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 solve this problem, we need to analyze the relationship between the mole fractions in liquid and vapor phases for a binary solution of two volatile components 1 and 2. This relationship can be derived using Raoult's Law and Dalton's Law for ideal solutions and vapors. 

1. The mole fraction of component 1 in the liquid phase is denoted as \(x_1\), and in the vapor phase as \(y_1\).
2. According to Raoult's Law, the partial pressure of component 1, \(p_1\), is given by: 

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

1.  where \(p_1^0\) is the vapor pressure of pure component 1.
2. Similarly, for component 2: 

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

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

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

1. The mole fraction in the vapor phase \(y_1\) can be represented 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 given question involves a linear plot of \(\frac{1}{x_1}\) vs \(\frac{1}{y_1}\). By rearranging the terms in step 5, we can write: 

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

1. Inverting the above equation to match the linear form in the question, we get: 

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

1. The slope of the line is derived as: 

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

1. The intercept of the line, considering the whole equation setup, is: 

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

Thus, the correct answer is the option with slope and intercept \(\frac{p_1^0}{p_2^0} - \frac{p_1^0}{p_2^0}\), which matches the interpretation of the question.

Answer 2

For a binary solution of volatile components, the relationship between the mole fractions of components in the liquid (\( x_1 \)) and vapor phases (\( y_1 \)) is governed by Raoult's Law: \[ y_1 = \frac{p_1^0}{p_1} x_1, \] Where \( p_1^0 \) is the vapor pressure of component 1, and \( p_1 \) is the partial pressure of component 1. Rearranging the equation gives a linear relationship: \[ \frac{1}{x_1} = \left( \frac{p_1^0}{p_2^0} - \frac{p_1^0}{p_2^0} \right) \frac{1}{y_1}. \] Thus, the slope is given by \( \frac{p_1^0}{p_2^0} - \frac{p_1^0}{p_2^0} \), and the correct answer is option (1).