Question

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

Short answer

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

Step-by-step solution

Determine Equilibrium for Cobalt Complex

For the complex ion \(\mathrm{Co}(\mathrm{NH}_{3})_{6}^{3+}\), the equilibrium involves the cobalt ion and ammonia molecules. The equilibrium is written as: \[\mathrm{Co^{3+}} + 6\mathrm{NH}_{3} \rightleftharpoons \mathrm{Co}(\mathrm{NH}_{3})_{6}^{3+}\]

Write the Equation for the Formation Constant for Cobalt Complex

The equilibrium constant expression for the formation of \(\mathrm{Co}(\mathrm{NH}_{3})_{6}^{3+}\) is written as: \[K_{\text{inst}} = \frac{[\mathrm{Co}(\mathrm{NH}_{3})_{6}^{3+}]}{[\mathrm{Co^{3+}}][\mathrm{NH}_{3}]^{6}}\]

Determine Equilibrium for Mercury Iodide Complex

For the complex ion \(\mathrm{HgI}_{4}^{2-}\), the equilibrium involves the mercury ion and iodide ions. The equilibrium is written as: \[\mathrm{Hg^{2+}} + 4\mathrm{I^{-}} \rightleftharpoons \mathrm{HgI}_{4}^{2-}\]

Write the Equation for the Formation Constant for Mercury Iodide Complex

The equilibrium constant expression for the formation of \(\mathrm{HgI}_{4}^{2-}\) is written as: \[K_{\text{inst}} = \frac{[\mathrm{HgI}_{4}^{2-}]}{[\mathrm{Hg^{2+}}][\mathrm{I^{-}}]^{4}}\]

Determine Equilibrium for Iron Cyanide Complex

For the complex ion \(\mathrm{Fe}(\mathrm{CN})_{6}^{4-}\), the equilibrium involves the iron ion and cyanide ions. The equilibrium is written as: \[\mathrm{Fe^{2+}} + 6\mathrm{CN^{-}} \rightleftharpoons \mathrm{Fe}(\mathrm{CN})_{6}^{4-}\]

Write the Equation for the Formation Constant for Iron Cyanide Complex

The equilibrium constant expression for the formation of \(\mathrm{Fe}(\mathrm{CN})_{6}^{4-}\) is written as: \[K_{\text{inst}} = \frac{[\mathrm{Fe}(\mathrm{CN})_{6}^{4-}]}{[\mathrm{Fe^{2+}}][\mathrm{CN^{-}}]^{6}}\]

Key concepts

Equilibrium Constant Expression

In chemistry, the equilibrium constant expression is a vital concept that quantifies the balance between products and reactants in a reversible reaction at equilibrium. It is expressed in a specific formula, depending on the reaction.

For a generic reaction where A and B react to form the complex AB, the equilibrium can be represented as: \[ A + B \rightleftharpoons AB \]. The equilibrium constant expression, denoted as \( K \), for this reaction is: \[ K = \frac{[AB]}{[A][B]} \], where the square brackets denote the concentration of each species in moles per liter (M).

Considering the complex ion equilibria shown in the exercise, we can see that the formation constant \( K_{\text{inst}} \) is a special type of equilibrium constant specifically for the formation of a complex ion from its constituent ions. For instance, the cobalt complex \( \mathrm{Co}(\mathrm{NH}_{3})_{6}^{3+} \) with an equilibrium constant expression \[ K_{\text{inst}} = \frac{[\mathrm{Co}(\rm{NH}_{3})_{6}^{3+}]}{[\mathrm{Co^{3+}}][\mathrm{NH}_{3}]^{6}} \] highlights the highly ordered structure of the complex and indicates how strongly the metal ion (Co) binds with the ligand (NH3).

Formation Constants for Complex Ions

Formation constants, often symbolized as \( K_f \) but also referred to as \( K_{\text{inst}} \) in the given exercise, are specific equilibrium constants for the formation of complex ions in solution. They provide insight into the stability of the complex ion formed from its metal ion and attached ligands.

The higher the value of the formation constant, the more stable the complex ion is, which means it is less likely to dissociate back into its constituent ions. These constants are crucial in predicting the behavior of ions in various chemical processes, including analytical chemistry, where complex formation is used to isolate or concentrate particular elements.

For example, in Step 4 of the solution, the formation constant for the mercury iodide complex is calculated as follows: \[ K_{\text{inst}} = \frac{[\mathrm{HgI}_{4}^{2-}]}{[\mathrm{Hg^{2+}}][\mathrm{I^{-}}]^{4}} \]. This tells us how strongly mercury ions bind with four iodide ions to create a stable complex, which is essential knowledge for processes such as environmental contaminant removal or pharmaceutical compound synthesis.

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry describing a state where the rate of the forward reaction equals the rate of the reverse reaction. This balance results in no net change in the concentrations of reactants and products over time, although both reactions are still occurring. It's important to understand that equilibrium does not mean the reactants and products are present in equal amounts, but rather that their ratios remain constant.

At chemical equilibrium, the reactions have reached a state of dynamic balance, and the equilibrium constant expression quantitatively describes this state. The values of equilibrium constants give us a measure of the proportions of reactants and products that will be present when the reaction has reached equilibrium.

Different factors such as temperature, pressure, and concentration can affect the position of equilibrium, and these are described by Le Chatelier's principle. This principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. In practice, this means that adding more of a reactant can shift the equilibrium position to produce more products, as seen within the context of complex ion equilibria.