Question

If a volatile solute is added to a volatile solvent, both substances contribute to the vapor pressure over the solution. Assuming an ideal solution, the vapor pressure of each is given by Raoult's law, and the total vapor pressure is the sum of the vapor pressures for each component. A solution, assumed to be ideal, is made from 1.0 mol of toluene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{3}\right)\) and \(2.0 \mathrm{mol}\) of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right) .\) The vapor pressures of the pure solvents are \(22 \mathrm{mm} \mathrm{Hg}\) and \(75 \mathrm{mm} \mathrm{Hg},\) respectively, at \(20^{\circ} \mathrm{C} .\) What is the total vapor pressure of the mixture? What is the mole fraction of each component in the liquid and in the vapor?

Short answer

The total vapor pressure is 57.33 mmHg. Mole fractions in liquid: 0.333 (toluene), 0.667 (benzene). Mole fractions in vapor: 0.128 (toluene), 0.872 (benzene).

Step-by-step solution

Calculate Total Moles of Solution

The solution is made up of 1.0 mol of toluene and 2.0 mol of benzene. Thus, the total moles in the solution is the sum of the moles of each component:\[ N_{ ext{total}} = N_{ ext{toluene}} + N_{ ext{benzene}} = 1.0 + 2.0 = 3.0 \]

Determine Mole Fractions in the Liquid Phase

The mole fraction of a component is the ratio of its moles to the total moles in the solution.- Mole fraction of toluene, \( X_{ ext{toluene}} \): \[ X_{ ext{toluene}} = \frac{N_{ ext{toluene}}}{N_{ ext{total}}} = \frac{1.0}{3.0} = 0.333 \]- Mole fraction of benzene, \( X_{ ext{benzene}} \): \[ X_{ ext{benzene}} = \frac{N_{ ext{benzene}}}{N_{ ext{total}}} = \frac{2.0}{3.0} = 0.667 \]

Calculate Partial Vapor Pressure of Each Component

According to Raoult's law, the partial vapor pressure of a component in a solution is given by the product of its mole fraction in the liquid phase and the vapor pressure of the pure component.- Partial pressure of toluene, \( P_{ ext{toluene}} \): \[ P_{ ext{toluene}} = X_{ ext{toluene}} \times P_{ ext{toluene, pure}} = 0.333 \times 22 \, \text{mmHg} = 7.33 \, \text{mmHg} \]- Partial pressure of benzene, \( P_{ ext{benzene}} \): \[ P_{ ext{benzene}} = X_{ ext{benzene}} \times P_{ ext{benzene, pure}} = 0.667 \times 75 \, \text{mmHg} = 50.0 \, \text{mmHg} \]

Calculate Total Vapor Pressure of the Solution

The total vapor pressure of the solution, according to Dalton's Law, is the sum of the partial pressures of each component:\[ P_{ ext{total}} = P_{ ext{toluene}} + P_{ ext{benzene}} = 7.33 + 50.0 = 57.33 \, \text{mmHg} \]

Determine Mole Fractions in the Vapor Phase

The mole fraction of each component in the vapor phase is calculated as the ratio of its partial pressure to the total vapor pressure.- Mole fraction of toluene in the vapor, \( Y_{ ext{toluene}} \): \[ Y_{ ext{toluene}} = \frac{P_{ ext{toluene}}}{P_{ ext{total}}} = \frac{7.33}{57.33} = 0.128 \]- Mole fraction of benzene in the vapor, \( Y_{ ext{benzene}} \): \[ Y_{ ext{benzene}} = \frac{P_{ ext{benzene}}}{P_{ ext{total}}} = \frac{50.0}{57.33} = 0.872 \]

Key concepts

ideal solution

An ideal solution is a fascinating concept in chemistry that refers to a mixture where the interactions between different components are similar to the interactions within the components themselves. This means that the behavior of the mixture closely resembles what you would expect if the molecules were alone. When we assume a solution is ideal, it means:

• Attraction forces between different molecules are equal.
• No volume change or heat exchange occurs on mixing.

In the context of Raoult's Law, this assumption simplifies our calculations because it considers both components in the solution to contribute to the overall behavior predictably. This predictability allows chemists to use simple equations to describe the system, as seen in the exercise with toluene and benzene.

vapor pressure

Vapor pressure is a crucial concept when discussing solutions because it deals with how a substance transitions from a liquid phase to a vapor phase. This pressure is the force exerted by the vapor released by a substance in a closed container over its own liquid. A volatile substance will contribute noticeably to vapor pressure, as the exercise demonstrates with toluene and benzene:

• Toluene has a vapor pressure of 22 mmHg.
• Benzene has a vapor pressure of 75 mmHg.

In an ideal solution, each component contributes to the total vapor pressure according to its mole fraction in the liquid phase, conforming with Raoult's Law around how each component's presence influences the total atmospheric makeup in terms of pressure.

mole fraction

Understanding the mole fraction helps simplify how we interpret solution compositions. It is the ratio of the moles of one component to the total moles of all components in the mixture. For the solution with toluene and benzene:

• Mole fraction of toluene = 0.333
• Mole fraction of benzene = 0.667

These values were calculated by dividing the moles of each component by the total number of moles in the solution. Inside the vapor phase, however, the mole fractions slightly differ since they depend on the relative contributions to the vapor pressure. These calculations are instrumental in predicting and understanding the behavior of solutions.

Dalton's Law

Dalton’s Law is an essential law in the realm of gases that helps us understand how multiple gases behave when present together. It essentially states that the total pressure exerted by a mixture of non-reacting gases is the sum of the partial pressures of individual gases. In liquid solutions of volatile substances, as demonstrated in the exercise:

• The total vapor pressure is the sum of the component’s partial pressures.
• This principle allows us to predict how the combination of gases will affect the pressure above a liquid.

Using this law, we calculated the pressure for an ideal solution and added the computed pressures from both toluene and benzene to get the total vapor pressure of the mix, employing this law to aid accurate measurement.