Problem 58 - A Complete Resource Book in Chemistry for JEE Main

Question

The elevation in boiling point for \(13.44 \mathrm{~g}\) of \(\mathrm{CuCl}_{2}\) dissolved in \(1 \mathrm{~kg}\) of water as solvent will be \(\left(\mathrm{K}_{\mathrm{b}}=0.52 \mathrm{~kg} / \mathrm{J}\right.\), molar mass of \(\left.\mathrm{CuCl}_{2}=134.4 \mathrm{~g} / \mathrm{mol}\right)\) (a) \(0.05\) (b) \(0.10\) (c) \(0.16\) (d) \(0.20\).

Step-by-Step Solution

Answer

The boiling point elevation is approximately 0.16.

Step 1: Calculating Moles of CuCl2

First, we need to find the number of moles of CuCl2. Use the formula \( ext{moles} = \frac{ ext{mass in grams}}{ ext{molar mass}} \). Here, the mass is 13.44 g and the molar mass is 134.4 g/mol. Thus, the number of moles \( = \frac{13.44}{134.4} = 0.1 \text{ mol} \).

Step 2: Determine the Boiling Point Elevation Formula

The boiling point elevation \( \Delta T_b \) is given by the formula \( \Delta T_b = i \cdot K_b \cdot m \), where \( i \) is the van't Hoff factor, \( K_b \) is the ebullioscopic constant, and \( m \) is the molality of the solution.

Step 3: Calculate Molality

The molality \( m \) is the number of moles of solute per kilogram of solvent. Since we have 0.1 moles of CuCl2 and 1 kg of water, the molality \( m = 0.1 \text{ mol/kg} \).

Step 4: Determine van't Hoff Factor (i)

For CuCl2, which dissociates into three ions (one Cu²⁺ and two Cl⁻), the van't Hoff factor \( i = 3 \).

Step 5: Calculate Boiling Point Elevation

Substitute \( i = 3 \), \( K_b = 0.52 \text{ kg/J} \), and \( m = 0.1 \text{ mol/kg} \) into the boiling point elevation formula: \( \Delta T_b = 3 \times 0.52 \times 0.1 = 0.156 \). Round this value to the nearest option, which is 0.16.

Key Concepts

Molality

Molality is a way to express the concentration of a solution. Unlike molarity, which is the number of moles of solute per liter of solution, molality is the number of moles of solute per kilogram of solvent.
It is denoted by the symbol \(m\).
To calculate molality, use the formula:

• \(m = \frac{\text{moles of solute}}{\text{kilograms of solvent}}\)

In our given scenario, we have 0.1 moles of copper chloride \(\text{CuCl}_2\) and 1 kilogram of water.
The calculation is straightforward:

• \(m = \frac{0.1}{1} = 0.1\text{ mol/kg}\)

Molality is particularly useful in colligative property calculations, such as boiling point elevation, where changes in boiling point depend on the number of particles in a solution rather than the volume.

van't Hoff Factor

The van't Hoff factor, symbolized as \(i\), is a measure of how many particles a compound disassociates into when dissolved in a solvent.
For non-electrolytes, \(i\) is typically 1 because they do not dissociate. However, for electrolytes like CuCl\(_2\), it’s a different story.
CuCl\(_2\) dissociates in water to form three ions: one \(\text{Cu}^{2+}\) and two \(\text{Cl}^-\). Therefore, the \(i\) value for CuCl\(_2\) is not 1 but 3.
This factor is crucial for accurate colligative property calculations.

• For instance, in the boiling point elevation, the formula \(\Delta T_b = i \cdot K_b \cdot m\) uses \(i\) to adjust for the number of particles that influence the boiling point.

The van't Hoff factor effectively accounts for dissociation and is essential for predicting the behavior of ionic solutions.

Ebullioscopic Constant

The ebullioscopic constant \(K_b\) is a property of the solvent that quantifies how much the boiling point will increase per molal concentration of the solute.
Each solvent has its own distinct \(K_b\). For water, \(K_b\) is typically given as 0.52 kg/J.
In the boiling point elevation formula \(\Delta T_b = i \cdot K_b \cdot m\), it determines the extent of boiling point rise:

• \(\Delta T_b\) is the increase in boiling point,
• \(i\) is the van't Hoff factor,
• \(m\) is the molality.

To solve our exercise, we knew \(K_b\) (0.52 kg/J), \(i\) (3, for CuCl\(_2\)), and \(m\) (0.1 mol/kg).
Substituting these values results in a calculated boiling point elevation of 0.156, rounded to 0.16 to match given options.
Understanding \(K_b\) is essential for predicting changes in the boiling point, which is a vital aspect of colligative properties.