Question

For an ideal binary liquid solution with \(P_{A}^{\circ}>P_{B}^{\circ}\), which relation between \(X_{A}\) (mole fraction of \(A\) in liquid phase) and \(Y_{A}\) (mole fraction of \(A\) in vapour phase) is correct ? (a) \(Y_{A}X_{B}\) (c) \(\frac{Y_{A}}{Y_{B}}>\frac{X_{A}}{X_{B}}\) (d) \(\frac{Y_{A}}{Y_{B}}<\frac{X_{A}}{X_{B}}\)

Short answer

(c) \(\frac{Y_{A}}{Y_{B}} > \frac{X_{A}}{X_{B}}\)

Step-by-step solution

Understanding Raoult's Law

For an ideal solution, Raoult's Law states that the partial pressure of each component of an ideal binary mixture is directly proportional to the mole fraction of the component in the liquid phase and the vapor pressure of the pure component. Thus, the vapor pressure of component A in the solution is given by \( P_A = X_A \cdot P_{A}^{\circ} \) and that of component B by \( P_B = X_B \cdot P_{B}^{\circ} \).

Relating Partial Pressure and Mole Fraction

The total pressure above the solution is the sum of the partial pressures of A and B. Thus, the total pressure is given by \( P = P_A + P_B \). Moreover, the mole fraction in the vapor phase is given by the ratio of the partial pressure of each component to the total pressure (Dalton's Law), so we have \( Y_A = \frac{P_A}{P} \) and \( Y_B = \frac{P_B}{P} \).

Calculating the Ratio of Mole Fractions

To find the correct relationship between the mole fractions \( X_A \), \( X_B \), \( Y_A \), and \( Y_B \), we can form ratios. Given that \( P_{A}^{\circ} > P_{B}^{\circ} \), we see that for every mole fraction of A in the liquid phase, there will be a greater increase in the mole fraction of A in the vapor phase compared to B. Hence, \( \frac{Y_A}{Y_B} > \frac{X_A}{X_B} \) because the component with higher vapor pressure (A in this case) is more volatile and will have a higher proportion in the vapor phase compared to the liquid phase. This comes from the ratios \( Y_A = \frac{X_A P_{A}^{\circ}}{P} \) and \( Y_B = \frac{X_B P_{B}^{\circ}}{P} \), which imply that \( \frac{Y_A}{Y_B} = \frac{X_A}{X_B} \cdot \frac{P_{A}^{\circ}}{P_{B}^{\circ}} \), and since \( P_{A}^{\circ} > P_{B}^{\circ} \), the ratio \( \frac{Y_A}{Y_B} \) should be larger than \( \frac{X_A}{X_B} \).

Key concepts

Ideal Binary Liquid Solution

In chemistry, an ideal binary liquid solution is one that combines two components in such a way that there are no interactions between molecules of different substances that affect their behaviors. This simplification allows us to assume that the properties of the solution only depend on the concentration of each component and the properties of the pure substances involved.

More specifically, the ideal solution follows Raoult's Law, meaning that each component's partial vapor pressure is proportional to its mole fraction in the solution. This linearity holds true throughout the range of possible concentrations. Ideal behavior is a theoretical construct and is closely approximated by real solutions under conditions of low pressures and when the component molecules are of similar size and intermolecular forces.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of that component to the total number of moles of all components of the mixture. The mole fraction is a dimensionless value, and his symbol is usually denoted by the letter X for the liquid phase or Y for the vapor phase.

Mathematically, it's given by the formula:
\[ X_A = \frac{n_A}{n_A + n_B} \] where \( n_A \) is the number of moles of component A and \( n_A + n_B \) is the total number of moles in the solution. Since mole fractions are ratios, they sum up to one. This concept is vital for predicting the behavior of a component in a mixture through laws like Raoult's or Dalton's.

Vapor Pressure

Vapor pressure is a critical concept in thermodynamics and chemistry, describing the pressure exerted by a vapor in thermal equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. For pure substances, this pressure is a fixed value at a certain temperature, reflecting the tendency of molecules to escape from the liquid or solid phase into the gaseous one.

For solutions, however, the vapor pressure of each component is affected by its interactions with other components, such as in an ideal binary liquid solution where the vapor pressure of each component is given by Raoult's Law: \( P_A = X_A \times P_A^0 \) where \( X_A \) is the mole fraction of component A and \( P_A^0 \) is the vapor pressure of the pure A.

Dalton's Law

Dalton's Law, or the Law of Partial Pressures, is an essential principle that relates to mixtures of gases. It states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. In the context of ideal binary liquid solutions, it helps us relate the total pressure above the solution to the pressures contributed by each volatile component in the vapor phase.

Expressed mathematically:
\[ P_{\text{total}} = P_A + P_B \] where \( P_A \) and \( P_B \) represent the partial pressures of components A and B, respectively. This- law is key to understanding the contributions each component makes to the overall pressure and helps in determining the composition of the vapor above a solution.

Partial Pressure

Partial pressure pertains to the pressure that a single component of a mixture of gases would exert if it occupied the entire volume of the mixture at the same temperature. It is an important concept when looking at vapor-liquid equilibrium in the field of chemistry. The partial pressure can be thought of as that component's contribution to the total pressure when mixed with other gases or vapors and can be calculated using Raoult's Law for ideal solutions.

The relationship between partial pressure and mole fraction is critical in determining the behavior of a substance within a mixture and is used in quantifying the way in which different components will separate or interact under varying conditions.