Question

To determine the \(K_{\mathrm{sp}}\) value of \(\mathrm{Hg}_{2} \mathrm{I}_{2},\) a chemist obtained a solid sample of \(\mathrm{Hg}_{2} \mathrm{I}_{2}\) in which some of the iodine is present as radioactive \(^{131}\) I. The count rate of the \(\mathrm{Hg}_{2} \mathrm{I}_{2}\) sample is \(5.0 \times 10^{11}\) counts per minute per mole of I. An excess amount of \(\mathrm{Hg}_{2} \mathrm{I}_{2}(s)\) is placed into some water, and the solid is allowed to come to equilibrium with its respective ions. A 150.0 -mL sample of the saturated solution is withdrawn and the radioactivity measured at 33 counts per minute. From this information, calculate the \(K_{\mathrm{sp}}\) value for \(\mathrm{Hg}_{2} \mathrm{I}_{2}\) $$\mathrm{Hg}_{22}(s) \rightleftharpoons \mathrm{Hg}_{2}^{2+}(a q)+2 \mathrm{I}^{-}(a q) \quad K_{\mathrm{sp}}=\left[\mathrm{Hg}_{2}^{2+}\right]\left[\mathrm{I}^{-}\right]^{2}$$

Short answer

The solubility product constant, \(K_{sp}\), for \(\mathrm{Hg}_{2}\mathrm{I}_{2}\) is approximately \(4.27 \times 10^{-29}\).

Step-by-step solution

Determine the concentration of iodide ions in the sample.

We can calculate the concentration of \(\mathrm{I}^{-}\) in the 150 mL sample based on the count rate of the radioactivity in the sample. The ratio of the count rate in the solution (33 counts per minute) to the count rate in the solid \(\mathrm{Hg}_2\mathrm{I}_{2}\) sample (5.0e11 counts per minute per mole of I) can help us find the concentration of iodide ions in the solution. \[ \frac{33\,\text{counts/min}}{5.0 \times 10^{11}\,\text{counts/min/mol}} = \frac{[\mathrm{I}^-]}{1\,\text{mol}} \] Now, we can calculate the concentration of \(\mathrm{I}^-\): \[ [\mathrm{I}^{-}] = 33 \div 5.0 \times 10^{11} = 6.6 \times 10^{-11}\,\text{mol} \] Since the volume of the sample is 150 mL, we can convert the amount (in moles) to concentration (in mol/L): \[ [\mathrm{I}^{-}] = \frac{6.6 \times 10^{-11}\,\text{mol}}{0.150\,\text{L}} = 4.4 \times 10^{-10}\,\text{M} \]

Determine the concentration of mercuric ions in the sample.

From the balanced chemical equation, we can see that for each mole of \(\mathrm{Hg}_{2}^{2+}\) ions, there are 2 moles of \(\mathrm{I}^{-}\) ions formed in the solution. Therefore, we can calculate the concentration of \(\mathrm{Hg}_{2}^{2+}\) using the following relationship: \[ [\mathrm{Hg}_{2}^{2+}] = \frac{1}{2}[\mathrm{I}^{-}] \] Now, we can calculate the concentration of \(\mathrm{Hg}_{2}^{2+}\): \[ [\mathrm{Hg}_{2}^{2+}] = \frac{1}{2}(4.4 \times 10^{-10}\,\text{M}) = 2.2 \times 10^{-10}\,\text{M} \]

Calculate the \(K_{sp}\) value for \(\mathrm{Hg}_{2}\mathrm{I}_{2}\).

