Problem 67 - Chemistry The Central Science

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?

Step-by-Step Solution

Answer

In summary: (a) The mole fraction of ethanol in the solution is approximately 0.617. (b) The vapor pressure of the solution at 63.5 °C is approximately 295.12 torr. (c) The mole fraction of ethanol in the vapor above the solution is approximately 0.786.

Step 1: Calculate moles of water and moles of ethanol

Since the masses of water and ethanol in the solution are equal, we can denote them as \(m_{H_2O} = m_{C_2H_5OH} = m\). In order to calculate the mole fraction of ethanol in the solution, we need to first find the moles of both water and ethanol present in the solution. We can do this using the molecular weights of water and ethanol: Molecular weight of water: \(M_{H_2O} = 18.015 \,g/mol\) Molecular weight of ethanol: \(M_{C_2H_5OH} = 46.07 \,g/mol\) Moles of water: \(n_{H_2O} = \frac{m}{M_{H_2O}}\) Moles of ethanol: \(n_{C_2H_5OH} = \frac{m}{M_{C_2H_5OH}}\)

Step 2: Calculate mole fraction of ethanol

Now, we can calculate the mole fraction of ethanol, denoted as \(x_{C_2H_5OH}\), in the solution using the moles of water and ethanol: \(x_{C_2H_5OH} = \frac{n_{C_2H_5OH}}{n_{H_2O} + n_{C_2H_5OH}}\) Substituting the expressions for \(n_{H_2O}\) and \(n_{C_2H_5OH}\): \(x_{C_2H_5OH} = \frac{\frac{m}{M_{C_2H_5OH}}}{\frac{m}{M_{H_2O}} + \frac{m}{M_{C_2H_5OH}}}\) Since m appears in both numerator and denominator, we can cancel it out: \(x_{C_2H_5OH} = \frac{1/M_{C_2H_5OH}}{1/M_{H_2O} + 1/M_{C_2H_5OH}}\) Plug in the molecular weights: \(x_{C_2H_5OH} = \frac{1/46.07}{1/18.015 + 1/46.07} \approx 0.617\) The mole fraction of ethanol in the solution is approximately 0.617.

Step 3: Calculate vapor pressure of the solution

We will now calculate the vapor pressure of the solution using Raoult's law, which states that the partial pressure of a component in an ideal solution is equal to the mole fraction of the component multiplied by its vapor pressure: \(P_{solution} = x_{H_2O} P_{H_2O}^* + x_{C_2H_5OH} P_{C_2H_5OH}^*\) where: - \(P_{H_2O}^* = 175 \,torr\) and \(P_{C_2H_5OH}^* = 400 \,torr\) are the vapor pressures of water and ethanol at 63.5 °C - \(x_{H_2O} = 1 - x_{C_2H_5OH}\) is the mole fraction of water \(P_{solution} = (1 - 0.617)(175) + 0.617(400) = 0.383(175) + 0.617(400) \approx 295.12 \,torr\) The vapor pressure of the solution at 63.5 °C is approximately 295.12 torr.

Step 4: Calculate mole fraction of ethanol in the vapor

Finally, we will calculate the mole fraction of ethanol in the vapor above the solution, denoted as \(y_{C_2H_5OH}\). This can be found using the partial pressures of water and ethanol in the vapor, which is equal to the product of mole fraction in the solution and the vapor pressure: \(P_{H_2O} = x_{H_2O} P_{H_2O}^* = 0.383(175) \approx 67.03 \,torr\) \(P_{C_2H_5OH} = x_{C_2H_5OH} P_{C_2H_5OH}^* = 0.617(400) \approx 246.8 \,torr\) Now, we can find the mole fraction of ethanol in the vapor: \(y_{C_2H_5OH} = \frac{P_{C_2H_5OH}}{P_{H_2O} + P_{C_2H_5OH}} = \frac{246.8}{67.03 + 246.8} \approx 0.786\) The mole fraction of ethanol in the vapor above the solution is approximately 0.786. In summary: (a) The mole fraction of ethanol in the solution is approximately 0.617. (b) The vapor pressure of the solution at 63.5 °C is approximately 295.12 torr. (c) The mole fraction of ethanol in the vapor above the solution is approximately 0.786.

Key Concepts

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture. For instance, in a solution composed of water and ethanol, the mole fraction of ethanol, denoted as \(x_{\text{C}_2\text{H}_5\text{OH}}\), is calculated by dividing the moles of ethanol by the sum of moles of both water and ethanol. This concept is useful in determining the proportion of substances within a solution, without reference to mass or volume.

• It is dimensionless, meaning it has no units.
• Mole fractions always add up to 1 for a solution.

Understanding mole fraction helps in analyzing how different substances interact within a mixture.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. In mixtures, it's used to understand how each component contributes to the overall pressure above the solution. For pure substances, vapor pressure depends only on temperature. However, for solutions, it involves mole fractions and the vapor pressures of pure components.

• In an ethanol-water solution, vapor pressure can be found using Raoult's Law.
• High vapor pressure means more molecules are escaping into the vapor phase.

This concept is crucial in explaining boiling points, evaporation rates, and chemical equilibrium in solutions.

Ideal Solution Behavior

Ideal solution behavior refers to the assumption that the interactions between different components of a mixture are similar to those occurring in pure substances. In ideal solutions, Raoult's Law is applicable, which states that the partial vapor pressure of each component is proportional to its mole fraction in the solution.

• Real solutions often deviate from ideal behavior but the law serves as a useful approximation.
• In an ethanol-water mixture, ideal behavior assumes no significant interaction differences between ethanol and water molecules.

Understanding this concept allows for easier calculations of various properties, like vapor pressure, in mixed solutions.

Ethanol-Water Solution

An ethanol-water solution is a common binary liquid mixture used in various applications, from laboratory settings to beverages. This mixture showcases particular physical behavior due to hydrogen bonding between its molecules. In terms of vapor pressure, ethanol has a higher vapor pressure compared to water at the same temperature, influencing the overall properties of the solution.

• Ethanol is a volatile liquid, contributing significantly to the vapor phase above the solution.
• Ideal and non-ideal solution behavior can occur, depending on specific conditions.

Studying such mixtures helps in understanding two-component systems and applies to processes like distillation and solvent extraction.