Problem 148 - A Complete Resource Book in Chemistry for JEE Main

Question

The solubility product of \(\mathrm{PbI}_{2}\) is \(7.47 \times 10^{-9}\) at \(15^{\circ} \mathrm{C}\) and \(1.39 \times 10^{8}\) at \(25^{\circ} \mathrm{C}\). The molar heat of solution of \(\mathrm{PbI}_{2}\) is (use \(\log 1.86=0.2695\) ) (a) \(44.29 \mathrm{~kJ} / \mathrm{mol}\) (b) \(46.25 \mathrm{~kJ} / \mathrm{mol}\) (c) \(29.37 \mathrm{~kJ} / \mathrm{mol}\) (d) \(21.15 \mathrm{~kJ} / \mathrm{mol}\)

Step-by-Step Solution

Answer

The molar heat of solution of \(\mathrm{PbI}_{2}\) is 46.25\, kJ/mol, which is option (b).

Step 1: Understand the Gibbs-Helmholtz Equation

The Gibbs-Helmholtz equation can be used to find the molar heat of solution from solubility data at two different temperatures. The formula is given by \[ \Delta H = -2.303R \left( \frac{T_1T_2}{T_2 - T_1} \right) . \log \left( \frac{K_{sp1}}{K_{sp2}} \right) \] where \( R \) is the ideal gas constant \(8.314 J/mol\,K\), \( T_1 \) and \( T_2 \) are temperatures, and \( K_{sp1} \) and \( K_{sp2} \) are solubility products at these temperatures.

Step 2: Convert Temperatures to Kelvin

Add 273.15 to each Celsius temperature to convert to Kelvin: \( T_1 = 15 + 273.15 = 288.15\, K \) and \( T_2 = 25 + 273.15 = 298.15\, K \).

Step 3: Calculate the Logarithm Term

Using the given solubility product values, calculate \( \log \left( \frac{K_{sp1}}{K_{sp2}} \right) \) as follows.Since \( \log 1.86 = 0.2695 \), substituting the given \( K_{sp} \) values:\[ \log \left( \frac{7.47 \times 10^{-9}}{1.39 \times 10^{-8}} \right) = \log (0.5377) = -0.2695 \].

Step 4: Plug Values into Gibbs-Helmholtz Equation

Substitute all known values and calculate:\[ \Delta H = -2.303 \times 8.314 \times \left( \frac{288.15 \times 298.15}{298.15 - 288.15} \right) \times (-0.2695) \]

Step 5: Simplify to Find the Molar Heat of Solution

Calculate the numerical value:\[ \Delta H = 2.303 \times 8.314 \times 8560.72 \times 0.2695 \]\[ \Delta H \approx 46,253\, J/mol = 46.25\, kJ/mol \].

Step 6: Identify the Correct Answer

Compare the calculated molar heat of solution with the options provided and select the correct answer. The calculated value of \(46.25\, kJ/mol\) corresponds to option (b).

Key Concepts

Gibbs-Helmholtz Equation

The Gibbs-Helmholtz equation is an essential tool in thermodynamics that relates changes in temperature to changes in the Gibbs free energy of a reaction. In our context, it's particularly useful for determining the molar heat of solution using solubility data at different temperatures.

The mathematical representation of the Gibbs-Helmholtz equation used in this scenario is:
\[ \Delta H = -2.303R \left( \frac{T_1T_2}{T_2 - T_1} \right) . \log \left( \frac{K_{sp1}}{K_{sp2}} \right) \] Here, \( \Delta H \) is the molar heat of solution, \( R \) is the ideal gas constant \(8.314 J/mol\,K\), and \( T_1 \) and \( T_2 \) are the temperatures at which the solubility products \( K_{sp1} \) and \( K_{sp2} \) are measured. This equation allows us to calculate the change in enthalpy (or heat change) when a substance dissolves across different temperatures by using the log of the ratio of the solubility products at these temperatures.

In simpler terms, if you know how soluble a compound is at two different temperatures, the Gibbs-Helmholtz equation explains how much heat is absorbed or released when that compound dissolves, giving you the molar heat of solution.

Molar Heat of Solution

The molar heat of solution refers to the amount of heat energy that is either absorbed or released when one mole of a substance dissolves in a solvent. It's a critical factor in understanding how temperature changes can affect a solute's solubility.

In our example, we calculate the molar heat of solution for lead (II) iodide \(\text{PbI}_2\) using the solubility product data given at two different temperatures. By substituting these values into the Gibbs-Helmholtz equation, we determine the heat change required to dissolve one mole of \(\text{PbI}_2\).

The process is essentially about finding how much energy is involved when the compound transitions from a solid to an aqueous state. If the calculated heat of solution is positive, the process is endothermic, meaning it absorbs heat. Conversely, if it's negative, the process is exothermic, meaning it releases heat.

By understanding this concept, we can predict how substances will dissolve under varying thermal conditions, which is invaluable in fields like chemical engineering, environmental science, and pharmaceuticals.

Temperature Conversion to Kelvin

Temperature plays a vital role in chemical reactions, including dissolving processes, which makes it crucial to express it accurately. The Kelvin scale is the standard unit of temperature in scientific measurements because it begins at absolute zero—the theoretical point where particles possess minimal thermal motion.

When working with the Gibbs-Helmholtz equation, converting Celsius to Kelvin is necessary to ensure that calculations remain consistent with other standard scientific constants and equations. To perform this conversion, simply add 273.15 to the Celsius temperature.

• For example, 15°C converts to 15 + 273.15 = 288.15 K.
• Similarly, 25°C converts to 25 + 273.15 = 298.15 K.

Using Kelvin ensures that temperatures are always positive, which is required for many thermodynamic equations. This consistency is critical for accurate calculations, enabling scientists and engineers to predict and evaluate the behavior of substances across different thermal conditions.