Question

Explain why the vapor pressure of a solvent is lowered by the presence of a nonvolatile solute.

Short answer

The presence of a nonvolatile solute reduces the solvent's surface molecules, lowering its vapor pressure as per Raoult's Law.

Step-by-step solution

Understanding Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid phase. It is affected by temperature and the nature of the liquid molecules. A higher vapor pressure implies that more molecules are escaping the liquid into the vapor phase.

Effect of Solute on Solvent

When a nonvolatile solute is added to a solvent, it occupies space on the liquid’s surface, replacing some solvent molecules and thereby reducing the number of solvent molecules that can escape into the vapor phase.

Raoult's Law Introduction

According to Raoult's Law, the presence of a nonvolatile solute lowers the vapor pressure of the solvent in a solution. It is a consequence of the reduced probability of solvent molecules escaping into the vapor phase due to the presence of solute particles.

Mathematical Expression of Raoult's Law

Raoult's Law can be represented mathematically as: \[ P_{ ext{solution}} = X_{ ext{solvent}} imes P_{ ext{solvent}}^0 \]where \( P_{ ext{solution}} \) is the vapor pressure of the solution, \( X_{ ext{solvent}} \) is the mole fraction of the solvent, and \( P_{ ext{solvent}}^0 \) is the vapor pressure of the pure solvent.

Conclusion

Since the mole fraction of the solvent, \( X_{ ext{solvent}} \), is less than 1 when a solute is present, the vapor pressure of the solution \( P_{ ext{solution}} \) is less than the vapor pressure of the pure solvent \( P_{ ext{solvent}}^0 \). Thus, the addition of a nonvolatile solute lowers the vapor pressure of the solvent.

Key concepts

Raoult's Law

Raoult's Law is a fundamental principle in chemistry that explains how the presence of a solute affects the vapor pressure of a solvent. Essentially, this law states that the vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent. This means that when you add a nonvolatile solute—something that doesn’t vaporize—into a solvent, the effective concentration of the solvent decreases. This has a direct impact on the vapor pressure.Raoult’s Law is mathematically expressed as:\[ P_{\text{solution}} = X_{\text{solvent}} \times P_{\text{solvent}}^0 \]Where:

• \( P_{\text{solution}} \) is the vapor pressure of the solution.
• \( X_{\text{solvent}} \) is the mole fraction of the solvent.
• \( P_{\text{solvent}}^0 \) is the vapor pressure of the pure solvent.

This formula shows that the vapor pressure of a solution is reduced by the presence of a solute since the mole fraction of the solvent \( X_{\text{solvent}} \) is less than 1 with the solute present.

Nonvolatile Solute

A nonvolatile solute is a substance that does not easily vaporize or become a gas. Unlike volatile substances, nonvolatile solutes do not contribute significantly to the vapor phase when mixed with a solvent. This characteristic is critical when understanding why such solutes lower the vapor pressure of a solvent. When a nonvolatile solute is added to a solvent:

• The solute particles take up space at the liquid’s surface. This limits the surface area available for the solvent molecules to escape into the vapor phase.
• Since fewer solvent molecules are able to escape into the vapor phase, the overall vapor pressure of the liquid decreases.

This is why solutions containing a nonvolatile solute generally exhibit a lower vapor pressure compared to the pure solvent.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture, making it particularly useful in the context of solutions. It is defined as the number of moles of a component divided by the total number of moles in the solution.The formula for calculating mole fraction \( X \) for the solvent in a solution is:\[ X_{\text{solvent}} = \frac{n_{\text{solvent}}}{n_{\text{solvent}} + n_{\text{solute}}} \]Where:

• \( n_{\text{solvent}} \) is the number of moles of the solvent.
• \( n_{\text{solute}} \) is the number of moles of the solute.

In a solution, the presence of a solute decreases the mole fraction of the solvent, \( X_{\text{solvent}} \), because part of the total moles in the denominator includes solute moles. A lower mole fraction of the solvent means the vapor pressure is reduced according to Raoult's Law.

Liquid-Vapor Equilibrium

Liquid-vapor equilibrium refers to the condition in which the rate of evaporation of a liquid equals the rate of condensation of its vapor. This equilibrium is important when considering vapor pressure, as it represents a state where the number of molecules escaping the liquid to become gas is equal to the number of gas molecules re-entering the liquid phase. In a closed system at a given temperature:

• The vapor pressure reflects this dynamic balance.
• Introducing a nonvolatile solute changes this equation, disrupting the equilibrium by lowering the vapor pressure.

This is because the solute particles interfere with the movement of solvent molecules, reducing the rate at which they enter the vapor phase, thus altering the equilibrium point and lowering the overall vapor pressure.