Now that we have the concentrations of \(\mathrm{Hg}_{2}^{2+}\) and \(\mathrm{I}^-\), we can calculate the \(K_{sp}\) using the following expression: \[ K_{sp} = [\mathrm{Hg}_{2}^{2+}][\mathrm{I}^-]^2 \] Substitute the values of the concentrations into the expression: \[ K_{sp} = (2.2 \times 10^{-10})(4.4 \times 10^{-10})^2 \] Now, we can calculate the \(K_{sp}\) value: \[ K_{sp} = 4.27 \times 10^{-29} \] So, the solubility product constant, \(K_{sp}\), for \(\mathrm{Hg}_{2}\mathrm{I}_{2}\) is approximately \(4.27 \times 10^{-29}\).

Key concepts

Radioactive Iodine

Radioactive iodine, specifically iodine-131 (denoted as \(^{131}\text{I}\)), is an isotope of iodine that emits radiation. It is utilized in various fields such as medicine and scientific research due to its radioactive properties. In the context of a chemistry calculation involving solubility, \(^{131}\text{I}\) serves as a tracer. This property makes it an invaluable tool for determining the concentration of iodine ions in a solution.

When a chemist analyzes a substance with radioactive iodine, they measure the radioactive count rate, which indicates the level of radioactivity associated with a given quantity of iodine. By comparing this measurement in a solid sample and a saturated solution, they can deduce important concentration values of the ions in the solution, which is essential for calculating the solubility product constant (\(K_{sp}\)). Fields such as radiochemistry and analytical chemistry often employ these techniques to monitor chemical behaviors and reactions.

Equilibrium Concentration

Equilibrium concentration refers to the concentrations of reactants and products in a chemical system that has reached a state of balance. In other words, at equilibrium, the rate of the forward chemical reaction equals the rate of the reverse reaction. For a saturated solution like \(\text{Hg}_2\text{I}_2\), the system is at equilibrium between dissolved ions and the undissolved solid.

In this particular exercise, finding the equilibrium concentration of iodide ions (\(\text{I}^-\)) in solution involves using the radioactive count rate to indirectly determine ion concentration. This is possible because the rate at which radiation is counted correlates to the number of radioactive iodine atoms present. By using this relationship, the chemist calculates the concentration of dissolved ions at equilibrium. This equilibrium concentration is crucial for further calculations, especially when finding the solubility product constant, \(K_{sp}\). Understanding these concentrations helps in evaluating how substances dissolve and in predicting the extent to which they form ions in solution.

Chemistry Calculation

Chemistry calculations are at the heart of determining various properties of substances in a chemical context. This exercise involves several calculations that require understanding solubility and equilibrium concepts. Initially, the concentration of iodide ions is deduced from radioactive measurements.

The calculation entails using the determined iodide ion concentration to find the concentration of mercury ions (\(\text{Hg}_2^{2+}\)) using stoichiometric relationships. According to the balanced reaction associated with \(\text{Hg}_2\text{I}_2\), one mole of \(\text{Hg}_2^{2+}\) corresponds to two moles of \(\text{I}^-\) ions. By employing this ratio, chemists can determine the concentration of other ions in the solution.

Ultimately, these concentrations enable the calculation of the solubility product constant, \(K_{sp}\), which represents the extent to which \(\text{Hg}_2\text{I}_2\) dissolves in water. Such calculations are vital as they provide insights into the solubility behavior of substances, helping predict their reactions and interactions in aqueous solutions.

Saturated Solution

A saturated solution is one in which no more solute can dissolve at a given temperature and pressure. It represents a state where the solution has reached its maximum concentration of dissolved ions, and additional solute will remain undissolved.

In the context of this exercise, the saturated solution of \(\text{Hg}_2\text{I}_2\) contains a dynamic equilibrium between the solid \(\text{Hg}_2\text{I}_2\) and its ions in the solution, \(\text{Hg}_2^{2+}\) and \(\text{I}^-\).

This state of saturation is pivotal when determining \(K_{sp}\), as the concentration of dissolved ions in a saturated solution directly relates to the solubility product constant. The presence of a saturated solution ensures the precision of equilibrium concentrations, which are required to perform accurate solubility calculations and to understand the dissolution behaviors of chemicals.