Problem 10 - A Complete Resource Book in Chemistry for JEE Main

Question

According to Raoult's law, relative lowering of vapour pressure for a solution is equal to (a) mole fraction of the solute (b) mole fraction of a solvent (c) moles of a solute (d) moles of a solvent

Step-by-Step Solution

Answer

(a) mole fraction of the solute

Step 1: Understanding Raoult's Law

Raoult's Law states that the vapour pressure of an ideal solution is directly proportional to the mole fraction of solvent present in the solution. It is given by the equation \( P_{ ext{solution}} = \chi_{ ext{solvent}} P^{ ext{pure}} \), where \( P_{ ext{solution}} \) is the vapour pressure of the solution, \( \chi_{ ext{solvent}} \) is the mole fraction of the solvent, and \( P^{ ext{pure}} \) is the vapour pressure of the pure solvent.

Step 2: Defining Relative Lowering of Vapour Pressure

The relative lowering of vapour pressure is defined as the decrease in vapour pressure of the solvent compared to the pure solvent, divided by the vapour pressure of the pure solvent. It can be expressed as \( \frac{\Delta P}{P^{\text{pure}}} = 1 - \chi_{\text{solvent}} = \chi_{\text{solute}} \), where \( \Delta P \) is the decrease in vapour pressure.

Step 3: Connection with Mole Fraction

According to the derived formula \( \frac{\Delta P}{P^{\text{pure}}} = \chi_{\text{solute}} \), the relative lowering of vapour pressure equals the mole fraction of the solute. This means that option (a) mole fraction of the solute is the correct answer for the problem.

Key Concepts

Vapour Pressure

Vapour pressure is a crucial concept in chemistry, especially when studying solutions and Raoult's Law. It is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. In simpler terms, it's the tendency of particles to escape from a liquid (or a solid) into the gaseous phase. This property is dependent on the nature of the liquid and temperature.

When we talk about solutions, the vapour pressure of the solution is often lower than that of the pure solvent due to the presence of solute particles. These solute particles occupy space at the surface, reducing the number of solvent particles that can escape, thus lowering the vapour pressure. Raoult's Law helps us predict this reduction by relating it to the composition of the solution.

This can be expressed mathematically as:

• For a solvent in a solution: \( P_{\text{solution}} = \chi_{\text{solvent}} \times P^{\text{pure}} \)
• For the relative lowering: \( \frac{\Delta P}{P^{\text{pure}}} = \chi_{\text{solute}} \)

Understanding vapour pressure and how it changes with the presence of a solute is fundamental to grasping the principles of solutions and their behaviors.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is a dimensionless quantity that represents the ratio of the number of moles of one component to the total number of moles of all components in the mixture. For a solvent, it is denoted as \( \chi_{\text{solvent}} \) and for a solute, it is \( \chi_{\text{solute}} \).

Mathematically, the mole fraction of the solute is given by:

• \( \chi_{\text{solute}} = \frac{n_{\text{solute}}}{n_{\text{solute}} + n_{\text{solvent}}} \)

This concept is essential in Raoult's Law because it provides a direct relationship between the vapour pressure of a solution and its composition. According to Raoult’s Law, as the mole fraction of the solvent decreases (due to more solute being present), the vapour pressure of the solution also decreases.

Thus, the relative lowering of vapour pressure in an ideal solution is equal to the mole fraction of the solute. It's a concise way to understand how adding a solute changes the physical properties of a solution.

Ideal Solution

An ideal solution is a theoretical concept where the interactions between all molecules are identical to those in the pure components. In ideal solutions, Raoult's Law holds true entirely. This means the total pressure of the solution is the sum of each component's partial pressure, calculated using their respective mole fractions and the pure component's vapour pressures.

In practice, real solutions may deviate from ideal behavior due to differences in intermolecular interactions. However, many dilute solutions behave nearly ideally, especially when the solute and solvent are similar in size and properties.

Characteristics of an ideal solution include:

• Strict adherence to Raoult's Law
• No enthalpy change upon mixing (i.e., no heat is absorbed or released)
• Volume of the mixture is equal to the sum of the volumes of pure components

The concept of an ideal solution is significant because it provides a simple model to understand how solutions behave under various conditions, allowing chemists to predict changes in vapour pressure, boiling points, and freezing points accurately.