Problem 14 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Vapour pressure of pure benzene is 119 torr and that of toluene is \(37.0\) torr at the same temperature. Mole fraction of toluene in vapour phase which is in equilibrium with a solution of benzene and toluene having a mole fraction of toluene \(0.50\), will be : (a) \(0.137\) (b) \(0.237\) (c) \(0.435\) (d) \(0.205\)

Step-by-Step Solution

Answer

(b) 0.237

Step 1: Calculate Partial Pressure of Benzene

To find the partial pressure of benzene, use Raoult's law, which states that the partial vapor pressure is equal to the mole fraction of the component in the liquid phase multiplied by the vapor pressure of the pure component.Given:- Vapor pressure of pure benzene = 119 torr- Mole fraction of benzene in the solution = 1 - 0.50 = 0.50Using Raoult's Law:\[P_{\text{benzene}} = \text{Mole fraction of benzene} \times \text{Vapor pressure of pure benzene} = 0.50 \times 119 = 59.5\text{ torr}\]

Step 2: Calculate Partial Pressure of Toluene

Similarly, calculate the partial pressure of toluene using Raoult's law:Given:- Vapor pressure of pure toluene = 37.0 torr- Mole fraction of toluene in the solution = 0.50Using Raoult's Law:\[P_{\text{toluene}} = \text{Mole fraction of toluene} \times \text{Vapor pressure of pure toluene} = 0.50 \times 37.0 = 18.5\text{ torr}\]

Step 3: Calculate Total Vapor Pressure of the Solution

The total vapor pressure of the solution is simply the sum of the partial pressures of benzene and toluene:\[P_{\text{total}} = P_{\text{benzene}} + P_{\text{toluene}} = 59.5 + 18.5 = 78.0\text{ torr}\]

Step 4: Calculate Mole Fraction of Toluene in Vapor Phase

The mole fraction of toluene in the vapor phase is given by the ratio of the partial pressure of toluene to the total vapor pressure of the solution.\[x_{\text{toluene, vapor}} = \frac{P_{\text{toluene}}}{P_{\text{total}}} = \frac{18.5}{78.0} \approx 0.237\]

Key Concepts

Vapour Pressure

Understanding vapour pressure is essential to grasp how different components in a solution behave. Vapour pressure refers to the pressure that a substance's vapor exerts when it is in equilibrium with its liquid or solid form. In simpler terms, it is the measure of a liquid's tendency to evaporate.

Every liquid has a specific vapour pressure at a given temperature. This is because the number of molecules escaping into the vapor phase increases with temperature, increasing pressure. For instance, in our exercise, the vapour pressure of pure benzene is 119 torr, whereas toluene has a vapour pressure of 37 torr at the same temperature.

In a mixture of liquids, each component contributes to the total vapour pressure based on its mole fraction and inherent vapour pressure. Therefore, knowing the vapour pressure of pure components helps calculate the pressure each component contributes in a solution.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of one component to the total number of moles in the mixture.

The concept is crucial because it helps determine how much a particular component contributes to properties like vapour pressure in a solution. In the given exercise, the mole fraction of toluene is initially provided as 0.50, meaning toluene constitutes half of the moles in the liquid mixture with benzene.

• Formula: \( \text{Mole fraction } x = \frac{\text{moles of component}}{\text{total moles of all components}} \)
• A mole fraction is always between 0 and 1, with the sum of all mole fractions in a mixture equaling 1.

This mole fraction is pivotal when applying Raoult's Law to determine partial pressures and further understand the behavior of substances in both liquid and vapor phases.

Partial Pressure

Partial pressure refers to the pressure exerted by a single gas component in a mixture of gases. According to Dalton’s law, each component in a gaseous mixture contributes to the total pressure in proportion to its percentage of the total moles.

To apply Raoult’s Law here, we need the mole fractions of benzene and toluene in the liquid phase to calculate their respective partial pressures in the vapor phase. In our example, the partial pressure of toluene is calculated using its mole fraction and its pure component vapour pressure.

• Partial Pressure Formula (using Raoult’s Law): \(P_{ ext{component}} = x_{ ext{component-liquid}} \times P_{ ext{component-pure}}\)
• Sum of all partial pressures (in this case benzene and toluene) equals the total vapour pressure of the solution.

It is important to compute partial pressures accurately as these are used in the final step to establish the mole fraction of each component in the vapor phase, giving insight into which component is more volatile and in what proportion they are present as vapor over the liquid mixture.