Problem 80 - Chemistry The Central Science In Si Units

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 Cr(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

(a) Draw d-orbital diagram for octahedral Cr(III) with t2g lower and eg higher. (b) Colors arise from light absorption corresponding to electron transitions between t2g and eg. (c) [Cr(NH3)6]^{3+} absorbs higher energy light.

Step 1: Understanding Octahedral Cr(III) Complexes

For Cr(III) complexes, the central chromium ion has a 3+ charge, meaning it loses three electrons. This results in a \(d^3\) electron configuration because chromium originally has the configuration \[3d^5 4s^1\]. By losing three electrons, it becomes \[3d^3\].

Step 2: Drawing the d-Orbital Energy Level for Octahedral Complexes

In an octahedral field, the \(d\)-orbitals split into two sets: the lower-energy \(t_{2g}\) (d_{xy}, d_{xz}, d_{yz}) and the higher-energy \(e_g\) (d_{x^2-y^2}, d_{z^2}) orbitals. For a \(d^3\) configuration, the three electrons fill the \(t_{2g}\) orbitals, leaving the \(e_g\) orbitals unoccupied. The energy difference between these two sets is called the crystal field splitting energy, \Delta_o\.

Step 3: Analyzing the Color of Complexes

The color of complexes arises from the absorption of light that corresponds to the energy difference \Delta_o\ between the \(t_{2g}\) and \(e_g\) orbitals. When light is absorbed, an electron can be excited from \(t_{2g}\) to \(e_g\). The complementary color of the absorbed light is what we observe.

Step 4: Comparing the Ligand Field Strength

Ammonia \((NH_3)\) is a stronger field ligand than bromide \((Br^-)\). This means that \[\Delta_o(\text{NH}_3) > \Delta_o(Br^-)\]. Therefore, \left[\text{Cr}(NH_3)_6\right]^{3+}\ will absorb higher energy (shorter wavelength) light compared to \left[\text{CrBr}_6\right]^{3-}\ because it has a larger \Delta_o\.

Key Concepts

Octahedral Complexes

Transition metal complexes can have different geometries depending on the ligands attached to the central metal ion. One common geometry is the octahedral complex, which involves six ligands symmetrically arranged around a central metal ion. This results in an octahedral shape, similar to two square pyramids joined at their bases.

In an octahedral complex, the coordination number is usually six, meaning there are six ligand atoms directly bonded to the metal center. These types of complexes are prevalent in chemistry because they can accommodate a wide range of ligands. The arrangement of ligands in an octahedral leads to significant interest in their electronic properties, particularly the distribution of the central metal's d-orbitals.

• Common in transition metals, especially with electron-rich metals.
• Helps stabilize the metal ion with its symmetrical arrangement.
• Often lead to interesting magnetic and electronic properties.

Crystal Field Theory

This theory is essential for understanding how ligands affect the electronic structure of metal ions in complexes. According to crystal field theory, when ligands approach a metal ion, they exert an electrostatic force that splits the metal's degenerate d-orbitals into different energy levels. For octahedral complexes, this splitting results in a specific pattern due to the symmetrical arrangement of the ligands around the metal ion.

In the case of an octahedral field, the five d-orbitals split into two distinct groups:

• The lower-energy set, known as the \(t_{2g}\) orbitals, includes \(d_{xy}, d_{xz}, \text{and} \,d_{yz}\) orbitals.
• The higher-energy set, known as the \(e_g\) orbitals, is composed of \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals.

The difference in energy levels created by this splitting is referred to as the crystal field splitting energy, \(\Delta_o\). This energy gap determines many of the complex's optical and magnetic properties. A larger \(\Delta_o\) can lead to the absorption of light at higher energies, influencing the color we observe.

d-Orbital Splitting

At the heart of understanding the behavior of transition metal complexes in an octahedral field is d-orbital splitting. When metal ions engage with ligands, the symmetrical distribution in an octahedral arrangement leads to the d-orbitals splitting into different energy states due to repulsion and the electrostatic environment created by the ligands.

For a metal ion with a \(d^n\) electron configuration, electrons will fill the lower-energy \(t_{2g}\) orbitals first, followed by the \(e_g\) orbitals if there are remaining electrons. This distribution plays a crucial role in the properties and reactivity of the complex.

• \(\Delta_o\), the energy difference between \(t_{2g}\) and \(e_g\) orbitals, influences electronic transitions.
• The arrangement affects how electrons are distributed among the \(d\)-orbitals, impacting magnetic properties.
• Ligands with stronger fields produce a larger \(\Delta_o\), leading to higher-energy absorption and often more intense colors.

Understanding the splitting of d-orbitals allows chemists to predict and explain the electronic behavior and physical properties of these complexes.