Problem 33 - Chemistry in 30 days

Question

A solution is prepared by mixing \(40 \mathrm{~g}\) of \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) and \(25.5 \mathrm{~g} \mathrm{CHCl}_{3} . \mathrm{P}^{\circ}\) for \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) is 200 \(\mathrm{nm}\) of \(\mathrm{Hg}\) and \(P^{\circ}\) for \(\mathrm{CHCl}_{3}\) is \(415 \mathrm{~nm}\) of \(\mathrm{Hg}\). Calculate the mole fraction of \(\mathrm{CHCl}_{3}\) in vapour phase. (a) \(0.28\) (b) \(0.18\) (c) \(0.38\) (d) \(0.48\)

Step-by-Step Solution

Answer

Mole fraction of \(\mathrm{CHCl}_{3}\) in vapor phase is \(0.48\).

Step 1: Calculate moles of each component

First, calculate the moles of \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) and \(\mathrm{CHCl}_{3}\). The molar mass of \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) is approximately \(84.93 \text{ g/mol}\) and for \(\mathrm{CHCl}_{3}\) it is \(119.38 \text{ g/mol}\). - Moles of \(\mathrm{CH}_{2} \mathrm{Cl}_{2} = \frac{40 \text{ g}}{84.93 \text{ g/mol}} \approx 0.471 \text{ mol}\)- Moles of \(\mathrm{CHCl}_{3} = \frac{25.5 \text{ g}}{119.38 \text{ g/mol}} \approx 0.213 \text{ mol}\)

Step 2: Calculate the Mole Fraction of each component in the liquid phase

Determine the total moles and the mole fraction of each component in the mixture:- Total moles = Moles of \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) + Moles of \(\mathrm{CHCl}_{3}\) = \(0.471 + 0.213 = 0.684 \text{ mol}\)- Mole fraction of \(\mathrm{CH}_{2} \mathrm{Cl}_{2} = \frac{0.471}{0.684} \approx 0.688\)- Mole fraction of \(\mathrm{CHCl}_{3} = \frac{0.213}{0.684} \approx 0.312\)

Step 3: Use Raoult's law to calculate partial pressures

According to Raoult's law, the partial pressure of a component is the product of its mole fraction in the liquid phase and its pure component vapor pressure:- Partial pressure of \(\mathrm{CH}_{2} \mathrm{Cl}_{2} = 0.688 \times 200 = 137.6 \text{ nm Hg}\)- Partial pressure of \(\mathrm{CHCl}_{3} = 0.312 \times 415 = 129.48 \text{ nm Hg}\)

Step 4: Calculate the total pressure and mole fraction in the vapor phase

Calculate the total pressure of the mixture and use it to find the mole fraction in the vapor phase:- Total pressure \( P_{\text{total}} = 137.6 + 129.48 = 267.08 \text{ nm Hg}\)- Mole fraction of \(\mathrm{CHCl}_{3}\) in vapor phase = \(\frac{129.48}{267.08} \approx 0.485\)

Step 5: Select the closest answer from given options

From the options provided, the mole fraction of \(\mathrm{CHCl}_{3}\) in the vapor phase is closest to 0.48.The correct answer is (d) \(0.48\).

Key Concepts

Mole Fraction

The mole fraction is an important concept in chemistry. It represents the ratio of moles of a component to the total moles present in a mixture. When working with solutions, understanding the mole fraction helps you understand the composition of the solution.
For example, to calculate the mole fraction of \({\mathrm{CHCl}_{3}}\) in this exercise, we divide the moles of \({\mathrm{CHCl}_{3}}\) by the total moles of the mixture:

• Moles of \({\mathrm{CH}_{2} \mathrm{Cl}_{2}}\) = 0.471 mol
• Moles of \({\mathrm{CHCl}_{3}}\) = 0.213 mol

The total moles in the mixture is the sum:\[\text{Total moles} = 0.471 + 0.213 = 0.684 \text{ mol}\]Then, the mole fraction of \({\mathrm{CHCl}_{3}}\) is:\[\text{Mole fraction of } \mathrm{CHCl}_{3} = \frac{0.213}{0.684} \approx 0.312\]This calculation helps determine how much of the solution is made up of \({\mathrm{CHCl}_{3}}\), which is crucial to further calculations in this solution.

Partial Pressure

Partial pressure is a vital concept when dealing with gas mixtures or vapor phases. It represents the pressure that individual gas components in a mixture contribute to the total pressure.According to Raoult's Law, the partial pressure is calculated by multiplying the mole fraction of each component in the liquid phase by its vapor pressure as a pure component. For our exercise:

• Partial pressure of \({\mathrm{CH}_{2} \mathrm{Cl}_{2}}\) is calculated as \(0.688 \times 200 = 137.6\) nm Hg
• Partial pressure of \({\mathrm{CHCl}_{3}}\) is calculated as \(0.312 \times 415 = 129.48\) nm Hg

These individual pressures then contribute to the total pressure of the system, allowing us to see how each component influences the environment at a molecular level.

Vapor Pressure

Vapor pressure is the pressure created by a vapor in equilibrium with its solid or liquid form. It depends primarily on temperature and the nature of the liquid or solid.For pure substances, vapor pressure measurements can predict how a liquid would behave in a mixture. In our exercise, the vapor pressures given are:

• 200 nm Hg for \({\mathrm{CH}_{2} \mathrm{Cl}_{2}}\)
• 415 nm Hg for \({\mathrm{CHCl}_{3}}\)

These values are used alongside Raoult’s Law to determine the partial pressures exerted by each component in a mixture, providing us with a deeper understanding of the behavior of volatile mixtures.

Chemistry Problem-Solving

Chemistry problem-solving often requires a methodical approach, addressing each part of a problem step-by-step. In this Raoult's Law problem, understanding each process is the key to a correct solution: 1. **Calculate Moles:** Recognizing that understanding the quantity of each component is foundational. 2. **Determine Mole Fraction:** Using the moles calculated to find the mole fraction gives insight into the relative amounts of each component in a solution. 3. **Apply Raoult's Law:** Understanding how to use the mole fraction to find the partial pressures shows how each component affects the vapor phase. 4. **Final Calculations:** Combining the partial pressures to find the total pressure and thus the mole fraction in the vapor phase completes the cycle. Each stage builds on the last, helping apply mathematical principles to a chemical context, and illustrating how these methods predict the behavior of chemical mixtures.