Question

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

Short answer

The equilibrium concentrations of Ag⁺ in parts per billion (ppb) when in contact with each slightly soluble salt are: (a) AgCl: 13,400 ppb, (b) AgBr: 707 ppb, and (c) AgI: 3.87 ppb.

Step-by-step solution

Write the solubility equilibria for the salts

We first need to write the solubility equilibria for the slightly soluble salts AgCl, AgBr, and AgI. Each equation consists of the dissociation of the salt into its constituent ions: \(AgCl_{(s)} ⇌ Ag^+_{(aq)} + Cl^-_{(aq)}\) \(AgBr_{(s)} ⇌ Ag^+_{(aq)} + Br^-_{(aq)}\) \(AgI_{(s)} ⇌ Ag^+_{(aq)} + I^-_{(aq)}\)

Write the solubility product expressions

Next, we write the solubility product expressions based on the solubility equilibria. The solubility product (Ksp) is the equilibrium constant for a solid substance dissolving in an aqueous solution. For each salt, Ksp is equal to the product of ion concentrations raised to their stoichiometric coefficients: \(K_{sp}(AgCl) = [Ag^+][Cl^-]\) \(K_{sp}(AgBr) = [Ag^+][Br^-]\) \(K_{sp}(AgI) = [Ag^+][I^-]\)

Use Ksp values to find the concentration of Ag⁺

Using the Ksp values for AgCl, AgBr, and AgI (found in Appendix D), we can determine the concentration of Ag⁺ in equilibrium with these salts. Let x be equal to the concentration of Ag⁺ in the aqueous solution. Since each salt is a 1:1 compound, the concentration of the corresponding anion (Cl⁻, Br⁻, or I⁻) must also be equal to x. Therefore, we can write the Ksp expressions as: \(K_{sp}(AgCl) = x^2\) \(K_{sp}(AgBr) = x^2\) \(K_{sp}(AgI) = x^2\) We can now solve for x (concentration of Ag⁺) for each salt using their respective Ksp values: (a) \(AgCl: K_{sp} = 1.8 × 10^{-10} = x^2\) (b) \(AgBr: K_{sp} = 5.0 × 10^{-13} = x^2\) (c) \(AgI: K_{sp} = 1.5 × 10^{-16} = x^2\)

Calculate the concentration of Ag⁺ in ppb

After solving for x in each case, we can convert the Ag⁺ concentration from moles per liter (M) to parts per billion (ppb) by using the following conversion factor: \(1\:M = 10^9\:ppb\) (a) For AgCl: \(x = \sqrt{1.8 × 10^{-10}} = 1.34 × 10^{-5}\: M\) \(1.34 × 10^{-5}\: M × 10^9\: ppb = 13,400\: ppb\) (b) For AgBr: \(x = \sqrt{5.0 × 10^{-13}} = 7.07 × 10^{-7}\: M\) \(7.07 × 10^{-7}\: M × 10^9\: ppb = 707\: ppb\) (c) For AgI: \(x = \sqrt{1.5 × 10^{-16}} = 3.87 × 10^{-9}\: M\) \(3.87 × 10^{-9}\: M × 10^9\: ppb = 3.87\: ppb\)

Present the final results

Finally, we can present the equilibrium concentrations of Ag⁺ in parts per billion when in contact with each salt: (a) AgCl: 13,400 ppb (b) AgBr: 707 ppb (c) AgI: 3.87 ppb

Key concepts

Solubility Product (Ksp)

The solubility product constant, known as \( K_{sp} \), is a vital concept in understanding solubility equilibria. It represents the equilibrium constant for a sparingly soluble ionic compound dissociating into its ions in a solution. In simple terms, \( K_{sp} \) quantifies how much of a salt can dissolve in a solution before reaching its saturation point under equilibrium conditions.
This constant is specific for each compound and changes at different temperatures.

• The lower the \( K_{sp} \), the less soluble the compound is.
• Conversely, a higher \( K_{sp} \) indicates greater solubility.
  

For instance, in the context of our exercise, we deal with three silver compounds: AgCl, AgBr, and AgI. Their respective \( K_{sp} \) values tell us how much silver ion \( \mathrm{Ag}^{+} \) solubility to expect when they are in equilibrium with their solid form in aqueous solutions. Calculating \( K_{sp} \) enables us to determine the concentrations of ions in equilibrium, which aids in assessing how a compound can affect environments like swimming pools when utilized as disinfectants.

Equilibrium Concentration

In chemical solutions, equilibrium concentration is when the chemical reaction and its reverse occur at the same rate.
This balance results in constant concentrations of reactants and products over time. When considering solubility equilibria, the equilibrium concentration of ions is crucial for predicting how a substance behaves when dissolved.
Generally, for a compound \( MX \), if \([M^+]\) and \([X^-]\) are the concentrations of ions, then at equilibrium, \( K_{sp} = [M^+][X^-] \).
This is typically seen in simple 1:1 ionic compounds like the silver salts (AgCl, AgBr, AgI) discussed in our scenario.

• Each pair of ions becomes equal at equilibrium, leading to equations like \( K_{sp} = x^2 \), where \( x \) represents the ion concentration in moles per liter.
• Through solving these equations, we transition from knowing the \( K_{sp} \) to finding the concentration \( x \) of \( \mathrm{Ag}^{+} \) in solution.
• Finally, this concentration can be converted to practical units like parts per billion (ppb) for real-world applications.
  

This approach helps maintain safe levels of ions in applications, ensuring that disinfectants don’t reach harmful concentrations.

Silver Compounds

Silver compounds are well-regarded for their antimicrobial properties, making them valuable in applications like water treatment and disinfection.
Despite their effectiveness, maintaining the correct concentration in solutions like swimming pools is essential to avoid toxic effects.
Let's take silver halides — AgCl, AgBr, and AgI — as examples.

• AgCl has more solubility compared to AgBr and AgI due to its higher \( K_{sp} \), leading to a higher equilibrium concentration of \( \mathrm{Ag}^{+} \).
• AgBr offers moderate solubility, featuring a unique balance in its applications.
• AgI, with its low \( K_{sp} \), is least soluble, making it the safest in terms of reduced \( \mathrm{Ag}^{+} \) content in solution.
  

It's critical to monitor concentrations to prevent the silver ions from exceeding safety thresholds in environments where these compounds are employed.
Such control ensures that their use as antimicrobial agents remains beneficial without posing health risks. By understanding these properties and calculations, especially around \( K_{sp} \), proper use of silver compounds is effectively managed.