Pentane (C₅H₁₂) and hexane (C₆H₁₄) form an ideal solution. The vapor pressures of pentane and hexane at 25° C are 511 torr and 150 torr, respectively. The mole fraction of hexane in a pentane-hexane solution is 0.50. Calculate the mole fraction of pentane in the vapor that is in equilibrium at 25°C with this solution. A) 0.23 B) 0.77 C) 0.89 D) 0.50 E) none of these

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Quizplus · Accepted Answer

Raoult's law states that the partial pressure of a component in a solution is equal to the product of its mole fraction in the solution and its vapor pressure in the pure state. For a two-component solution, this can be expressed as:P_A = X_A * P_A^0P_B = X_B * P_B^0where P_A and P_B are the partial pressures of components A and B, respectively, X_A and X_B are their mole fractions in the solution, and P_A^0 and P_B^0 are their vapor pressures in the pure state.In this problem, we are given the vapor pressures of pentane and hexane at 25° C:P_pentane^0 = 511 torrP_hexane^0 = 150 torrWe are also given the mole fraction of hexane in the solution:X_hexane = 0.50We can use Raoult's law to calculate the partial pressure of hexane in the solution:P_hexane = X_hexane * P_hexane^0 = 0.50 * 150 torr = 75 torrThe total pressure of the solution is the sum of the partial pressures of the two components:P_total = P_pentane + P_hexaneWe can use Dalton's law to calculate the mole fraction of pentane in the vapor:Y_pentane = P_pentane / P_totalwhere P_pentane is the partial pressure of pentane in the vapor.We can rearrange this equation to solve for P_pentane:P_pentane = Y_pentane * P_totalWe can substitute the expression for P_total into this equation:P_pentane = Y_pentane * (P_pentane + P_hexane)We can solve this equation for Y_pentane:Y_pentane = P_pentane / (P_pentane + P_hexane)We can substitute the given values into this equation:Y_pentane = (P_total - P_hexane) / (P_total - P_hexane + P_hexane) = (P_total - 75 torr) / P_totalWe do not know the value of P_total, but we can use the fact that the mole fraction of hexane in the solution is 0.50 to find a relationship between P_total and P_pentane:X_hexane = P_hexane / P_total = 0.50We can rearrange this equation to solve for P_total:P_total = P_hexane / 0.50 = 75 torr / 0.50 = 150 torrWe can substitute this value into the equation for Y_pentane:Y_pentane = (P_total - 75 torr) / P_total = (150 torr - 75 torr) / 150 torr = 0.77Therefore, the mole fraction of pentane in the vapor that is in equilibrium at 25°C with this solution is 0.77.

Pentane (C5H12) and Hexane (C6H14) Form an Ideal Solution

Pentane (C₅H₁₂) and Hexane (C₆H₁₄) Form an Ideal Solution