Problem 26 - A Complete Resource Book in Chemistry for JEE Main

Question

The vapour pressure of a solution (P) and the vapour pressure of the solvent \(\left(\mathrm{P}^{0}\right)\) are related to each other as (a) \(\mathrm{P}=\mathrm{P}^{0} \mathrm{x}_{2}\) (b) \(\mathrm{P}=\mathrm{P}^{0} \mathrm{x}_{1}\) (c) \(\mathrm{P}^{0}=\mathrm{Px}_{2}\) (d) \(\mathrm{P}^{0}=\mathrm{Px}_{1}\)

Step-by-Step Solution

Answer

The correct formula is (b) \(P = P^0 x_1\).

Step 1: Understanding Vapor Pressure

Vapor pressure is the pressure exerted by the vapor of a liquid when the liquid and vapor are in equilibrium. In solutions, vapor pressure is influenced by the presence of a solute.

Step 2: Understanding Mole Fractions

The mole fraction of a component in a solution is the ratio of the number of moles of that component to the total number of moles in the solution. Given two components, solute (
) and solvent (), in a binary solution, the mole fraction of the solvent is \(x_1\), and the mole fraction of the solution is \(x_2\).

Step 3: Raoult’s Law for Ideal Solutions

Raoult’s Law states that the vapor pressure of a solvent in an ideal solution is directly proportional to its mole fraction (\(x_1\)) in the solution and the vapor pressure of the pure solvent (\(P^0\)). This is given by the formula: \[ P = P^0 x_1 \]

Step 4: Identifying the Correct Formula

Using Raoult’s Law, we identify that the expression for vapor pressure of a solution in terms of the vapor pressure of the pure solvent follows the equation \(P = P^0 x_1\). This matches option (b).

Key Concepts

Understanding Vapor Pressure

Vapor pressure refers to the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phase (liquid or solid) at a given temperature. When a liquid is enclosed in a container, its molecules escape into the vapor phase. This occurs until a dynamic equilibrium is reached, where the rate of evaporation equals the rate of condensation. At this point, the pressure of the vapor above the liquid is known as vapor pressure.
In solutions, vapor pressure is particularly interesting because it changes with the addition of a solute. When a non-volatile solute is added to a solvent, the vapor pressure of the solution is lower than that of the pure solvent. This happens because the solute molecules take up space at the surface, reducing the number of solvent molecules capable of escaping into the vapor phase. This concept is crucial in understanding how solutions behave differently from pure substances.

Exploring Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture or solution. It is the ratio of the number of moles of a particular substance to the total number of moles of all substances present. For a binary solution containing a solute and a solvent:

• Let the mole fraction of the solute be denoted as \( x_2 \).
• Let the mole fraction of the solvent be denoted as \( x_1 \).

It is important to remember that the sum of the mole fractions of all components in a solution must equal one:
\[ x_1 + x_2 = 1 \]
This concept helps in calculating various properties of solutions, including their vapor pressures. The mole fraction is a marker of relative amounts and does not depend on the total amount of substances, making it a convenient measure in thermodynamics and chemistry.

Ideal Solutions and Raoult's Law

Raoult's Law is fundamental in understanding how the vapor pressure of a solution is determined. It defines ideal solutions, which are mixtures that obey Raoult's Law over a range of concentrations. In an ideal solution, the interactions between unlike molecules (solute-solvent) are similar to those between like molecules (solute-solute or solvent-solvent).
Raoult’s Law states that the vapor pressure of the solvent above a solution \( P \) is directly proportional to the mole fraction of the solvent \( x_1 \) in the solution. The formula is given as:
\[ P = P^0 x_1 \]
where \( P^0 \) is the vapor pressure of the pure solvent. This relationship relies heavily on the assumption that the solution behaves ideally, meaning it has perfect interactions without any volume or enthalpy change upon mixing. While real solutions might slightly deviate, Raoult's Law is a useful approximation for many scenarios and provides predictions about how the concentration and volatility of components affect the overall vapor pressure.