Question

In acidic aqueous solution, the complex trans-[ \(\left.\mathrm{Co}(\mathrm{en})_{2} \mathrm{Cl}_{2}\right]^{+}\) undergoes the following substitution reaction: trans- \(\left[\mathrm{Co}(\mathrm{en})_{2} \mathrm{Cl}_{2}\right]^{+}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow\) trans- \(\left[\mathrm{Co}(\mathrm{en})_{2}\left(\mathrm{H}_{2} \mathrm{O}\right) \mathrm{Cl}\right]^{2+}(a q)+\mathrm{CI}(a q)\) The reaction is first order in trans- \(\left[\mathrm{Co}(\mathrm{en})_{2} \mathrm{Cl}_{2}\right]^{+}\), and the rate constant at \(25^{\circ} \mathrm{C}\) is \(3.2 \times 10^{-5} \mathrm{~s}^{-1}\). (a) What is the half-life of the reaction in hours? (b) If the initial concentration of trans-[Co(en) \(\left._{2} \mathrm{Cl}_{2}\right]^{+}\) is \(0.138\) \(\mathrm{M}\), what is its molarity after a reaction time of \(16.5 \mathrm{~h}\) ? (c) Devise a possible reaction mechanism with a unimolecular rate-determining step. (d) Is the reaction product chiral or achiral? Explain. (e) Draw a crystal field energy-level diagram for trans-[Co(en) \(\left._{2} \mathrm{Cl}_{2}\right]^{+}\) that takes account of the fact that \(\mathrm{Cl}\) is a weaker-field ligand than ethylenediamine.

Short answer

(a) The half-life is approximately 6.02 hours. (b) Molarity after 16.5 h is approximately 0.0207 M. (c) Unimolecular mechanism with Cl dissociation as slow step. (d) The product is achiral. (e) Draw octahedral CFT with lower split for Cl.

Step-by-step solution

Calculate Half-Life

For a first-order reaction, the half-life \( t_{1/2} \) is given by the formula:\[t_{1/2} = \frac{0.693}{k}\]Substitute \( k = 3.2 \times 10^{-5} \, \text{s}^{-1} \):\[t_{1/2} = \frac{0.693}{3.2 \times 10^{-5}} = 21656.25 \, \text{s}\]Convert seconds to hours:\[21656.25 \, \text{s} \times \frac{1 \, \text{hour}}{3600 \, \text{s}} = 6.015 \, \text{hours}\]So, the half-life of the reaction is approximately 6.02 hours.

Determine Molarity After 16.5 Hours

For a first-order reaction, the concentration after time \( t \) can be found using the equation:\[[A] = [A]_0 \cdot e^{-kt}\]Given \([A]_0 = 0.138 \, \text{M}\), \( k = 3.2 \times 10^{-5} \, \text{s}^{-1} \), and \( t = 16.5 \, \text{h} = 59400 \, \text{s} \), we substitute:\[[A] = 0.138 \cdot e^{-3.2 \times 10^{-5} \times 59400}\]\[[A] = 0.138 \cdot e^{-1.9008}\]\[[A] = 0.138 \cdot 0.1498 = 0.02067 \, \text{M}\]So, the molarity after 16.5 hours is approximately 0.0207 M.

Propose Reaction Mechanism

A possible unimolecular mechanism involves the dissociation of the chloride ligand in the slow, rate-determining step:1. \(\text{trans-}[\text{Co}( ext{en})_2\text{Cl}_2]^+ \rightarrow \text{trans-}[\text{Co}( ext{en})_2\text{Cl}]^{2+} + \text{Cl}^-\) (slow)2. \(\text{trans-}[\text{Co}( ext{en})_2\text{Cl}]^{2+} + \text{H}_2\text{O} \rightarrow \text{trans-}[\text{Co}( ext{en})_2(\text{H}_2\text{O})\text{Cl}]^{2+}\) (fast)\, This mechanism fits the first-order kinetics as the slow step only involves the concentration of the complex.

Determine Chirality of the Product

The product, trans-[Co(en)\( _2(\text{H}_2\text{O})\text{Cl}]^{2+}\), has a center of symmetry, making it achiral. Trans complexes with coordination number 6 and bidentate ligands like ethylenediamine typically don't exhibit optical isomerism.

