Problem 75 - A Complete Resource Book in Chemistry for JEE Main

Question

The vapour pressure of two liquids ' \(\mathrm{P}\) ' and ' \(\mathrm{Q}\) ' are 80 and 60 torr respectively. The total vapour pressure of solution obtained by mixing 3 mole of \(\mathrm{P}\) and \(2 \mathrm{~mol}\) of Q would be (a) 20 torr (b) 72 torr (c) 68 torr (d) 140 torr

Step-by-Step Solution

Answer

The total vapour pressure of the solution is 72 torr.

Step 1: Calculate Mole Fraction of Each Component

To start, we need to determine the mole fractions of liquids P and Q in the solution. The mole fraction \( X_P \) for P is given by the formula: \( X_P = \frac{n_P}{n_P + n_Q} = \frac{3}{3 + 2} = 0.6 \). Similarly, the mole fraction \( X_Q \) for Q is \( X_Q = \frac{n_Q}{n_P + n_Q} = \frac{2}{3 + 2} = 0.4 \).

Step 2: Find Partial Vapour Pressures

Using Raoult's Law, the partial vapour pressure of each component is found by multiplying the mole fraction by its pure component vapour pressure. Thus, \( P_P = X_P \times P^0_P = 0.6 \times 80 = 48 \text{ torr} \) for P, and \( P_Q = X_Q \times P^0_Q = 0.4 \times 60 = 24 \text{ torr} \) for Q.

Step 3: Calculate Total Vapour Pressure

The total vapour pressure of the solution is the sum of the partial vapour pressures of P and Q. Therefore, total vapour pressure \( P_{total} = P_P + P_Q = 48 + 24 = 72 \text{ torr} \).

Key Concepts

Vapour Pressure

Vapour pressure is an essential concept in understanding how liquids evaporate and form vapour. When a liquid is placed in a closed container, some molecules at the surface gain enough energy to escape into the air as vapour. Over time, a balance is achieved where an equal number of molecules are evaporating and condensing, creating a pressure exerted by the vapour above the liquid's surface. This pressure is the vapour pressure.
Here are some key points about vapour pressure:

• Every liquid has a characteristic vapour pressure depending on its temperature. Higher temperatures typically increase vapour pressure because molecules have more energy to escape the liquid phase.
• Vapour pressure is independent of the volume of the liquid or the size of the container, as long as some liquid remains to keep the equilibrium intact.

Understanding vapour pressure is crucial when dealing with solutions, as it directly impacts the boiling point and evaporation rates.

Mole Fraction

Mole fraction is a way of describing the concentration of a component in a mixture. It gives insight into how much of each substance is present compared to the total amount. Calculating the mole fraction involves dividing the moles of the individual component by the total moles of all components in the mixture.
More on mole fraction:

• The formula for mole fraction for a component A is: \( X_A = \frac{n_A}{n_{total}} \), where \( n_A \) is moles of A and \( n_{total} \) is the total moles of all components.
• The sum of mole fractions in a mixture always equals one: \( X_A + X_B + X_C + \, \ldots = 1 \).
• It is a dimensionless quantity and can provide a simple way to express mixture compositions, both in liquid and gas phases.

Understanding mole fraction helps in applying Raoult's Law, where mixture behaviour can be tied to the properties of individual components.

Partial Vapour Pressure

Partial vapour pressure is a pivotal concept in understanding how mixtures behave. According to Raoult's Law, the partial vapour pressure of a component in a liquid mixture is the product of the mole fraction of the component and the vapour pressure of the pure component.
Insights about partial vapour pressure include:

• Formula: \( P_i = X_i \times P^0_i \), where \( P_i \) is the partial vapour pressure, \( X_i \) is the mole fraction, and \( P^0_i \) is the vapour pressure of the pure component.
• This concept is crucial in predicting the total vapour pressure in a solution, as it is simply the sum of all partial vapour pressures: \( P_{total} = P_1 + P_2 + \, \ldots \).
• Partial vapour pressure helps in understanding how individual components influence the overall behaviour of solutions, such as boiling points and evaporation rates.

Grasping partial vapour pressure is essential for exploring and predicting the behaviours of solutions, especially in multi-component systems.