Question

Liquids \(\mathrm{A}\) and \(\mathrm{B}\) form an ideal solution. The plot of \(\frac{1}{X_{\mathrm{A}}}\) ( \(Y\) -axis) versus \(\frac{1}{Y_{\mathrm{A}}}\) \(\left(X\right.\) -axis) \(\left(\right.\) where \(X_{\mathrm{A}}\) and \(Y_{\mathrm{A}}\) are the mole fractions of A in liquid and vapour phases at equilibrium, respectively) is linear whose slope and intercept, respectively, are given as (a) \(\frac{P_{\Lambda}^{o}}{P_{\mathrm{B}}^{\circ}}, \frac{\left(P_{\mathrm{A}}^{0}-P_{\mathrm{B}}^{\circ}\right)}{P_{\mathrm{B}}^{\circ}}\) (b) \(\frac{P_{\mathrm{A}}^{\circ}}{P_{\mathrm{B}}^{\circ}}, \frac{\left(P_{\mathrm{B}}^{0}-P_{\mathrm{A}}^{\circ}\right)}{P_{\mathrm{B}}^{\circ}}\) (c) \(\frac{P_{\mathrm{B}}^{\circ}}{P_{\mathrm{A}}^{\circ}}, \frac{\left(P_{\mathrm{A}}^{\mathrm{o}}-P_{\mathrm{B}}^{\circ}\right)}{P_{\mathrm{B}}^{\circ}}\) (d) \(\frac{P_{\mathrm{B}}^{\circ}}{P_{\mathrm{A}}^{\circ}}, \frac{\left(P_{\mathrm{B}}^{0}-P_{\mathrm{A}}^{\circ}\right)}{P_{\mathrm{B}}^{\circ}}\)

Short answer

The correct option is (c) \(\frac{P_{\mathrm{B}}^{\text{circ}}}{P_{\mathrm{A}}^{\text{circ}}}, \frac{\left(P_{\mathrm{A}}^{\text{o}}-P_{\mathrm{B}}^{\text{circ}}\right)}{P_{\mathrm{B}}^{\text{circ}}}\).

Step-by-step solution

Understand Raoult's Law

According to Raoult's Law for an ideal solution, the partial vapor pressure of each component is directly proportional to its mole fraction in the liquid phase. This can be expressed as: For liquid A, \(P_{\mathrm{A}} = P_{\mathrm{A}}^{\circ} X_{\mathrm{A}}\) and for liquid B, \(P_{\mathrm{B}} = P_{\mathrm{B}}^{\circ} X_{\mathrm{B}}\), where \(P_{\mathrm{A}}^{\circ}\) and \(P_{\mathrm{B}}^{\circ}\) are the vapor pressures of the pure components.

Apply Dalton's Law

Dalton's Law of Partial Pressures states that the total pressure above the solution is the sum of the partial pressures of the components. This can be written as: \(P_{\text{total}} = P_{\mathrm{A}} + P_{\mathrm{B}}\).

Express the Mole Fraction in Vapor Phase

The mole fraction of A in the vapor phase can be expressed using the partial pressures by \(Y_{\mathrm{A}} = \frac{P_{\mathrm{A}}}{P_{\text{total}}}\). Substituting the expressions for \(P_{\mathrm{A}}\) and \(P_{\text{total}}\) using Raoult's and Dalton's Law yields: \(Y_{\mathrm{A}} = \frac{P_{\mathrm{A}}^{\circ} X_{\mathrm{A}}}{P_{\mathrm{A}}^{\circ} X_{\mathrm{A}} + P_{\mathrm{B}}^{\circ} (1 - X_{\mathrm{A}})}\).

Derive the Equation for the Ideal Solution

To represent \(\frac{1}{Y_{\mathrm{A}}}\) versus \(\frac{1}{X_{\mathrm{A}}}\), rearrange the previous expression to give \(\frac{1}{Y_{\mathrm{A}}} = \frac{P_{\text{total}}}{P_{\mathrm{A}}^{\circ}}\frac{1}{X_{\mathrm{A}}} = \frac{1}{P_{\mathrm{A}}^{\circ}X_{\mathrm{A}}} + \frac{P_{\mathrm{B}}^{\circ}}{P_{\mathrm{A}}^{\circ}}\frac{1}{P_{\mathrm{B}}^{\circ}X_{\mathrm{A}}}\).

