Question

Sketch and explain the most likely pattem for the crystal field diagram for the complex ion trans-diamminetetracyanonickelate(II), where \(\mathrm{CN}^{-}\) produces a much stronger crystal field than \(\mathrm{NH}_{3}\). Explain completely and label the \(d\) orbitals in your diagram. Assume the \(\mathrm{NH}_{3}\) ligands lie on the \(z\) axis.

Short answer

In the trans-diamminetetracyanonickelate(II) complex with an octahedral geometry, the five d-orbitals split into \(t_{2g}\) (containing \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\)) and \(e_{g}\) (containing \(d_{x^2-y^2}\) and \(d_{z^2}\)) groups. Due to the stronger CN⁻ ligands along the \(x\) and \(y\) axes, the \(d_{x^2-y^2}\) orbital has the highest energy, while the weaker NH₃ ligands along the \(z\) axis result in a slightly lower energy for the \(d_{z^2}\) orbital. The three \(t_{2g}\) orbitals remain non-bonding and unchanged in energy.

Step-by-step solution

Determine the geometry of the complex ion.

We are given that the complex ion is trans-diamminetetracyanonickelate(II). In this complex, there are two NH₃ ligands (ammine) and four CN⁻ ligands (cyanide). The trans prefix indicates that the two ammine ligands are opposite each other in the complex. This suggests that the geometry of the complex is octahedral, with the four cyanide ligands occupying the other four vertices of the octahedron.

Determine the position and strength of ligands in the crystal field.

Now that we have determined the geometry of the complex, we need to consider the strengths and positions of the ligands in the crystal field. We are told that CN⁻ produces a much stronger crystal field than NH₃. Therefore, the greatest energy difference will be between the d-orbitals interacting with these stronger ligands. We are also given that the \(\mathrm{NH}_{3}\) ligands lie on the \(z\) axis. Therefore, the CN⁻ ligands lie on the \(x\) and \(y\) axes.

Draw the crystal field diagram.

To get the crystal field diagram for the complex ion, we need to sketch the splitting of the d-orbitals in an octahedral crystal field. In an octahedral field, the five d-orbitals (\(d_{xy}\), \(d_{xz}\), \(d_{yz}\), \(d_{x^2-y^2}\), and \(d_{z^2}\)) are split into two groups: \(t_{2g}\) (composed of \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\)) and \(e_{g}\) (composed of \(d_{x^2-y^2}\) and \(d_{z^2}\)). Since the CN⁻ ligands are stronger and lie along the \(x\) and \(y\) axes, the greatest energy difference will be between \(d_{x^2-y^2}\) and the other d-orbitals. Therefore, the \(d_{x^2-y^2}\) orbital will be at a higher energy level than the other orbitals in the \(e_{g}\) group. The \(d_{z^2}\) orbital will have a slightly lower energy since it interacts with the weaker NH₃ ligands along the \(z\) axis. The three \(t_{2g}\) orbitals will remain non-bonding and unchanged in energy.

Label and explain the crystal field diagram.

Now that we have sketched the crystal field diagram, we need to label the d-orbitals and energy levels. Starting from the lowest energy level, we have: - The \(t_{2g}\) group, consisting of \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals, which are non-bonding and unchanged in energy. - The \(d_{z^2}\) orbital, which interacts with the weak field NH₃ ligands along the \(z\) axis and has slightly increased energy. - The \(d_{x^2-y^2}\) orbital, which interacts with the strong field CN⁻ ligands along the \(x\) and \(y\) axes and has the highest energy. In this crystal field diagram, the splitting of the d-orbitals can be attributed to the difference in strength and position of the ligands in the crystal field. The CN⁻ ligands create a much stronger crystal field than NH₃, causing the \(d_{x^2-y^2}\) orbital to have a higher energy. The weaker NH₃ ligands cause the \(d_{z^2}\) orbital to be at a lower energy level than the \(d_{x^2-y^2}\) orbital. Overall, this crystal field diagram illustrates the energies of the d-orbitals in the complex ion trans-diamminetetracyanonickelate(II).

Key concepts

Octahedral Complex

In crystal field theory, an octahedral complex refers to a type of structure where a central metal ion is surrounded by six ligands positioned at the corners of an octahedron. This spatial arrangement significantly influences the electronic structure of the metal ion, especially the distribution of its d-orbitals. Octahedral complexes are prevalent in coordination chemistry and are known for their symmetrical arrangement and the ability to form strong metal-ligand bonds.
When considering the example of the trans-diamminetetracyanonickelate(II), it's essential to note that the layout of ligands in octahedral geometry directly affects the electronic environment of the central nickel ion. The 'trans' configuration indicates ligands are directly opposite each other, presenting a unique crystal field impact compared with other geometric orientations such as 'cis'. Understanding this geometry is the foundation for predicting electronic configurations and consequent magnetic and spectral properties.

D-Orbital Splitting

The d-orbital splitting in an octahedral complex arises due to the electrostatic interactions between the central metal ion's d-electrons and the electric fields produced by surrounding ligands. In a perfectly symmetrical octahedral field, the five d-orbitals split into two groups with different energy levels: the lower-energy trio called t_{2g} (consisting of the d_xy, d_xz, and d_yz orbitals) and the higher-energy pair called e_{g} (composed of the d_x^2-y^2 and d_z^2 orbitals).
In the case of our complex ion trans-diamminetetracyanonickelate(II), the splitting pattern is influenced by the presence of strong-field CN⁻ ligands and weak-field NH₃ ligands. The CN⁻ ligands, lying on the x and y axes, elevate the energy of the d_x^2-y^2 orbital more so compared to d_z^2, which is only slightly affected by the weaker NH₃ on the z axis. This intricate splitting of the d-orbitals, as demonstrated in customizable crystal field diagrams, is essential for understanding the electronic transitions and chemical behavior of the complex.

Ligand Field Strength

The ligand field strength is a measure of a ligand's ability to split the d-orbitals of a central metal ion in a coordination complex. Ligands that cause larger splitting are referred to as strong-field ligands, whereas those causing smaller splitting are known as weak-field ligands. Ligand field strength can be predicted using the spectrochemical series, which ranks ligands from weak to strong field based on the splitting energy they induce.
In the problem at hand, cyanide (CN⁻) is identified as a much stronger ligand than ammonia (NH₃). As such, CN⁻ has a more significant impact on the d-orbital energy levels, particularly on the d_x^2-y^2 orbital aligned with the x and y axes. This difference in field strength between CN⁻ and NH₃ is crucial for predicting the electronic arrangement of octahedral complexes, such as our example complex, and also for explaining their physical and chemical properties.

Crystal Field Stabilization Energy

The concept of crystal field stabilization energy (CFSE) is pivotal when analyzing transition metal complexes. CFSE is the energy difference between the energy of the d-orbitals in the presence of a ligand field compared to a hypothetical situation where no splitting occurs. A complex with a negative CFSE is considered more stable because the electrons are in a lower energy state than they would be without the influence of the ligands.
For the trans-diamminetetracyanonickelate(II) complex, the crystal field created by the strong CN⁻ ligands drastically raises the energy of the d_x^2-y^2 orbital, which translates into a different CFSE than if only NH₃ ligands were present. The specific pattern of d-orbital splitting and the resulting CFSE provide insights into the behavior of the complex, including its color, magnetism, and reactivity. It also explains why certain ligand arrangements lead to more stable complexes than others.