Question

A saturated solution of \(\mathrm{AgI}\) has \(\left[\mathrm{Ag}^{+}\right]=9.2 \times 10^{-9} \mathrm{M}\) and \(\left[\mathrm{I}^{-}\right]=9.2 \times 10^{-9} \mathrm{M}\). What is the value of \(K_{\mathrm{sp}}\) for \(\mathrm{AgI}\) ?

Short answer

The value of Ksp for AgI is approximately 84.64 × 10−18.

Step-by-step solution

Understand the Concept of Solubility Product Constant (Ksp)

The solubility product constant, denoted as Ksp, is the product of the concentrations of the ions in a saturated solution, raised to the power of their respective stoichiometric coefficients in the balanced equation. For the dissolution of silver iodide (AgI), the equation can be written as: AgI(s) ⇌ Ag+(aq) + I−(aq)The Ksp would be the product of the molar concentrations of Ag+ and I−.

Calculate the Ksp

The Ksp for AgI can be calculated using the concentrations of its ions in the saturated solution: Ksp = [Ag+][I−]Given that the concentration of both Ag+ and I− ions in the solution is the same (9.2 × 10−9 M), substitute the values directly into the equation to compute the Ksp.

Find Ksp Value

Plug in the values into the Ksp expression obtained from the balanced dissolution equation: Ksp = (9.2 × 10−9)(9.2 × 10−9) = 84.64 × 10−18Finalize the Ksp value by performing the multiplication.

Key concepts

Saturated Solution

In chemistry, a saturated solution is a solution in which the maximum amount of solvent has been dissolved. At this point, any additional solute will not dissolve and excess solute will remain undissolved in the solid state. This balance point, where the rate of dissolution equals the rate of precipitation, is essential in understanding solubility product.

When we talk about a saturated solution of AgI (silver iodide), we mean that the solution contains the maximum concentration of Ag⁺ and I^- ions that can exist in equilibrium with solid AgI. These concentrations represent a delicate balance, and if disturbed, the system will shift to reestablish equilibrium according to Le Chatelier's principle, a key concept in chemical equilibrium. The importance of knowing the saturated levels of ions lies in predicting product formation and its solubility in various conditions.

Ionic Product

The ionic product, often represented by Q, is the product of the concentrations of the ions involved in a dissolved ionic compound, each raised to the power of their stoichiometric coefficient. It is important to note that the ionic product can be calculated for any solution, whether saturated or not.

In the case of a saturated solution, if the ionic product is equal to the solubility product constant, K_sp, the solution is at equilibrium and no more solute will dissolve. If the ionic product is less than K_sp, the solution can still dissolve more solute, whereas if it is greater than K_sp, the solution is supersaturated and precipitation will likely occur to restore equilibrium. This concept is central to understanding the conditions under which a precipitate will form in solution.

Chemical Equilibrium

The state of chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, leading to no net change in the concentration of reactants and products over time. This does not mean that the reactions have stopped, but that they are occurring at the same rate in both directions.

In the context of solubility, equilibrium is reached in a saturated solution where the dissolved ions and the undissolved solute coexist. The K_sp value we determine is essentially a quotient representing this equilibrium state for a specific temperature and pressure. Understanding equilibrium concepts helps us predict the behavior of substances in various chemical processes, including solubility equilibria and more complex reactive systems.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It is derived from the balanced chemical equation and provides the proportions in which chemicals combine or react. For solubility product calculations, stoichiometry dictates the powers to which the ion concentrations are raised when calculating the K_sp.

In the dissolution of AgI, the stoichiometry is 1:1 for Ag⁺ and I^- as indicated by the balanced equation, AgI(s) ⇌ Ag⁺(aq) + I^-(aq). Therefore, when we calculate the solubility product constant, we take the concentration of Ag⁺ and I^-, each raised to the power of their stoichiometric coefficients, which in this case is 1. Stoichiometry is the backbone of predictive chemistry allowing us to understand and calculate reactant-use, product yield, and other important aspects of chemical reactions.