Question

At \(350 .\) K, pure toluene and hexane have vapor pressures of \(3.57 \times 10^{4}\) Pa and \(1.30 \times 10^{5}\) Pa, respectively. a. Calculate the mole fraction of hexane in the liquid mixture that boils at \(350 . \mathrm{K}\) at a pressure of \(1 \mathrm{atm}\). b. Calculate the mole fraction of hexane in the vapor that is in equilibrium with the liquid of part (a).

Short answer

The mole fraction of hexane in the liquid mixture that boils at 350K and 1 atm is 0.219. The mole fraction of hexane in the vapor above the liquid is 0.5056.

Step-by-step solution

Convert pressure to Pascal

First, we need to convert the given pressure from atm to Pascal, since the vapor pressures are given in Pascal. Use the conversion factor 1 atm = 101325 Pa. \(P_{total} = 1 \thinspace atm \times \frac{101325 \thinspace Pa}{1 \thinspace atm} = 101325 \thinspace Pa \)

Write Raoult's law equation

Raoult's law states that the partial pressure of a component in a solution is the product of its mole fraction in the solution and its vapor pressure. The total pressure is the sum of the partial pressures of each component. So, write the equation using mole fractions and vapor pressures for hexane (\(x_H\)) and toluene (\(x_T\)): \(P_{total} = P_{Hexane} \times x_{H} + P_{Toluene} \times x_{T}\), where \(P_{Hexane} = 1.30 \times 10^5\) Pa, \(P_{Toluene} = 3.57 \times 10^4\) Pa, and \(x_{H} + x_{T} = 1\).

Solve the equation for the mole fraction of hexane

Rewrite the equation to isolate the variable \(x_{H}\): \(x_{H} = \frac{P_{total} - P_{Toluene} \times x_{T}}{P_{Hexane}}\). Substitute the relationship between mole fractions (\(x_{T} = 1 - x_{H}\)) and the given values for the pressures: \(x_{H} = \frac{101325 \thinspace Pa - 3.57 \times 10^4 \thinspace Pa \times (1 - x_{H})}{1.30 \times 10^5 \thinspace Pa}\).

Solve for the mole fraction of hexane in the liquid mixture

Solve the equation for \(x_{H}\): \(x_{H} = 0.219\). The mole fraction of hexane in the liquid mixture is 0.219.

Use mole fractions to find the partial pressures of the components

Use the mole fractions and vapor pressures of each component to find the partial pressures in the vapor phase: \(P_{T,vapor} = P_{Toluene} \times (1 - x_{H}) = 3.57 \times 10^4 \thinspace Pa \times (1 - 0.219) = 27862.9 \thinspace Pa\) \(P_{H,vapor} = P_{Hexane} \times x_{H} = 1.30 \times 10^5 \thinspace Pa \times 0.219 = 28470 \thinspace Pa\)

Calculate the mole fraction of hexane in the vapor

Add the partial pressures of the components to find the total pressure in the vapor phase: \(P_{vapor} = P_{T,vapor} + P_{H,vapor} = 27862.9 \thinspace Pa + 28470 \thinspace Pa = 56332.9 \thinspace Pa\) Now, find the mole fraction of hexane in the vapor using the partial pressure: \(x_{H,vapor} = \frac{P_{H,vapor}}{P_{vapor}} = \frac{28470 \thinspace Pa}{56332.9 \thinspace Pa} = 0.5056\) The mole fraction of hexane in the vapor is 0.5056.

Key concepts

Vapor Pressure

Understanding vapor pressure is crucial when studying the physical properties of liquids. It is defined as the pressure exerted by a vapor when it is in equilibrium with its liquid or solid form, at a given temperature. Essentially, it's a measure of how much a substance tends to vaporize. In the context of Raoult's law, the vapor pressure of a pure substance is significant because it allows us to predict how the substance will behave when mixed with other components.

For substances in a mixture, the vapor pressure of each component is proportional to its mole fraction, a concept explained by Raoult's law. This relationship becomes pivotal when trying to determine the composition of a liquid mixture and the corresponding vapor based on their boiling points and overall pressure.

Mole Fraction Calculation

The mole fraction is a way of expressing the concentration of a component in a mixture. It is calculated by dividing the number of moles of that component by the total number of moles of all components in the mixture. The mole fraction is dimensionless and always ranges between 0 and 1. For example, in a binary mixture of hexane and toluene, if the mole fraction of hexane is 0.219, this means that hexane constitutes 21.9% of the total moles in the mixture.

Mole fraction calculations are at the heart of Raoult's law; they allow us to determine the partial pressures of each component within a mixture. In Raoult's law, the vapor pressure of each pure component is multiplied by its mole fraction to find the pressure it contributes to the overall vapor pressure above the mixture. The proper calculation of mole fractions enables accurate predictions of vapor-liquid equilibrium behavior.

Partial Pressure

Partial pressure is the pressure that each gas in a mixture of gases would exert if it alone occupied the entire volume of the mixture at the same temperature. According to Raoult's law, the partial pressure of each component in a liquid mixture is directly proportional to its mole fraction in the liquid phase and its pure component vapor pressure at the same temperature. This means that for a mixture of liquids, we can calculate the pressure contribution of each component based on its composition and vapor pressure.

In our textbook example, the mole fraction of hexane in the liquid phase and its pure vapor pressure are used to determine the pressure it contributes to the total vapor pressure. This concept is vital as it forms the basis for predicting how a mixture of substances will vaporize and is important in applications such as distillation and separating components in industrial processes.

Liquid-Vapor Equilibrium

Liquid-vapor equilibrium occurs when the rates of evaporation and condensation are equal, leading to a stable system where the vapor pressure above a liquid remains constant. At this point, the liquid and its vapor exist in a dynamic balance; molecules are continually escaping from the liquid into the vapor phase and returning at the same rate.

Raoult's law helps us determine liquid-vapor equilibrium in mixtures. By using the mole fractions and vapor pressures of each component, we can calculate the equilibrium vapor pressure of the mixture. In the exercise, the liquid mixture's vapor pressure equaled the atmospheric pressure at the boiling point. Calculating mole fractions in both the liquid and vapor phases tells us the composition of both phases at equilibrium. Understanding this equilibrium is essential across various scientific fields, including chemistry and environmental science, and has practical applications in distillation, perfumery, and many other industries.