Question

Which of the following behaviour is true about the ideal binary liquid solution of liquids 'A' and ' \(B\) ', if \(P_{A}^{\circ}

Short answer

The correct behavior about the ideal binary liquid solution, given that \( P_{A}^{\text{°}} < P_{B}^{\text{°}} \), is option (b): the plot of \( P_{\text{total}} \) vs \( X_{B} \) is linear with a positive slope.

Step-by-step solution

Understanding Raoult's Law

Raoult's Law states that for an ideal solution, the partial vapor pressure of each component is directly proportional to its mole fraction in the solution. Mathematically, it is given as: \( P_A = X_A \times P_A^{\text{°}} \) and \( P_B = X_B \times P_B^{\text{°}} \), where \( P_A \) and \( P_B \) are the partial vapor pressures of components A and B respectively, \( X_A \) and \( X_B \) are the mole fractions of components A and B respectively, and \( P_A^{\text{°}} \) and \( P_B^{\text{°}} \) are the equilibrium vapor pressures of the pure components A and B respectively.

Total Vapor Pressure of the Solution

The total vapor pressure of the solution (\( P_{\text{total}} \)) is given by the sum of the partial pressures of the components. Mathematically, it can be written as: \( P_{\text{total}} = P_A + P_B = X_A \times P_A^{\text{°}} + X_B \times P_B^{\text{°}} \). Since \( X_A + X_B = 1 \), we can substitute \( X_B = 1 - X_A \) into the equation to get \( P_{\text{total}} = X_A \times P_A^{\text{°}} + (1 - X_A) \times P_B^{\text{°}} \).

Analysis of the Slope

By rearranging the equation for \( P_{\text{total}} \), we get \( P_{\text{total}} = P_B^{\text{°}} + X_A \times (P_A^{\text{°}} - P_B^{\text{°}}) \). The first term \( P_B^{\text{°}} \) is a constant, and the second term is linear in \( X_A \) with a slope of \( (P_A^{\text{°}} - P_B^{\text{°}}) \). Given that \( P_A^{\text{°}} < P_B^{\text{°}} \), it follows that the slope \( (P_A^{\text{°}} - P_B^{\text{°}}) \) is negative. Therefore, the plot for \( P_{\text{total}} \) versus \( X_A \) is linear with a negative slope, and consequently, the plot for \( P_{\text{total}} \) versus \( X_B \) will also be linear with a positive slope simply because \( X_B = 1 - X_A \).

Comparing to the Given Options

Option (b), which states 'plot of \( P_{\text{total}} \) vs \( X_{B} \) is linear with a positive slope,' matches our conclusion. Since \( P_{\text{total}} \) will increase with increasing \( X_{B} \) in a positive linear manner, option (b) is the correct one. Options (a), (c), and (d) are incorrect because option (a) suggests a non-linear behavior, which contradicts Raoult's Law for ideal solutions, and options (c) and (d) propose either no slope or a negative slope, which are not consistent with the given condition \( P_{A}^{\text{°}} < P_{B}^{\text{°}} \).

Key concepts

Vapor Pressure

Vapor pressure is an essential 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 (either liquid or solid) at a given temperature in a closed system. The vapor pressure of a substance is influenced by temperature; as the temperature increases, so does the vapor pressure.

In context of the Raoult's Law, the vapor pressure of each component of an ideal binary liquid solution is proportional to the mole fraction of the component in the solution. This relationship is crucial for predicting how the overall vapor pressure will change as the composition of the solution changes. The ideal nature of the solution implies that the intermolecular forces between unlike molecules are similar to those between like molecules, leading to a predictable, linear change in vapor pressure with composition.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture or solution. It is defined as the ratio of the number of moles of one component to the total number of moles of all components in the mixture. Represented by the symbol 'X', the mole fraction is dimensionless and falls within the range of 0 to 1, where a value of 1 represents a pure substance.

For the ideal binary liquid solution described by Raoult's Law, the mole fraction of each component directly determines its partial vapor pressure. A higher mole fraction of a component means a greater partial vapor pressure of that component, assuming the temperature remains constant. The relationship between mole fraction and vapor pressure forms the basis for the graphical representations used to analyze solution behavior in Raoult's Law scenarios.

Binary Liquid Solution

Binary liquid solutions are mixture of two liquids. For simplicity, they are usually discussed in terms of an 'ideal' behavior where the solution follows Raoult's Law closely. An ideal binary liquid solution is characterized by several assumptions: there are no volume changes upon mixing, the intermolecular forces between different molecules are similar to those between similar molecules, and the solution is perfectly miscible at all proportions of the component liquids.

An understanding of binary liquid solutions is important for predicting how different liquid mixtures will behave when combined, which has practical applications in fields like chemistry, pharmacy, and chemical engineering. The interplay between the mole fractions of the two components in a binary liquid solution determines the total vapor pressure and the characteristics of phase diagrams for the solution.

Ideal Solution

An ideal solution is one that obeys Raoult's Law at all concentrations, which states that the partial vapor pressure of each component is equal to the product of the pure component's vapor pressure and its mole fraction in the mixture. These solutions exhibit several properties: the enthalpy of mixing is zero (which implies no heat is absorbed or evolved when the solution forms), and the volume of mixing is also zero (meaning the total volume is the sum of the individual volumes of the components before mixing).

For an ideal solution, the interaction forces between different particles are equivalent to those between like particles. This allows us to predict and understand the solution behavior with reasonable accuracy. In the context of the given exercise, because the ideal solution's vapor pressure plot is linear, we can confidently determine the mole fractions and partial pressures of the components in a mixture based on this linearity. However, it is important to note that not all solutions behave ideally; deviations from ideality do occur due to differences in intermolecular forces among the components.