Question

Carbon tetrachloride (CCl\(_4\)) and benzene (C \(_{6} \mathrm{H}_{6}\) ) form ideal solutions. Consider an equimolar solution of \(\mathrm{CCl}_{4}\) 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 of the condensed vapor is: - Mole fraction of Benzene: \(y_{C_{6}H_{6}} = 0.4536\) - Mole fraction of Carbon Tetrachloride: \(y_{CCl_{4}} = 0.5464\)

Step-by-step solution

Write down the given information and variables

Temperature: \(25^{\circ} C\) Benzene: \(C_{6}H_{6}\) Carbon Tetrachloride: \(CCl_{4}\) \[x_{C_{6}H_{6}} = x_{CCl_4} = 0.5\] Vapor Pressures: - Pure Benzene: \(P_{C_{6}H_{6}}^{sat}= 12.7\ kPa\) - Pure Carbon Tetrachloride: \(P_{CCl_{4}}^{sat}= 15.3\ kPa\)

Apply Raoult's Law to both components

We'll use Raoult's Law for each component to find the partial pressures. Raoult's Law for Benzene: \(P_{C_{6}H_{6}} = x_{C_{6}H_{6}}P_{C_{6}H_{6}}^{sat}\) Raoult's Law for Carbon Tetrachloride: \(P_{CCl_{4}} = x_{CCl_4}P_{CCl_{4}}^{sat}\)

Calculate partial pressures for both components

For Benzene: \(P_{C_{6}H_{6}} = 0.5 \times 12.7\ kPa = 6.35\ kPa\) For Carbon Tetrachloride: \(P_{CCl_{4}} = 0.5 \times 15.3\ kPa = 7.65\ kPa\)

Calculate total pressure of the mixture

The total pressure of the mixture can be calculated by adding the partial pressures. \(P_{total} = P_{C_{6}H_{6}} + P_{CCl_{4}}\) \(P_{total} = 6.35\ kPa + 7.65\ kPa = 14\ kPa\)

Calculate the mole fractions of Benzene and Carbon Tetrachloride in the vapor

Mole fraction of Benzene in the vapor: \(y_{C_{6}H_{6}} = \frac{P_{C_{6}H_{6}}}{P_{total}} = \frac{6.35\ kPa}{14\ kPa} = 0.4536\) Mole fraction of Carbon Tetrachloride in the vapor: \(y_{CCl_{4}} = \frac{P_{CCl_{4}}}{P_{total}} = \frac{7.65\ kPa}{14\ kPa} = 0.5464\)

Express the results

The composition of the condensed vapor: - Mole fraction of Benzene: \(y_{C_{6}H_{6}} = 0.4536\) - Mole fraction of Carbon Tetrachloride: \(y_{CCl_{4}} = 0.5464\)

Key concepts

Raoult's Law

This law plays a crucial role in understanding how mixtures behave. Developed by Francois-Marie Raoult, it states that the partial vapor pressure of each component in an ideal solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution.

This means that for a component A in a solution, its partial pressure (\(P_A\)) is given by \(P_A = x_A P_A^{\text{sat}}\), where \(x_A\) is the mole fraction of A and \(P_A^{\text{sat}}\) is its vapor pressure when pure. Raoult's Law assumes ideal behavior, where interactions between molecules are similar whether they’re in a solution or in pure form.

• Useful for dilute solutions
• Helps in predicting how solvent and solute interact
• Key for determining phase equilibrium in mixtures

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form in a closed system. Each pure substance has a specific vapor pressure at any given temperature. For example, benzene and carbon tetrachloride have different vapor pressures at 25°C, reflecting their volatility differences.

In a mixture, the total vapor pressure comprises the sum of the partial pressures of the components in the vapor phase. This total pressure is crucial for understanding boiling points and the behavior of solvents. More volatile substances have higher vapor pressures, tending to evaporate more readily.

• Dependent on temperature
• Important in distillation processes
• Varies between substances

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is calculated by dividing the number of moles of a substance by the total number of moles in the solution. Unlike other concentration measures like molarity, mole fraction is dimensionless and always adds up to 1 for all components in a solution.

In the context of Raoult's Law, the mole fraction of a component directly influences its vapor pressure contribution in a mixture. Mole fraction provides a simple way to express quantitative relationships between different substances in a chemical equilibrium.

• Dimensionless measure of concentration
• Used in determining vapor pressures and other colligative properties
• Sum of all mole fractions in a solution is always 1

Ideal Solutions

An ideal solution is one where the enthalpy of mixing is zero, implying that intermolecular forces between different components are equal to those within the same component. Under these conditions, solutions often obey Raoult's Law very accurately.

Ideal behavior is a simpler model of real-world interactions and provides a baseline for understanding deviations. In these solutions, we assume no volume change upon mixing and similar molecular interactions across components. Ideal solutions serve as the conceptual groundwork for understanding how real mixtures may deviate under similar conditions.

• Follows Raoult's Law perfectly
• No energy change upon mixing
• Useful as a model for more complex systems