Question

At \(63.5^{\circ} \mathrm{C}\) the vapor pressure of \(\mathrm{H}_{2} \mathrm{O}\) is 175 torr, and that of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) is 400 torr. A solution is made by mixing equal masses of \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\). (a) What is the mole fraction of ethanol in the solution? (b) Assuming ideal-solution behavior, what is the vapor pressure of the solution at \(63.5^{\circ} \mathrm{C} ?\) (c) What is the mole fraction of ethanol in the vapor above the solution?

Short answer

The mole fraction of ethanol (C2H5OH) in the solution is approximately 0.281. The vapor pressure of the solution at 63.5°C is approximately 238.23 torr. The mole fraction of ethanol in the vapor above the solution is approximately 0.472.

Step-by-step solution

Calculate moles of H2O and C2H5OH in the solution

To determine the mole fraction of ethanol, we first need to find the moles of both ethanol and water in the solution. We know equal masses of both substances are mixed. Let's assume 1 gram of each is taken for simplicity. We now need to find the moles of both substances: Moles of H2O = mass / molar mass = 1 g / 18.015 g/mol ≈ 0.0555 mol Moles of C2H5OH = mass / molar mass = 1 g / 46.068 g/mol ≈ 0.0217 mol

Calculate mole fraction of ethanol in the solution

Mole fraction is defined as the ratio of moles of a solute to the total moles in the solution. In this case, the mole fraction of ethanol (C2H5OH) is: Mole fraction of C2H5OH = Moles of C2H5OH / (Moles of H2O + Moles of C2H5OH) = 0.0217 / (0.0555 + 0.0217) = 0.0217 / 0.0772 ≈ 0.281

Apply Raoult's Law to find the vapor pressure of the solution

Raoult's Law states that the partial pressure of a component in a solution is equal to the product of the mole fraction of that component in the solution and its vapor pressure in the pure state. Using Raoult's law, we can determine the partial pressures of both H2O and C2H5OH in the solution. Partial pressure of H2O = Mole fraction of H2O × Vapor pressure of pure H2O = (1 - 0.281) × 175 torr ≈ 0.719 × 175 torr ≈ 125.83 torr Partial pressure of C2H5OH = Mole fraction of C2H5OH × Vapor pressure of pure C2H5OH = 0.281 × 400 torr ≈ 112.4 torr

Calculate the vapor pressure of the solution

Since we have the partial pressures of H2O and C2H5OH, we can sum them up to find the total vapor pressure of the solution. Vapor pressure of the solution = Partial pressure of H2O + Partial pressure of C2H5OH = 125.83 torr + 112.4 torr ≈ 238.23 torr

Calculate the mole fraction of ethanol in the vapor

Since the vapor is in equilibrium with the solution, we can use the partial pressures to find the mole fraction of ethanol in the vapor. Mole fraction of C2H5OH in the vapor = Partial pressure of C2H5OH / Vapor pressure of the solution = 112.4 torr / 238.23 torr ≈ 0.472 The answers are as follows: (a) The mole fraction of ethanol (C2H5OH) in the solution is approximately 0.281. (b) The vapor pressure of the solution at 63.5°C is approximately 238.23 torr. (c) The mole fraction of ethanol in the vapor above the solution is approximately 0.472.

Key concepts

Mole Fraction

The concept of mole fraction is essential in chemistry, particularly when dealing with solutions. It is a way of expressing the concentration of a component in a mixture. The mole fraction, represented by the symbol \( X \), is the ratio of the number of moles of a particular substance to the total number of moles of all substances present in the mixture.

For instance, when you have a mixture of two substances, A and B, and you are interested in the mole fraction of A, you calculate it using the formula: \[ X_A = \frac{n_A}{n_A + n_B} \] where \( n_A \) and \( n_B \) are the moles of substance A and B, respectively. Importantly, the sum of mole fractions in a binary mixture is always equal to 1. This calculation is crucial for understanding the composition of solutions and predicting their behavior in chemical processes, such as vapor pressure as determined by Raoult's Law.

Raoult's Law

Raoult's Law is a principle that provides a link between the composition of a liquid solution and its vapor pressure. According to Raoult's Law, the partial vapor pressure of each component in an ideal solution is equal to the product of the mole fraction of that component in the liquid phase and its pure vapor pressure.

In mathematical terms, for a component A in a solution, Raoult's Law is expressed as: \[ P_A = X_A \times P^\circ_A \] where \( P_A \) is the partial vapor pressure of component A in the solution, \( X_A \) is the mole fraction of A in the liquid phase, and \( P^\circ_A \) is the vapor pressure of pure A. This relationship assumes that the solution behaves ideally, meaning that there are no significant interactions between different molecules that would affect the vapor pressure. This concept is fundamental when calculating the vapor pressure of solutions.

Partial Pressure

In a mixture of gases, each gas exerts a pressure as if it were alone in the container. This is known as its partial pressure. The total pressure exerted by the mixture is the sum of the individual partial pressures of all the gases present.

The partial pressure of a gas can be calculated when knowing the mole fraction of the gas in the mixture and the total pressure. The formula is: \[ P_{\text{gas}} = X_{\text{gas}} \times P_{\text{total}} \] Here, \( P_{\text{gas}} \) is the partial pressure we’re trying to find, \( X_{\text{gas}} \) is the mole fraction of the gas in question, and \( P_{\text{total}} \) is the total pressure of the mixture. Understanding partial pressures is crucial when studying gas mixtures and can be applied directly in understanding vapor-liquid equilibria through Raoult's Law.

Solution Equilibrium

In the context of vapor pressures, solution equilibrium refers to the dynamic balance between the evaporation and condensation rates of a substance. At equilibrium, the rate at which molecules evaporate from the liquid phase to enter the vapor phase is equal to the rate at which molecules condense from the vapor phase back into the liquid phase.

This equilibrium is crucial when solutions are involved because the vapor above the solution becomes saturated when the equilibrium is reached. At this point, for any component of the solution, its partial pressure in the vapor phase will become constant, and it is this constant vapor pressure that can be used in conjunction with Raoult's Law to find the mole fraction of the components in the vapor phase as long as the temperature remains constant. This understanding allows chemists to predict the composition of vapors in contact with liquid mixtures and can be critical in applications such as distillation or predicting the behavior of volatile components in various environmental conditions.