Problem 175 - A Complete Resource Book in Chemistry for JEE Main

Question

A \(5.2\) molal aqueous solution of methyl alcohol, \(\mathrm{CH}_{3} \mathrm{OH}\), is supplied. What is the mole fraction of methyl alcohol in the solution? (a) \(0.86\) (b) \(0.086\) (c) \(0.043\) (d) \(1.0\)

Step-by-Step Solution

Answer

The mole fraction of methyl alcohol is 0.086.

Step 1: Understand Molality

Molality is defined as the number of moles of solute per kilogram of solvent. Here, we have a 5.2 molal solution of methyl alcohol ( CH_3OH), meaning there are 5.2 moles of methyl alcohol in 1 kg (1000 g) of water (solvent).

Step 2: Convert Water Mass to Moles

The molar mass of water (H_2O) is approximately 18 g/mol. To find the moles of water in 1000 g, divide the mass by the molar mass: \( \text{moles of } H_2O = \frac{1000 \, \text{g}}{18 \, \text{g/mol}} \approx 55.56 \, \text{moles} \).

Step 3: Calculate Total Moles in Solution

Add the moles of methyl alcohol to the moles of water to get the total moles of substances in the solution: \( \text{Total moles} = 5.2 + 55.56 = 60.76 \).

Step 4: Calculate Mole Fraction of Methyl Alcohol

The mole fraction of methyl alcohol is found by dividing the moles of methyl alcohol by the total moles of all components in the solution: \( \text{Mole fraction of CH}_3\text{OH} = \frac{5.2}{60.76} \approx 0.0855 \).

Step 5: Select the Correct Answer

Round the calculated mole fraction to two decimal places: 0.0855 rounds to 0.086. Therefore, the correct answer is option (b) 0.086.

Key Concepts

Molality

Molality is a way to express the concentration of a solution. It focuses on the amount of solute present in a given amount of solvent, typically measured in kilograms. This helps in understanding how concentrated a solution is without being influenced by temperature changes, which can alter volume.
To calculate molality:

• Find the number of moles of solute, which in this case is methyl alcohol ( \(\mathrm{CH}_{3}\mathrm{OH}\)).
• Divide the moles of solute by the mass of the solvent in kilograms. In our example, we have a 5.2 molal aqueous solution, meaning there are 5.2 moles of methyl alcohol for every kilogram of water.

Molality is particularly useful in calculating properties like boiling point elevation and freezing point depression in solutions.

Moles Conversion

Converting masses to moles is a basic yet essential process in chemistry. This conversion allows us to relate mass in grams to the quantity of entities (atoms, molecules, etc.) in a substance.
The formula for converting mass to moles is:

• Moles = \(\frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\)

In the given exercise, we have 1000 grams (1 kg) of water. Using the molar mass of water, around 18 g/mol, the conversion to moles tells us that there are about 55.56 moles of water.
This calculation is crucial for finding the total moles in a solution, which then helps in determining the mole fraction or other concentration measures.

Aqueous Solution

An aqueous solution is a solution where water acts as the solvent. These solutions are pivotal in various chemical reactions because water is a common, effective solvent.
Understanding properties of aqueous solutions allows scientists to predict and control chemical behavior in a liquid state. For instance, this concept helps in determining how substances will interact or dissolve. In our problem, methyl alcohol is dissolved in water forming an aqueous solution.
The properties of this type of solution, such as concentration (useful for molality and molarity calculations), boiling point, and freezing point, are affected by the presence of the solute. By knowing the concentration, especially in terms of molality, we can better understand the solution's physical properties and behavior.