Problem 47 - Chemistry: An Atoms-Focused Approach

Question

At \(20^{\circ} \mathrm{C},\) the vapor pressure of ethanol \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\right)\) is 45 torr and the vapor pressure of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) is 92 torr. What is the vapor pressure at \(20^{\circ} \mathrm{C}\) of a solution prepared by mixing \(25 \mathrm{g}\) of methanol and \(75 \mathrm{g}\) of ethanol?

Step-by-Step Solution

Answer

Answer: The vapor pressure at 20°C of the solution prepared by mixing 25g of methanol and 75g of ethanol is approximately 60.2 torr.

Step 1: Calculate the mole fraction of each component in the solution

We first need to calculate the mole fraction of the two components, methanol, and ethanol. To do so, we need the molecular weight of each component. The molecular weight of methanol (CH3OH) is 12.01g/mol + 4(1.008g/mol) + 16.00g/mol = 32.04g/mol, and the molecular weight of ethanol (CH3CH2OH) is 12.01g/mol + 12.01g/mol + 6(1.008g/mol) + 16.00g/mol + 1.008g/mol = 46.0688g/mol. Now, let's calculate the number of moles of each component: Number of moles of methanol = 25g / 32.04g/mol ≈ 0.7803 mol Number of moles of ethanol = 75g / 46.0688g/mol ≈ 1.6274 mol The mole fraction (X) is determined as the number of moles of a component divided by the total moles in the solution: X_methanol = 0.7803 / (0.7803 + 1.6274) ≈ 0.324 X_ethanol = 1.6274 / (0.7803 + 1.6274) ≈ 0.676

Step 2: Use Raoult's Law to calculate the partial pressures of the components

Raoult's Law states that the partial pressure of each component in a solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution. We are given the vapor pressures of methanol and ethanol at 20°C. Partial pressure of methanol: P_methanol = X_methanol * Vapor pressure of methanol = 0.324 * 92 torr ≈ 29.8 torr Partial pressure of ethanol: P_ethanol = X_ethanol * Vapor pressure of ethanol = 0.676 * 45 torr ≈ 30.4 torr

Step 3: Calculate the total vapor pressure of the solution

To find the total vapor pressure of the solution, simply add the partial pressures calculated in step 2. Total vapor pressure = P_methanol + P_ethanol ≈ 29.8 torr + 30.4 torr ≈ 60.2 torr In conclusion, the vapor pressure at 20°C of the solution prepared by mixing 25g of methanol and 75g of ethanol is approximately 60.2 torr.

Key Concepts

Vapor Pressure

Vapor pressure is a fundamental concept in solution chemistry. It represents the pressure exerted by the vapor of a liquid in equilibrium with its liquid phase at a given temperature. When a solution contains more than one volatile component, each component contributes to the total vapor pressure of the solution. This contribution is detailed further in the law that describes this behavior: Raoult's Law.

Understanding vapor pressure is crucial for determining how a solution will behave under different temperature and concentration conditions. In solutions like those composed of ethanol and methanol, knowing the vapor pressures of the pure components helps us predict how the combined solution behaves.

Mole Fraction

The mole fraction is a way to express the concentration of one component in a mixture. It is the ratio of the number of moles of a particular component to the total number of moles in the mixture. To calculate the mole fraction of methanol in our solution, use its number of moles divided by the total moles of all components:

\( X_{\text{methanol}} = \frac{n_{\text{methanol}}}{n_{\text{methanol}} + n_{\text{ethanol}}} \)

In this exercise, methanol's mole fraction plays a key role in calculating its partial pressure. The same process applies to ethanol:

• Mole fraction of methanol helps determine how much the methanol component influences the solution's vapor pressure.
• The mole fraction sum of all components always equals 1.

Partial Pressure

Partial pressure is the pressure contributed by a single component in a mixture of gases or vapor, as if it alone occupied the entire volume. In solutions, the partial pressure of a volatile component can be determined using Raoult's Law, which multiplies the mole fraction by the pure substance's vapor pressure:
\( P_{\text{component}} = X_{\text{component}} \times P^{\circ}_{\text{component}} \)
For instance, the partial pressure of methanol can be found by multiplying its mole fraction by its pure vapor pressure. Both calculated partial pressures (ethanol and methanol) are then used to find the total vapor pressure of the solution by summing them up:

• Partial pressures inform us how each component's presence in the solution affects the overall pressure.
• Summing up partial pressures gives the total pressure exerted by the vapor above the solution.

Solution Chemistry

Solution chemistry involves understanding how components interact within a solution. Key areas include how substances mix, their concentrations, and how they impact physical properties like vapor pressure. Raoult's Law is central when dealing with mixtures of volatile liquids, explaining how each substance in a solution influences the total pressure:

• Simplifies complex mixtures by considering individual contributions to properties like vapor pressure.
• Helps predict outcomes in chemical reactions and physical processes involving solutions.
• Determines behavior of solutions under different conditions, crucial in designing industrial processes like distillation.

By understanding these principles, we can solve practical problems, like calculating the total vapor pressure of a methanol-ethanol mixture, helping highlight the importance of mastering solution chemistry.