Problem 80 - Chemistry: The Central Science in SI Units, Global Edition

Question

The complexes \(\left[\mathrm{CrBr}_{6}\right]^{3-}\) and \(\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\) are both known. (a) Draw the \(d\) -orbital energy-level diagram for octahedral \(\mathrm{Cr}(\mathrm{III})\) complexes. (b) What gives rise to the colors of these complexes? (c) Which of the two complexes would you expect to absorb light of higher energy?

Step-by-Step Solution

Answer

The d-orbital energy-level diagram for octahedral Cr(III) complexes shows three spin-up electrons in the lower-energy t₂g orbitals and empty e_g orbitals. The colors of these complexes arise from electronic transitions between the d-orbitals when light is absorbed. Comparing the ligand-field splitting energies for both complexes, we find that [Cr(NH₃)₆]³⁺ absorbs light of higher energy due to a larger energy gap between the t₂g and e_g orbitals, as NH₃ causes a larger splitting than Br⁻ according to the spectrochemical series.

Step 1: (a) d-orbital energy-level diagram for octahedral Cr(III) complexes

To draw the d-orbital energy-level diagram for octahedral Cr(III) complexes, we must first know the electron configuration of Cr(III). Chromium has an atomic number of 24, and its ground state electron configuration is [Ar] 3d^5 4s^1. In the Cr(III) state, it loses 3 electrons, resulting in the following electron configuration: [Ar] 3d^3. In an octahedral complex, the d-orbitals split into two sets due to the crystal field: the t₂g set (dxy, dyz, dxz) and the e_g set (dz^2, dx^2-y^2). The t₂g orbitals are lower in energy than the e_g orbitals. The Cr(III) ion has 3 electrons to distribute among the d-orbitals. These electrons will fill the t₂g orbitals first, following Hund's rule. The resulting d-orbital energy-level diagram is as follows: (1) t₂g: ↑↑↑ (three spin-up electrons) (2) e_g : (empty)

Step 2: (b) Cause of the colors of these complexes

The colors of these complexes arise from the electronic transitions between the d-orbitals. When light is absorbed, an electron from a lower energy d-orbital (t₂g) is promoted to a higher energy d-orbital (e_g). The energy difference between these orbitals (∆E) corresponds to the energy of the absorbed photon and depends on the ligands surrounding the metal ion. Since visible light covers a range of energies (and hence colors), the absorbed light's energy (color) will determine the color of the complex as perceived by the human eye. The complementary color of the absorbed light will be observed as the color of the complex.

Step 3: (c) Comparing the energy of absorbed light for both complexes

To determine which of the two complexes would absorb light of higher energy, we must compare the ligand-field splitting energies in both cases. Different ligands cause different degrees of splitting, which is summarized in the spectrochemical series, a list of ligands ordered by their ability to split d-orbitals: I^- < Br^- < Cl^- < F^- < OH^- < H_2O < NH_3 < en < NO_2^- < CN^- In the given complexes, the Cr ion is coordinated to Br^- in the first complex and NH_3 in the second complex. According to the spectrochemical series, NH_3 causes a larger splitting than Br^-. A larger splitting means that the energy gap between the t₂g and e_g orbitals will be greater in the [Cr(NH₃)₆]³⁺ complex compared to the [CrBr₆]³⁻ complex. Therefore, the [Cr(NH₃)₆]³⁺ complex would absorb light of higher energy, as the electronic transitions in this complex require a higher energy to promote an electron from the t₂g orbitals to the e_g orbitals.

Key Concepts

d-Orbital Energy-Level Diagram

One of the essential aspects of understanding transition metal complexes is the splitting of the d-orbitals in different geometric arrangements. In the case of octahedral complexes, the d-orbitals experience an interaction with ligands, leading to energy level splitting. This phenomenon is deeply connected to the electron configuration of the metal center.
For chromium(III), which has the electron configuration \ ext{[Ar] 3d}^3\, we observe that in an octahedral field, the five degenerate d-orbitals split into two groups: the lower-energy \(t_{2g}\) orbitals \((d_{xy}, d_{yz}, d_{xz})\) and the higher-energy \(e_g\) orbitals \((d_{z^2}, d_{x^2-y^2})\).
In general:

• The \(t_{2g}\) set is lower in energy because these orbitals point between the axes where ligands are located.
• The \(e_g\) set is higher in energy as these orbitals point directly at the ligands along the octahedral axes.

In an octahedral Cr(III) complex, you will place the three d-electrons in the \(t_{2g}\) orbitals, according to the Aufbau principle and Hund's rule, which minimizes electron repulsion.

Octahedral Coordination

When it comes to coordination chemistry, octahedral complexes are among the most studied due to their prevalence in transition metal chemistry. In an octahedral complex, six ligands symmetrically surround a central metal ion. This arrangement influences both the geometry and the electronic properties of the metal.
The nature of the ligands and the geometry contribute significantly to the properties of the complex:

• The six ligands coordinate in a way that they maximize their distance from each other, forming an octahedral shape.
• This arrangement creates two different environments for the d-orbitals, leading to the crystal field splitting observed.

In terms of symmetry, octahedral complexes have a high degree of symmetry, which affects vibrational and optical properties. These complexes provide insights into the electronic transitions that depend on ligand interactions and geometry.

Ligand-Field Splitting

Ligand-field splitting is a core concept in transition metal chemistry, explaining the behavior of electrons in split energy levels within a complex. This splitting dictates many properties including color and reactivity. The size of \(\Delta\) (the energy gap between \(t_{2g}\) and \(e_g\) orbitals) is influenced by the ligands surrounding the metal – described by the spectrochemical series.
The spectrochemical series is a ranking that shows a ligand's ability to cause d-orbital splitting:

• Weak field ligands like \(I^-\) and \(Br^-\) cause small splitting \(\Delta\).
• Strong field ligands such as \(NH_3\) and \(CN^-\) lead to significant splitting.

In the exercise, we compared the complexes \([\text{CrBr}_6]^{3-}\) and \([\text{Cr(NH}_3\text{)}_6]^{3+}\). \(NH_3\), being a stronger field ligand than \(Br^-\), results in a larger \(\Delta\), leading to the absorption of higher energy light with \([\text{Cr(NH}_3\text{)}_6]^{3+}\). This principle is an essential explanation for the difference in coloration and absorption properties of complexes with different ligands.