Question

Carbon tetrachloride \(\left(\mathrm{CCl}_{4}\right)\) and benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) form ideal solutions. Consider an equimolar solution of \(\mathrm{CC}_{4}\) and \(\mathrm{C}_{6} \mathrm{H}_{6}\) at \(25^{\circ} \mathrm{C}\) . The vapor above the solution is collected and condensed. Using the following data, determine the composition in mole fraction of the condensed vapor.

Short answer

The composition in mole fraction of the condensed vapor is: - Carbon tetrachloride (CCl\(_4\)): \(\frac{1}{6}\) - Benzene (C\(_6\)H\(_6\)): \(\frac{5}{6}\)

Step-by-step solution

1: Given data

The given data includes the following information: 1. Carbon tetrachloride (CCl\(_4\)): \(P^{*}_{\mathrm{CCl}_4} = 4 \times 10^{4}\) Pa 2. Benzene (C\(_6\)H\(_6\)): \(P^{*}_{\mathrm{C}_6 \mathrm{H}_6} = 20 \times 10^{4}\) Pa 3. Equimolar solution of CCl\(_4\) and C\(_6\)H\(_6\) at \(25^{\circ}\) C.

2: Calculate the partial pressures

Using Raoult's Law to calculate the partial pressures of CCl\(_4\) and C\(_6\)H\(_6\) in the solution: \(P_{\mathrm{CCl}_4} = x_{\mathrm{CCl}_4} * P^{*}_{\mathrm{CCl}_4}\) Since it is an equimolar solution, the mole fractions of CCl4 and C6H6 are equal (0.5), therefore: \(P_{\mathrm{CCl}_4} = 0.5 * 4 \times 10^{4} \thinspace \mathrm{Pa} = 2 \times 10^{4} \thinspace \mathrm{Pa}\) \(P_{\mathrm{C}_6 \mathrm{H}_6} = x_{\mathrm{C}_6 \mathrm{H}_6} * P^{*}_{\mathrm{C}_6 \mathrm{H}_6}\) \(P_{\mathrm{C}_6 \mathrm{H}_6} = 0.5 * 20 \times 10^{4} \thinspace \mathrm{Pa} = 10 \times 10^{4} \thinspace \mathrm{Pa}\)

3: Calculate the total pressure of the solution

We'll now calculate the total pressure of the solution by adding the partial pressures of CCl\(_4\) and C\(_6\)H\(_6\): \(P = P_{\mathrm{CCl}_4} + P_{\mathrm{C}_6 \mathrm{H}_6}\) \(P = 2 \times 10^{4} \thinspace \mathrm{Pa} + 10 \times 10^{4} \thinspace \mathrm{Pa}\) \(P = 12 \times 10^{4} \thinspace \mathrm{Pa}\)

4: Determine the condensed vapor composition

Find the mole fractions of CCl\(_4\) and C\(_6\)H\(_6\) in the condensed vapor using their partial pressures. \(x_{\mathrm{CCl}_4} = \frac{P_{\mathrm{CCl}_4}}{P} = \frac{2 \times 10^{4} \thinspace \mathrm{Pa}}{12 \times 10^{4} \thinspace \mathrm{Pa}} = \frac{1}{6}\) \(x_{\mathrm{C}_6 \mathrm{H}_6} = \frac{P_{\mathrm{C}_6 \mathrm{H}_6}}{P} = \frac{10 \times 10^{4} \thinspace \mathrm{Pa}}{12 \times 10^{4} \thinspace \mathrm{Pa}} = \frac{5}{6}\) The composition in mole fraction of the condensed vapor is: - Carbon tetrachloride (CCl\(_4\)): \(\frac{1}{6}\) - Benzene (C\(_6\)H\(_6\)): \(\frac{5}{6}\)

Key concepts

Ideal Solutions

When we talk about ideal solutions, we're referring to solutions that perfectly follow Raoult's Law. In such solutions, the interactions between different molecules are similar to the interactions among like molecules.
An ideal solution experiences no change in volume or enthalpy when mixed, meaning:

• The chemical properties remain balanced.
• The solutions don't show deviations from expected behaviors.

For CCl\(_4\) and C\(_6\)H\(_6\), their ideal nature simplifies calculations. By assuming ideality, predicting their behavior becomes easier as they adhere to theoretical expectations without surprises.

Partial Pressures

In mixtures, gases exert partial pressures, which is literally the pressure each gas would exert if it occupied the whole volume alone. According to Raoult's Law, the partial pressure in a solution can be calculated using:

• \( P_i = x_i \cdot P_i^* \)

Where:

• \( P_i \) is the partial pressure of the component.
• \( x_i \) is the mole fraction of the component in the solution.
• \( P_i^* \) is the vapor pressure of the pure component.

For CCl\(_4\), its partial pressure is \( 2 \times 10^4 \ \mathrm{Pa} \), and for C\(_6\)H\(_6\), it's \( 10 \times 10^4 \ \mathrm{Pa} \), showing their proportional contribution to the total pressure.

Mole Fraction

A mole fraction is a way to express concentration. It represents the ratio of moles of one component to the total moles in a mixture:

• \( x_i = \frac{n_i}{n_{total}} \)

Where:

• \( x_i \) is the mole fraction.
• \( n_i \) is the moles of component \( i \).
• \( n_{total} \) is the total moles in the mixture.

In our example, each substance in an equimolar solution initially has a mole fraction of 0.5. Later, in the vapor phase, due to varying volatilities, these fractions change to \( \frac{1}{6} \) for CCl\(_4\) and \( \frac{5}{6} \) for C\(_6\)H\(_6\).

Vapor-Liquid Equilibrium

Vapor-liquid equilibrium (VLE) describes a state where a liquid and its vapor coexist without any net evaporation or condensation.
At equilibrium:

• The rate of evaporation equals the rate of condensation.
• The system's temperature and pressure remain stable.

This concept is crucial for understanding how mixtures separate into different phases. The results for CCl\(_4\) and C\(_6\)H\(_6\) show that despite starting as an equimolar mixture, the more volatile C\(_6\)H\(_6\) dominates in the vapor phase (\( \frac{5}{6} \)), highlighting differences in volatility and affinity at equilibrium.