Question

Write the equilibria that are associated with the equations for \(K_{\text {inst }}\) for each of the following complex ions. Write also the equations for the \(K_{\text {inst }}\) of each: (a) \(\mathrm{Hg}\left(\mathrm{NH}_{3}\right)_{4}^{2+},\) (b) \(\mathrm{SnF}_{6}^{2-}\), (c) \(\mathrm{Fe}(\mathrm{CN})_{6}^{3-}\).

Short answer

Equilibria and equations for the instability constants: (a) \(K_{\text{inst}} = \frac{[\mathrm{Hg}(\mathrm{NH}_{3})_{4}^{2+}]}{[\mathrm{Hg}^{2+}][\mathrm{NH}_{3}]^4}\), (b) \(K_{\text{inst}} = \frac{[\mathrm{SnF}_{6}^{2-}]}{[\mathrm{Sn}^{4+}][\mathrm{F}^{-}]^6}\), (c) \(K_{\text{inst}} = \frac{[\mathrm{Fe}(\mathrm{CN})_{6}^{3-}]}{[\mathrm{Fe}^{3+}][\mathrm{CN}^{-}]^6}\).

Step-by-step solution

- Write Equilibrium for Mercury Ammonia Complex

For the complex ion \(\mathrm{Hg}(\mathrm{NH}_{3})_{4}^{2+}\), the equilibrium equation shows mercury(II) ions reacting with ammonia molecules to form the complex ion: \[\mathrm{Hg}^{2+}(aq) + 4\mathrm{NH}_{3}(aq) \rightleftharpoons \mathrm{Hg}(\mathrm{NH}_{3})_{4}^{2+}(aq).\] The corresponding equilibrium constant expression for the formation of this complex ion is given by \[ K_{\text{inst}} = \frac{[\mathrm{Hg}(\mathrm{NH}_{3})_{4}^{2+}]}{[\mathrm{Hg}^{2+}][\mathrm{NH}_{3}]^4}.\]

- Write Equilibrium for Tin Fluoride Complex

For the complex ion \(\mathrm{SnF}_{6}^{2-}\), the equilibrium equation is based on tin(IV) ions and fluoride ions reacting to form the complex ion: \[\mathrm{Sn}^{4+}(aq) + 6\mathrm{F}^{-}(aq) \rightleftharpoons \mathrm{SnF}_{6}^{2-}(aq).\] The equilibrium constant for this reaction is then defined as: \[ K_{\text{inst}} = \frac{[\mathrm{SnF}_{6}^{2-}]}{[\mathrm{Sn}^{4+}][\mathrm{F}^{-}]^6}.\]

- Write Equilibrium for Iron Cyanide Complex

For the complex ion \(\mathrm{Fe}(\mathrm{CN})_{6}^{3-}\), the equilibrium process involves iron(III) ions reacting with cyanide ions: \[\mathrm{Fe}^{3+}(aq) + 6\mathrm{CN}^{-}(aq) \rightleftharpoons \mathrm{Fe}(\mathrm{CN})_{6}^{3-}(aq).\] The equilibrium constant expression for the formation of the iron cyanide complex is: \[ K_{\text{inst}} = \frac{[\mathrm{Fe}(\mathrm{CN})_{6}^{3-}]}{[\mathrm{Fe}^{3+}][\mathrm{CN}^{-}]^6}.\]

Key concepts

Equilibrium Constant

The equilibrium constant, denoted by the symbol K, is a crucial concept in understanding chemical reactions that reach a state of balance, where the rate of the forward reaction equals that of the reverse reaction. Specifically, for complex ion formation, the equilibrium constant reflects the strength of the interaction between the central metal ion and the ligands surrounding it.
In the context of complex ions such as \(\mathrm{Hg}(\mathrm{NH}_3)_{4}^{2+}\), \(\mathrm{SnF}_{6}^{2-}\), and \(\mathrm{Fe}(\mathrm{CN})_{6}^{3-}\), the equilibrium constants are referred to as K_inst, or formation constants. These constants quantify the extent to which complex ions are formed from their constituent metal ions and ligands in solution. A higher K_inst value indicates a more stable complex ion, meaning it is less likely to dissociate into its components.
When using these constants to solve problems, it's imperative to carefully write down the equilibrium expressions based on the stoichiometry of the reaction. For instance, \(K_{\text{inst}}\) for the mercury-ammonia complex would be calculated using the concentration of the complex ion in the numerator and the concentrations of the mercury and ammonia (raised to the fourth power due to the 4:1 stoichiometry) in the denominator. Understanding this concept enables students to predict the position of equilibrium and the concentrations of species in complex ion equilibria.

Chemical Equilibrium

Chemical equilibrium is the state of a reaction in which the rate of the forward reaction equals the rate of the backward reaction, thereby resulting in no net change in the concentration of reactants and products over time. It is a dynamic process, meaning that the reactions continue to occur, but because they happen at the same rate, the system appears static. The state of equilibrium is described by the equilibrium constant (K), which, in the case of complex ions, is specifically called K_inst.
Equilibrium concepts apply to the formation of complex ions when a metal ion in solution reacts with ligands to form a stable arrangement. This process is reversible as the complex ion could dissociate back to the metal ion and ligands under certain conditions. For students to effectively analyze these systems, they must understand that changing the concentration of reactants or products, or changing the temperature or pressure (for gases), can shift the position of equilibrium in favor of the formation of products or reactants according to Le Chatelier's Principle.
In solving problems related to chemical equilibria, it is important to recognize that only species in the aqueous state (denoted by aq) or gaseous state (denoted by g) are included in the equilibrium expressions. Solid and liquid pure substances are omitted. Understanding the principles behind equilibria can help students predict the outcome of reactions and manipulate conditions to shift the equilibrium position in the desired direction.

Ligand Coordination

Ligand coordination is a fundamental aspect of transition metal chemistry. It involves the interaction of ligands, which are ions or molecules with a lone pair of electrons, with a central metal ion to form a coordinate complex. The metal and ligands act as Lewis acids and bases, respectively, with the ligands donating electron pairs to the metal, which accepts them.
In the exercises involving complex ions such as \(\mathrm{Hg}(\mathrm{NH}_3)_{4}^{2+}\), \(\mathrm{SnF}_{6}^{2-}\), and \(\mathrm{Fe}(\mathrm{CN})_{6}^{3-}\), the ammonia (\(NH_3\)), fluoride (\(F^-\)), and cyanide (\(CN^-\)) act as ligands coordinating to the metal ions mercury, tin, and iron, respectively. The number of ligands attached to the metal ion is referred to as the coordination number, which is important in determining the three-dimensional shape of the complex.
A deeper comprehension of ligand coordination allows students to predict the properties and reactivities of complex ions. For example, in biochemical systems, ligand coordination enables crucial processes such as oxygen transport by hemoglobin. When discussing ligand coordination, it is also significant to distinguish between different kinds of ligands - monodentate ligands that bind through a single point, polydentate ligands that bind through multiple points, and ambidentate ligands that can bind through multiple different atoms - based on their binding behavior.
Understanding the principles of ligand coordination gives students the tools to explore complex formation and stability, essential for areas ranging from industrial catalysis to pharmacology.