A solution contains a mixture of pentane and hexane at room temperature. The solution has a vapor pressure of 258 torr. Pure pentane and hexane have vapor pressures of 425 torr and 151 torr, respectively, at room temperature. What is the mole fraction composition of the mixture? (Assume ideal behavior.)

Verified step by step guidance

1. Identify the vapor pressures of the pure components. For pentane, it is 425 torr, and for hexane, it is 151 torr.
2. Let the mole fraction of pentane in the mixture be represented as $x_{ ext{pentane}}$ and that of hexane as $x_{ ext{hexane}}$. Note that $x_{ ext{pentane}} + x_{ ext{hexane}} = 1$.
3. Apply Raoult's Law to find the total vapor pressure of the mixture. According to Raoult's Law, the total vapor pressure $P_{ ext{total}}$ is given by $P_{ ext{total}} = x_{ ext{pentane}} imes P_{ ext{pentane}} + x_{ ext{hexane}} imes P_{ ext{hexane}}$.
4. Substitute the given total vapor pressure of the mixture (258 torr) and the vapor pressures of the pure components into the equation from step 3. Solve for $x_{ ext{pentane}}$.
5. Calculate $x_{ ext{hexane}}$ using the relationship $x_{ ext{hexane}} = 1 - x_{ ext{pentane}}$.

Key Concepts

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 a mixture of volatile components, the total vapor pressure is the sum of the partial pressures of each component, which can be calculated using their respective mole fractions and pure component vapor pressures.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture. This concept is crucial for applying Raoult's Law to determine the contributions of each component to the overall vapor pressure.

Ideal Gas Behavior

Ideal gas behavior refers to the assumption that gases behave according to the ideal gas law, where interactions between gas molecules are negligible, and the volume of the gas molecules themselves is also negligible. In the context of solutions, assuming ideal behavior allows for the simplification of calculations involving vapor pressures and mole fractions, making it easier to predict the properties of the mixture.