Draw Crystal Field Energy Diagram

For \([\text{Co(en)}_2\text{Cl}_2]^+\), which is in an octahedral field, Cl being a weaker-field ligand than en, the \(\Delta_o\) is split less for chlorides:- The splitting pattern is \( t_{2g} (d_{xy}, d_{yz}, d_{zx}) \) lower in energy.- \( e_g (d_{x^2-y^2}, d_{z^2}) \) is higher.- Place the electrons according to \(\text{Co}^{3+}\)'s d6 configuration: \( t_{2g}^6 \).

Key concepts

First-order Reactions

In coordination chemistry, a *first-order reaction* is a type of chemical reaction where the rate depends linearly on only one reactant's concentration. You can express the rate of the reaction as \[ \text{Rate} = k[A] \]where \(k\) is the rate constant and \([A]\) is the concentration of the reactant. First-order reactions are common in the substitution of ligands on metal complexes, as seen in our exercise.

To calculate the half-life for a first-order reaction, use the formula:\[ t_{1/2} = \frac{0.693}{k} \] This formula indicates that the half-life is constant and does not depend on the initial concentration of the reactant. This makes first-order reactions particularly straightforward to analyze and predict.

• For the given reaction, the half-life is governed by the rate constant, \(3.2 \times 10^{-5} \, \text{s}^{-1}\), resulting in a half-life of approximately 6.02 hours.

Crystal Field Theory

*Crystal Field Theory (CFT)* is an essential concept in coordination chemistry, helping to understand the electronic structures and color of metal complexes. In CFT, we consider how ligands affect the degenerate "d" orbitals of the central metal ion by splitting them in energy.

When ligands approach the metal, they create an electrostatic field that causes the normally degenerate \(d\) orbitals to split. In an octahedral complex, such as trans-\([\text{Co(en)}_2\text{Cl}_2]^+\), the \(d\) orbitals split into two sets:

• \(t_{2g}\) (\(d_{xy}, d_{yz}, d_{zx}\))
• \(e_g\) (\(d_{x^2-y^2}, d_{z^2}\))

The splitting depends on the strength of the ligand field. In our example, chloride (Cl) is a weaker-field ligand compared to ethylenediamine (en), resulting in less energy splitting. Hence, the \(\Delta_o\) (octahedral splitting energy) is smaller. For the \(\text{Co}^{3+}\) complex with a \(d^6\) configuration, electrons fill the lower-energy \(t_{2g}\) orbitals first, consistent with the higher-spin state as influenced by CFT.

Chemical Chirality

*Chemical chirality* refers to the property of a molecule that is not superimposable on its mirror image, much like human hands. Chiral molecules have "handedness" and usually lack a plane or center of symmetry.

In coordination chemistry, chirality can be observed in complexes with specific arrangements of ligands. However, for molecules like the product trans-\([\text{Co(en)}_2(\text{H}_2\text{O})\text{Cl}]^{2+}\), chirality is absent.

• Trans complexes often exhibit a plane or center of symmetry, making them achiral.
• The ligands in a trans configuration create a symmetric arrangement, preventing optical isomerism.
• Complexes with bidentate ligands such as ethylenediamine in this setup typically do not result in chiral centers due to symmetric distribution.

Reaction Mechanism

Understanding the *reaction mechanism* is crucial for identifying the step-by-step sequence of elementary reactions leading to a product. For the substitution reaction in coordination chemistry, proposing a mechanistic pathway helps illustrate how the reactants transform into products over time.

For the given coordination complex, a unimolecular mechanism involving a rate-determining step is proposed:

• First step (slow): The chloride ligand dissociates from the metal center, forming a reactive intermediate: \[\text{trans-}[\text{Co}(\text{en})_2\text{Cl}_2]^+ \rightarrow \text{trans-}[\text{Co}(\text{en})_2\text{Cl}]^{2+} + \text{Cl}^-\]
• Second step (fast): The water molecule coordinates with the intermediate forming the final product: \[\text{trans-}[\text{Co}(\text{en})_2\text{Cl}]^{2+} + \text{H}_2\text{O} \rightarrow \text{trans-}[\text{Co}(\text{en})_2(\text{H}_2\text{O})\text{Cl}]^{2+}\]

This mechanism is consistent with the first-order kinetics as the rate of the reaction is determined by the slowest step, which involves only the concentration of the trans-\([\text{Co(en)}_2\text{Cl}_2]^+\) complex.