Problem 168 - A Complete Resource Book in Chemistry for JEE Main

Question

On mixing, heptane and octane form an ideal solution. At \(373 \mathrm{~K}\), the vapour pressures of the two liquid components (heptane and octane) are \(105 \mathrm{kPa}\) and \(45 \mathrm{kPa}\) respectively. Vapour pressure of the solution obtained by mixing \(25.0 \mathrm{~g}\) of heptane and \(35 \mathrm{~g}\) of octane will be (molar mass of heptane \(=100 \mathrm{~g} \mathrm{~mol}^{-1}\) and of octane \(=\) \(114 \mathrm{~g} \mathrm{~mol}^{-1}\) ) (a) \(72.0 \mathrm{kPa}\) (b) \(36.1 \mathrm{kPa}\) (c) \(96.2 \mathrm{kPa}\) (d) \(144.5 \mathrm{kPa}\)

Step-by-Step Solution

Answer

(a) 72.0 kPa.

Step 1: Calculate moles of heptane and octane

To determine the number of moles of heptane, use the formula \( n = \frac{m}{M} \) where \( n \) is the number of moles, \( m \) is the mass, and \( M \) is the molar mass.For heptane: \( n_{heptane} = \frac{25.0 \, \text{g}}{100 \, \text{g/mol}} = 0.25 \, \text{mol} \).For octane: \( n_{octane} = \frac{35.0 \, \text{g}}{114 \, \text{g/mol}} \approx 0.307 \, \text{mol} \).

Step 2: Calculate the mole fractions

The mole fraction of a component is given by \( x_i = \frac{n_i}{n_{total}} \).Total moles \( n_{total} = n_{heptane} + n_{octane} = 0.25 + 0.307 = 0.557 \, \text{mol} \).Mole fraction of heptane \( x_{heptane} = \frac{0.25}{0.557} \approx 0.449 \).Mole fraction of octane \( x_{octane} = \frac{0.307}{0.557} \approx 0.551 \).

Step 3: Apply Raoult's Law

According to Raoult's Law, the partial vapor pressures are given by \( P_i = x_i \, P_i^{\text{pure}} \), where \( P_i^{\text{pure}} \) is the vapor pressure of the pure component.Partial vapor pressure of heptane: \( P_{heptane} = 0.449 \times 105 \, \text{kPa} \approx 47.145 \, \text{kPa} \).Partial vapor pressure of octane: \( P_{octane} = 0.551 \times 45 \, \text{kPa} \approx 24.795 \, \text{kPa} \).

Step 4: Calculate the total vapor pressure

The total vapor pressure of the solution is the sum of the partial vapor pressures: \( P_{total} = P_{heptane} + P_{octane} \).\( P_{total} \approx 47.145 \, \text{kPa} + 24.795 \, \text{kPa} = 71.94 \, \text{kPa} \).

Step 5: Check the closest option

Compare the calculated total vapor pressure to the given options. The one closest to \( 71.94 \, \text{kPa} \) is option (a) \( 72.0 \, \text{kPa} \).

Key Concepts

Ideal Solution

When two liquids like heptane and octane are mixed, they can form an ideal solution. An ideal solution is a special case where the interaction between different molecules of the mixed substances is the same as the interaction between molecules of each individual substance.
This means that the properties of the mixture can be calculated based on the properties of the individual components without any corrections.

• There should be no volume change or heat exchange during mixing.
• Each component of the mixture should contribute to the overall vapour pressure in proportion to its mole fraction.

An ideal solution closely follows Raoult's Law, which we will explore further in the sections below.

Mole Fraction

The mole fraction represents the ratio of moles of a particular component to the total number of moles in the solution. It is a way of expressing the concentration of a component in a mixture.
The formula to calculate the mole fraction is: \[ x_i = \frac{n_i}{n_{total}} \]where \( n_i \) is the number of moles of the component and \( n_{total} \) is the sum of moles of all components in the solution.

• Moles tell us how many particles are present in a given weight of a substance.
• Mole fractions are dimensionless numbers that help in calculating properties like partial pressures.

In essence, the mole fraction helps in understanding the composition of the solution, critical for applying Raoult's Law.

Vapour Pressure

Vapour pressure is the pressure exerted by the vapour in equilibrium with its liquid phase at a given temperature.
For a pure liquid, vapour pressure depends solely on the nature of the liquid and the temperature.

• A higher vapour pressure indicates that more liquid is evaporating into the vapour state.
• In an ideal solution, each component's vapour pressure is modified by its mole fraction.

Given a mixture of heptane and octane at a specific temperature, each component contributes to the total vapour pressure proportionally to its mole fraction as per Raoult's Law.

Partial Vapor Pressure

Partial vapor pressure refers to the contribution of each component in a mixture to the total pressure exerted by the vapour above the solution.
Raoult's Law suggests that this contributes in a way dictated by both the vapour pressure of the pure component and its mole fraction within the solution.
The equation used to calculate partial vapor pressure:\[ P_i = x_i \times P_i^{\text{pure}} \]

• \( P_i \) is the partial vapor pressure of component \( i \).
• \( x_i \) is the mole fraction of the component in the solution.
• \( P_i^{\text{pure}} \) is the vapor pressure of the pure component.

Together, the sum of all partial vapor pressures provides the total vapour pressure of the solution, which can predict the behavior of the solution in various conditions.