Question

A solution is prepared by mixing \(0.0300 \mathrm{~mol} \mathrm{CH}_{2} \mathrm{Cl}_{2}\) and \(0.0500\) \(\mathrm{mol} \mathrm{CH}_{2} \mathrm{Br}_{2}\) at \(25^{\circ} \mathrm{C}\). Assuming the solution is ideal, calculate the composition of the vapor (in terms of mole fractions) at \(25^{\circ} \mathrm{C}\). At \(25^{\circ} \mathrm{C}\), the vapor pressures of pure \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) and pure \(\mathrm{CH}_{2} \mathrm{Br}_{2}\) are 133 and \(11.4\) torr, respectively.

Short answer

The composition of the vapor at 25°C is: Mole fraction of CH₂Cl₂ (Y₁) = 0.875 Mole fraction of CH₂Br₂ (Y₂) = 0.125

Step-by-step solution

Calculate the mole fractions of CH₂Cl₂ and CH₂Br₂ in the solution.

First, we need to find the mole fractions of CH₂Cl₂ and CH₂Br₂ in the solution. To do this, we divide the moles of each component by the total moles in the solution. Total moles = moles of CH₂Cl₂ + moles of CH₂Br₂ Total moles = 0.0300 mol + 0.0500 mol = 0.0800 mol Now, we find the mole fractions: Mole fraction of CH₂Cl₂ (X₁) = (moles of CH₂Cl₂) / (total moles) = 0.0300 mol / 0.0800 mol = 0.375 Mole fraction of CH₂Br₂ (X₂) = (moles of CH₂Br₂) / (total moles) = 0.0500 mol / 0.0800 mol = 0.625

Calculate the partial pressures of CH₂Cl₂ and CH₂Br₂.

Next, we will use Raoult's Law to calculate the partial pressures (P₁ and P₂) of CH₂Cl₂ and CH₂Br₂ in the vapor phase. Raoult's Law states: P_i = X_i * P_i° where P_i is the partial pressure of component i, X_i is the mole fraction of component i in the solution, and P_i° is the vapor pressure of the pure component i. For CH₂Cl₂ (component 1): P₁ = X₁ * P₁° = 0.375 * 133 torr = 49.875 torr For CH₂Br₂ (component 2): P₂ = X₂ * P₂° = 0.625 * 11.4 torr = 7.125 torr

Calculate the total pressure of the vapor phase.

Now we will find the total pressure (P_total) of the vapor phase by adding the partial pressures of each component: P_total = P₁ + P₂ = 49.875 torr + 7.125 torr = 57.000 torr

Calculate the mole fractions of CH₂Cl₂ and CH₂Br₂ in the vapor phase.

Finally, we will use the partial pressures to calculate the mole fractions of each component in the vapor phase: Mole fraction of CH₂Cl₂ in vapor (Y₁) = P₁ / P_total = 49.875 torr / 57.000 torr = 0.875 Mole fraction of CH₂Br₂ in vapor (Y₂) = P₂ / P_total = 7.125 torr / 57.000 torr = 0.125 The composition of the vapor at 25°C is: Mole fraction of CH₂Cl₂ (Y₁) = 0.875 Mole fraction of CH₂Br₂ (Y₂) = 0.125

Key concepts

Ideal Solution

An ideal solution is a theoretical concept where the mixing of components does not result in any volume change or heat exchange. This means that the interactions between unlike molecules are similar to those between like molecules. In other words, the intermolecular forces in the mixture are essentially uniform.

Raoult's Law is often applicable to ideal solutions. When two liquids form an ideal solution, each liquid behaves as if it were in its pure form but reduced by its mole fraction.

• No expansion or contraction occurs upon mixing.
• The solution displays no heat of mixing. This means there is no absorption or release of energy.
• Raoult's Law holds true strictly.

In our exercise, the solution of dichloromethane (CH₂Cl₂) and dibromomethane (CH₂Br₂) is assumed to be ideal, hence Raoult's Law can be effectively applied.

Partial Pressure

To understand partial pressure, imagine a gas mixture where each gas exerts a pressure independently. The partial pressure of any component is the pressure it would exert if it occupied the entire volume alone.

Raoult's Law helps us calculate this in an ideal solution: \[ P_i = X_i \cdot P_i^\circ \]where:

• \( P_i \) is the partial pressure of component \( i \).
• \( X_i \) is the mole fraction of component \( i \) in the solution.
• \( P_i^\circ \) is the vapor pressure of the pure component \( i \).

In the solution involving CH₂Cl₂ and CH₂Br₂, partial pressures were calculated for each component. These values depend on each component's mole fraction in the solution and their respective vapor pressures in their pure states.

Vapor Pressure

Vapor pressure is the pressure at which a liquid's molecules are in equilibrium in the gaseous state. For any liquid, vapor pressure signifies how volatile the liquid is.

A higher vapor pressure means a higher tendency for the molecules to escape into the gas phase.

• Pure CH₂Cl₂ with a vapor pressure of 133 torr indicates more molecules escapable to the vapor phase compared to CH₂Br₂ with only 11.4 torr.
• The difference in vapor pressures reflects different volatilities between the two substances.

In our solution scenario, these values are used to compute the component's partial pressures via Raoult's Law, integrating directly into the understanding of how the vapor phase will form over the solution mixture.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is calculated as the ratio of the moles of a component to the total moles present. This dimensionless quantity plays a critical role in applying Raoult's Law.

Formulaically, for any substance \( A \) in a mixture:\[ X_A = \frac{\text{moles of } A}{\text{total moles in solution}} \]

• The mole fraction of CH₂Cl₂ in the solution was found to be 0.375, indicating that 37.5% of the total moles in the solution are CH₂Cl₂.
• Conversely, the mole fraction of CH₂Br₂ was 0.625, signifying that 62.5% are CH₂Br₂.

Mole fractions are pivotal for determining the partial pressures of each component in an ideal solution, which further allows us to determine their respective compositions in the vapor phase as seen in the exercise.