Compare with the Given Linear Equation

A general linear equation has the form \(y = mx + c\). Comparing this with our derived equation, we get the slope \(m = \frac{P_{\mathrm{B}}^{\circ}}{P_{\mathrm{A}}^{\circ}}\) and the intercept \(c = \frac{1}{P_{\mathrm{B}}^{\circ}}\). The expression with this slope and intercept is option (c) \(\frac{P_{\mathrm{B}}^{\circ}}{P_{\mathrm{A}}^{\circ}}, \frac{\left(P_{\mathrm{A}}^{\mathrm{o}}-P_{\mathrm{B}}^{\text{circ}}\right)}{P_{\mathrm{B}}^{\text{circ}}}\).

Key concepts

Vapor Pressure

Vapor pressure is a fundamental concept when discussing the physical properties of liquids and solutions. It refers to the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. For any liquid, the tendency to transition into the vapor phase is measured as its vapor pressure. An essential characteristic of vapor pressure is that it is dependent on the temperature; as temperature increases, so does vapor pressure.

When considering a solution of two ideal liquids, such as in the given exercise, Raoult's Law provides the basis for understanding vapor pressure. Raoult's Law states that the vapor pressure of an ideal solution component is directly proportional to its mole fraction in the solution. In simpler terms, if you increase the amount of a substance in a solution, the vapor pressure due to that substance will also increase, assuming the solution behaves ideally, which means it follows Raoult's Law without deviation.

Each substance in a binary solution contributes to the total vapor pressure of the solution by exerting what is called a partial vapor pressure, which can be calculated as the vapor pressure of the pure component multiplied by its mole fraction in the solution. Through combining the partial vapor pressures of all components, we can determine the total vapor pressure exerted by the mixture.

Mole Fraction

Mole fraction is a way to express the composition of a mixture or solution in terms of the amount of its constituents. It is defined as the ratio of the number of moles of a particular component to the total number of moles of all substances present in the mixture. Represented by the symbol 'X', the mole fraction is a unitless quantity that provides a measure of concentration without the need for specifying volumes, which may change with temperature or pressure.

In the context of the given exercise, mole fractions play a crucial role in applying Raoult's Law. They are used to predict the partial vapor pressures of the components of an ideal solution. The mole fraction is used as it ensures that the influence of each component on the solution's properties is accounted for purely based on the quantity of that component, regardless of its identity. This trait is particularly useful in ideal solutions where interactions between different molecules do not affect the overall properties of the solution significantly. By knowing the mole fractions, we can calculate both the partial pressures and the total pressure in a system, leading to a wide range of thermodynamic predictions.

Partial Pressure

Partial pressure is the pressure that would be exerted by one of the gases in a mixture of gases if it occupied the entire volume of the mixture on its own at the same temperature. Described by Dalton's Law of Partial Pressures, it asserts that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases.

Applying this to an ideal solution, as the exercise demonstrates, when components are volatile, the vapor phase above the solution will be composed of both substances contributing to the total pressure. Here, each component's partial pressure is determined by its mole fraction in the liquid phase, as per Raoult's Law, and by its pure substance vapor pressure. Dalton's Law is essential when dealing with vapors above solutions, as it allows us to understand how different components contribute to the overall pressure and hence, to the dynamics of vaporization and condensation in closed systems.

Ideal Solution Thermodynamics

Ideal solution thermodynamics is a theoretical framework used to describe the behavior of mixtures where the interactions between different particles are similar to those between like particles. In ideal solutions, the physical properties are directly related to the properties of the individual substances and their proportions in the mixture. Thermodynamically, these solutions follow Raoult's Law and Dalton's Law without any deviation, indicating that the enthalpy change on mixing is zero and the volume change on mixing is also negligible.

For an ideal solution, the thermodynamic activity (a measure of the effective concentration of a species in a mixture) of each component is equal to its mole fraction, which in turn reflects the idea that there is no significant difference in interactions among different molecules in the solution. This simplification is often used to predict vapor-liquid equilibrium and phase diagrams, which are important in separation processes, such as distillation. As seen in the provided exercise, understanding ideal solution behavior enables us to derive relationships between vapor and liquid phase compositions, allowing us to make quantitative predictions about the behavior of the solution.