Problem 157 - A Complete Resource Book in Chemistry for JEE Main

Question

Benzene and toluene form nearly ideal solutions. At \(20^{\circ} \mathrm{C}\), the vapour pressure of benzene is 75 torr and that of toluene is 22 torr. The partial vapour pressure of benzene at \(20^{\circ} \mathrm{C}\) for a solution containing \(78 \mathrm{~g}\) of benzene and \(46 \mathrm{~g}\) of toluene in torr is (a) 25 (b) 50 (c) \(37.5\) (d) \(53.5\)

Step-by-Step Solution

Answer

The partial vapour pressure of benzene is (b) 50 torr.

Step 1: Calculate Moles of Benzene and Toluene

First, determine the number of moles of benzene and toluene. The molar mass of benzene (C₆H₆) is approximately 78 g/mol, and the molar mass of toluene (C₇H₈) is approximately 92 g/mol. Number of moles of benzene = \( \frac{78 \, \text{g}}{78 \, \text{g/mol}} = 1 \) mole.Number of moles of toluene = \( \frac{46 \, \text{g}}{92 \, \text{g/mol}} \approx 0.5 \) moles.

Step 2: Calculate the Mole Fraction of Benzene

The mole fraction of a component is calculated using the formula:\[ x_B = \frac{\text{moles of benzene}}{\text{total moles}} \]Here, moles of benzene = 1 and total moles (benzene + toluene) = 1 + 0.5 = 1.5.\[ x_B = \frac{1}{1.5} \approx 0.67 \]

Step 3: Apply Raoult's Law for Partial Vapour Pressure of Benzene

Raoult's Law states that the partial vapour pressure of a component in a solution is the product of the mole fraction of that component and its pure component vapour pressure:\[ P_B = x_B \times P^0_B \]Where \( P^0_B \) is the vapour pressure of pure benzene (75 torr) and \( x_B \) is the mole fraction of benzene (0.67).\[ P_B = 0.67 \times 75 \, \text{torr} \approx 50 \, \text{torr} \]

Step 4: Conclusion

The partial vapour pressure of benzene in the solution at \(20^{\circ} \mathrm{C}\) is approximately 50 torr. Therefore, the correct answer is (b) 50.

Key Concepts

Raoult's Law

Raoult's Law is fundamental in understanding the behavior of components in ideal solutions. It provides a way to calculate the partial vapor pressure of each component in a mixture based on the mole fraction and the pure component vapor pressure. According to Raoult's Law:- The partial vapor pressure of a component is directly proportional to its mole fraction in the solution.- The law is expressed as: \[P_i = x_i \times P^0_i\] where \(P_i\) represents the partial vapor pressure of component \(i\), \(x_i\) is the mole fraction of component \(i\), and \(P^0_i\) is the pure component vapor pressure.- Raoult's Law is applicable to ideal solutions, where the interactions between molecules are similar.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is a dimensionless quantity that provides the ratio of the moles of one component to the total moles in the solution:- The formula for mole fraction \(x_i\) is: \[ x_i = \frac{\text{moles of component } i}{\text{total moles in solution}} \]- It ranges from 0 to 1, where a value close to 1 indicates a high concentration of that component.- To calculate the mole fraction in a binary solution of benzene and toluene, determine the moles for each and sum them for total moles. This step is crucial when applying Raoult’s Law.

Partial Vapour Pressure

Partial vapor pressure is an important concept when dealing with liquid mixtures and solutions. It refers to the pressure that a component contributes to the total pressure of a vapor above the solution:- Each component in a solution exerts its own vapor pressure, contributing to the overall pressure.- The total vapor pressure is the sum of all partial pressures.- Partial vapor pressure is calculated using the mole fraction and the pure component vapor pressure via Raoult’s Law.- For example, if benzene has a mole fraction of 0.67 and a pure vapor pressure of 75 torr, the partial vapor pressure is \(0.67 \times 75 = 50\) torr.

Benzene and Toluene

Benzene (C₆H₆) and toluene (C₇H₈) are hydrocarbons known for forming nearly ideal solutions. This means their molecular interactions in the mixture are similar to those in their pure forms: - Both are aromatic compounds with similar chemical properties. - They are often used together in research and industry due to their compatible physical properties. - At a temperature of 20°C, benzene has a vapor pressure of 75 torr, while toluene has a vapor pressure of 22 torr. - Knowing these values is crucial when using Raoult's Law to determine the behavior of their solutions.

Vapor Pressure Calculation

Calculating vapor pressure in a mixture involves understanding the properties of each component and using Raoult's Law: - First, determine the moles of each component to find the mole fractions. - Then, use the mole fraction and the pure component vapor pressures to find each partial pressure. - The steps include: - Calculate the number of moles from given mass and molar mass. - Use the mole fractions in Raoult’s equation to find pressures. - For example, in a solution with benzene and toluene, if benzene contributes 50 torr, it is essential to know its share in the total pressure, which would include toluene’s contribution.