Problem 160 - A Complete Resource Book in Chemistry for JEE Main

Question

\(18 \mathrm{~g}\) of glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) is added to \(178.2 \mathrm{~g}\) of water. The vapour pressure of water for this aqueous solution at \(100^{\circ} \mathrm{C}\) is (a) \(759.00\) torr (b) \(7.60\) torr (c) \(76.00\) torr (d) \(752.40\) torr

Step-by-Step Solution

Answer

The vapour pressure of water for this aqueous solution is 752.40 torr (option d).

Step 1: Calculate Moles of Glucose

First, we need to find the number of moles of glucose (\(\mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6\)). The molar mass of glucose is approximately \(180 \text{ g/mol}\). Therefore, the moles of glucose are calculated as follows: \( \text{Moles of glucose} = \frac{\text{mass of glucose}}{\text{molar mass of glucose}} = \frac{18 \text{ g}}{180 \text{ g/mol}} = 0.1 \text{ mol}\).

Step 2: Calculate Moles of Water

The molecular weight of water (\(\mathrm{H}_2\mathrm{O}\)) is approximately \(18 \text{ g/mol}\). Calculate the moles of water using: \( \text{Moles of water} = \frac{\text{mass of water}}{\text{molar mass of water}} = \frac{178.2 \text{ g}}{18 \text{ g/mol}} = 9.9 \text{ mol} \).

Step 3: Calculate the Mole Fraction of Water

The mole fraction of water in the solution is the ratio of the moles of water to the total number of moles (glucose + water). Therefore, \( x_{\text{water}} = \frac{\text{moles of water}}{\text{moles of glucose} + \text{moles of water}} = \frac{9.9}{9.9+0.1} = \frac{9.9}{10} = 0.99 \).

Step 4: Use Raoult's Law to Calculate Vapour Pressure

Raoult's law states that the vapour pressure of the solvent (water) in the solution is equal to the mole fraction of the solvent times the vapour pressure of the pure solvent. The vapour pressure of pure water at \(100^{\circ}\text{C}\) is approximately \(760\text{ torr}\). Therefore, the vapour pressure of the solution is \(P_{\text{solution}} = x_{\text{water}} \times P^{\circ}_{\text{water}} = 0.99 \times 760\text{ torr} = 752.4\text{ torr}\).

Key Concepts

Vapor Pressure

Vapor pressure is an important concept in chemistry that deals with the pressure exerted by a vapor in equilibrium with its liquid or solid form in a closed system. When you have a liquid in a container, some of its molecules have enough energy to escape into the air above, creating vapor pressure. At a given temperature, the vapor pressure is a measure of how easily molecules escape into the gaseous state.

When a solute is added to a solvent, the vapor pressure of the solvent is altered, a principle described by Raoult's Law. The vapor pressure of the pure solvent is reduced when a non-volatile solute is dissolved in it, like glucose in water. This decrease occurs because the solute molecules take up space at the surface of the liquid, meaning fewer solvent molecules can escape into vapor, thereby lowering the vapor pressure of the solvent.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a component to the total number of moles in the mixture. In a solution of water and glucose, the mole fraction of water is crucial in determining the vapor pressure of the solution via Raoult's Law.

For example, if the mole fraction of water ( x_{ ext{water}} ) in the solution is calculated to be 0.99, it suggests that water still constitutes virtually the entire solvent component of the solution. This near purity means the solution's vapor pressure is just slightly lower than the vapor pressure of pure water.

Aqueous Solution

An aqueous solution is a solution in which water is the solvent. These types of solutions are fundamental in chemistry and biology. When a substance dissolves in water, it results in an aqueous solution. In the context of glucose and water, the glucose dissolves in the water to form such a solution.

Aqueous solutions are significant because water is an excellent solvent for many compounds. Its polarity allows it to dissolve various substances. Moreover, the properties of the solution, such as vapor pressure, boiling point, and freezing point, can change depending on the concentration and nature of the solute.

Glucose

Glucose is a simple sugar and an important carbohydrate in biology. Its chemical formula is C6H12O6. It is a non-volatile solute, meaning it doesn't easily vaporize, which is why its addition to water lowers the vapor pressure of the solution according to Raoult's Law.

Glucose's role in solutions is often to exemplify how solutes affect solvent properties. Since it doesn't contribute to the vapor phase, it allows us to focus on how the solvent's behavior changes. When understanding solutions, glucose serves as a model compound for demonstrating principles in chemistry like solubility, colligative properties, and the application of Raoult's Law.