Problem 4 - Chemistry in 30 days

Question

The vapour pressure at a given temperature of an ideal solution containing \(0.2 \mathrm{~mol}\) of a non-volatile solute and \(0.8 \mathrm{~mol}\) of a solvent is \(60 \mathrm{~mm}\) of \(\mathrm{Hg}\). The vapour pressure of the pure solvent at the same temperature will be (a) \(120 \mathrm{~m}\) of \(\mathrm{Hg}\) (b) \(150 \mathrm{~mm}\) of \(\mathrm{Hg}\) (c) \(60 \mathrm{~mm}\) of \(\mathrm{Hg}\) (d) \(75 \mathrm{~mm}\) of \(\mathrm{Hg}\)

Step-by-Step Solution

Answer

The vapor pressure of the pure solvent is 75 mm of Hg (option d).

Step 1: Understand Raoult's Law

Raoult's law states that the vapor pressure of a solution is directly proportional to the mole fraction of the solvent. For an ideal solution, the formula is given by \( P = X_{A} \cdot P_{A}^{0} \), where \( P \) is the vapor pressure of the solution, \( X_{A} \) is the mole fraction of the solvent, and \( P_{A}^{0} \) is the vapor pressure of the pure solvent.

Step 2: Calculate Mole Fraction of the Solvent

The mole fraction of the solvent \( X_{A} \) is given by the ratio of moles of solvent to the total moles of the solution. Here, the moles of non-volatile solute is \( 0.2 \) mol and the moles of solvent is \( 0.8 \) mol. Thus, \( X_{A} = \frac{0.8}{0.2 + 0.8} = 0.8 \).

Step 3: Apply Raoult’s Law

According to Raoult's Law, the vapor pressure of the solution \( P \) is equal to the product of the mole fraction of the solvent \( X_{A} \) and the vapor pressure of the pure solvent \( P_{A}^{0} \). Rearranging the formula give us \( P_{A}^{0} = \frac{P}{X_{A}} \).

Step 4: Solve for Vapor Pressure of Pure Solvent

Substitute the given values into the rearranged formula to find \( P_{A}^{0} \). Here, \( P = 60 \) mm of Hg and \( X_{A} = 0.8 \), so \( P_{A}^{0} = \frac{60}{0.8} = 75 \) mm of Hg.

Key Concepts

Vapor Pressure

Vapor pressure 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. This occurs when the rate of evaporation_of a liquid equals the rate of condensation. Vapor pressure plays a key role in the boiling point of a liquid as a liquid boils when its vapor pressure equals the atmospheric pressure.

In terms of solutions, the presence of a non-volatile solute reduces the solvent's ability to escape into the vapor phase. Consequently, the vapor pressure of the solution is lower than that of the pure solvent. Understanding vapor pressure helps us predict and explain the behavior of solutions under various conditions, including how they evaporate and condense.

Raoult's Law ties into this concept by explaining how the vapor pressure of an ideal solution depends on the vapor pressure of each component and their respective mole fractions.

Ideal Solution

An ideal solution refers to a solution that follows Raoult's Law over the entire range of concentrations. Such solutions are characterized by interactions between particles that are similar to the interactions in the pure components. In simpler terms, the forces between like and unlike molecules are the same.

For an ideal solution:

• The enthalpy of mixing is zero, which means there is no heat absorbed or released during the mixing process.
• Changes in volume upon mixing are negligible.

The concept of an ideal solution simplifies the study of solutions by providing a model that can be mathematically described easily. While real solutions might deviate from ideality due to various factors, understanding ideal solutions sets a foundation for understanding more complex behaviors in mixtures.

When discussing vapor pressure in ideal solutions, we use Raoult's Law to calculate how much each component contributes to the total vapor pressure.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a solution or mixture. It represents the ratio of the number of moles of a component to the total number of moles of all components in the mixture.

For a solution of a solute in a solvent, the mole fraction of the solvent is given by:\[ X_{A} = \frac{\text{moles of solvent}}{\text{total moles in solution}} \]
For example, if a solution contains 0.8 moles of solvent and 0.2 moles of a non-volatile solute, the mole fraction of the solvent would be \( X_{A} = \frac{0.8}{0.8 + 0.2} = 0.8 \).

Utilizing mole fractions allows us to accurately determine how much each component affects the properties of the solution, such as its vapor pressure. This concept is crucial in applying Raoult's Law, as it dictates the proportional contribution of each component to the total vapor pressure.

Non-Volatile Solute

A non-volatile solute is a term used to describe a solute that does not easily vaporize at a given temperature, meaning it has an extremely low vapor pressure compared to the solvent. In other words, the non-volatile solute does not contribute to the vapor above the solution.

The addition of a non-volatile solute to a volatile solvent decreases the overall vapor pressure of the solution. This occurs because the presence of the solute lowers the number of solvent molecules at the surface available to escape into the vapor phase. As such, the solution will boil at a higher temperature than the pure solvent.

This principle is useful in processes like evaporation and distillation, where the separation of solvent from solute involves changes in vapor pressure and boiling points. Additionally, understanding the effects of non-volatile solutes helps in designing solutions with specific boiling or freezing points, such as antifreeze.