Question

The three common silver halides (AgCl, AgBr, and 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. $$ \text {Silver Halide } \quad \quad\quad\quad \text {\(K_{\mathrm{sp}}\)} $$ $$ \text {\(\mathrm{AgCl}\) } \quad \quad \text {\(1.8 \times 10^{-10}\)} $$ $$ \text {\(\mathrm{AgBr}\) } \quad \quad \text {\(5.0 \times 10^{-13}\)} $$ $$ \text {\(\mathrm{Agl}\) } \quad \quad \text {\(8.3 \times 10^{-17}\)} $$

Short answer

The concentration of silver ions (Ag+) in saturated solutions of AgCl, AgBr, and AgI are \( 1.34 \times 10^{-5} \) M, \( 7.07 \times 10^{-7} \) M, and \( 2.88 \times 10^{-9} \) M, respectively.

Step-by-step solution

Write the balanced chemical equation for each silver halide dissolving in water

For each silver halide, the balanced chemical equation for dissolution in water can be written as follows: \( \text{AgCl } (s) \rightleftharpoons \text{Ag}^+ (aq) + \text{Cl}^- (aq) \) \( \text{AgBr } (s) \rightleftharpoons \text{Ag}^+ (aq) + \text{Br}^- (aq) \) \( \text{AgI } (s) \rightleftharpoons \text{Ag}^+ (aq) + \text{I}^- (aq) \)

Write the solubility product constant expression for each silver halide

The solubility product constant expression for a salt can be written based on its chemical equation. Since the stoichiometric coefficient for each ion is one, the following expressions can be used for each silver halide: \( K_{\mathrm{sp}} (\mathrm{AgCl}) = [\mathrm{Ag}^+][\mathrm{Cl}^-]\) \( K_{\mathrm{sp}} (\mathrm{AgBr}) = [\mathrm{Ag}^+][\mathrm{Br}^-]\) \( K_{\mathrm{sp}} (\mathrm{AgI}) = [\mathrm{Ag}^+][\mathrm{I}^-]\)

Solve for the silver ion concentration in each saturated silver halide solution

In a saturated solution of each salt, the concentrations of the corresponding anions (Cl-, Br-, and I-) are equal to the concentration of silver ions (Ag+). Therefore, we can write the solubility product constant expression for each silver halide using variable x, which represents the concentration of silver ions. For AgCl: \( K_{\mathrm{sp}} (\mathrm{AgCl}) = [x][x] = x^2 = 1.8 \times 10^{-10} \) For AgBr: \( K_{\mathrm{sp}} (\mathrm{AgBr}) = [x][x] = x^2 = 5.0 \times 10^{-13} \) For AgI: \( K_{\mathrm{sp}} (\mathrm{AgI}) = [x][x] = x^2 = 8.3 \times 10^{-17} \) Now, we'll solve for x in each equation to find the concentration of silver ions in each saturated solution.

Calculate the silver ion concentration in a saturated solution of AgCl

Solve for x in the equation for AgCl: \( x^2 = 1.8 \times 10^{-10} \) \( x = \sqrt{1.8 \times 10^{-10}} = 1.34 \times 10^{-5} \mathrm{M} \) Therefore, the concentration of Ag+ in a saturated solution of AgCl is \( 1.34 \times 10^{-5} \) M.

Calculate the silver ion concentration in a saturated solution of AgBr

Solve for x in the equation for AgBr: \( x^2 = 5.0 \times 10^{-13} \) \( x = \sqrt{5.0 \times 10^{-13}} = 7.07 \times 10^{-7} \mathrm{M} \) Therefore, the concentration of Ag+ in a saturated solution of AgBr is \( 7.07 \times 10^{-7} \) M.

Calculate the silver ion concentration in a saturated solution of AgI

Solve for x in the equation for AgI: \( x^2 = 8.3 \times 10^{-17} \) \( x = \sqrt{8.3 \times 10^{-17}} = 2.88 \times 10^{-9} \mathrm{M} \) Therefore, the concentration of Ag+ in a saturated solution of AgI is \( 2.88 \times 10^{-9} \) M. In conclusion, the concentration of silver ions (Ag+) in saturated solutions of AgCl, AgBr, and AgI are \( 1.34 \times 10^{-5} \) M, \( 7.07 \times 10^{-7} \) M, and \( 2.88 \times 10^{-9} \) M, respectively.

Key concepts

Silver Halides

The term 'silver halides' refers to compounds formed by the combination of silver (Ag) with halogen elements such as chlorine (Cl), bromine (Br), and iodine (I). These compounds, specifically silver chloride (AgCl), silver bromide (AgBr), and silver iodide (AgI), are known to be sparingly soluble in water. This means that only a small amount of these salts can dissolve in water to form a solution.

Silver halides have practical applications in photography and are often used in photographic film and paper. This is due to their light-sensitive properties. When exposed to light, they undergo a photochemical reaction that is crucial in image capture. However, from a chemistry perspective, understanding their solubility is important when determining their behavior in various solutions.

Even though they are sparingly soluble, the concentrations of ions produced in saturated solutions are significant because they determine the Ksp values for each salt.

Saturated Solution

A saturated solution is a chemical solution in which the maximum amount of solute has been dissolved. In the context of silver halides, it refers to a solution that contains as many dissolved ions as possible at a given temperature and pressure. Once the solution reaches this point, it cannot dissolve any more solute; instead, excess solute will start settling as a solid.

For silver halides like AgCl, AgBr, and AgI, in water, saturation occurs when the reversible process of dissolution reaches equilibrium. This means that the rate at which silver halide dissolves equals the rate at which it precipitates back out. Equilibrium here is represented by the reaction:

• For AgCl: \( ext{AgCl (s)} \rightleftharpoons \text{Ag}^+ (aq) + \text{Cl}^- (aq) \ \)
• Similarly for AgBr and AgI.

The dissolved ions are present in their highest stable concentrations, known as the solubility equilibrium point, and this is what defines the solubility product constant, or Ksp.

Silver Ion Concentration

In the given exercise, calculating the silver ion concentration for a saturated solution involves using the solubility product constant (Ksp). The Ksp is crucial because it helps predict how much a compound will dissolve. When a salt like AgCl dissolves in water, it dissociates into its constituting ions. The concentration of these dissociated ions at equilibrium determines its Ksp.

For example, when silver chloride (\( ext{AgCl}\)dissolves, the following occurs:

• It forms silver ions (\([ ext{Ag}^+]\)) and chloride ions (\([ ext{Cl}^-]\)).

In a saturated solution, these concentrations can be represented by the variable \(x\), leading to the expression:\[K_{sp} = [ ext{Ag}^+][ ext{Cl}^-] = x^2\]Solving for \(x\) gives you the concentration of silver ions in the solution. By using these principles, you can find that the concentrations of silver ions (Ag+) in a saturated solution of AgCl, AgBr, and AgI are \(1.34 \times 10^{-5} \text{ M}\), \(7.07 \times 10^{-7} \text{ M}\), and \(2.88 \times 10^{-9} \text{ M}\) respectively. This exercise shows the relationship between solubility, Ksp, and ion concentration in solutions.