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 \(0.086\), so the answer is (b).

Step 1: Understand Molality

Molality is defined as the number of moles of solute (methyl alcohol, \(\mathrm{CH}_3\mathrm{OH}\)) per kilogram of solvent (water, \(\mathrm{H}_2\mathrm{O}\)). A 5.2 molal solution means there are 5.2 moles of \(\mathrm{CH}_3\mathrm{OH}\) in 1 kilogram of water.

Step 2: Calculate Moles of Water

Given that the molality is calculated per kilogram of solvent, convert the mass of 1 kg of water to moles using the molar mass of water, which is approximately 18 g/mol. Therefore, the moles of water are \(\frac{1000\, \text{g}}{18\, \text{g/mol}} \approx 55.56\, \text{mol}\).

Step 3: Calculate Total Moles in Solution

Add the moles of methyl alcohol to the moles of water to find the total moles in the solution: \(5.2\, \text{mol (\(\mathrm{CH}_3\mathrm{OH}\))} + 55.56\, \text{mol (\(\mathrm{H}_2\mathrm{O}\))} = 60.76\, \text{mol (total)}\).

Step 4: Calculate Mole Fraction of Methyl Alcohol

The mole fraction of methyl alcohol is the ratio of the moles of methyl alcohol to the total moles in the solution: \(\frac{5.2}{60.76} \approx 0.086\).

Step 5: Choose Correct Option

Given the calculated mole fraction of 0.086, the correct answer to the problem is option (b) \(0.086\).

Key Concepts

Molality

Molality is an important property in chemistry that's used to describe the concentration of a solution. It's defined as the number of moles of solute divided by the mass of the solvent in kilograms. This measure of concentration is particularly useful when dealing with temperature-dependent experiments because it doesn't change with temperature fluctuations. When you see a solution labeled as 5.2 molal, it specifically means that there are 5.2 moles of the solute, in this case, methyl alcohol (\(\mathrm{CH}_3\mathrm{OH}\)), dissolved in 1 kilogram of solvent, usually water.

• Molality focuses solely on the ratio between the moles of solute and the kilograms of solvent.
• It is a temperature-independent concentration measure, unlike molarity.
• Useful when mixing temperatures are not constant.

When calculating molality, pay attention to ensure that the solvent's weight is in kilograms and the amount of solute is in moles.

Moles of Solute

Understanding the concept of moles is crucial in chemistry. A mole is just a unit of measure that represents a specific number - Avogadro's number (\(6.022 \times 10^{23}\)). It's analogous to saying a dozen eggs, where 'dozen' always means twelve. Similarly, when we talk about moles of a substance, we mean that many molecules or atoms of that substance.
In the given problem, the solute is methyl alcohol (\(\mathrm{CH}_3\mathrm{OH}\)), and it's important to note how many moles of this compound are present in the solution. This is directly given by the molality of the solution. So, with 5.2 molal, you have 5.2 moles of methyl alcohol per kilogram of water.
Calculating this correctly is crucial for determining how substances will react together, as reactions occur based on the number of particles, not the mass or volume of the substance.

• Moles relate to Avogadro's number for a clear count of molecules.
• Necessary for stoichiometric calculations in reactions.
• Ensure correct representation when dealing with chemical equations.

Mole Calculation

Performing mole calculations in solutions is often necessary to understand the properties and behaviors of solutions. These calculations usually involve determining the total number of moles in the solution to find properties like mole fraction or concentration. In our exercise, to find the mole fraction of the methyl alcohol, we needed both the moles of the solute and the solvent.
The total moles in the solution were calculated by adding the moles of methyl alcohol (\(5.2 \, \text{mol}\)) to the moles of water, obtained by converting the water's mass to moles as it has a molar mass of 18 g/mol:

• The conversion of 1 kg of water equals \(55.56 \, \text{mol}\).
• Total moles are thus \(5.2 + 55.56 = 60.76 \, \text{mol}\).

Once you have the total moles, the mole fraction can be found by dividing the moles of the solute by the total moles. These calculations make it easier to understand how components interact and predict the behavior of substances in different scenarios.