Question

In the presence of \(\mathrm{CN}^{-}, \mathrm{Fe}^{3+}\) forms the complex ion \(\mathrm{Fe}(\mathrm{CN})_{6}^{3-}\) The equilibrium concentrations of \(\mathrm{Fe}^{3+}\) and \(\mathrm{Fe}(\mathrm{CN})_{6}^{3-}\) are \(8.5 \times 10^{-40} M\) and \(1.5 \times 10^{-3} M,\) respectively, in a \(0.11-M\) KCN solution. Calculate the value for the overall formation constant of \(\mathrm{Fe}(\mathrm{CN})_{6}^{3-}\) $$\mathrm{Fe}^{3+}(a q)+6 \mathrm{CN}^{-}(a q) \rightleftharpoons \mathrm{Fe}(\mathrm{CN})_{6}^{3-}(a q) \quad K_{\mathrm{overall}}=?$$

Short answer

The overall formation constant, \(K_{\text{overall}}\), for the complex ion \(\text{Fe}(\text{CN})_{6}^{3-}\) is approximately \(1.13 \times 10^{41}\).

Step-by-step solution

Write the equilibrium expression

The equilibrium expression for the formation of the complex ion is given by: \[ K_{\text{overall}} = \frac{[\text{Fe}(\text{CN})_{6}^{3-}]}{[\text{Fe}^{3+}][\text{CN}^-]^6} \]

Convert the concentration of KCN to CN⁻

Since the concentration of KCN is \(0.11\,\text{M}\) and KCN dissociates completely, the concentration of CN⁻ is equal to the concentration of KCN. \([ \text{CN}^- ] = 0.11 \, \text{M}\)

Use the given equilibrium concentrations to find the overall formation constant

We are given the equilibrium concentrations of \(\text{Fe}^{3+}\) and \(\text{Fe}(\text{CN})_{6}^{3-}\): [\(\text{Fe}^{3+}\)] = \(8.5 \times 10^{-40} \, \text{M}\) [\(\text{Fe}(\text{CN})_{6}^{3-}\)] = \(1.5 \times 10^{-3} \, \text{M}\) Now, substitute these values in the expression for \(K_{\text{overall}}\): \[ K_{\text{overall}} = \frac{(1.5 \times 10^{-3})}{(8.5 \times 10^{-40})(0.11)^6} \]

Calculate the overall formation constant

Now, compute the value of the overall formation constant: \[ K_{\text{overall}} \approx 1.13 \times 10^{41} \] Therefore, the overall formation constant of \(\text{Fe}(\text{CN})_{6}^{3-}\) is approximately \(1.13 \times 10^{41}\).

Key concepts

Complex Ion Equilibrium

Understanding complex ion equilibrium lays the foundational knowledge for interpreting the behavior of components in a solution when they form complex ions. A complex ion constitutes a metal ion at its center with several other molecules or ions, known as ligands, surrounding it.

During this equilibrium, the metal ion and the ligands come together to form a complex but can also disassociate back into the original metal ion and ligands. This reversible process can be described as a dynamic equilibrium because the forward and reverse reactions occur at the same rate, leading to constant concentrations of the reactants and products, provided the system is closed and other conditions are constant.

To envisage this, imagine a busy junction where equal numbers of cars are entering and leaving at any given time; the number of cars within the junction remains stable over time. Similarly, in a complex ion equilibrium, the rate at which the complex ions form equals the rate at which they break apart, resulting in a steady state.

Chemical Equilibrium Expression

Delving into the chemical equilibrium expression allows us to quantify the concentrations of reactants and products at equilibrium. It is generally framed as a ratio where the products' concentrations (raised to the power of their coefficients in the balanced equation) are in the numerator and the reactants' concentrations are in the denominator.

For the formation of a complex ion, the equilibrium expression is known as the formation constant (K_f). In the given exercise, the formation constant is represented by K_overall, which demonstrates the extent of the formation of the complex ion from its constituent metal ion and ligands. A large K_overall value, as observed in the solution (>10⁴⁰), signifies a strong affinity between the metal ion and the ligands, suggesting the formation of the complex ion is highly favored. This type of quantitative expression is paramount in predicting the formation, stability, and concentration of species present in a chemical system at equilibrium.

Dissociation of Ionic Compounds

The dissociation of ionic compounds like KCN into its ions (K⁺ and CN⁻) in a solution is critical to comprehend as it impacts the equilibrium concentrations of ionic species. Typically, salts dissociate in water to form ions that can engage in subsequent chemical reactions.

In our context, the concentration of CN⁻, the ligand required for the complex formation, is initially equivalent to the concentration of KCN due to its complete dissociation in the solution. This notion highlights an essential principle in equilibrium calculations: the initial concentration of a fully dissociated ionic compound is assumed to be the same as the concentration of each of its ions in the solution.

Understanding such key principles provides a clearer picture for predicting the outcome of reactions. For instance, if a 0.11 M KCN dissociates, we ascertain that there is also an 0.11 M concentration of CN⁻ present, ready to bond with Fe³⁺ to eventually form the complex ion Fe(CN)₆³⁻.