Question

A concentration of 10–100 parts per billion (by mass) of Ag+ is an effective disinfectant in swimming pools. However, if the concentration exceeds this range, the Ag+ can cause adverse health effects. One way to maintain an appropriate concentration of Ag+ is to add a slightly soluble salt to the pool. Using \(K_{s p}\) values from Appendix \(\mathrm{D},\) calculate the the equilibrium concentration of Ag+ in parts per billion that would exist in equilibrium with (a) \(\mathrm{AgCl},(\mathbf{b}) \mathrm{AgBr},(\mathbf{c}) \mathrm{AgI}\)

Short answer

The equilibrium concentrations of Ag+ in parts per billion (ppb) that exist in equilibrium with the slightly soluble salts, AgCl, AgBr, and AgI can be calculated using their respective Ksp values and the following steps: 1. Write the Ksp expression for each salt and determine the concentration of Ag+: [Ag+] = sqrt(Ksp) 2. Convert the molar concentrations of Ag+ to ppb using the formula: ppb(Ag+) = ([Ag+(salt)] × 1 mol Ag × 107.87 g/mol Ag × 10^9 μg/g) / (1 L × 1000 g/L) After performing these calculations, you will find the equilibrium concentration of Ag+ in ppb that exists in equilibrium with each salt (AgCl, AgBr, and AgI).

Step-by-step solution

Find Ksp values for the salts

Consult Appendix D in your textbook or search online sources to find the Ksp values for AgCl, AgBr, and AgI.

Write the balanced chemical equations and expressions for Ksp

For each of the salts, we have the following balanced chemical equations and expressions for Ksp: (a) AgCl(s) -> Ag+(aq) + Cl-(aq) Ksp = [Ag+][Cl-] (b) AgBr(s) -> Ag+(aq) + Br-(aq) Ksp = [Ag+][Br-] (c) AgI(s) -> Ag+(aq) + I-(aq) Ksp = [Ag+][I-]

Determine the equilibrium concentrations of Ag+

Since these salts are slightly soluble, we can assume that the concentrations of Ag+ and the corresponding halide ions are approximately equal at equilibrium, i.e., [Ag+] ≈ [X-], where X- represents Cl-, Br-, or I-. Thus, we can rewrite the Ksp expressions as: (a) Ksp(AgCl) = [Ag+]^2 (b) Ksp(AgBr) = [Ag+]^2 (c) Ksp(AgI) = [Ag+]^2 Now, we can calculate the equilibrium concentrations of Ag+ for each salt: (a) [Ag+(AgCl)] = sqrt(Ksp(AgCl)) (b) [Ag+(AgBr)] = sqrt(Ksp(AgBr)) (c) [Ag+(AgI)] = sqrt(Ksp(AgI))

Convert molar concentrations to parts per billion (ppb)

To convert the equilibrium concentrations of Ag+ from molarity (M) to parts per billion (ppb), use the following formula: ppb = (molarity × 1 mol Ag × 107.87 g/mol Ag × 10^9 μg/g) / (1 L × 1000 g/L) Calculate the ppb values for Ag+ from each salt: (a) For AgCl: ppb(Ag+) = ([Ag+(AgCl)] × 1 mol Ag × 107.87 g/mol Ag × 10^9 μg/g) / (1 L × 1000 g/L) (b) For AgBr: ppb(Ag+) = ([Ag+(AgBr)] × 1 mol Ag × 107.87 g/mol Ag × 10^9 μg/g) / (1 L × 1000 g/L) (c) For AgI: ppb(Ag+) = ([Ag+(AgI)] × 1 mol Ag × 107.87 g/mol Ag × 10^9 μg/g) / (1 L × 1000 g/L) After calculating the ppb values, you will find the equilibrium concentration of Ag+ in parts per billion that exist in equilibrium with AgCl, AgBr, and AgI.

Key concepts

Solubility Product Constant (Ksp)

The solubility product constant, or Ksp, is a special type of equilibrium constant that measures the solubility of a sparingly soluble ionic compound in a solution. It is expressed as the product of the concentrations of the ions produced when the compound dissolves, each raised to the power of its coefficient in the balanced equation.

For example, in the case of silver chloride (AgCl), which dissociates into silver ions (Ag+) and chloride ions (Cl-) in water, the Ksp expression is given by Ksp = [Ag+][Cl-]. Ksp is a fixed value at a given temperature for each ionic compound and indicates the extent to which the compound can dissolve. In other words, a higher Ksp value means more of the compound can dissolve in the solution.

Understanding Ksp is crucial for predicting whether a precipitate will form in a reaction and for calculating the equilibrium concentrations of ions in solution, a concept which we'll examine next.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of reactants and products at equilibrium in a chemical reaction. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time.

In the context of solubility equilibrium for ionic compounds like silver halides, equilibrium concentrations are determined by the Ksp of the compound. By setting up an equation using the Ksp value and the stoichiometry of the dissolution process, we can calculate the equilibrium concentrations of the ions in solution. For the silver halides in the given exercise, assuming equal concentrations for the silver ions and the corresponding halide ions simplifies the Ksp expression to Ksp = [Ag+]^2. This allows us to solve for the silver ion concentration by taking the square root of the Ksp. These calculated concentrations play a vital role in applications such as maintaining the desired level of disinfection in swimming pools while preventing adverse health effects from excessive silver ion presence.

Silver Halides

Silver halides are a group of chemical compounds formed between silver and halogen elements such as chlorine, bromine, and iodine. Notable for their roles in photographic films and antibacterial agents, silver halides such as AgCl, AgBr, and AgI are known for their low solubility in water. This low solubility is quantified by their Ksp values.

Silver halides can be used to maintain the concentration of silver ions in solutions, such as in swimming pools for disinfection purposes. It's crucial to control the concentration of Ag+ to avoid toxicity. The solubility equilibrium comes into play here, where the silver halides act as a source of silver ions that dissolve until the equilibrium is established as dictated by the particular Ksp of the halide.

For instance, AgCl has a different Ksp than AgBr and AgI, affecting their respective contributions to the equilibrium concentration of Ag+ ions in the water. Understanding the properties of these compounds, including their Ksp values, allows for precise control of the silver ion concentration for safe and effective use in various applications.