Question

The vapor pressure of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) at \(20^{\circ} \mathrm{C}\) is \(44 \mathrm{mmHg},\) and the vapor pressure of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) at the same temperature is \(94 \mathrm{mmHg} .\) A mixture of \(30.0 \mathrm{~g}\) of methanol and \(45.0 \mathrm{~g}\) of ethanol is prepared (and can be assumed to behave as an ideal solution). (a) Calculate the vapor pressure of methanol and ethanol above this solution at \(20^{\circ} \mathrm{C}\). (b) Calculate the mole fraction of methanol and ethanol in the vapor above this solution at \(20^{\circ} \mathrm{C}\). (c) Suggest a method for separating the two components of the solution.

Short answer

Methanol vapor pressure is 45.97 mmHg; ethanol is 22.48 mmHg. Methanol vapor fraction is 0.671. Use fractional distillation to separate them.

Step-by-step solution

Calculate moles of methanol and ethanol

First, calculate the number of moles for each component. The molar mass of methanol (CH₃OH) is approximately 32.04 g/mol and for ethanol (C₂H₅OH) is approximately 46.08 g/mol.For methanol: \( n_{\text{methanol}} = \frac{30.0 \text{ g}}{32.04 \text{ g/mol}} \approx 0.936 \text{ mol} \).For ethanol: \( n_{\text{ethanol}} = \frac{45.0 \text{ g}}{46.08 \text{ g/mol}} \approx 0.977 \text{ mol} \).

Calculate mole fractions in the liquid

Calculate the mole fraction of each component in the liquid phase. Total moles = \( n_{\text{methanol}} + n_{\text{ethanol}} = 0.936 + 0.977 = 1.913 \text{ mol} \).\( X_{\text{methanol}} = \frac{n_{\text{methanol}}}{n_{\text{total}}} = \frac{0.936}{1.913} \approx 0.489 \).\( X_{\text{ethanol}} = \frac{n_{\text{ethanol}}}{n_{\text{total}}} = \frac{0.977}{1.913} \approx 0.511 \).

Calculate partial vapor pressures above the solution

Using Raoult's Law, the partial vapor pressure of each component can be calculated.For methanol: \( P_{\text{methanol}}^{\text{solution}} = X_{\text{methanol}} \times P_{\text{methanol}}^{\text{pure}} = 0.489 \times 94 \text{ mmHg} = 45.966 \text{ mmHg} \).For ethanol: \( P_{\text{ethanol}}^{\text{solution}} = X_{\text{ethanol}} \times P_{\text{ethanol}}^{\text{pure}} = 0.511 \times 44 \text{ mmHg} = 22.484 \text{ mmHg} \).

Calculate total vapor pressure of the solution

The total vapor pressure is the sum of the partial pressures.\( P_{\text{total}} = P_{\text{methanol}}^{\text{solution}} + P_{\text{ethanol}}^{\text{solution}} = 45.966 + 22.484 = 68.45 \text{ mmHg} \).

Calculate mole fractions in the vapor

The mole fraction of each component in the vapor phase can be calculated using the partial and total pressures.\( Y_{\text{methanol}} = \frac{P_{\text{methanol}}^{\text{solution}}}{P_{\text{total}}} = \frac{45.966}{68.45} \approx 0.671 \).\( Y_{\text{ethanol}} = \frac{P_{\text{ethanol}}^{\text{solution}}}{P_{\text{total}}} = \frac{22.484}{68.45} \approx 0.329 \).

Suggest a method for separation

The mixture can be separated using fractional distillation. This method relies on the difference in boiling points of methanol and ethanol, allowing methanol to vaporize first due to its higher vapor pressure and lower boiling point.

Key concepts

Raoult's Law

Raoult's Law is a fundamental principle in chemistry that helps us understand how the vapor pressure of an ideal solution is affected by its components. According to this law, the partial vapor pressure of any component in an ideal solution is equal to the product of the mole fraction of the component in the liquid phase and the vapor pressure of the pure component. In formula terms, it is expressed as:\[ P_{i} = X_{i} \times P_{i}^{\text{pure}} \]where:

• \( P_{i} \) is the partial vapor pressure of component \( i \) in the solution.
• \( X_{i} \) is the mole fraction of component \( i \) in the liquid phase.
• \( P_{i}^{\text{pure}} \) is the vapor pressure of the pure component \( i \).

Raoult's Law assumes that the solution behaves ideally, meaning there are no interactions between the different types of molecules, other than simple mixing. This makes it very useful for calculating the vapor pressures of solutions when the behavior is close to ideal. You can see this law in action in our example when we calculated the vapor pressures of methanol and ethanol above the solution.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a solution. It tells us how many moles of a certain component are present in relation to the total number of moles in the solution. Mathematically, it's given by the formula:\[ X_{i} = \frac{n_{i}}{n_{\text{total}}} \]where:

• \( X_{i} \) is the mole fraction of the component \( i \).
• \( n_{i} \) is the number of moles of the component \( i \).
• \( n_{\text{total}} \) is the total number of moles in the solution.

In calculations, the sum of the mole fractions of all components in a solution is always equal to 1. In our exercise, we calculated the mole fractions for methanol and ethanol in both the liquid and vapor phases. This was essential for applying Raoult's Law to find out the respective vapor pressures of each component.

Ideal Solution

An ideal solution is a theoretical concept in chemistry where the individual components mix perfectly without any change in volume or enthalpy. In such solutions, Raoult's Law applies perfectly because the intermolecular forces between unlike molecules are identical to those between like molecules. Ideal solutions assume:

• No heat is evolved or absorbed during the mixing of the components.
• The size and shape of all molecule types are similar, leading to similar forces between particles.
• The properties of each component are purely additive.

This concept is helpful for simplifying complex solutions into manageable calculations, as seen in the original exercise when dealing with the mixture of methanol and ethanol. Real solutions often show some deviation from ideal behavior, especially when the solution components interact more or less favorably than they do in pure states.

Fractional Distillation

Fractional distillation is a technique used to separate liquid mixtures into individual components based on their boiling points. It exploits the differences in vapor pressures of the components at a given temperature to selectively evaporate and condense the substance with the lowest boiling point first. The method involves:

• Heating the mixture gradually until the component with the lower boiling point boils.
• Continuous vaporization and condensation by passing steamy vapors through a fractionating column.
• Collectors remove the components as they condense back into a liquid state at different levels in the fractionating column.

For the separation of methanol and ethanol, fractional distillation relies on methanol's higher vapor pressure and lower boiling point, meaning it will vaporize first. This allows for efficient separation, which is an essential aspect of many industrial and laboratory processes.