Question

A mixture of ethanol and 1 -propanol behaves ideally at \(36^{\circ} \mathrm{C}\) and is in equilibrium with its vapor. If the mole fraction of ethanol in the solution is \(0.62,\) calculate its mole fraction in the vapor phase at this temperature. (The vapor pressures of pure ethanol and 1 -propanol at \(36^{\circ} \mathrm{C}\) are 108 and \(40.0 \mathrm{mmHg}\), respectively.)

Short answer

The mole fraction of ethanol in the vapor phase is approximately 0.8155.

Step-by-step solution

Identify Given Information

We are given that the mole fraction of ethanol (let's denote it as \( x_{1} \)) in the liquid mixture is 0.62. The vapor pressure of pure ethanol \( P^{*}_{ ext{ethanol}} \) is 108 mmHg and for 1-propanol \( P^{*}_{ ext{1-propanol}} \) is 40 mmHg. We need to find the mole fraction of ethanol in the vapor phase (denoted as \( y_{1} \)).

Apply Raoult's Law for Liquid Phase

Raoult's law states that the partial pressure of a component in the vapor is the product of its mole fraction in the liquid and its vapor pressure as a pure component: \( P_{ ext{ethanol}} = x_{1} \times P^{*}_{ ext{ethanol}} \) and \( P_{ ext{1-propanol}} = x_{2} \times P^{*}_{ ext{1-propanol}} \).

Calculate Partial Pressures

Substitute \( x_{1} = 0.62 \) into the equation to find \( P_{ ext{ethanol}} = 0.62 \times 108 = 66.96 \) mmHg. The mole fraction of 1-propanol \( x_{2} \) is \( 1 - 0.62 = 0.38 \), so \( P_{ ext{1-propanol}} = 0.38 \times 40 = 15.2 \) mmHg.

Calculate Total Pressure

The total pressure \( P_{ ext{total}} \) is the sum of the partial pressures: \( P_{ ext{total}} = P_{ ext{ethanol}} + P_{ ext{1-propanol}} = 66.96 + 15.2 = 82.16 \) mmHg.

Calculate Mole Fraction in Vapor Phase

Using Dalton's law, which relates partial pressures to mole fractions in the vapor, the mole fraction of ethanol in the vapor \( y_{1} \) is given by \( y_{1} = \frac{P_{ ext{ethanol}}}{P_{ ext{total}}} = \frac{66.96}{82.16} \approx 0.8155 \).

Key concepts

Vapor-Liquid Equilibrium

Vapor-liquid equilibrium is a crucial concept in understanding how mixtures separate into phases. At a given temperature and pressure, a system is in vapor-liquid equilibrium when the rates of evaporation and condensation of each component are equal.
This means each component's vapor and liquid phases have identical conditions overall, leading to a stable system without net change.
In an ideal mixture, each component contributes to the vapor phase according to its properties. If you have ethanol and 1-propanol mixed, both will exist in liquid and vapor phases once equilibrium is achieved. The proportions of each substance in these phases depend on various factors, including temperature and pressure.
Understanding vapor-liquid equilibrium is essential for predicting the behavior of mixtures in various industries, particularly in chemical engineering and distillation processes.

Mole Fraction

Mole fraction represents how much of a substance is present in a mixture relative to the total amount of all substances. To calculate it, you divide the moles of one component by the total moles of the mixture. The sum of all mole fractions in the mixture is always 1.
In the given problem, the mole fraction of ethanol in the liquid is 0.62, which means that 62% of the mixture is ethanol, while 38% is 1-propanol.
Calculating the mole fraction is crucial because it allows us to apply Raoult's and Dalton's laws effectively, helping us to determine other properties of the mixture. Using mole fractions, we can find out how much of each component contributes to the vapor phase of a mixture, which is critical in processes like distillation.

Partial Pressure

Partial pressure refers to the pressure exerted by a single component in a mixture of gases. In a mixture, each gas contributes to the total pressure independently, according to its proportion in the mixture and its tendency to evaporate.
Using Raoult's Law, partial pressures in the liquid phase are calculated by multiplying the mole fraction of a component by its vapor pressure as a pure substance. For example, ethanol's partial pressure in the liquid phase is calculated using its mole fraction of 0.62 and its pure vapor pressure of 108 mmHg.
Partial pressure helps us understand how each component in a mixture behaves separately. By knowing partial pressures, we can calculate the total pressure and examine how mixtures respond under different conditions.

Dalton's Law

Dalton's Law of Partial Pressures states that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. This principle is widely used to find the composition of gases in vapor mixtures.
In the problem given, we find the total pressure from the partial pressures of ethanol and 1-propanol. Once calculated, Dalton's Law helps us find the mole fraction in the vapor phase.
By taking the partial pressure of ethanol (66.96 mmHg) and dividing it by the total pressure (82.16 mmHg), we determine its vapor-phase mole fraction to be approximately 0.8155.
This application is instrumental in making sense of how gases distribute themselves in a mixture and how each component's properties can predict the overall behavior of the mixture.