Problem 83 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(\mathrm{P}^{0}\) and \(\mathrm{P}\) are the vapour pressure of solvent and solution and \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) are the moles of solute and solvent respectively (a) \(\mathrm{P}^{0}=\mathrm{P}\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{1}+\mathrm{n}_{2}}\right]\) (b) \(\mathrm{P}^{0}=\mathrm{P}\left[\frac{\mathrm{n}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}\right]\) (c) \(\mathrm{P}=\mathrm{P}^{0}\left[\frac{\mathrm{n}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}\right]\) (d) \(\mathrm{P}^{0}=\mathrm{P} \times \mathrm{n}_{1}\)

Step-by-Step Solution

Answer

Option (c) is correct.

Step 1: Identify What is Asked

We need to determine which given option correctly represents the relationship between the vapor pressures of pure solvent and solution, as well as the moles of solute and solvent.

Step 2: Review Raoult's Law

According to Raoult's Law, the vapor pressure of a solvent in a solution is proportional to the mole fraction of the solvent in the solution. The formula is given by:\[\mathrm{P} = \mathrm{P}^{0} \times \frac{n_{2}}{n_{1} + n_{2}}\]where \(\mathrm{P}\) is the vapor pressure of the solvent in the solution, \(\mathrm{P}^{0}\) is the vapor pressure of the pure solvent, \(n_{2}\) is the moles of solvent, and \(n_{1}\) is the moles of solute.

Step 3: Analyze Each Option

Evaluate the options:- (a) \(\mathrm{P}^{0} = \mathrm{P}\left[\frac{n_{1}}{n_{1} + n_{2}}\right]\): This does not align with Raoult's Law.- (b) \(\mathrm{P}^{0} = \mathrm{P}\left[\frac{n_{2}}{n_{1} + n_{2}}\right]\): This places the mole fraction of solvent with \(\mathrm{P}\) incorrectly.- (c) \(\mathrm{P} = \mathrm{P}^{0}\left[\frac{n_{2}}{n_{1} + n_{2}}\right]\): Correct, aligns with Raoult's Law.- (d) \(\mathrm{P}^{0} = \mathrm{P} \times n_{1}\): Incorrect representation of relationship.

Step 4: Select the Correct Answer

Option (c) is the correct representation since it aligns with Raoult's Law, showing the correct relationship between the vapor pressure of the solution and the mole fraction of the solvent.

Key Concepts

Vapor Pressure

Vapor pressure is an important concept to understand when studying solutions. It refers to the pressure exerted by the vapor present above a liquid in a closed container. This pressure results from the equilibrium between molecules evaporating from the liquid and those condensing back into it. When a liquid is pure, its vapor pressure depends solely on the temperature.
The presence of a solute in the solvent alters this equilibrium, generally lowering the vapor pressure compared to that of the pure solvent. This reduction occurs because the solute particles take up space at the surface of the liquid, diminishing the overall number of solvent molecules capable of escaping into the vapor phase. Understanding vapor pressure is crucial for various applications, such as predicting boiling points and sublimation processes.

Mole Fraction

One of the key ways to measure the composition of a solution is through the concept of mole fraction. Mole fraction is the ratio of the number of moles of a component to the total number of moles in the solution. This concept is unique because it is a dimensionless quantity, meaning it does not have units and provides a clear proportion of components.

For example, if a solution has 2 moles of solute and 8 moles of solvent, the mole fraction of the solute (_{1}) would be \( rac{2}{2 + 8} = 0.2 \), and for the solvent (_{2}), it would be \( rac{8}{2 + 8} = 0.8 \). These fractions help in applying Raoult's Law, which predicts the changes in vapor pressure due to the presence of a solute.
Using mole fraction makes it easier to calculate other properties, like the partial pressure of components in gas mixtures, especially in chemistry and thermodynamics.

Solute and Solvent Relationship

In a solution, the relationship between solute and solvent is essential for understanding how solutions form and behave. The solute is the substance dissolved in another substance, the solvent. The interplay between these two components defines many of the solution's properties, including its vapor pressure and boiling point.
According to Raoult's Law, the addition of a non-volatile solute to a volatile solvent will lower the vapor pressure of the resulting solution. This is because the solute particles occupy the surface of the liquid, preventing some solvent molecules from escaping into the vapor phase. This relationship is not just theoretical but can be statistically broad, affecting fields such as food science, pharmacology, and even materials engineering.

Understanding how the solute and solvent interact also helps in predicting behaviors like mixing, solubility, and the effects of concentration changes. These insights are crucial for practical applications such as creating solutions with desired concentrations and tailoring the properties of products in chemical industries.