Problem 111 - A Complete Resource Book in Chemistry for JEE Main

Question

The vapour pressure of pure benzene and toluene are 160 and 60 torr respectively. The mole fraction of toluene in vapour phase in contact with equimolar solution of benzene and toluene is (a) \(0.6\) (b) \(0.50\) (c) \(0.27\) (d) \(0.74\)

Step-by-Step Solution

Answer

The mole fraction of toluene in the vapor phase is approximately 0.27 (option c).

Step 1: Understand Raoult’s Law

When two volatile liquids are mixed, the partial pressure of each component is proportional to its mole fraction in the mixture and its vapor pressure when pure, according to Raoult's Law. This can be expressed as \( P_A = x_A \cdot P^0_A \) and \( P_B = x_B \cdot P^0_B \), where \( P_A \) and \( P_B \) are the partial pressures of the components A and B, \( x_A \) and \( x_B \) are their mole fractions in the liquid mixture, and \( P^0_A \) and \( P^0_B \) are their vapor pressures when pure.

Step 2: Calculate Mole Fractions in Liquid

Since the solution is equimolar, the mole fraction of benzene (B) and toluene (T) in the liquid phase are both \( x_B = x_T = 0.5 \). Therefore, using Raoult’s Law, we can calculate the partial pressures: \[ P_B = x_B \cdot P^0_B = 0.5 \times 160 = 80 \text{ torr} \] \[ P_T = x_T \cdot P^0_T = 0.5 \times 60 = 30 \text{ torr} \]

Step 3: Calculate Total Vapor Pressure

The total vapor pressure \( P_{total} \) of the solution is the sum of the partial pressures of benzene and toluene:\[ P_{total} = P_B + P_T = 80 + 30 = 110 \text{ torr} \]

Step 4: Determine Mole Fraction in Vapor Phase

The mole fraction of toluene in the vapor phase, \( y_T \), is given by the ratio of the partial pressure of toluene to the total vapor pressure:\[ y_T = \frac{P_T}{P_{total}} = \frac{30}{110} \approx 0.27 \]

Key Concepts

Vapor Pressure

Vapor pressure is an essential concept in understanding how substances in a liquid form evaporate into a gas under certain conditions. Vapor pressure is the pressure that is exerted by the vapor of a liquid when it is in equilibrium with its liquid phase. For a pure substance, this pressure is determined only by the temperature. As temperature increases, vapor pressure increases as well, because more molecules have the energy to escape the liquid phase.
In a mixture of two volatile liquids, each component contributes to the total vapor pressure, each depending on its nature and quantity.
When experimenting or solving problems with solutions, it's crucial to understand how vapor pressure behaves and how it's influenced by different substances present in a solution. This is where Raoult's Law plays a significant role.

Mole Fraction

The mole fraction is a mathematical term used to describe the concentration of a component within a mixture. 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.
Mathematically, for a mixture with a component A:\[x_A = \frac{n_A}{n_{total}}\]Where:

• \(x_A\) = mole fraction of component A
• \(n_A\) = number of moles of component A
• \(n_{total}\) = total number of moles in the mixture

The mole fraction gives insight into the composition of a mixture, which is crucial when determining how each component affects the properties of the mixture, such as its vapor pressure. In the exercise, an equimolar solution implies that both components have the same mole fraction.

Partial Pressure

The partial pressure of a gas is the pressure that gas would exert if it occupied the space alone. In a mixture of gases, each component contributes to the total pressure according to its own properties and amount.For solutions containing volatile liquids, like benzene and toluene, partial pressure is a key aspect of Raoult's Law:\[P_A = x_A \cdot P^0_A\]Here:

• \(P_A\) is the partial pressure of component A
• \(x_A\) is its mole fraction in the liquid phase
• \(P^0_A\) is its pure vapor pressure

Each component's partial pressure is added to find the total vapor pressure of the mixture.
Understanding partial pressures is crucial when determining the behavior of components in a solution, particularly in calculations of how each component evaporates.

Equimolar Solution

An equimolar solution is a mixture where the components are present in equal molar amounts. This means that the mole fraction for each component in the solution is the same, meaning they are both\(x_B = x_T = 0.5\). In practice, having an equimolar solution implies easier calculations since you don't have to worry about disproportionate contributions from components in terms of mole fraction.In problems involving Raoult's Law, an equimolar solution will simplify calculations as each component contributes equally relative to their pure vapor pressures. This can affect results, such as the composition of the vapor phase, where components may not remain equimolar due to different evaporation rates and vapor pressures.An equimolar concept makes solving problems regarding vapor and liquid equilibria straightforward and less complex.