Problem 51 - A Complete Resource Book in Chemistry for JEE Main

Question

The vapours pressure of water at \(23^{\circ} \mathrm{C}\) is \(19.8 \mathrm{~mm}\). of Hg. \(0.1\) mole of glucose is dissolved in \(178.2 \mathrm{~g}\) of water. What is the vapour pressure (in \(\mathrm{mm}\) of \(\mathrm{Hg}\) ) of the resultant solution? (a) \(19.0\) (b) \(19.602\) (c) \(19.402\) (d) \(19.202\)

Step-by-Step Solution

Answer

The vapour pressure of the resultant solution is approximately 19.602 mm Hg, which is option (b).

Step 1: Understanding Raoult's Law

Raoult's Law states that the vapour pressure of a solvent over a solution is proportional to the mole fraction of the solvent. Mathematically, it can be expressed as \( P_{solution} = P_{solvent} \times \chi_{solvent} \), where \( P_{solution} \) is the vapour pressure of the solution, \( P_{solvent} \) is the vapour pressure of the pure solvent, and \( \chi_{solvent} \) is the mole fraction of the solvent.

Step 2: Molar Mass of Water

To find the mole fraction of water, we first need to determine the number of moles of water. The molar mass of water (\( H_2O \)) is approximately \(18.02 \text{ g/mol}\).

Step 3: Calculating Moles of Water

Given \(178.2\text{ g}\) of water, we calculate the moles of water as follows: \[\text{moles of water} = \frac{178.2\text{ g}}{18.02 \text{ g/mol}} = 9.89 \text{ moles}\].

Step 4: Calculating Mole Fraction of Water

The mole fraction of water \( \chi_{water} \) is given by \( \chi_{water} = \frac{\text{moles of water}}{\text{moles of water} + \text{moles of glucose}} \). Thus, \( \chi_{water} = \frac{9.89}{9.89 + 0.1} \approx 0.99 \).

Step 5: Applying Raoult's Law

Using Raoult's Law: \[ P_{solution} = P_{solvent} \times \chi_{water} = 19.8 \text{ mm Hg} \times 0.99 \approx 19.602 \text{ mm Hg}\].

Step 6: Selecting the Correct Answer

From the given options, the closest value to our calculated vapour pressure for the solution is \( 19.602 \text{ mm Hg} \) which corresponds to option (b).

Key Concepts

Vapour Pressure

Vapour pressure is a crucial concept in understanding how liquids behave when a solute is dissolved in them. It refers to the pressure exerted by a vapor in equilibrium with its liquid phase.
When a solute is added to a solvent, the vapor pressure of the solvent decreases, due to the reduced number of solvent molecules available at the surface to escape into the vapor phase.

This phenomenon can be observed when a non-volatile solute, such as glucose, is dissolved in a volatile solvent like water.

• Simplified: Imagine pressing down on the amount of gas or vapor in a balloon; this is similar to what happens when a solute decreases the vapor pressure of a solvent.
• Vapour pressure is measured in units like millimeters of mercury (mm Hg), providing a standardized way to express these changes.

Mole Fraction

In solution chemistry, the mole fraction is essential for understanding how many moles of each component are present in the solution. It is defined as the ratio of the number of moles of one component to the total number of moles of all components in the solution.

• The formula for mole fraction of the solvent is given by \( \chi_{solvent} = \frac{n_{solvent}}{n_{solute} + n_{solvent}} \).
• It's a dimensionless number, meaning it doesn't have any units. It simply expresses the proportion of one component relative to the total.

For example, if you dissolve 0.1 moles of glucose in 9.89 moles of water, the mole fraction of water becomes approximately \(0.99\). This high mole fraction indicates that water remains the predominant component in the solution, which is crucial when applying Raoult's Law.

Solution Chemistry

Solution chemistry is the study of how substances dissolve and interact at a molecular level within a solvent. It concerns itself with the fundamental properties and behaviors of solutions.
Solutions can be liquid, solid, or gaseous, but in most solution chemistry cases, we discuss liquids. Important aspects include:

• The solute: the substance being dissolved; in this case, glucose.
• The solvent: the substance that dissolves the solute; here, it's water.
• Concentration and saturation points: key properties determining how much solute can dissolve in a given amount of solvent.

When understanding solution chemistry, it is crucial to comprehend how properties like vapor pressure and mole fraction influence the outcome of dissolving processes. Mastering these basic principles provides greater insight into more complex chemical reactions and interactions.

Colligative Properties

Colligative properties are special characteristics of solutions that change when a solute is added but are dependent only on the number of solute particles, not their identity. When glucose is added to water, several properties are affected, such as boiling point elevation, freezing point depression, and vapor pressure reduction.

These properties give insight into how mixtures behave differently than pure substances, and each of these properties have practical implications and applications:

• Vapour pressure lowering: A decrease in vapor pressure results from the addition of solute, as explored with Raoult's Law in our exercise.
• Boiling point elevation: Adding a solute raises the boiling point of the solvent due to intermolecular forces.
• Freezing point depression: Solutes lower the freezing point, and this principle is used in de-icing roads or antifreeze in vehicles.
• Osmotic pressure: An important concept in biology, where the flow of solvent through a semi-permeable membrane depends on solute concentration.

These properties can fundamentally alter the behavior of a solution, making collocative properties an exciting area of study within solution chemistry.