At 20 °C, the vapor pressure of benzene (C₆H₆) is 75 torr, and that of toluene (C₇H₈) is 22 torr. Assume that benzene and toluene form an ideal solution. (a) What is the composition in mole fraction of a solution that has a vapor pressure of 35 torr at 20 °C?

Verified step by step guidance

1. Start by understanding Raoult's Law, which states that the vapor pressure of an ideal solution is the sum of the partial pressures of its components. The partial pressure of each component is the product of its mole fraction in the solution and its vapor pressure as a pure substance.
2. Let x_benzene be the mole fraction of benzene and x_toluene be the mole fraction of toluene. Since these are the only two components in the solution, their mole fractions must add up to 1: x_benzene + x_toluene = 1.
3. According to Raoult's Law, the total vapor pressure (P_total) of the solution is given by: P_total = x_benzene * P_benzene + x_toluene * P_toluene, where P_benzene = 75 torr and P_toluene = 22 torr.
4. Substitute the given total vapor pressure (35 torr) into the equation: 35 = x_benzene * 75 + x_toluene * 22.
5. Use the equation x_benzene + x_toluene = 1 to express x_toluene in terms of x_benzene (x_toluene = 1 - x_benzene) and substitute it into the vapor pressure equation to solve for x_benzene. Once you have x_benzene, you can find x_toluene.

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 context, the vapor pressures of benzene and toluene indicate how readily each substance evaporates, which is crucial for determining the composition of the solution at a specific vapor pressure.

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 solution. For an ideal solution, the total vapor pressure is the sum of the partial pressures of each component, calculated using their respective mole fractions and pure component vapor pressures. This law is essential for solving the problem, as it allows us to relate the vapor pressure of the solution to its composition.

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 question, calculating the mole fractions of benzene and toluene will help determine the composition of the solution that results in a specific vapor pressure, which is key to finding the answer.