Question

In the presence of \(\mathrm{NH}_{3}, \mathrm{Cu}^{2+}\) forms the complex ion \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}\). If the equilibrium concentrations of \(\mathrm{Cu}^{2+}\) and \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}\) are \(1.8 \times 10^{-17} M\) and \(1.0 \times 10^{-3} \mathrm{M}\), respec- tively, in a \(1.5-M \mathrm{NH}_{3}\) solution, calculate the value for the overall formation constant of \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}\). \(\mathrm{Cu}^{2+}(a q)+4 \mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}^{2+}(a q) \quad K_{\text {overall }}=?\)

Short answer

The overall formation constant for the complex ion Cu(NH₃)₄²⁺ can be calculated using the given equilibrium concentrations of Cu²⁺, Cu(NH₃)₄²⁺, and NH₃ using the expression \(K_{\text{overall}} = \frac{[\text{Cu}(\text{NH}_3)_4^{2+}]}{[\text{Cu}^{2+}] \times [\text{NH}_3]^4}\). Substituting the given values, we find \(K_{\text{overall}} = 1.21 \times 10^{13}\).

Step-by-step solution

1. Write down the balanced equation and expression for the equilibrium constant K_overall.

The balanced chemical equation for the reaction is given by: Cu²⁺(aq) + 4 NH₃(aq) ⇌ Cu(NH₃)₄²⁺(aq) The expression for the overall formation constant, K_overall, can be written as: \(K_{\text{overall}} = \frac{[\text{Cu}(\text{NH}_3)_4^{2+}]}{[\text{Cu}^{2+}] \times [\text{NH}_3]^4}\)

2. Substitute the given equilibrium concentrations into the K_overall expression.

We are given the following equilibrium concentrations: Cu²⁺ = \(1.8 \times 10^{-17}\) M Cu(NH₃)₄²⁺ = \(1.0 \times 10^{-3}\) M NH₃ = 1.5 M Substitute these values into the K_overall expression: \(K_{\text{overall}} = \frac{(1.0 \times 10^{-3})}{(1.8 \times 10^{-17}) \times (1.5)^4}\)

3. Solve for K_overall.

Now, simplify the expression and calculate the overall formation constant: \(K_{\text{overall}} = \frac{(1.0 \times 10^{-3})}{(1.8 \times 10^{-17}) \times (5.0625)} = 1.21 \times 10^{13}\) So, the overall formation constant for the complex ion Cu(NH₃)₄²⁺ is approximately \(1.21 \times 10^{13}\).

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is fundamental when studying reactions, such as the formation of complex ions. It's essentially the state in which the rate of the forward reaction equals the rate of the reverse reaction, meaning that the concentrations of the reactants and products no longer change. However, this doesn't imply that the reactants and products are equal in concentration, but rather that they have reached a stability point.

When you're given a problem involving equilibrium, like calculating the formation constant, it is integral to first write down the balanced chemical equation and the expression for the equilibrium constant, known as K. In the provided exercise, the K — termed as K_overall for complex formation — represents the ratio of product concentration to the multiplied concentrations of the reactants, each raised to the power of their stoichiometric coefficients in the balanced equation. This mathematical representation is crucial as it allows us to quantify the position of equilibrium and predict how the system will respond to changes in conditions based on Le Chatelier's Principle.

Complex Ion Formation

Complex ions consist of a central metal ion bonded to one or more molecules or ions, known as ligands. The ligands can donate electrons to the metal, establishing coordination bonds and creating a more stable entity known as a complex ion. In this exercise, ammonia, NH₃, serves as the ligand for the copper ion, Cu²⁺.

During complex ion formation, the equilibrium constant reflects the stability of the complex. The higher the value of the formation constant, the more stable the complex ion is. This stability is influenced by several factors, including the nature of the metal ion, the ligands, and the overall charge of the complex. Highlighting these details also helps students to develop a deeper understanding of the factors affecting the stability of complex ions and can provide insights into their behavior in various chemical contexts.

Equilibrium Concentration

The equilibrium concentration is the amount of each reactant and product present in a reaction mixture when it has reached equilibrium. These concentrations remain constant over time unless the system is disturbed. In the case of the complex ion formation being studied, equilibrium concentrations for Cu²⁺ and Cu(NH₃)₄²⁺ are crucial for calculating the formation constant.

To use these equilibrium concentrations properly in calculations, it's important to know not only their values but how to apply them correctly in the context of the equilibrium constant formula. As noticed in the problem, even though ammonia is in large excess, its concentration is still factored into the calculation—raised to the fourth power due to its stoichiometric coefficient in the balanced equation. This emphasizes the role each reaction component plays in dictating the position of equilibrium and thus the composition of the system at equilibrium.