Calcium ion present in water supplies is easily precipitated as calcite \(\left(\mathrm{CaCO}_{3}\right)\) $$ \mathrm{Ca}^{2+}(a q)+\mathrm{CO}_{3}^{2-}(a q) \rightleftharpoons \mathrm{CaCO}_{3}(s) $$ Because the \(K_{\mathrm{sp}}\) decreases with temperature, heating hard water forms a calcite "scale," which clogs pipes and water heaters. Find the solubility of calcite in water (a) at \(10^{\circ} \mathrm{C}\left(K_{\mathrm{sp}}=4.4 \times 10^{-9}\right)\) and (b) at \(30^{\circ} \mathrm{C}\left(K_{\mathrm{sp}}=3.1 \times 10^{-9}\right)\)

The solubility product constant, denoted as \(K_{\text{sp}}\), is a unique value for each sparingly soluble compound at a given temperature. It indicates the level at which a solute will dissolve in water to form a saturated solution, beyond which it will start precipitating. For a compound like calcite (\(CaCO_3\)), which dissociates into calcium ions (\(Ca^{2+}\)) and carbonate ions (\(CO_3^{2-}\)), the \(K_{\text{sp}}\) expression is:

\[ K_{\text{sp}} = [\text{Ca}^{2+}][\text{CO}_3^{2-}] \]
This product remains constant at equilibrium in a saturated solution. Using \(K_{\text{sp}}\), we can determine the solubility (maximum concentration) of the ions in solution before they start to precipitate, as shown in the steps above.

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction. For calcite in water, this is represented by the equation:

\[ \text{Ca}^{2+}(aq) + \text{CO}_3^{2-}(aq) \rightleftharpoons \text{CaCO}_3(s) \]
At equilibrium, the concentration of dissolved calcium and carbonate ions remains constant because their rates of dissolution and precipitation are equal. This balance is quantified by the \(K_{\text{sp}}\) value, helping us to calculate how much of the calcite can dissolve in water at a specific temperature.

A precipitation reaction occurs when the product of the ionic concentrations in a solution exceeds the \(K_{\text{sp}}\) of a sparingly soluble compound, causing it to form a solid. In calcite's case:

\[ \text{Ca}^{2+}(aq) + \text{CO}_3^{2-}(aq) \rightarrow \text{CaCO}_3(s) \]
When the combined concentration of \(Ca^{2+}\) and \(CO_3^{2-}\) ions exceeds the solubility product constant \(K_{\text{sp}}\), calcite precipitates out of the solution. This principle is crucial in understanding how heating water can lead to the formation of calcite scale in pipes and heaters.