Problem 44 - Chemistry and Chemical Reactivity

Question

At \(20^{\circ} \mathrm{C},\) a saturated aqueous solution of silver acetate, \(\mathrm{AgCH}_{3} \mathrm{CO}_{2}\), contains \(1.0 \mathrm{g}\) of the silver compound dissolved in \(100.0 \mathrm{mL}\) of solution. Calculate \(K_{\mathrm{sp}}\) for silver acetate. $$\mathrm{AgCH}_{3} \mathrm{CO}_{2}(\mathrm{s}) \rightleftharpoons \mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{CH}_{3} \mathrm{CO}_{2}^{-}(\mathrm{aq})$$

Step-by-Step Solution

Answer

The \(K_{sp}\) for silver acetate is 0.00359.

Step 1: Understand the Dissolution Reaction

The dissolution of silver acetate can be expressed by the equation \(\mathrm{AgCH}_{3} \mathrm{CO}_{2}(\mathrm{s}) \rightleftharpoons \mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{CH}_{3} \mathrm{CO}_{2}^{-}(\mathrm{aq})\). This indicates that, when dissolved, one mole of silver acetate (\mathrm{AgCH}_{3} \mathrm{CO}_{2}) dissociates into one mole of silver ions (\(\mathrm{Ag}^{+}\)) and one mole of acetate ions (\(\mathrm{CH}_{3}\mathrm{CO}_{2}^{-}\)).

Step 2: Calculate Molar Mass of Silver Acetate

First, determine the molar masses: \(\text{Ag} = 107.87\text{ g/mol}\), \(\text{C} = 12.01\text{ g/mol}\), \(\text{H} = 1.01\text{ g/mol}\), \(\text{O} = 16.00\text{ g/mol}\). Therefore, the molar mass of silver acetate \(\mathrm{AgCH}_3\mathrm{CO}_2\) is \(\text{Ag} + \text{C} + 3\text{H} + 2\text{O} = 107.87 + 12.01 + 3(1.01) + 2(16.00) = 166.91\text{ g/mol}\).

Step 3: Convert Mass to Moles

With the given, 1.0 gram of silver acetate is dissolved in 100.0 mL of solution. Convert this mass to moles using its molar mass: \(\text{moles of AgCH}_3\text{CO}_2 = \frac{1.0 \text{ g}}{166.91 \text{ g/mol}} = 0.00599 \text{ moles}\).

Step 4: Calculate the Concentration of Ions

Since the solution volume is 100 mL, convert this to liters, obtaining 0.1 L. The concentration of \(\mathrm{Ag}^{+}\) and \(\mathrm{CH}_3\mathrm{CO}_2^-\) ions in the solution is \(\frac{0.00599 \text{ moles}}{0.1 \text{ L}} = 0.0599 \text{ mol/L}\).

Step 5: Calculate the Solubility Product Constant, \(K_{sp}\)

For the dissolution equilibrium of \(\mathrm{AgCH}_{3} \mathrm{CO}_{2}\), the expression for the solubility product \(K_{sp}\) is \([\mathrm{Ag}^+][\mathrm{CH}_3\mathrm{CO}_2^-]\). Since both ions have the same concentration, \(K_{sp} = (0.0599)^2 = 0.00359\).

Key Concepts

Equilibrium Calculations

Equilibrium calculations are essential for understanding how substances dissolve and reach a state of balance in solution. When a solid like silver acetate is added to water, it partially dissolves, and an equilibrium is established between the solid and its ions in solution. This equilibrium can be represented by its dissolution reaction: \[\text{AgCH}_3\text{CO}_2 (s) \rightleftharpoons \text{Ag}^+ (aq) + \text{CH}_3\text{CO}_2^- (aq)\] Here, one mole of silver acetate dissociates into one ion each of silver \(\text{Ag}^+\) and acetate \(\text{CH}_3\text{CO}_2^-\). At equilibrium, the concentration of these ions remains constant. To calculate the solubility product constant, \(K_{sp}\), you multiply the concentrations of the ions at equilibrium. This constant helps predict whether a precipitate will form when two solutions are mixed. Understanding and calculating equilibrium conditions is crucial for fields like chemistry and environmental science, as these calculations can model processes like reactions in natural waters or industrial systems.

Molar Mass Calculation

Calculating the molar mass of compounds enables us to convert between grams and moles, an essential step in many chemistry solutions. Molar mass is the sum of the atomic masses of all atoms in a molecule, often expressed in grams per mole (g/mol). For silver acetate, the atomic masses are:

• Silver (Ag): 107.87 g/mol
• Carbon (C): 12.01 g/mol
• Hydrogen (H): 1.01 g/mol
• Oxygen (O): 16.00 g/mol

The molar mass of silver acetate \(\text{AgCH}_3\text{CO}_2\) is therefore calculated as: \[107.87 + 12.01 + 3(1.01) + 2(16.00) = 166.91 \text{ g/mol}\] Once we know the molar mass, we can determine how many moles are in a given mass of the substance. In this specific example, we take 1.0 g of silver acetate and convert it to moles, which is essential for calculating concentrations later on.

Dissolution Reaction

Through dissolution reactions, we can predict how solid compounds dissolve in solvents and turn into their respective ions. A dissolution reaction is a chemical reaction where a solid dissolves in a solvent to yield its chemicals in ionic form. For silver acetate, this can be expressed by: \[\text{AgCH}_3\text{CO}_2 (s) \rightleftharpoons \text{Ag}^+ (aq) + \text{CH}_3\text{CO}_2^- (aq)\]This reaction is a reversible process where the solute material disassociates into ionic components. It's crucial in predicting and calculating the resultant ion concentrations when the solution reaches equilibrium. Understanding this helps chemists in formulating solutions, predicting reactions, and assessing the stability of solutions under various conditions. Dissolution reactions are omnipresent in chemistry, illustrating the uniqueness of substances to dissociate in specific patterns.

Ion Concentration

Ion concentration, or the amount of a specific ion in a solution, is essential for assessing the solution's properties and behavior. When silver acetate dissolves, it dissociates into silver \(\text{Ag}^+\) and acetate \(\text{CH}_3\text{CO}_2^-\) ions with equal concentrations because of their (1:1) stoichiometric ratio. For example, if 0.00599 moles of silver acetate are dissolved in 100 mL (or 0.1 L), the concentration of each ion is calculated by dividing the moles by the volume:\[\text{Ion concentration} = \frac{0.00599 \text{ moles}}{0.1 \text{ L}} = 0.0599 \text{ mol/L}\] Accurate ion concentration measurement allows chemists to predict outcomes like precipitation or reaction rates. Knowing these concentrations enables the calculation of \(K_{sp}\), by multiplying the ion concentrations at equilibrium. This understanding is integral across numerous practical applications, from pharmaceuticals to environmental chemistry, where ion behavior dictates many solution-based processes.