Problem 55 - Chemistry The Central Science In Si Units

Question

A 1.50-L solution saturated at \(25^{\circ} \mathrm{C}\) with cobalt carbonate \(\left(\mathrm{CoCO}_{3}\right)\) contains \(2.71 \mathrm{mg}\) of \(\mathrm{CoCO}_{3} .\) Calculate the solubility-product constant for this salt at \(25^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The solubility-product constant \( K_{sp} \) for \( \mathrm{CoCO}_3 \) at \( 25^{\circ} \mathrm{C} \) is \( 2.31 \times 10^{-10} \).

Step 1: Understand Solubility Product

The solubility product constant, denoted as \( K_{sp} \), is used to describe the equilibrium between a solid and its respective ions in a solution. It is specific to the temperature provided.

Step 2: Write the Dissolution Equation

The dissolution of cobalt carbonate (\( \mathrm{CoCO}_3 \)) in solution can be written as: \[ \mathrm{CoCO}_3(s) \rightleftharpoons \mathrm{Co}^{2+}(aq) + \mathrm{CO}_3^{2-}(aq) \] Each \( \mathrm{CoCO}_3 \) that dissolves will produce one \( \mathrm{Co}^{2+} \) ion and one \( \mathrm{CO}_3^{2-} \) ion.

Step 3: Calculate the Molarity of \( \mathrm{CoCO}_3 \)

First, determine the number of moles of \( \mathrm{CoCO}_3 \) dissolved. Convert mass to grams: \( 2.71 \text{ mg} = 0.00271 \text{ g} \). The molar mass of \( \mathrm{CoCO}_3 \) is approximately \( 118.94 \text{ g/mol} \). Then, calculate moles: \[ \text{moles of } \mathrm{CoCO}_3 = \frac{0.00271 \text{ g}}{118.94 \text{ g/mol}} \approx 2.28 \times 10^{-5} \text{ mol} \] Next, compute the molarity (mol/L): \[ \text{Molarity} = \frac{2.28 \times 10^{-5} \text{ mol}}{1.50 \text{ L}} \approx 1.52 \times 10^{-5} \text{ M} \]

Step 4: Write the Expression for \( K_{sp} \) and Substitute Values

The \( K_{sp} \) expression for \( \mathrm{CoCO}_3 \) is: \[ K_{sp} = [\mathrm{Co}^{2+}][\mathrm{CO}_3^{2-}] \] Assuming complete dissolution, each species has a concentration of \( 1.52 \times 10^{-5} \text{ M} \). Substitute the values into the expression: \[ K_{sp} = (1.52 \times 10^{-5})(1.52 \times 10^{-5}) = 2.31 \times 10^{-10} \]

Key Concepts

Cobalt Carbonate

Cobalt carbonate, with the chemical formula \( \text{CoCO}_3 \), is an inorganic compound that appears as a reddish solid. This compound is noteworthy in the study of solubility because it does not completely dissolve in water, thereby reaching a state of equilibrium. When cobalt carbonate is added to water, it dissociates into cobalt ions (\( \text{Co}^{2+} \)) and carbonate ions (\( \text{CO}_3^{2-} \)). The extent to which cobalt carbonate dissolves in water is limited, which is where the solubility product constant becomes relevant. Understanding cobalt carbonate's behavior in a solution helps us to calculate its solubility and further explore equilibrium expressions.

Molarity Calculation

Molarity is a measure of the concentration of a solute in a solution, expressed as moles of solute per liter of solution (mol/L). Calculating molarity begins with determining the number of moles of the solute, in this case, cobalt carbonate. To do this, convert the mass of cobalt carbonate (given in milligrams) to grams. Then, use the compound's molar mass to convert grams to moles.

• Convert 2.71 mg of cobalt carbonate to grams: \( 0.00271 \text{ g} \).
• Find the number of moles by dividing the mass in grams by the molar mass (118.94 g/mol): \( \text{moles of } \text{CoCO}_3 = \frac{0.00271 \text{ g}}{118.94 \text{ g/mol}} \approx 2.28 \times 10^{-5} \text{ mol} \).

Finally, to calculate molarity, divide the number of moles by the volume of the solution in liters. This straightforward method of calculation applies to many chemical scenarios, enhancing your understanding of concentration measurements.

Equilibrium Expression

When cobalt carbonate dissolves in water, it forms an equilibrium with its ions. The equilibrium expression describes the relationship between the concentrations of these ions at equilibrium. For cobalt carbonate, this expression is known as the solubility product constant (\( K_{sp} \)). The equilibrium reaction for cobalt carbonate dissolution is:\[ \text{CoCO}_3 (s) \rightleftharpoons \text{Co}^{2+} (aq) + \text{CO}_3^{2-} (aq) \]Each dissolved cobalt carbonate molecule produces one cobalt ion and one carbonate ion. Thus, the \( K_{sp} \) expression is:\[ K_{sp} = [\text{Co}^{2+}][\text{CO}_3^{2-}] \]In this problem, we assume complete dissociation, so the concentrations of both ions are equal to the molarity of the solute, \( 1.52 \times 10^{-5} \text{ M} \). Therefore, \( K_{sp} \) becomes:\[ K_{sp} = (1.52 \times 10^{-5})^2 = 2.31 \times 10^{-10} \]This constant is specific to the temperature of the solution and provides insights into how much of the solid can dissolve before the solution becomes saturated. Studying equilibrium expressions is crucial as they apply to various contexts in chemistry, helping to predict the behavior and stability of solutions.