At 60 °C, compound X has a vapor pressure of 96 mm Hg, benzene 1C6H62 has a vapor pressure of 395 mm Hg, and a 50:50 mixture by mass of benzene and X has a vapor pres- sure of 299 mm Hg. What is the molar mass of X?

Verified step by step guidance

1. Determine the mole fraction of benzene in the mixture. Use the formula for mole fraction, \( x_{benzene} = \frac{m_{benzene}}{M_{benzene}} / (\frac{m_{benzene}}{M_{benzene}} + \frac{m_X}{M_X}) \), where \( m_{benzene} \) and \( m_X \) are the masses of benzene and compound X, respectively, and \( M_{benzene} \) and \( M_X \) are their molar masses.
2. Calculate the expected vapor pressure of the mixture using Raoult's Law, which states that the total vapor pressure of the solution is the sum of the vapor pressures of each component multiplied by their respective mole fractions: \( P_{total} = x_{benzene} \times P^\circ_{benzene} + x_X \times P^\circ_X \), where \( P^\circ_{benzene} \) and \( P^\circ_X \) are the vapor pressures of pure benzene and compound X, respectively.
3. Set up an equation with the observed vapor pressure of the mixture (299 mm Hg) equal to the calculated vapor pressure from step 2. Solve this equation for \( M_X \), the molar mass of compound X.
4. Rearrange the equation to isolate \( M_X \) on one side. This might involve algebraic manipulation such as multiplying through by \( M_X \) and solving for \( M_X \) explicitly.
5. Check the units and make sure they are consistent across all terms in your equations to ensure the correctness of your calculation.

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. Higher vapor pressure indicates a greater volatility of the substance. Understanding vapor pressure is crucial for analyzing mixtures and predicting how components behave under varying conditions.

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 present. This law is fundamental in understanding how the addition of a solute affects the vapor pressure of a solvent. In this question, Raoult's Law can be applied to determine the contribution of compound X to the overall vapor pressure of the mixture with benzene.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of all atoms in a molecule. In this problem, the molar mass of compound X can be determined by using the vapor pressures of the pure components and the mixture, along with the mass ratio of the components, to find the mole fraction and subsequently the molar mass.