^0\) is the vapor pressure of pure component 2.
2. Dalton's Law for total pressure is expressed as:

\[P_{\text{total}} = p_1 + p_2\]

1. The mole fraction in the vapor phase, \(y_1\), can be expressed as:

\[y_1 = \frac{p_1}{P_{\text{total}}} = \frac{x_1 \cdot p_1^0}{x_1 \cdot p_1^0 + x_2 \cdot p_2^0}\]

1. The current question involves a linear plot of \(\frac{1}{x_1}\) against \(\frac{1}{y_1}\). By rearranging the terms from step 5, we obtain:

\[\frac{y_1}{x_1} = \frac{p_1^0}{p_1^0 + \frac{x_2}{x_1} \cdot p_2^0}\]

1. Inverting this equation to align with the linear form presented in the question yields:

\[\frac{1}{y_1} = \frac{1}{x_1} \cdot \frac{p_1^0 + x_2/x_1 \cdot p_2^0}{p_1^0}\]

1. The slope of this line is determined to be:

\[\frac{p_2^0}{p_1^0}\]

1. Considering the entire equation setup, the intercept of the line is:

\[1 - \frac{p_1^0}{p_2^0}\]

Consequently, the correct answer corresponds to the option presenting a slope of \(\frac{p_1^0}{p_2^0}\) and an intercept of \(1 - \frac{p_1^0}{p_2^0}\), aligning with the question's interpretation.