Question

If a saturated solution prepared by dissolving \(\mathrm{Ag}_{2} \mathrm{CO}_{3}\) in water has \(\left[\mathrm{Ag}^{+}\right]=2.56 \times 10^{-4} \mathrm{M}\), what is the value of \(K_{\mathrm{sp}}\) for \(\mathrm{Ag}_{2} \mathrm{CO}_{3} ?\)

Short answer

The value of \(K_{\mathrm{sp}}\) for \(\mathrm{Ag}_2 \mathrm{CO}_3\) is \(8.37 \times 10^{-12}\).

Step-by-step solution

Write the Dissolution Equation

First, write the dissolution equation for the compound. The dissolution of silver carbonate can be depicted as: \( \mathrm{Ag}_2\mathrm{CO}_3 (s) \rightleftharpoons 2\mathrm{Ag}^+ (aq) + \mathrm{CO}_3^{2-} (aq) \). This helps us understand the stoichiometry of the dissolution reaction.

Express the Equilibrium Concentrations

Given \([\mathrm{Ag}^+] = 2.56 \times 10^{-4} \mathrm{M}\), and from the dissolution equation, we can determine that \([\mathrm{CO}_3^{2-}] = \frac{1}{2} [\mathrm{Ag}^+] = \frac{1}{2} \times 2.56 \times 10^{-4} \mathrm{M} = 1.28 \times 10^{-4} \mathrm{M}\).

Write the Solubility Product Expression

For the dissolution reaction \(\mathrm{Ag}_2\mathrm{CO}_3 (s) \rightleftharpoons 2\mathrm{Ag}^+ (aq) + \mathrm{CO}_3^{2-} (aq)\), the solubility product expression is \(K_{\mathrm{sp}} = [\mathrm{Ag}^+]^2 [\mathrm{CO}_3^{2-}]\).

Calculate the K_sp Value

Substitute the given concentration of \([\mathrm{Ag}^+]\) and the calculated \([\mathrm{CO}_3^{2-}]\) into the \(K_{\mathrm{sp}}\) expression to find: \(K_{\mathrm{sp}} = (2.56 \times 10^{-4})^2 \times 1.28 \times 10^{-4} = 8.37 \times 10^{-12}\).

Key concepts

Saturated Solution

In chemistry, a solution is called 'saturated' when it contains the maximum amount of a solute that can be dissolved in a given amount of solvent at a specific temperature. When this balance is achieved, any additional solute will not dissolve, resulting in undissolved particles at the bottom of the container.
For silver carbonate \( \mathrm{Ag}_2\mathrm{CO}_3 \), reaching a saturated solution means that the dissolved ions \( 2\mathrm{Ag}^+ \) and \( \mathrm{CO}_3^{2-} \) are at their maximum concentrations. This state is crucial for calculating the solubility product \( K_{\mathrm{sp}} \) because it signifies the point where the dissolution process is in equilibrium.
Saturated solutions provide critical insights into the solubility characteristics of substances, which is essential for various applications, from pharmaceuticals to environmental science.

Dissolution Equation

The dissolution equation is a chemical equation that represents how a compound separates into its constituent ions in a solution. For silver carbonate, the equation is:
\[ \mathrm{Ag}_2\mathrm{CO}_3 (s) \rightleftharpoons 2\mathrm{Ag}^+ (aq) + \mathrm{CO}_3^{2-} (aq) \]
This equation is vital because it shows the stoichiometric relationships between the compound and its ions in the solution.

• Ionic Breakdown: Silver carbonate breaks into two silver ions \( 2\mathrm{Ag}^+ \) for every carbonate ion \( \mathrm{CO}_3^{2-} \).
• Dynamic Equilibrium: The double arrow indicates an equilibrium state where dissolution and precipitation occur at the same rate.

Understanding the dissolution equation helps with determining the mathematical framework needed to calculate the solubility product \( K_{\mathrm{sp}} \).

Silver Carbonate

Silver carbonate (\( \mathrm{Ag}_2\mathrm{CO}_3 \)) is an ionic compound composed of silver and carbonate ions. It is known for its low solubility in water. This means that only a small amount will dissolve to form a saturated solution under equilibrium conditions.
In practical applications, the low solubility of silver carbonate is beneficial for processes requiring precise control of silver ions in solutions, such as in photographic processes or certain chemical syntheses.
When studying \( \mathrm{Ag}_2\mathrm{CO}_3 \), one must understand how its dissolution is represented in the solubility product (\( K_{\mathrm{sp}} \)) expression, which helps predict whether precipitation will occur in a given situation.

Equilibrium Concentrations

Equilibrium concentrations refer to the concentrations of ions in a saturated solution at equilibrium. In the context of silver carbonate dissolution, these are the concentrations of \( \mathrm{Ag}^+ \) and \( \mathrm{CO}_3^{2-} \) ions when any additional solute will not further dissolve.
Knowing the equilibrium concentration of one ion allows for calculation of the other ions' concentrations using stoichiometry from the dissolution equation. For example, if \( [\mathrm{Ag}^+] = 2.56 \times 10^{-4} \mathrm{M} \), then \( [\mathrm{CO}_3^{2-}] = \frac{1}{2} \times [\mathrm{Ag}^+] \), due to the 1:2 ratio in the dissolution equation.
These concentrations are key to computing the solubility product \( K_{\mathrm{sp}} \), which is derived from the expression:

• \( K_{\mathrm{sp}} = [\mathrm{Ag}^+]^2 [\mathrm{CO}_3^{2-}] \)

Substituting the equilibrium concentrations into this formula helps chemists determine the solubility characteristics of the compound.