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. Understand that the problem involves Raoult's Law, which is used to determine the vapor pressure of an ideal solution. Raoult's Law states that the partial vapor pressure of each component in the solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution.
2. Set up the equation for the total vapor pressure of the solution using Raoult's Law: P_total = (X_benzene * P_benzene) + (X_toluene * P_toluene), where X_benzene and X_toluene are the mole fractions of benzene and toluene, respectively, and P_benzene and P_toluene are their respective vapor pressures.
3. Since the solution is composed of only benzene and toluene, the sum of their mole fractions must equal 1: X_benzene + X_toluene = 1.
4. Substitute the known values into the equation: 35 torr = (X_benzene * 75 torr) + (X_toluene * 22 torr). Use the relationship X_toluene = 1 - X_benzene to express X_toluene in terms of X_benzene.
5. Solve the system of equations to find the mole fractions X_benzene and X_toluene. This involves substituting X_toluene = 1 - X_benzene into the total vapor pressure equation and solving for X_benzene, then using the result to find X_toluene.