Problem 164 - A Complete Resource Book in Chemistry for JEE Main

Question

The vapour pressure of water at \(20^{\circ} \mathrm{C}\) is \(17.5 \mathrm{~mm} \mathrm{Hg}\). If \(18 \mathrm{~g}\) of glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) is added to \(178.2 \mathrm{~g}\) of water at \(20^{\circ} \mathrm{C}\), the vapour pressure of the resulting solution will be (a) \(17.675 \mathrm{~mm} \mathrm{Hg}\) (b) \(15.750 \mathrm{~mm} \mathrm{Hg}\) (c) \(16.500 \mathrm{~mm} \mathrm{Hg}\) (d) \(17.325 \mathrm{~mm} \mathrm{Hg}\).

Step-by-Step Solution

Answer

The vapour pressure of the solution is \( 17.325 \text{ mm Hg} \), corresponding to option (d).

Step 1: Determine the Moles of Solute and Solvent

To find the moles of glucose, use its molecular weight: \( \text{molecular weight of glucose} = 180 \text{ g/mol} \). Therefore, the moles of glucose is \( \frac{18}{180} = 0.1 \text{ mol} \). For water, with a molecular weight of 18 \( \text{ g/mol} \), there are \( \frac{178.2}{18} = 9.9 \text{ mol} \).

Step 2: Calculate the Mole Fraction of Water

The mole fraction of water is given by the formula \( X_{\text{water}} = \frac{\text{moles of water}}{\text{moles of water} + \text{moles of solute}} \). Substituting the known values yields: \( X_{\text{water}} = \frac{9.9}{9.9 + 0.1} = \frac{9.9}{10} = 0.99 \).

Step 3: Apply Raoult's Law to Find Vapour Pressure

Raoult's Law states that the vapour pressure of a solution \( P \) is the product of the mole fraction of the solvent and its pure vapour pressure. Thus, \( P = X_{\text{water}} \times P^0_{\text{water}} \). Substituting in the values gives \( P = 0.99 \times 17.5 = 17.325 \text{ mm Hg} \).

Step 4: Compare with Provided Options

The calculated vapour pressure of the solution is \( 17.325 \text{ mm Hg} \). Comparing with the options, it matches option (d) \( 17.325 \text{ mm Hg} \).

Key Concepts

Mole Fraction

Understanding the formula for mole fraction is essential in solution chemistry. The mole fraction is a way of expressing the concentration of components in a mixture, including solutions. It is represented by the Greek letter chi (X) and is defined as the ratio of the moles of a component to the total moles in the solution.
For instance, if we consider a solution made by dissolving a solute in a solvent, the mole fraction of the solvent, like water, is calculated by dividing the moles of the solvent by the total moles (solvent plus solute). The formula is:

• \( X_{ ext{water}} = \frac{ ext{moles of water}}{ ext{moles of water} + ext{moles of glucose}} \)

Calculating the mole fraction helps us determine how dilution affects the vapour pressure of solutions. In this specific example, the mole fraction of water is very close to 1, indicating that water is the predominant component in the solution.

Vapour Pressure

Vapour pressure is an important concept in physical chemistry, particularly when dealing with mixtures and solutions. It is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system.
The principle of vapour pressure reveals that every liquid exerts a certain pressure in the form of vapor. When a solute is added to a solvent, like in our solution exercise, the vapour pressure of the solvent is typically reduced. This is because the solute molecules replace some of the solvent molecules at the surface, reducing the solvent's tendency to evaporate. Vapour pressure can be predicted and calculated through Raoult's Law, which provides the relationship between the vapor pressure of a solution and the concentration of its components.

Solution Chemistry

Solution chemistry involves the study of how substances dissolve in a solvent to form solutions and the properties of those solutions. When substances dissolve, they interact at the molecular level, resulting in a homogenous mixture.
There are several key components of solutions that are important to understand:

• Solute: The substance that is dissolved in a solvent, such as glucose in water.
• Solvent: The substance that dissolves the solute, typically present in a larger amount. In this example, it's water.
• Concentration: The amount of solute in a given amount of solvent or solution, affecting the solution's properties like boiling point, freezing point, and vapour pressure.

The properties of solutions, such as lowering of vapor pressure upon adding a solute, are crucial for applications like cooking, chemical reactions, and various industrial processes. Understanding how different factors influence solutions can lead to better manipulation and usage of chemical reactions and material properties.