Problem 24 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The mole fraction of glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) in an aqueous binary solution is \(0.1\). The mass percentage of water in it, to the nearest integer, is [Main Sep. \(\mathbf{0 3}, \mathbf{2 0 2 0}\) (I)]

Step-by-Step Solution

Answer

The mass percentage of water is approximately 47%.

Step 1: Understand Mole Fraction

The mole fraction of a component in a solution is the ratio of the number of moles of that component to the total number of moles of all components in the solution. For glucose, its mole fraction is given as 0.1.

Step 2: Calculate Moles of Water and Glucose

Assume 1 mole of the entire solution. For glucose, with a mole fraction of 0.1, this means 0.1 moles of glucose. Therefore, there will be 0.9 moles of water (since the solution is binary: water and glucose).

Step 3: Calculate Molar Masses

The molar mass of glucose \(\mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6\) is calculated as: \[6(12) + 12(1) + 6(16) = 180 \text{ g/mol}\]The molar mass of water (\(\mathrm{H}_2\mathrm{O}\)) is: \[2(1) + 16 = 18 \text{ g/mol}\]

Step 4: Calculate Masses of Glucose and Water

The mass of 0.1 moles of glucose is calculated as: \[0.1 \times 180 = 18 \text{ g}\]The mass of 0.9 moles of water is calculated as:\[0.9 \times 18 = 16.2 \text{ g}\]

Step 5: Calculate Total Mass and Mass Percentage of Water

The total mass of the solution is the sum of the masses of glucose and water:\[18 + 16.2 = 34.2 \text{ g}\]To find the mass percentage of water:\[\frac{16.2}{34.2} \times 100 \approx 47.37\%\]Rounding to the nearest integer gives 47\%.

Key Concepts

Mole Fraction

The concept of mole fraction is crucial in chemistry, particularly when dealing with solutions. It represents the proportion of moles of a solute relative to the total moles present in the solution.
Mole fraction is defined as the ratio of the number of moles of a given component to the total number of moles in the solution.
For example, if the mole fraction of glucose in an aqueous solution is 0.1, it means that out of all the moles present, 10% are glucose and the remaining 90% are water, considering it's a binary solution.
The following formula is used to calculate mole fraction:

• Mole fraction = \( \frac{n_i}{n_\text{total}} \)

where \( n_i \) is the moles of the component of interest, and \( n_\text{total} \) is the total moles of all components.
Understanding mole fraction is fundamental when calculating other properties such as mass percentages or when preparing solutions with specific concentrations.

Molar Mass

Molar mass is the mass of a given substance (chemical element or compound) divided by its amount of substance in moles.
In the periodic table, the relative atomic mass of each element helps us find molar mass.
For example, glucose (\(\mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6}\)) is made up of carbon, hydrogen, and oxygen atoms.
The molar mass is calculated by adding together the atomic masses of all atoms in a molecule:

• Carbon: \(12 \text{ g/mol} \times 6 \text{ (atoms)} = 72 \text{ g/mol}\)
• Hydrogen: \(1 \text{ g/mol} \times 12 \text{ (atoms)} = 12 \text{ g/mol}\)
• Oxygen: \(16 \text{ g/mol} \times 6 \text{ (atoms)} = 96 \text{ g/mol}\)

Adding these gives us a molar mass for glucose:\[180 \text{ g/mol}\]
Similarly, water (\(\mathrm{H}_{2}\mathrm{O}\)) has a molar mass calculated as:

• Hydrogen: \(1 \times 2 = 2 \text{ g/mol}\)
• Oxygen: \(16 \text{ g/mol}\)

Thus, the molar mass of water is \(18 \text{ g/mol}\). Calculating the molar mass is essential when determining the mass of a specific amount of substance in moles.

Binary Solution

A binary solution consists of two components, often classified as a solute and a solvent.
In our example, the solution in question is made up of glucose and water. Glucose acts as the solute, whereas water is the solvent, creating what is known as an aqueous solution.
Binary solutions simplify calculations of mass percentage and mole fraction since there are only two components to consider. This is easier than dealing with multi-component systems.
When discussing binary solutions, it's crucial to understand the relationship between the two components.

• For instance, knowing glucose's mole fraction (0.1) immediately tells us that water makes up the remaining 0.9 of the mole fraction.
• This understanding simplifies determining the amount in moles for each component.

Overall, binary solutions provide a straightforward model for studying more complex behaviors in chemistry.

Aqueous Solution

An aqueous solution is a type of binary solution where water is the solvent.
This is common in many chemical reactions because water is a polar solvent capable of dissolving a wide range of substances.
In an aqueous solution like our glucose-water example, water allows the glucose to disperse uniformly across the mixture.
Some important characteristics of aqueous solutions include:

• Water’s ability to dissolve ionic and polar compounds.
• The potential for hydrogen bonding with solutes.
• Aqueous solutions are essential in biological and chemical processes due to their ability to transport substances.

Knowing that a solution is aqueous gives insight into the potential interactions between solute and solvent, affecting the solution's chemical behavior and properties.
Aqueous solutions play a fundamental role in everyday phenomena like precipitation reactions and acid-base reactions.