Question

The molar solubility of \(\mathrm{MnCO}_{3}\) is \(4.2 \times 10^{-6} \mathrm{M}\). What is \(K_{\mathrm{sp}}\) for this compound?

Short answer

\(K_{\mathrm{sp}} = 1.764 \times 10^{-11}\)

Step-by-step solution

Identify the Dissociation Equation

Manganese(II) carbonate (\(\mathrm{MnCO}_3\)) dissociates in water according to the equation: \[ \mathrm{MnCO}_3 (s) \rightleftharpoons \mathrm{Mn}^{2+} (aq) + \mathrm{CO}_3^{2-} (aq) \] This means that one mole of \(\mathrm{MnCO}_3\) produces one mole of \(\mathrm{Mn}^{2+}\) ions and one mole of \(\mathrm{CO}_3^{2-}\) ions in solution.

Express Molar Solubility

The molar solubility of \(\mathrm{MnCO}_3\) is given as \(4.2 \times 10^{-6} \mathrm{M}\). This means that \([\mathrm{Mn}^{2+}] = 4.2 \times 10^{-6} \mathrm{M}\) and \([\mathrm{CO}_3^{2-}] = 4.2 \times 10^{-6} \mathrm{M}\) at equilibrium.

Write the Expression for \(K_{\mathrm{sp}}\)

The expression for the solubility product \(K_{\mathrm{sp}}\) is \[ K_{\mathrm{sp}} = [\mathrm{Mn}^{2+}][\mathrm{CO}_3^{2-}] \] where \([\mathrm{Mn}^{2+}]\) and \([\mathrm{CO}_3^{2-}]\) are the molar concentrations of the ions at equilibrium.

Substitute Values into \(K_{\mathrm{sp}}\) Expression

Substitute the molar solubility into the \(K_{\mathrm{sp}}\) expression: \[ K_{\mathrm{sp}} = (4.2 \times 10^{-6})(4.2 \times 10^{-6}) \] Calculate the value: \[ K_{\mathrm{sp}} = 17.64 \times 10^{-12} \] Simplifying, \[ K_{\mathrm{sp}} = 1.764 \times 10^{-11} \]

Key concepts

Molar Solubility

Molar solubility is an important concept in the study of solutions, particularly when dealing with sparingly soluble salts. It is the number of moles of a solute that can dissolve per liter of solution before the solution becomes saturated. In the case of manganese(II) carbonate, \(\mathrm{MnCO}_3\), the molar solubility is given as \(4.2 \times 10^{-6} \, \mathrm{M}\). This means that at equilibrium, only \(4.2 \times 10^{-6} \, \, \mathrm{mol/L}\) of \(\mathrm{MnCO}_3\) can dissolve in water to form manganese ions, \(\mathrm{Mn}^{2+}\), and carbonate ions, \(\mathrm{CO}_3^{2-}\), before reaching the saturation point.
Understanding molar solubility helps predict the concentration of ions in a solution which is critical for determining the solubility product (\(K_{\mathrm{sp}}\)). Only knowing the molar solubility allows us to calculate the concentration of each ion at equilibrium. This information also assists in determining whether a precipitate will form under specific conditions.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of solutes in a solution when the rates of dissolution and precipitation are equal. This is a fundamental aspect of chemical equilibrium, especially in systems involving sparingly soluble salts such as manganese(II) carbonate, \(\mathrm{MnCO}_3\).
The current equilibrium concentrations of \(\mathrm{Mn}^{2+}\) and \(\mathrm{CO}_3^{2-}\) ions are both equal to the molar solubility of \(\mathrm{MnCO}_3\), which is \(4.2 \times 10^{-6} \, \mathrm{M}\).
This concept is critical when assessing the solubility product \(K_{\mathrm{sp}}\), because \(K_{\mathrm{sp}}\) relies on these concentrations to determine the product's overall solubility.

• At equilibrium, the concentration of dissolved ions remains constant unless disturbed by changes in temperature, pressure, or the addition of other substances.
• If additional \(\mathrm{MnCO}_3\) is added to the solution, the concentrations will not change, as any extra \(\mathrm{MnCO}_3\) will remain undissolved at the bottom of the container.

Equilibrium concentrations provide crucial information for applications such as predicting whether a reaction will proceed to the formation of a precipitate.

Dissociation Equation

A dissociation equation shows how an ionic compound separates into its individual ions when dissolved in water. For \(\mathrm{MnCO}_3\), the dissociation is represented by the equation:
\[ \mathrm{MnCO}_3 (s) \rightleftharpoons \mathrm{Mn}^{2+} (aq) + \mathrm{CO}_3^{2-} (aq) \]
This equation indicates that one mole of solid manganese(II) carbonate dissociates into one mole of manganese ions \(\mathrm{Mn}^{2+}\) and one mole of carbonate ions \(\mathrm{CO}_3^{2-}\) in solution. In simpler terms, every molecule of \(\mathrm{MnCO}_3\) that dissolves yields an equal number of \(\mathrm{Mn}^{2+}\) and \(\mathrm{CO}_3^{2-}\) ions.
The dissociation equation is vital for calculating the solubility product \(K_{\mathrm{sp}}\).

• By knowing the dissociation equation, you can easily determine the molar ratio between solute and its ions.
• It simplifies calculations of equilibrium concentrations and aids in predicting the outcome of dissolution reactions.

Thus, the dissociation equation acts as a roadmap, showing how the compound behaves in water and providing insights into its solubility characteristics.