(a) How would you prepare a saturated aqueous solution of copper(I) iodide at \(25^{\circ} \mathrm{C}\) ? (b) What is the mass in milligrams of CuI in \(400.0 \mathrm{~mL}\) of the saturated solution? (Hint: Begin with a \(K_{\mathrm{sp}}\) value from Table 14.1.) (c) Suppose you add some CuI* to this saturated solution, where \(\mathrm{I}^{*}\) is a radioactive form of iodide ion. A student says, "Because the solution is already saturated, the added CuI* won't dissolve and there's no danger of getting any radioactive iodide ion in solution." What is wrong with his thinking?

Understanding the solubility product constant, commonly known as the Ksp, is crucial in chemistry, particularly when dealing with the dissolution and precipitation of ionic compounds in solutions. It is a numerical value that represents the extent to which a compound can dissolve in water; specifically, Ksp is the product of the concentrations of the ions in a saturated solution, each raised to the power of their respective stoichiometric coefficients in the dissolution equation.

For copper(I) iodide (CuI), the dissolution equation in water at equilibrium can be shown as: \[CuI \rightleftharpoons Cu^{+}(aq) + I^{-}(aq)\]. Here, the Ksp expression would be \[Ksp = [Cu^+][I^-]\]. Since the stoichiometry of Cu+ and I- ions is 1:1 in the dissolution reaction, the Ksp is simply the square of the molar solubility (x) of CuI.

It's important to note that the Ksp value is constant for a given compound at a particular temperature, and it's not influenced by the presence of other ions in the solution. This means that no matter how much CuI is initially added to the water, once the solution reaches saturation, the product of the ion concentrations will always equal the Ksp value. Understanding this helps in predicting whether a precipitate will form when two solutions are mixed.

Molar solubility, often represented by the symbol 'x' in chemistry problems, is a term that defines how many moles of a solute can be dissolved in a liter of solution before the solution becomes saturated. It is a measure of a compound's solubility in a specific solvent at a given temperature.

To calculate the molar solubility, we often use the Ksp value and set up an expression based on the equilibrium reaction of the solute in water. In the case of CuI, the equilibrium equation shows that each mole of dissolved CuI yields one mole of Cu+ ions and one mole of I- ions.

Finding Molar Solubility from Ksp

To calculate molar solubility from Ksp:

1. Set up the Ksp expression based on the dissolution equation.
2. Substitute molar solubility into the Ksp expression.
3. Solve for molar solubility.

This value not only tells us the maximum amount of the compound that can dissolve but also can be used to calculate the maximum concentration of ions that will be in solution and, consequently, to determine the mass of the dissolved solute, as seen in the step-by-step solution.

Radioactive isotopes are atoms that have an unstable nucleus and thus emit radiation as they decay to more stable forms. When working with solutions, it's essential to understand how radioactive isotopes can affect the equilibrium and safety of the system.

In the exercise, the introduction of a radioactive isotope of iodide, denoted as I*, challenges the notion that a saturated solution cannot accept more of a solute. However, radioisotopes have unique properties due to their radioactive nature, and their introduction into a solution can lead to the displacement of stable isotopes.

Radioactive Isotopes Displacing Stable Ones

The misconception that the solution cannot hold more iodide ions because it is saturated fails to account for isotopic exchange. Radioactive iodide ions can replace stable iodide ions in the solution, maintaining the equilibrium concentrations of the ions while introducing radioactivity into the solution. This isotopic exchange is a critical concept in environmental chemistry, nuclear medicine, and other fields where the safety and behavior of radioactive substances in solutions must be carefully managed.