Some octahedral complexes have distorted shapes. In some, two metal-ligand bonds that are \(180^{\circ}\) apart are shorter than the other four. \(\operatorname{In}\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+},\) for example, two \(\mathrm{Cu}-\mathrm{N}\) bonds are \(207 \mathrm{pm}\) long, and the other four are \(262 \mathrm{pm}\) long. (a) Calculate the longest distance between two \(\mathrm{N}\) atoms in this complex. (b) Calculate the shortest distance between two \(\mathrm{N}\) atoms.

Octahedral complexes are a fundamental concept in coordination chemistry. Imagine an octahedron as a geometric shape with eight faces and six vertices. In an octahedral complex, six ligands (molecules or ions that can donate a pair of electrons to the metal) are symmetrically arranged around a central metal ion at the vertices of an octahedron.
In the octahedral arrangement, the angles between the ligands are all 90 degrees, contributing to the stability and predictability of these compounds.

For example, in the complex \(\text{[Cu(NH}_3\text{)}_6]^{2+}\), six ammonia (NH3) molecules serve as ligands, surrounding a central copper ion (Cu). These positions resemble the shape of an octahedron, giving the complex its name. However, not all octahedral complexes are perfectly symmetrical. Distortions commonly occur and are important for understanding the geometry and reactivity of these complexes.

Metal-ligand bonds are the connections formed between the metal ion in a complex and the surrounding ligands. These can be covalent, ionic, or a combination of both, depending on the metal and the ligand involved.
In an octahedral complex like \(\text{[Cu(NH}_3\text{)}_6]^{2+}\), metal-ligand bonds are particularly significant as they determine the geometry and stability of the molecule. The strength of these bonds can vary, leading to different bond lengths and creating distortions in the ideal octahedral shape.

In our example of \(\text{[Cu(NH}_3\text{)}_6]^{2+}\), two \(\text{Cu-N}\) bonds are shorter, measuring 207 pm, while the other four \(\text{Cu-N}\) bonds are longer, measuring 262 pm. This distortion might be due to factors like electronic configuration and ligand properties, influencing the overall symmetry of the complex. Distortions affect the chemical and physical properties of the complex, impacting reactivity, color, and magnetic behavior.

Bond lengths refer to the distances between the nuclei of two bonded atoms. They are a crucial aspect in determining the shape and size of a molecule. In coordination complexes, bond lengths can provide insights into the nature of metal-ligand interactions and the degree of distortion from idealized shapes.

In the octahedral complex \(\text{[Cu(NH}_3\text{)}_6]^{2+}\), the \(\text{Cu-N}\) bond lengths vary significantly. The two shorter bonds are 207 pm long, while the four longer bonds are 262 pm. This variation leads to a distorted octahedral shape, affecting the distances between the nitrogen atoms.
To calculate the longest and shortest distances between nitrogen atoms in this complex, we use the bond lengths:
- **Longest distance:** Occurs across the octahedron's center along the axis defined by the longer bonds. Thus, \(d_{longest} = 2 \times 262 \text{ pm} = 524 \text{ pm}\).
- **Shortest distance:** Occurs between two nitrogen atoms bonded to copper by the shorter bonds, forming a triangular geometry. Calculated using the sine function for the bond angle, \(d_{shortest} = 2 \times 207 \text{ pm} \times \text{sin}(45^{\text{o}}) = 292.63 \text{ pm}\).