Question

The three common silver halides ( \(\mathrm{AgCl}, \mathrm{AgBr},\) and \(\mathrm{AgI}\) ) are all sparingly soluble salts. Given the values for \(K_{\mathrm{sp}}\) for these salts below, calculate the concentration of silver ion, in \(\mathrm{mol} / \mathrm{L},\) in a saturated solution of each salt.

Short answer

The concentrations of silver ions in the saturated solutions of the silver halides are: - AgCl: \(1.33 \times 10^{-5} \mathrm{mol}/\mathrm{L}\) - AgBr: \(7.07 \times 10^{-7} \mathrm{mol}/\mathrm{L}\) - AgI: \(2.92 \times 10^{-9} \mathrm{mol}/\mathrm{L}\)

Step-by-step solution

Write the balanced chemical equation for the dissolution of each salt.

For each silver halide, the balanced chemical equation is: 1. Silver chloride (AgCl): \[AgCl \rightleftharpoons Ag^{+} + Cl^{-}\] 2. Silver bromide (AgBr): \[AgBr \rightleftharpoons Ag^{+} + Br^{-}\] 3. Silver iodide (AgI): \[AgI \rightleftharpoons Ag^{+} + I^{-}\]

Write the Ksp expressions for each reaction

Next, we write the solubility product constant (Ksp) expressions for each silver halide: 1. Silver chloride (AgCl): \[K_{sp} (AgCl) = [Ag^{+}][Cl^{-}]\] 2. Silver bromide (AgBr): \[K_{sp} (AgBr) = [Ag^{+}][Br^{-}]\] 3. Silver iodide (AgI): \[K_{sp} (AgI) = [Ag^{+}][I^{-}]\]

Determine the concentrations of silver ions in saturated solutions

Since the stoichiometry of the silver halides is equal, the concentration of the silver ions is equal to that of the halide ions. Therefore, we can use the following expression to find the concentrations of silver ions: \[K_{sp} = [Ag^{+}][X^{-}] = [Ag^{+}]^2\] \[ [Ag^{+}] = \sqrt{K_{sp}} \] Given the Ksp values for the different silver halides, we can now calculate the concentration of silver ions for each salt: 1. Silver chloride (AgCl): \[ [Ag^{+}] = \sqrt{K_{sp} (AgCl)} = \sqrt{1.77 \times 10^{-10} \mathrm{mol}^2 / \mathrm{L}^2} = 1.33 \times 10^{-5} \mathrm{mol} / \mathrm{L} \] 2. Silver bromide (AgBr): \[ [Ag^{+}] = \sqrt{K_{sp} (AgBr)} = \sqrt{5.0 \times 10^{-13} \mathrm{mol}^2 / \mathrm{L}^2} = 7.07 \times 10^{-7} \mathrm{mol} / \mathrm{L} \] 3. Silver iodide (AgI): \[ [Ag^{+}] = \sqrt{K_{sp} (AgI)} = \sqrt{8.51 \times 10^{-17} \mathrm{mol}^2 / \mathrm{L}^2} = 2.92 \times 10^{-9} \mathrm{mol} / \mathrm{L} \] Thus, the concentrations of silver ions in the saturated solutions are: - AgCl: \(1.33 \times 10^{-5} \mathrm{mol}/\mathrm{L}\) - AgBr: \(7.07 \times 10^{-7} \mathrm{mol}/\mathrm{L}\) - AgI: \(2.92 \times 10^{-9} \mathrm{mol}/\mathrm{L}\)

Key concepts

Saturated Solution

A saturated solution is a mixture where a solute is dissolved in a solvent to the maximum extent possible at a given temperature and pressure. When a solution is saturated, it has reached equilibrium; no more solute can dissolve unless conditions change. For example, when adding sugar to tea, only a certain amount will dissolve. After that, any additional sugar will remain undissolved at the bottom of the cup. For the silver halides like \( \text{AgCl} \), a saturated solution means the dissolved ions \([\text{Ag}^+]\) and \([\text{Cl}^-]\) are in balance with the undissolved solid, establishing a constant concentration of ions in the solution.

Chemical Equilibrium

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentration of reactants and products over time. In the dissolution of silver halides, equilibrium is reached when silver ions \( \text{Ag}^+ \) and chloride ions \( \text{Cl}^- \) have a stable concentration.
This equilibrium can be represented by the equation: \[AgCl_{(s)} \rightleftharpoons Ag^+_{(aq)} + Cl^-_{(aq)}\] At this point, the concentration of ions remains constant, and the system can be described by the solubility product constant \( K_{sp} \). This value is unique for each compound and is used to quantify the solubility of the compound in water.

Silver Halides

Silver halides, including \( \text{AgCl} \), \( \text{AgBr} \), and \( \text{AgI} \), are compounds formed by silver and halogen elements. These halides are known for their low solubility in water, making them sparingly soluble salts. In solutions, they dissociate into their respective ions. For example, \( \text{AgCl} \) dissociates into \( \text{Ag}^+ \) and \( \text{Cl}^- \) ions.
These properties are utilized in various applications such as photography, where silver halides react to light. Understanding their solubility is crucial in predicting their behavior in different chemical reactions and solutions.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It is fundamental in determining how substances interact and transform. When dealing with silver halides, stoichiometry helps in deriving the solubility product expression \( K_{sp} \).
For \( \text{AgCl} \), the reaction can be expressed as: \[AgCl_{(s)} \rightleftharpoons Ag^+_{(aq)} + Cl^-_{(aq)}\]The stoichiometry shows that the dissolution of one mole of \( \text{AgCl} \) produces one mole of \( \text{Ag}^+ \) and one mole of \( \text{Cl}^- \), meaning their concentrations are equal. Hence, the \( K_{sp} \) is calculated as the product of the concentrations of these ions, facilitating the determination of solubility for each silver halide.