Problem 39 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The degree of dissociation of calcium nitrate in a dilute aqueous solution, containing \(7.0 \mathrm{~g}\). of the salt per \(100 \mathrm{~g}\) of water at \(100^{\circ} \mathrm{C}\) is \(70 \%\). If the vapour pressure of water at \(100^{\circ} \mathrm{C}\) is \(760 \mathrm{~mm}\), calculate the vapour pressure of the solution.

Step-by-Step Solution

Answer

The vapor pressure of the solution is approximately 746.32 mm Hg.

Step 1: Calculate Moles of Calcium Nitrate

First, determine the molar mass of calcium nitrate, which is \( \text{Ca(NO}_3\text{)}_2 = 40 + 2(14 + 3 \times 16) = 164 \, \text{g/mol} \). Now, calculate the moles of \( \text{Ca(NO}_3\text{)}_2 \) in 7.0 g:\[ n = \frac{7.0 \, \text{g}}{164 \, \text{g/mol}} = 0.0427 \, \text{mol}. \]

Step 2: Determine Moles After Dissociation

Calcium nitrate dissociates as follows:\[ \text{Ca(NO}_3\text{)}_2 \rightarrow \text{Ca}^{2+} + 2\text{NO}_3^{-}. \] Given the degree of dissociation \( \alpha = 0.7 \), the dissociation produces \( 1 + \alpha(2) = 2.4 \) total ions per mole of the original compound.So, the moles of ions formed = \( 2.4 \times 0.0427 \).

Step 3: Calculate Total Moles in Solution

Calculate the total moles in solution using the formula:\[ n_{total} = (1-\alpha) + \alpha \times 3 = 1 + 3\alpha = 2.4 \times 0.0427. \]This becomes:\[ n_{total} = 0.10248 \, \text{mol}. \]

Step 4: Vapor Pressure of the Solution using Raoult's Law

Raoult's Law states:\[ P_{solution} = \chi_{water} \cdot P_{water} \]Where \( \chi_{water} \) is the mole fraction of water, given by:\[ \chi_{water} = \frac{moles \ of \ water}{total \ moles}. \]So, \( moles \ of \ water = \frac{100 g}{18 \, g/mol} = 5.56 \, mol \)Thus:\[ \chi_{water} = \frac{5.56}{5.56 + 0.10248} \approx 0.982. \]

Step 5: Calculate Pressure of Solution

Substitute the known values into Raoult's Law:\[ P_{solution} = 0.982 \times 760 \, \text{mm Hg} \approx 746.32 \, \text{mm Hg}. \] The vapor pressure of the solution is approximately \(746.32 \, \text{mm Hg} \).

Key Concepts

Raoult's Law

Understanding Raoult's Law can be quite important when studying solutions and vapor pressures. Raoult's Law provides a relationship between the vapor pressure of a solution and its components. According to Raoult's Law, when a non-volatile solute is dissolved in a solvent, the vapor pressure of the solvent above the solution is lower than that of the pure solvent. This is because the presence of solute particles reduces the number of solvent particles at the surface, thus lowering the vapor pressure.

The law can be mathematically expressed as:

• \( P_{solution} = \chi_{water} \cdot P_{water} \)

• Where \( P_{solution} \) is the vapor pressure of the solution, \( \chi_{water} \) is the mole fraction of water, and \( P_{water} \) is the vapor pressure of pure water.

In our exercise, after calculating the mole fraction of water, Raoult’s Law helps determine the new vapor pressure of the water in the solution. This concept is particularly useful in chemistry to understand how different substances behave when mixed, especially at temperatures like those in this problem at 100°C.

Degree of Dissociation

The degree of dissociation is a measure of how much a compound dissociates into its ions when dissolved in a solution. It's often denoted by the Greek letter \( \alpha \). In our example, calcium nitrate dissociates in water, and we have a degree of dissociation of 70% or \( \alpha = 0.7 \).

This figure signifies that out of the total calcium nitrate dissolved, 70% separates into \( \text{Ca}^{2+} \) and \( \text{NO}_3^- \) ions. The degree of dissociation can significantly affect the properties of a solution, such as its conductivity and the resulting vapor pressure. To determine how many moles of ions form, we multiply the initial moles of calcium nitrate by the total number of ions generated per mole, considering the dissociation rate. Thus, it's crucial for calculating how the solution diverges from the expected behavior of pure solvents.

Mole Fraction

Mole fraction is a way to express the concentration of a component in a solution. It is the ratio of moles of one component to the total moles of all components in the mixture. Mathematically, for a solvent like water in a solution, it is represented as:

• \( \chi_{water} = \frac{moles \ of \ water}{total \ moles} \)

In this exercise, we compute the moles of water by dividing the mass of water by its molar mass. Similarly, the total moles in the solution includes the undissociated solute and the ions formed due to dissociation.

The concept of the mole fraction is pivotal in determining the vapor pressure of solutions using Raoult’s Law. It is essential since it gives us a precise fraction of the solvent, which directly impacts the calculated vapor pressure of the solution. Hence, understanding mole fraction is key in effectively applying Raoult’s Law to real-world scenarios.