Question

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.)

Short answer

The mole fraction of pentane (\( X_{pentane} \)) in the mixture is calculated to be about 0.40, and the mole fraction of hexane (\( X_{hexane} \)) is about 0.60.

Step-by-step solution

Understand Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures states that in a mixture of non-reacting gases, the total pressure exerted by the mixture is equal to the sum of the partial pressures of individual gases. For an ideal solution, this can be expressed as: \( P_{total} = X_{pentane} \times P_{pentane} + X_{hexane} \times P_{hexane} \), where \( X_{pentane} \) and \( X_{hexane} \) are the mole fractions of pentane and hexane, and \( P_{pentane} \) and \( P_{hexane} \) are the vapor pressures of pure pentane and hexane, respectively.

Set Up the Equation Using the Given Data

Substitute the given vapor pressures and total pressure into Dalton's Law equation: 258 torr = \( X_{pentane} \times 425 \) torr + \( X_{hexane} \times 151 \) torr. Since in a binary mixture the mole fractions must sum up to 1, we have \( X_{hexane} = 1 - X_{pentane} \).

Solve for the Mole Fraction of Pentane

Substitute \( X_{hexane} \) with \( 1 - X_{pentane} \) in the equation to get: 258 torr = \( X_{pentane} \times 425 \) torr + (1 - \( X_{pentane} \)) \times 151 torr. Solve this linear equation for \( X_{pentane} \).

Calculate the Mole Fraction of Pentane

Rearrange and simplify the equation to solve for \( X_{pentane} \): 258 = 425\( X_{pentane} \) + 151 - 151\( X_{pentane} \). This simplifies to 425\( X_{pentane} \) - 151\( X_{pentane} \) = 258 - 151. Solve for \( X_{pentane} \) to find its value.

Determine the Mole Fraction of Hexane

Substitute the value of \( X_{pentane} \) back into the equation \( X_{hexane} = 1 - X_{pentane} \) to find the mole fraction of hexane.

Key concepts

Vapor Pressure

When a liquid is placed in a closed container, the molecules escape from the surface of the liquid and enter the gas phase, creating pressure above the liquid. This exerted pressure by the gas phase is known as the vapor pressure. Vapor pressure is dependent on the substance's inherent properties and temperature. At a given temperature, different liquids have different vapor pressures due to variations in their molecular interactions.

For instance, in the exercise, pentane and hexane have different vapor pressures at room temperature: pentane has a higher vapor pressure (425 torr) compared to hexane (151 torr), indicating that pentane molecules escape into the gas phase more readily than hexane molecules. Understanding the concept of vapor pressure is crucial when applying Dalton's Law of Partial Pressures to find the composition of a mixture in gaseous or liquid state.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. Defined as the ratio of the number of moles of a given component to the total number of moles of all components in the mixture, the mole fraction is a dimensionless number and is very useful in the study of solution properties.

In the context of the provided exercise, the mole fraction for pentane (\( X_{pentane} \)) and hexane (\( X_{hexane} \)) can help to determine the overall composition of the mixture. Since the total mole fraction must sum to one, this relationship forms the basis for setting up the mathematical equations needed to solve Dalton's Law and other problems in chemistry.

Ideal Solution Behavior

An ideal solution is one where the intermolecular forces between dissimilar molecules are equal to those between similar molecules. This means a solution exhibits ideal behavior when the enthalpy change of the mixing process is zero, and the solution components obey Raoult's Law throughout the composition range.

In the exercise, it is assumed that the mixture of pentane and hexane behaves ideally, which greatly simplifies the calculation. When a solution behaves ideally, the partial pressure exerted by a component is directly proportional to its mole fraction. This assumption allows us to apply Dalton's Law to calculate the mole fractions of each component using their given vapor pressures and the total pressure of the mixture.