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Equilibrium concentrations are the amounts of each species within a chemical reaction when a dynamic balance between forward and reverse reactions is achieved. In our example, we are examining the concentrations of copper ions (\(\mathrm{Cu}^{2+}\)), ammonia (\(\mathrm{NH}_{3}\)), and the complex ion \(\mathrm{Cu(NH}_3)_4^{2+}\) at equilibrium.To find the equilibrium concentrations, begin with the known initial concentrations of \(0.10\, \text{M}\) for \(\mathrm{Cu}^{2+}\) and \(0.40\, \text{M}\) for \(\mathrm{NH}_{3}\). Initially, the concentration of \(\mathrm{Cu(NH}_3)_4^{2+}\) is \(0\, \text{M}\), because it hasn’t formed yet. At equilibrium, changes are represented with \(x\), where \(x\) describes the amount of \(\mathrm{Cu}^{2+}\) converted into \(\mathrm{Cu(NH}_3)_4^{2+}\). Hence, the equilibrium concentrations are given by:

• \([\mathrm{Cu}^{2+}] = 0.10 - x\)
• \([\mathrm{NH}_3] = 0.40 - 4x\)
• \([\mathrm{Cu(NH}_3)_4^{2+}] = x\)

Understanding how these concentrations change helps in calculating the final state of each component.

The dissociation constant, denoted as \(K_d\), is a special equilibrium constant that measures the tendency of a complex ion to dissociate into its constituent ions. For the reaction we are considering, the complex \(\mathrm{Cu(NH}_3)_4^{2+}\) can dissociate back into \(\mathrm{Cu}^{2+}\) and \(\mathrm{NH}_3\). The expression for \(K_d\) is:\[K_d = \frac{[\text{Cu}^{2+}][\text{NH}_3]^4}{[\text{Cu(NH}_3\text{)}_4^{2+}]}.\]The very small value of \(2.1 \times 10^{-13}\) indicates that the complex ion is very stable, with minimal dissociation under normal conditions. Thus, the larger the dissociation constant, the more likely the complex will break down into its individual ions.

Complex ions are ions that consist of a central metal ion bonded to one or more molecules or ions, termed ligands. These form a coordinate covalent bond with the metal. In our context, \(\mathrm{Cu(NH}_3)_4^{2+}\) is the complex ion formed when \(\mathrm{Cu}^{2+}\) ions interact with \(\mathrm{NH}_3\) molecules. The central metal ion here is the copper ion, and it is surrounded by four ammonia ligands.Characteristics of complex ions include:

• Their ability to distribute charge over a larger volume, affecting stability and reactivity.
• Formation of specific geometric configurations around the central metal atom.
• Significant role in a variety of chemical reactions, especially in catalysis and material science.

The stability of \(\mathrm{Cu(NH}_3)_4^{2+}\) greatly influences the reactions it partakes in, as evidenced by its low dissociation constant.

Equilibrium calculations involve determining the concentrations of different species when a chemical reaction reaches a state of balance. For our reaction of interest, we follow several steps:Following the setup of the reaction equation \(\text{Cu}^{2+} + 4\text{NH}_3 \rightleftharpoons \text{Cu(NH}_3\text{)}_4^{2+}\), we used the dissociation constant to form an expression for calculating \(x\). This expression relates the equilibrium concentrations to \(K_d\):\[2.1 \times 10^{-13} = \frac{(0.10 - x)(0.40 - 4x)^4}{x}.\]Given the minuscule \(K_d\), it's valid to approximate \(0.10 - x \approx 0.10\) and \(0.40 - 4x \approx 0.40\), simplifying the calculation to:\[2.1 \times 10^{-13} \approx \frac{(0.10)(0.40)^4}{x}.\]This simplification allows us to solve for \(x\), yielding \(x \approx 1.22 \times 10^{-14}\). Subsequently, we use this value to find the equilibrium concentrations:

• \([\text{Cu}^{2+}] \approx 0.10 \text{ M}\)
• \([\text{NH}_3] \approx 0.40 \text{ M}\)
• \([\text{Cu(NH}_3\text{)}_4^{2+}] \approx 1.22 \times 10^{-14} \text{ M}\)

Equilibrium calculations are crucial for predicting the outcomes of reactions and determining how conditions such as concentration and temperature will affect the system.