Question

Calculate the solubility of \(\mathrm{AgCN}(s)\left(K_{\mathrm{sp}}=2.2 \times 10^{-12}\right)\) in a solution containing \(1.0 M \mathrm{H}^{+} .\left(K_{\mathrm{a}}\right.\) for \(\mathrm{HCN}\) is \(6.2 \times 10^{-10}\).)

Short answer

The solubility of silver cyanide (AgCN) in a solution containing 1.0 M hydrogen ions (H₊) is approximately \( s \approx 1.48 \times 10^{-6} \,\mathrm{M} \).

Step-by-step solution

Write the solubility equilibrium reaction for AgCN

The solubility equilibrium reaction for silver cyanide can be represented as follows: \[ \mathrm{AgCN}(s) \rightleftharpoons \mathrm{Ag^{+}}(aq) + \mathrm{CN^{-}}(aq) \]

Write the dissociation reaction for HCN

The dissociation reaction for hydrogen cyanide can be represented as: \[ \mathrm{HCN}(aq) + \mathrm{H_2O}(l) \rightleftharpoons \mathrm{H_3O^{+}}(aq) + \mathrm{CN^{-}}(aq) \]

Write the expressions for solubility product constant Kₛₚ and acid dissociation constant Kₐ

For the solubility equilibrium reaction of AgCN: \[ K_{\mathrm{sp}} = [\mathrm{Ag^{+}}][\mathrm{CN^{-}}]\] For the dissociation reaction of HCN: \[ K_{\mathrm{a}} = \frac {[\mathrm{H_3O^{+}}][\mathrm{CN^{-}}]}{[\mathrm{HCN}]} \]

Relate the concentration of H₊ with the concentration of Ag₊ in the solution

Since the solution contains 1.0 M H₊ ions and Kₐ of HCN is given, we can use the following relationship: \[ [\mathrm{Ag^{+}}] = [\mathrm{CN^{-}}] \]

Substitute the expressions in the Kₛₚ equation and solve for the solubility of AgCN

Let the solubility of AgCN be denoted by 's'. Based on Step 4, we can rewrite the Kₛₚ equation as: \[ K_{\mathrm{sp}} = (s)(s) \] Now, substitute the given value of Kₛₚ and solve for 's': \[ 2.2 \times 10^{-12} = (s)(s) \] \[ s^2 = 2.2 \times 10^{-12} \] \[ s = \sqrt{2.2 \times 10^{-12}} \approx 1.48 \times 10^{-6} \]

Present the final answer

The solubility of silver cyanide (AgCN) in a solution containing 1.0 M hydrogen ions (H₊) is approximately: \[ s \approx 1.48 \times 10^{-6} \,\mathrm{M} \]

Key concepts

Solubility equilibrium

Solubility equilibrium refers to the balance established between a solid substance and its dissolved ions in a solution. It's a dynamic process where solid and dissolved ions continuously dissolve and precipitate, although at equilibrium, their concentrations remain constant.
For silver cyanide (AgCN), when it dissolves in water, it dissociates into silver ions (\( \mathrm{Ag^{+}} \)) and cyanide ions (\( \mathrm{CN^{-}} \)).
The equilibrium we consider can be written as:

• \( \mathrm{AgCN}(s) \rightleftharpoons \mathrm{Ag^{+}}(aq) + \mathrm{CN^{-}}(aq) \)

The solubility product constant (\( K_{\mathrm{sp}} \)) is an expression of this equilibrium, representing the product of the concentrations of ions in their molar proportions. For \( \mathrm{AgCN} \), \( K_{\mathrm{sp}} = [\mathrm{Ag^{+}}][\mathrm{CN^{-}}] \) is extremely small (\( 2.2 \times 10^{-12} \)), indicating low solubility. Thus, only a tiny amount of \( \mathrm{AgCN} \) can dissolve before reaching saturation and establishing equilibrium.

Acid dissociation constant

The acid dissociation constant (\( K_{\mathrm{a}} \)) helps us understand the strength of an acid in a solution. It measures the tendency of an acid to donate protons to the solution. A larger \( K_{\mathrm{a}} \) value indicates a stronger acid, which more completely dissociates in water.
For hydrogen cyanide (\( \mathrm{HCN} \)), its dissociation in water can be written as:

• \( \mathrm{HCN}(aq) + \mathrm{H_2O}(l) \rightleftharpoons \mathrm{H_3O^{+}}(aq) + \mathrm{CN^{-}}(aq) \)

The \( K_{\mathrm{a}} \) expression for \( \mathrm{HCN} \) is:

• \( K_{\mathrm{a}} = \frac{[\mathrm{H_3O^{+}}][\mathrm{CN^{-}}]}{[\mathrm{HCN}]} \)

In this exercise, \( \mathrm{HCN} \) acts as a weak acid, with a \( K_{\mathrm{a}} \) of \( 6.2 \times 10^{-10} \). This small value shows relatively poor dissociation in water, which influences the ion concentrations in the solution. Thus, understanding \( K_{\mathrm{a}} \) is essential for predicting the behavior of weak acids.

Silver cyanide

Silver cyanide (\( \mathrm{AgCN} \)) is an inorganic compound known for its low solubility in water. In the context of chemistry and solubility calculations, it is used to illustrate how equilibrium constants can help determine solubility in complex environments.
When placed in water, \( \mathrm{AgCN} \) slowly dissolves into silver ions (\( \mathrm{Ag^{+}} \)) and cyanide ions (\( \mathrm{CN^{-}} \)). The low solubility arises from the small \( K_{\mathrm{sp}} \) value, which poses a significant limits to its solubility in aqueous solutions.
Understanding the behavior of \( \mathrm{AgCN} \) in different solutions, like those containing \( \mathrm{H^+} \) ions or other competing equilibria, aids in comprehending interactions in ionic compounds. This knowledge helps predict and calculate the solubility of \( \mathrm{AgCN} \) under various conditions, like those in this exercise, where other ions influence the ionic balance and solubility.