Question

An ideal solution has two components \(A\) and \(B . A\) is more volatile than \(B\), i.e., \(P_{A}^{\circ}>P_{B}^{\circ}\) and also \(P_{A}^{\circ}>P_{\text {total }}\). If \(X_{A}\) and \(Y_{A}\) are mole fractions of components \(A\) in liquid and vapour phases, then: (a) \(X_{A}=Y_{A}\) (b) \(X_{A}>Y_{A}\) (c) \(X_{A}

Short answer

\(X_{A} < Y_{A}\)

Step-by-step solution

Understanding Raoult's Law

Raoult's Law expresses that the partial vapor pressure of a component in a mixture is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. This law is represented by the formula: \( P_i = X_i \cdot P_i^\circ \), where \( P_i \) is the partial vapor pressure of the component, \( X_i \) is the mole fraction of the component in the liquid phase, and \( P_i^\circ \) is the vapor pressure of the pure component.

Understanding Volatility

Volatility indicates how readily a substance vaporizes. In this context, since component A is more volatile than component B, \( P_A^\circ > P_B^\circ \). Therefore, component A has a greater tendency to escape into the vapor phase than component B.

Applying Dalton's Law of Partial Pressures

According to Dalton's Law, the total pressure in a system is the sum of the partial pressures of all the individual components. For a two-component system, this is represented as \( P_{total} = P_A + P_B \). Given that \( P_A^\circ > P_{total} \), component A in its pure form has a vapor pressure higher than the combined pressure of the solution, indicating an even stronger tendency to be present in the vapor phase.

Relating Volatility to Mole Fractions

For a more volatile component A, its mole fraction in the vapor phase \( Y_A \) will be greater than its mole fraction in the liquid phase \( X_A \) since it has a stronger tendency to vaporize compared to the less volatile component B. Thus, a greater proportion of A will be found in the vapor phase relative to the liquid phase, considering partial pressures and the behavior of volatile substances.

Choosing the Correct Option

Based on the volatility and the behaviors expressed through Raoult's and Dalton's laws, we can deduce that the mole fraction of A in the vapor phase \( Y_A \) is greater than the mole fraction of A in the liquid phase \( X_A \). Therefore, the correct answer is \( X_{A} < Y_{A} \).

Key concepts

Volatility in Solutions

Volatility refers to how readily a substance can vaporize, which directly affects the composition of a solution's vapor phase. More volatile substances have higher vapor pressures even at lower temperatures, causing them to escape into the gaseous phase more easily compared to less volatile substances. This concept helps in explaining why some liquids evaporate faster than others.

In the context of solutions, the volatility of each component has a direct impact on their respective concentration in the vapor phase. For instance, in a binary solution where component A is more volatile than component B, component A will have a higher presence in the vapor that develops above the solution. Accordingly, in the provided exercise, it's stated that component A is indeed more volatile than component B, meaning that the concentration of A in the vapor phase will be higher than that in the liquid phase.

This results in a discrepancy between the mole fractions of the corresponding components in the liquid and vapor phases. Consequently, if we were to look at a collection of vapor above the solution, we would find a greater proportion of the more volatile component A compared to its proportion in the liquid phase.

Partial Vapor Pressure

The partial vapor pressure of a component within a mixture is a vital concept when discussing solutions in chemistry. It represents the pressure exerted by the vapor of that component above the liquid solution when in equilibrium at a given temperature. For a component in a solution, Raoult's Law defines its partial vapor pressure as directly proportional to its mole fraction in the liquid phase multiplied by the vapor pressure of the pure substance, shown in the equation as \( P_i = X_i \times P_i^\text{o} \).

The vapor pressure of the pure substance, \( P_i^\text{o} \), is a fixed value at a given temperature, but the actual partial pressure varies depending on the component's presence in the solution (given by its mole fraction, \( X_i \)). Regarding our problem, since the vapor pressure of A as a pure substance (\( P_A^\text{o} \)) is greater than the total pressure of the solution and higher than that of component B, reflecting component A's greater propensity to form vapor and its volatility.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a cornerstone in the study of gaseous mixtures. It states that the total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of individual gases. The formula for this law is expressed as \( P_{\text{total}} = P_A + P_B + \text{... for all components} \).

This law allows us to understand the behavior of gases in a mixture and, by extension, the vapor above a liquid solution. In the scenario we're analyzing, Dalton's Law tells us that the total vapor pressure above the solution (\( P_{\text{total}} \)) is the sum of the partial pressures of components A and B. The exercise indicates that the pure vapor pressure of component A (\( P_A^\text{o} \)) is greater than the total vapor pressure. This means that when A is part of a mixture, its strong tendency to vaporize will lead to it disproportionately influencing the total vapor pressure, resulting in a vapor phase richer in component A than predicted by the mole fraction of A in the liquid phase.