two components A & B FORM AN IDEAL solution.the mole fractions of A & B in ideal solutions are XA&XB while that of in vapour phase ,these components have their mole fractions as YA & YB .then the swlope and intercept of plot of 1/ya vs 1/xa will be

To answer your question about the relationship between the mole fractions of components A and B in an ideal solution, we need to delve into Raoult's Law and some basic thermodynamics concepts. In an ideal solution, the properties of the solution behave predictably based on the individual components' mole fractions. Let's break down the specifics of your query regarding the plot of \( \frac{1}{Y_A} \) versus \( \frac{1}{X_A} \).

Understanding the Relationships

In an ideal solution, Raoult's Law states that the partial vapor pressure of each component is directly proportional to its mole fraction in the liquid phase. For components A and B, we can express this relationship as:

• For component A: \( P_A = X_A \cdot P^0_A \)
• For component B: \( P_B = X_B \cdot P^0_B \)

Here, \( P^0_A \) and \( P^0_B \) are the vapor pressures of the pure components A and B, respectively. The total vapor pressure \( P \) of the solution can be written as:

\( P = P_A + P_B = X_A \cdot P^0_A + X_B \cdot P^0_B \)

Deriving the Plots

Now, when we consider the mole fractions in the vapor phase, \( Y_A \) and \( Y_B \), we can also express them using Raoult's Law:

• \( Y_A = \frac{P_A}{P} \)
• \( Y_B = \frac{P_B}{P} \)

Substituting from our earlier equations, we find:

\( Y_A = \frac{X_A \cdot P^0_A}{X_A \cdot P^0_A + X_B \cdot P^0_B} \)

Now, to analyze the plot of \( \frac{1}{Y_A} \) against \( \frac{1}{X_A} \), we can rearrange the equation for \( Y_A \) to express it in terms of \( X_A \):

\( Y_A = \frac{X_A \cdot P^0_A}{X_A \cdot P^0_A + (1 - X_A) \cdot P^0_B} \)

From here, we can find \( \frac{1}{Y_A} \) and rewrite it:

\( \frac{1}{Y_A} = \frac{X_A \cdot P^0_A + (1 - X_A) \cdot P^0_B}{X_A \cdot P^0_A} \)

Breaking this down results in:

\( \frac{1}{Y_A} = \frac{P^0_B}{X_A \cdot P^0_A} + \frac{1}{P^0_A} \)

Slope and Intercept

When we plot \( \frac{1}{Y_A} \) against \( \frac{1}{X_A} \), the relationship resembles a linear equation of the form:

\( y = mx + b \)

Where:

• The slope \( m \) is equal to \( \frac{P^0_B}{P^0_A} \)
• The y-intercept \( b \) is \( \frac{1}{P^0_A} \)

Thus, analyzing the plot yields important insights into the vapor-liquid equilibrium of the ideal solution. The linear nature allows us to determine the relative volatility of the components based on their vapor pressures.

This understanding is crucial in fields such as chemical engineering, where the design of distillation columns and separation processes relies heavily on these principles. By grasping these concepts, you can better predict how changes in composition will affect the vapor and liquid phases in an ideal solution.