Question

Write equilibria that correspond to \(K_{\text {form }}\) for each of the following complex ions and write the equations for \(K_{\text {form }}:\) (a) \(\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}^{3+}\) (b) \(\mathrm{HgI}_{4}^{2-}\), (c) \(\mathrm{Fe}(\mathrm{CN})_{6}^{4-}\).

Short answer

Equilibrium equations: (a) \(K_{\text{form}} = \frac{[\mathrm{Co}(\textrm{NH}_{3})_{6}^{3+}]}{[\textrm{Co}^{3+}][\textrm{NH}_{3}]^6}\), (b) \(K_{\text{form}} = \frac{[\textrm{HgI}_{4}^{2-}]}{[\textrm{Hg}^{2+}][\textrm{I}^{-}]^4}\), (c) \(K_{\text{form}} = \frac{[\textrm{Fe}(\textrm{CN})_{6}^{4-}]}{[\textrm{Fe}^{2+}][\textrm{CN}^{-}]^6}\).

Step-by-step solution

Write the Equilibrium for a Cobalt Complex Ion

The equilibrium for the formation of the complex ion \(\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}^{3+}\) involves cobalt ion and ammonia ligands coming together. The balanced chemical equation is:\[\mathrm{Co}^{3+} + 6\mathrm{NH}_{3} \rightleftharpoons \mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}^{3+}\]. The equilibrium constant expression for this formation, \(K_{\text {form }}\), is then:\[K_{\text {form }} = \frac{[\mathrm{Co}(\mathrm{NH}_{3})_{6}^{3+}]}{[\mathrm{Co}^{3+}][\mathrm{NH}_{3}]^6}\].

Write the Equilibrium for a Mercury Iodide Complex Ion

The equilibrium for the formation of the complex ion \(\mathrm{HgI}_{4}^{2-}\) involves the mercury ion and iodide ions coming together. The balanced chemical equation is:\[\mathrm{Hg}^{2+} + 4 \mathrm{I}^{-} \rightleftharpoons \mathrm{HgI}_{4}^{2-}\]. The equilibrium constant expression for this formation, \(K_{\text {form }}\), is then:\[K_{\text {form }} = \frac{[\mathrm{HgI}_{4}^{2-}]}{[\mathrm{Hg}^{2+}][\mathrm{I}^{-}]^4}\].

Write the Equilibrium for an Iron Cyanide Complex Ion

The equilibrium for the formation of the complex ion \(\mathrm{Fe}(\mathrm{CN})_{6}^{4-}\) involves the iron ion and cyanide ligands coming together. The balanced chemical equation is:\[\mathrm{Fe}^{2+} + 6 \mathrm{CN}^{-} \rightleftharpoons \mathrm{Fe}(\mathrm{CN})_{6}^{4-}\]. The equilibrium constant expression for this formation, \(K_{\text {form }}\), is then:\[K_{\text {form }} = \frac{[\mathrm{Fe}(\mathrm{CN})_{6}^{4-}]}{[\mathrm{Fe}^{2+}][\mathrm{CN}^{-}]^6}\].

Key concepts

Equilibrium Constant Expression

In the context of complex ion formation, an equilibrium constant expression quantitatively describes the balance between the reactants and the products in a reversible chemical reaction. For instance, when metal ions combine with ligands to form a complex ion, the reaction reaches a point of equilibrium, where the rate of the forward reaction, forming the complex ion, equals the rate of the reverse reaction, breaking it apart.

The equilibrium constant expression, typically denoted as K_form, is calculated by dividing the concentration of the product (the complex ion) by the multiplied concentrations of the reactants (the metal ion and the ligands), each raised to the power of its stoichiometric coefficient. This relationship is captured in the mathematical formula
\[K_{\text{form}} = \frac{[\text{Complex Ion}]}{[\text{Metal Ion}][\text{Ligand}]^\text{n}}\]
where n represents the number of ligand molecules. It's critical to note that only species in aqueous solution or in the gas phase are included in the equilibrium constant expression; solids and pure liquids are omitted.

Chemical Equilibrium

The concept of chemical equilibrium is central to understanding the dynamics of reactions involving complex ions. At equilibrium, the concentrations of the reactants and products remain constant over time, not because the reactions have ceased, but due to a balance between the forward and reverse processes.

This state can be recognized by the constant physical properties of the system, such as color or pressure, in case of gases. Also, the equilibrium does not imply that the concentrations of reactants and products are equal, but rather that their ratios are constant, as defined by the equilibrium constant.

If the equilibrium constant is large, the formation of the complex ion is favored, suggesting a high concentration of the complex ion at equilibrium. Conversely, a small equilibrium constant indicates that the reactants are more predominant at equilibrium. Changing conditions such as temperature, pressure, or concentrations can 'shift' the equilibrium, as predicted by Le Chatelier's principle, adjusting the balance between reactants and products accordingly.

Formation Constant

The formation constant, also known as the stability constant, is a specific type of equilibrium constant that applies to complex ions. It is denoted as K_form or K_f and represents the strength by which a complex ion holds together in a solution.

A high formation constant indicates a stable complex ion, suggesting that once formed, it tends not to dissociate back into the metal ion and its ligands. This is crucial in coordination chemistry, as it dictates the feasibility of a complex ion in various chemical environments.

The calculation of the formation constant follows the general rules for equilibrium constants, taking into account only the aqueous or gaseous species present in the reaction. Since formation constants can vary significantly with different metals and ligands, they provide essential information for the design of complexes for industrial, environmental, or medicinal applications.

Coordination Chemistry

At the heart of coordination chemistry lies the study of complex ions, which are formed when central metal atoms or ions bond with surrounding molecules or ions known as ligands. The nature of the metal-ligand bonding can range from ionic to covalent, and these interactions define the geometric and electronic structure of the complex ions.

Factors such as the identity of the metal ion, the type of ligands, and their charge and electronegativity greatly influence the properties of the resulting complex. The coordination number, reflecting the number of ligand attachment sites on the metal ion, and the chelation, where ligands form multiple bonds to a single metal ion, are key aspects in designing novel compounds with specific functions.

These complex ions often exhibit characteristic colors, magnetic properties, and reactivities, which are harnessed in a range of applications, including catalysis, materials science, and medicine. Understanding the fundamentals of coordination chemistry is thus essential for the advancement of many technological and scientific fields.