Question

Complete the table. Assume that all concentrations are equilibrium concentrations in moles per liter, \(M\). $$ \begin{array}{llll} \text { Compound } & \text { [Cation] } & \text { [Anion] } & \boldsymbol{K}_{\text {eq }} \\ \mathrm{CdS} & 3.7 \times 10^{-15} & 3.7 \times 10^{-15} & \\ \mathrm{BaF}_{2} & & 7.2 \times 10^{-3} & 1.9 \times 10^{-7} \\ \mathrm{Ag}_{2} \mathrm{SO}_{4} & 2.8 \times 10^{-2} & & 1.1 \times 10^{-5} \\\ \hline \end{array} $$

Short answer

Keq for CdS is 1.37 x 10^{-28}. [Ba2+] for BaF2 is 3.7 x 10^{-3} M. [SO4^2-] for Ag2SO4 is 1.4 x 10^{-4} M.

Step-by-step solution

Identify the Dissociation Reactions

For each compound, identify its dissociation reaction in water. For CdS: CdS(s) ⇌ Cd^{2+}(aq) + S^{2-}(aq).For BaF2: BaF2(s) ⇌ Ba^{2+}(aq) + 2F^{-}(aq).For Ag2SO4:Ag2SO4(s) ⇌ 2Ag^{+}(aq) + SO4^{2-}(aq).

Calculate the Missing Equilibrium Constant for CdS

Using the equilibrium concentrations given for CdS and the balanced equation, calculate the equilibrium constant (Keq) using the formula: Keq = [Cation]^[Coefficient of Cation] * [Anion]^[Coefficient of Anion]. So for CdS, Keq = [Cd^{2+}]*[S^{2-}]. Substitute the concentration values to get the Keq.

Calculate the Missing Cation Concentration for BaF2

For BaF2, use the equilibrium constant and the known anion concentration to calculate the cation concentration. Because the dissociation produces 1 mole of Ba^{2+} for every 2 moles of F^{-}, we can write: Keq = [Ba^{2+}] * [F^{-}]^2.We can then rearrange to solve for [Ba^{2+}]: [Ba^{2+}] = Keq / [F^{-}]^2. Substitute the known values to find [Ba^{2+}].

Calculate the Missing Anion Concentration for Ag2SO4

For Ag2SO4, use the equilibrium constant and the known cation concentration to calculate the anion concentration. Here, for every mole of Ag2SO4 that dissociates, 2 moles of Ag^{+} and 1 mole of SO4^{2-} are produced. Keq = [Ag^{+}]^2 * [SO4^{2-}].Rearranging for [SO4^{2-}], we get: [SO4^{2-}] = Keq / [Ag^{+}]^2. Insert the known values to find [SO4^{2-}].

Key concepts

Equilibrium Constant

The equilibrium constant (\( K_{eq} \) is a fundamental concept in chemistry that quantifies the ratio of products to reactants at equilibrium for a reversible chemical reaction. It is crucial for understanding how different chemicals interact in a state of balance where the rate of the forward reaction is equal to the rate of the reverse reaction.

Understanding the equilibrium constant involves recognizing the balanced chemical equation for the reaction. As seen in the exercise, dissociation reactions occur when a compound such as CdS (CdS(s) ⇌ Cd^{2+}(aq) + S^{2-}(aq)) breaks down into its constituent ions in water. The equilibrium constant for such a reaction would be expressed as the product of the concentrations of the ions raised to the power of their stoichiometric coefficients. For instance, the expression for the equilibrium constant (Keq) of CdS would be Keq = [Cd^{2+}] * [S^{2-}].

Calculation of Equilibrium Constants

To calculate the missing Keq for CdS, as demonstrated in the solution steps, we take the given ion concentrations and simply multiply them, as both have a coefficient of one in the balanced equation. However, it's important to note that solids and pure liquids are not included in the equilibrium constant expression, only aqueous solutions and gases are considered.

Dissociation Reactions

Dissociation reactions play a pivotal role in chemical equilibrium as they involve the separation of molecules into smaller particles, usually ions, upon interaction with a solvent like water. These reactions are reversible, and at equilibrium, the rate at which the molecules dissociate into ions equals the rate at which the ions recombine to form the original molecules.

The exercise presents the dissociation reactions of ionic compounds like BaF2 (BaF2(s) ⇌ Ba^{2+}(aq) + 2F^{-}(aq)) and Ag2SO4 (Ag2SO4(s) ⇌ 2Ag^{+}(aq) + SO4^{2-}(aq)). The coefficients in these reactions are crucial because they directly affect the expression and calculation of the equilibrium constant, Keq.

Understanding Coefficients in Reactions

In the case of BaF2, the stoichiometric coefficient of F− is 2, which means for every one Ba^{2+} ion formed, two F− ions are formed. This 1:2 ratio must be reflected in the Keq calculation as Keq = [Ba^{2+}] * [F^{-}]^2. The square on the anion concentration indicates that it is raised to the power of its coefficient. This principle is a key aspect of solving equilibrium problems.

Concentration Calculations

Concentration calculations are essential for determining the amounts of reactants and products in a chemical reaction at equilibrium. As shown in the solution steps, these calculations often involve solving for an unknown concentration of an ion using the equilibrium constant and the concentrations of other ions in the reaction.

For example, to find the missing concentration of Ba^{2+} for BaF2, we employ the equilibrium constant and the known concentration of the anion F−. The mathematical manipulation involves isolating the unknown concentration and then substituting the known values to solve for it. In this case, the formula [Ba^{2+}] = Keq / [F^{-}]^2 is used, derived from the equation of the dissociation reaction and its Keq expression.

Adjusting for Stoichiometry

A similar calculation is done for Ag2SO4, but this time it is the anion concentration of SO4^{2-} that is unknown. Since the coefficient of Ag+ is 2 in the balanced equation, the expression becomes Keq = [Ag^{+}]^2 * [SO4^{2-}], which is rearranged to [SO4^{2-}] = Keq / [Ag^{+}]^2. These concentration calculations are not only fundamental for equilibrium problems but also for a variety of applications in chemical analysis and research.