Question

The complexion of \(\mathrm{Fe}^{2+}\) with the chelating agent dipyridyl has been studied kinetically in both the forward and reverse directions. \(\mathrm{Fe}^{2+}+3\) dipy \(\rightleftharpoons\left[\mathrm{Fe}(\mathrm{dipy})_{3}\right]^{2+}\) For this reaction, the rates of forward and reverse reactions are \(\left(1.45 \times 10^{13}\right.\) \(\left.\mathrm{M}^{-3} \mathrm{~s}^{-1}\right)\left[\mathrm{Fe}^{2+}\right][\mathrm{dipy}]^{3}\) and \(\left(1.22 \times 10^{-4} \mathrm{~s}^{-1}\right)\) \(\left[\mathrm{Fe}(\mathrm{dipy})_{3}{ }^{2+}\right]\), at \(25^{\circ} \mathrm{C}\). What is the stability constant of the complex? (a) \(1.77 \times 10^{9}\) (b) \(8.4 \times 10^{-18}\) (c) \(1.18 \times 10^{17}\) (d) \(5.65 \times 10^{-10}\)

Short answer

\(1.18 \times 10^{17}\)

Step-by-step solution

Write down the equilibrium expression

For the given complexation equilibrium \(\mathrm{Fe}^{2+}+3 \, \mathrm{dipy} \rightleftharpoons \left[\mathrm{Fe}(\mathrm{dipy})_{3}\right]^{2+}\), we will first write down the equilibrium constant expression, often denoted as \(K\), in terms of the concentrations of the reactants and products at equilibrium. The equilibrium constant, also known as the stability constant for complexation reactions, is defined as: \[K = \frac{\left [\left[\mathrm{Fe}(\mathrm{dipy})_{3}\right]^{2+}\right ]}{\left[\mathrm{Fe}^{2+}\right]\left[\mathrm{dipy}\right]^{3}}.\]

Relate the rate constants to the equilibrium constant

The rate of the forward reaction is given by the rate law \(k_{\text{fwd}}[\mathrm{Fe}^{2+}][\mathrm{dipy}]^{3}\), and for the reverse reaction it is \(k_{\text{rev}}\left[\mathrm{Fe}(\mathrm{dipy})_{3}{ }^{2+}\right]\). At equilibrium, the rates of the forward and reverse reactions are equal, so we have \[k_{\text{fwd}}[\mathrm{Fe}^{2+}][\mathrm{dipy}]^{3} = k_{\text{rev}}\left[\left[\mathrm{Fe}(\mathrm{dipy})_{3}\right]^{2+}\right].\] Dividing both sides by \([\mathrm{Fe}^{2+}][\mathrm{dipy}]^{3}\) and \(\left[\mathrm{Fe}(\mathrm{dipy})_{3}{ }^{2+}\right]\) gives us the equilibrium constant in terms of rate constants: \[K = \frac{k_{\text{fwd}}}{k_{\text{rev}}}.\]

Calculate the stability constant

Substituting the given rate constants into the above expression for the equilibrium constant yields: \[K = \frac{1.45 \times 10^{13} \, \text{M}^{-3} \, \text{s}^{-1}}{1.22 \times 10^{-4} \, \text{s}^{-1}} = 1.18 \times 10^{17}.\] So, the stability constant of the complex \(\left[\mathrm{Fe}(\mathrm{dipy})_{3}\right]^{2+}\) is \(1.18 \times 10^{17}\.\)

Key concepts

Chemical Equilibrium

When discussing chemical equilibrium, we are referring to a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. As a result, the concentrations of the reactants and products remain constant over time, although they are not necessarily equal to each other.

Understanding the concept of equilibrium is crucial when considering complexation reactions, as it helps to determine the ratio of product to reactants at a given point when the system is at rest. An important characteristic of the equilibrium state is the equilibrium constant, denoted as K. This constant is a quantitative measure of the position of equilibrium and is defined by the ratio of the concentration of products to the concentration of reactants, each raised to the power of their coefficients in the balanced chemical equation.

In the calculation of the stability constant of a complex, such as \(\left[\mathrm{Fe}(\mathrm{dipy})_{3}\right]^{2+}\), the equilibrium constant is particularly referred to as the stability constant (K), signifying the stability of the complex ion in solution. A high value of the stability constant indicates a more stable complex which dissociates less easily to form free ions.

Reaction Kinetics

Moving on to reaction kinetics, this field of chemistry deals with the rate of a chemical reaction and the factors affecting it. The kinetics of a reaction are determined by measuring the speed at which reactants transform into products. Specifically, the rate constants for the forward and reverse reactions (\(k_{\text{fwd}}\) and \(k_{\text{rev}}\), respectively) play a pivotal role.

The rate of the forward reaction can be represented by \(k_{\text{fwd}}[\mathrm{Fe}^{2+}][\mathrm{dipy}]^{3}\), emphasizing how it depends on the concentration of reactants and the rate constant. Similarly, the rate of the reverse reaction can be denoted by \(k_{\text{rev}}\left[\left[\mathrm{Fe}(\mathrm{dipy})_{3}\right]^{2+}\right]\).

At chemical equilibrium, these two rates are balanced, which provides a direct relationship between the rate constants and the equilibrium constant: \(K = \frac{k_{\text{fwd}}}{k_{\text{rev}}}\). This relationship is indispensable as it allows chemists to calculate the stability constant from the rates of the forward and reverse reactions.

Complexation Reactions

Finally, let's delve into the realm of complexation reactions. These reactions involve the association between a central metal ion and one or more ligands to form a complex ion. In our exercise, the central metal ion \(\mathrm{Fe}^{2+}\) reacts with the ligand dipyridyl, often abbreviated as dipy, to form the complex \(\left[\mathrm{Fe}(\mathrm{dipy})_{3}\right]^{2+}\). The formation of a complex is crucial in many areas of chemistry, including biochemical systems and industrial catalysis.

Complexation reactions are reversible, and the extent to which a metal binds to ligands is described by the stability constant. This constant gives insight into the formation strength and therefore the stability of the complex under given conditions. A complex with a larger stability constant indicates a stronger affinity between the metal ion and the ligands, signifying a higher likelihood of the complex ion existing in solution rather than dissociating into its components.

To help students understand complexation reactions and their equilibria, it's beneficial to connect the concepts of chemical equilibrium and reaction kinetics. It sheds light on how equilibrium is reached and provides a mathematical way of evaluating the strength of the resultant complex ion.