Question

By using the values of \(K_{s p}\) for \(\mathrm{AgI}\) and \(K_{f}\) for \(\mathrm{Ag}(\mathrm{CN})_{2}^{-}\), calculate the equilibrium constant for the reaction $$ \mathrm{AgI}(\mathrm{s})+2 \mathrm{CN}^{-}(a q) \rightleftharpoons \mathrm{Ag}(\mathrm{CN})_{2}^{-}(a q)+\mathrm{I}^{-}(a q) $$

Short answer

The equilibrium constant (K) for the given reaction can be found using the formula \( K = \frac{K_{f}}{K_{sp}} \), where Kf is the formation constant for \(Ag(CN)_{2}^{-}\), and Ksp is the solubility product constant for AgI. Insert the given values for Kf and Ksp and solve for the equilibrium constant (K).

Step-by-step solution

Write the expressions for Ksp and Kf

In this step, we'll write the solubility product constant (Ksp) for AgI and the formation constant (Kf) for Ag(CN)₂⁻. For AgI, the dissociation reaction is: \( AgI_{(s)} \rightleftharpoons Ag^{+}(aq) + I^{-}(aq) \) The solubility product constant (Ksp) expression is: \( K_{sp} = [Ag^{+}][I^{-}] \) For Ag(CN)₂⁻, the formation reaction is: \( Ag^{+}(aq) + 2CN^{-}(aq) \rightleftharpoons Ag(CN)_{2}^{-}(aq) \) The formation constant (Kf) expression is: \( K_{f} = \frac{[Ag(CN)_{2}^{-}]}{[Ag^{+}][CN^{-}]^{2}} \)

Write the expression for the equilibrium constant of the given reaction

Next, we'll write the expression for the equilibrium constant of the given reaction: \( AgI(s) + 2CN^{-}(aq) \rightleftharpoons Ag(CN)_{2}^{-}(aq) + I^{-}(aq) \) The equilibrium constant (K) expression for the given reaction is: \( K = \frac{[Ag(CN)_{2}^{-}][I^{-}]}{[AgI][CN^{-}]^{2}} \)

Rewrite the K expression using Ksp and Kf expressions

Now, we need to rewrite the equilibrium constant (K) expression in terms of Ksp and Kf. We'll substitute the expressions for Ksp and Kf from Step 1 into the K expression: \( K = \frac{K_{f}[Ag^{+}][CN^{-}]^{2}}{K_{sp}[Ag^{+}][I^{-}]} \)

Simplify the expression and solve for K

Now, we can simplify the expression by canceling out the terms of [Ag⁺] and rearranging the equation: \( K = \frac{K_{f}}{K_{sp}} \) Plug in the given values of Ksp for AgI and Kf for Ag(CN)₂⁻, and calculate K: \( K = \frac{K_{f}}{K_{sp}} \) At this point, you should insert the given values for Ksp and Kf to find the equilibrium constant (K) for the given reaction.

Key concepts

Ksp (Solubility Product Constant)

Understanding the concept of solubility and how it contributes to chemical reactions in solutions is crucial in the study of chemistry. The solubility product constant, commonly known as Ksp, provides vital information about the solubility of a compound. Specifically, Ksp is an equilibrium constant that applies to the dissolution of a sparingly soluble ionic compound.

Consider a generic solid compound AB, which dissociates into A⁺ and B⁻ ions in a solution. The dissolution can be represented as: \[ AB_{(s)} \rightleftharpoons A^{+}(aq) + B^{-}(aq) \] The Ksp expression for this reaction would be given by the product of the concentrations of the ions: \[ K_{sp} = [A^{+}][B^{-}] \]
Therefore, Ksp is a quantitative measure of a compound's solubility in water, reflecting the maximum amount of compound that can dissolve in a given volume of solvent at a specific temperature. A higher Ksp value indicates greater solubility.

For a concrete example, let's examine the dissolution of silver iodide (AgI): \[ AgI_{(s)} \rightleftharpoons Ag^{+}(aq) + I^{-}(aq) \] The Ksp would be the product of the molar concentrations of silver and iodide ions at equilibrium. This constant is invaluable for predicting whether a precipitate will form under certain conditions.

Kf (Formation Constant)

In tandem with Ksp, another important equilibrium constant is the formation constant, denoted as Kf. It is associated with the formation of complex ions in solution. A complex ion consists of a central metal ion bonded to one or more ligands, which are molecules or ions that donate one or more pairs of electrons to the metal ion.

The formation of a complex ion can be depicted by the general reaction: \[ M^{+}(aq) + nL^{z-}(aq) \rightleftharpoons ML_{n}^{(z-n)+}(aq) \] In this case, the Kf expression would be: \[ K_{f} = \frac{[ML_{n}^{(z-n)+}]}{[M^{+}][L^{z-}]^{n}} \]
Here, Kf quantifies the equilibrium favorability of complex ion formation and is influenced by various factors including the nature of the metal ion and ligands. A large Kf value signifies a strong tendency for complex formation, which can have implications for solubility and can affect various analytical procedures in chemistry.

Equilibrium Expressions

At the heart of predicting the direction and extent of chemical reactions lies the concept of equilibrium expressions. An equilibrium expression is a mathematical representation of a chemical reaction at equilibrium, relating the concentrations of reactants to products using equilibrium constants like Ksp and Kf.

For a balanced chemical reaction: \[ aA + bB \rightleftharpoons cC + dD \] The equilibrium expression would be written as: \[ K = \frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}} \]
Here, K is the equilibrium constant, and the concentrations are raised to the power corresponding to their stoichiometric coefficients in the balanced equation. The law of mass action states that at a fixed temperature, K will remain constant for a given reaction.

In the specific case of complex ion formation, we use the formation constant (Kf) to describe the equilibrium state. If dealing with a solubility equilibrium, we employ the solubility product constant (Ksp). By mastering these expressions, chemists can predict the outcome of reactions, calculate concentrations at equilibrium, and engineer conditions to shift equilibria as desired.

Chemical Equilibrium

The status of a chemical reaction when the rates of the forward and reverse reactions are equal, and the concentrations of reactants and products cease to change is referred to as chemical equilibrium. It is a dynamic process, meaning that the reactions continue to occur, but there is no net change in concentration of the reactants and products over time.

Consider the reversible reaction: \[ aA + bB \rightleftharpoons cC + dD \] At equilibrium, the rate at which A and B react to form C and D is exactly balanced by the rate at which C and D revert to A and B. The constant state of these competing reactions is captured by the equilibrium constant (K), a value that embodies the propensity of a reaction mixture to favor the formation of products or reactants under specific conditions.

Chemical equilibrium is a fundamental concept not only in theoretical studies but also in practical applications like synthesizing compounds and pharmaceuticals. By manipulating equilibrium conditions, such as temperature, pressure, and concentration, chemists can steer reactions to yield desired products more efficiently.