Cyclopentane 1C5H102 and cyclohexane 1C6H122 are vola- tile, nonpolar hydrocarbons. At 30.0 °C, the vapor pres- sure of cyclopentane is 385 mm Hg, and the vapor pressure of cyclohexane is 122 mm Hg. What is Xpentane in a mixture of C5H10 and C6H12 that has a vapor pressure of 212 mm Hg at 30.0 °C?

Verified step by step guidance

1. Identify the given values: Vapor pressure of pure cyclopentane (P°_pentane) is 385 mm Hg, vapor pressure of pure cyclohexane (P°_hexane) is 122 mm Hg, and the total vapor pressure of the mixture (P_total) is 212 mm Hg.
2. Use Raoult's Law which states that the total vapor pressure of a solution is the sum of the partial pressures of each component. The partial pressure of each component is the product of the mole fraction of the component and the vapor pressure of the pure component: P_total = X_pentane * P°_pentane + X_hexane * P°_hexane.
3. Set up the equation for the mole fraction of cyclohexane (X_hexane) in terms of the mole fraction of cyclopentane (X_pentane) since the sum of the mole fractions must equal 1: X_hexane = 1 - X_pentane.
4. Substitute the expression for X_hexane from step 3 into the equation from step 2 and solve for X_pentane: P_total = X_pentane * P°_pentane + (1 - X_pentane) * P°_hexane.
5. Isolate X_pentane in the equation to find its value: X_pentane = (P_total - P°_hexane) / (P°_pentane - P°_hexane).

Key Concepts

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. It reflects the tendency of particles to escape from the liquid phase into the vapor phase. In this question, understanding vapor pressure is crucial as it helps determine the composition of the mixture based on the observed vapor pressure of the solution.

Raoult's Law

Raoult's Law states that the vapor pressure of a solvent in a solution is directly proportional to the mole fraction of the solvent in the mixture. This principle allows us to calculate the vapor pressure of a mixture based on the individual vapor pressures of its components and their respective mole fractions. It is essential for solving the problem of finding the mole fraction of cyclopentane in the given mixture.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture, defined as the ratio of the number of moles of that component to the total number of moles of all components in the mixture. In this context, calculating the mole fraction of cyclopentane is necessary to relate the vapor pressures of the individual components to the overall vapor pressure of the mixture.