Problem 16 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Benzene and toluene form an ideal solution. The vapour pressures of benzene and toluene are \(75 \mathrm{~mm}\) and \(25 \mathrm{~mm}\), respectively, at \(20^{\circ} \mathrm{C}\). If the mole fractions of benzene and toluene in vapour are \(0.75\) and \(0.25\), respectively, the vapour pressure of the ideal solution is (a) \(62.5 \mathrm{~mm}\) (b) \(50 \mathrm{~mm}\) (c) \(30 \mathrm{~mm}\) (d) \(40 \mathrm{~mm}\)

Step-by-Step Solution

Answer

The vapour pressure of the ideal solution is 62.5 mm.

Step 1: Understanding Raoult's Law

Raoult's Law states that the partial vapor pressure of a component in a mixture is equal to the mole fraction of the substance in the liquid phase times the vapor pressure of the pure substance. Mathematically, for a component A it is expressed as: \( P_A = x_A \times P_A^\circ \), where \( P_A \) is the partial vapor pressure of component A, \( x_A \) is the mole fraction of component A in the liquid phase, and \( P_A^\circ \) is the vapor pressure of pure component A.

Step 2: Calculate partial pressures

For benzene (component A) with a mole fraction in vapor of \( x_A = 0.75 \) and vapor pressure of \( 75 \, \mathrm{mm} \), its partial pressure is \( P_A = 0.75 \times 75 \, \mathrm{mm} \). Similarly, for toluene (component B) with a mole fraction in vapor of \( x_B = 0.25 \) and vapor pressure of \( 25 \, \mathrm{mm} \), its partial pressure is \( P_B = 0.25 \times 25 \, \mathrm{mm} \).

Step 3: Find the total vapor pressure of the solution

The total vapor pressure \( P_{\text{total}} \) is the sum of the partial pressures of benzene and toluene. Using the partial pressures calculated in Step 2, sum them to find the total vapor pressure: \( P_{\text{total}} = P_A + P_B \).

Key Concepts

Vapor Pressure

Vapor pressure is an essential concept in chemistry, particularly when studying solutions and their behaviors. It refers to the pressure exerted by a vapor in equilibrium with its liquid or solid form. Higher vapor pressure indicates a substance will evaporate—or boil—at a lower temperature, meaning it is more volatile.

At a given temperature, every pure substance possesses a characteristic vapor pressure that depends on intermolecular forces; stronger intermolecular forces usually result in lower vapor pressure. In the context of solutions, understanding vapor pressure is crucial for predicting how a mixture will behave when exposed to different environmental conditions.

Ideal Solution

An ideal solution is a mixture where the interactions between different molecules are the same as the interactions between like molecules. This concept is important because it simplifies the prediction of how a solution will behave. In an ideal solution, the components obey Raoult's Law, meaning that the presence of each component does not significantly alter the properties of the solvent.

Ideal solutions are theoretical constructs because most real mixtures exhibit some deviation from ideal behavior. However, systems that closely approximate ideal solutions are often assumed for the sake of calculation simplicity, particularly in educational settings and when the deviation from ideality is not significant for the conclusion being drawn.

Partial Vapor Pressure

The partial vapor pressure is the pressure contributed by a single component in a mixture of gases or a solution. According to Raoult's Law, for an ideal solution, the partial vapor pressure of a component is directly proportional to its mole fraction in the liquid phase and its vapor pressure when pure.

Calculating the partial vapor pressures of each component allows us to determine the total pressure exerted by the mixture. The idea is that each component independently contributes to the total pressure, without interference from the other components, which is a characteristic of an ideal solution.

Mole Fraction

The mole fraction is a way of expressing the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture. Represented by the symbol 'x', mole fractions are dimensionless numbers that provide a convenient way to describe concentrations in multi-component systems, such as solutions.

Mole fraction is used in Raoult's Law to calculate the partial vapor pressure because it is a measure of the relative amount of each component present in a solution. Since mole fractions take into consideration the ratio of all components, they are particularly useful in mixtures with many components, allowing for a straightforward way to represent composition without the need for units.