Pentane \(\left(\mathrm{C}_{5} \mathrm{H}_{12}\right)\) and hexane \(\left(\mathrm{C}_{6} \mathrm{H}_{14}\right)\) form an ideal solution. At \(25^{\circ} \mathrm{C}\) the vapor pressures of pentane and hexane are 511 and 150\. torr, respectively. A solution is prepared by mixing 25 mL pentane (density, \(0.63 \mathrm{~g} / \mathrm{mL}\) ) with \(45 \mathrm{~mL}\) hexane (density, \(0.66 \mathrm{~g} / \mathrm{mL})\) a. What is the vapor pressure of the resulting solution? b. What is the composition by mole fraction of pentane in the vapor that is in equilibrium with this solution?

To find the mass of pentane and hexane, use the given volumes and densities: Mass of pentane = Volume of pentane × Density of pentane Mass of hexane = Volume of hexane × Density of hexane

Use the molar mass of each component to find the number of moles for each: Moles of pentane = Mass of pentane / Molar mass of pentane Moles of hexane = Mass of hexane / Molar mass of hexane #Step 2: Determine the mole fractions of pentane and hexane#

Add the moles of pentane and hexane to find the total moles of the solution: Total moles = Moles of pentane + Moles of hexane

Divide the moles of each component by the total moles to find the mole fraction of each: Mole fraction of pentane = Moles of pentane / Total moles Mole fraction of hexane = Moles of hexane / Total moles #Step 3: Calculate the vapor pressure of the resulting solution#

Raoult's Law states that the vapor pressure of an ideal solution is the product of the mole fraction of the component and its pure vapor pressure. Calculate the vapor pressure of each component in the solution: Vapor pressure of pentane in solution = Mole fraction of pentane × Vapor pressure of pure pentane Vapor pressure of hexane in solution = Mole fraction of hexane × Vapor pressure of pure hexane

Add the vapor pressures of pentane and hexane in the solution to find the total vapor pressure of the solution: Total vapor pressure = Vapor pressure of pentane in solution + Vapor pressure of hexane in solution #Step 4: Determine the composition of pentane in the vapor in equilibrium with the solution#

The mole fraction of pentane in the vapor is equal to the vapor pressure of pentane in the solution divided by the total vapor pressure of the solution: Mole fraction of pentane in vapor = Vapor pressure of pentane in solution / Total vapor pressure Now, let's calculate the values in each step: #Step 1#: Mass of pentane = 25 mL × 0.63 g/mL = 15.75 g Mass of hexane = 45 mL × 0.66 g/mL = 29.7 g Moles of pentane = 15.75 g / (5*12.01 + 12*1.01) g/mol ≈ 0.3068 mol Moles of hexane = 29.7 g / (6*12.01 + 14*1.01) g/mol ≈ 0.2583 mol #Step 2#: Total moles = 0.3068 mol + 0.2583 mol ≈ 0.5651 mol Mole fraction of pentane = 0.3068 mol / 0.5651 mol ≈ 0.5429 Mole fraction of hexane = 0.2583 mol / 0.5651 mol ≈ 0.4571 #Step 3#: Vapor pressure of pentane in solution = 0.5429 × 511 torr ≈ 277.33 torr Vapor pressure of hexane in solution = 0.4571 × 150 torr ≈ 68.57 torr Total vapor pressure = 277.33 torr + 68.57 torr ≈ 345.9 torr #Step 4#: Mole fraction of pentane in vapor = 277.33 torr / 345.9 torr ≈ 0.8016 The vapor pressure of the resulting solution is approximately 345.9 torr, and the composition by mole fraction of pentane in the vapor that is in equilibrium with this solution is approximately 0.8016.