Animated Solution for Chemistry - Coordination Compounds: Homoleptic octahedral complexes of a metal ion $M^{3+}$ with three monodentate ligands $L_1,L_2$ and $L_3$ absorb wavelengths in the region of green, blue and red respectively. The increasing order of the ligand strength is

Imagine you are looking at three different solutions, each glowing with a distinct color: green, blue, and red. These aren't just pretty colors; they are secret messages from the molecules themselves, telling us about their internal structure and the strength of their chemical bonds. Let's decode this message!

Colors and Complexes

We are given three homoleptic octahedral complexes of the same metal ion, $M^{3+}$. The only difference between them is the ligand attached: $L_1$, $L_2$, and $L_3$. We are told that they absorb green, blue, and red light, respectively.

Why does a complex absorb light? When white light passes through a solution of a coordination compound, the molecules absorb a specific wavelength of light to excite an electron from a lower energy d-orbital to a higher energy d-orbital. The color we see is the complementary color of the light absorbed. However, in this problem, we are directly given the color of the absorbed light, which makes our job even easier!

Energy and Wavelength

To understand what these colors mean, we need to take a quick trip to the electromagnetic spectrum. Remember the VIBGYOR pattern? As we move from Violet to Red, the wavelength ($\lambda$) of the light increases. Therefore, among the colors given, red has the longest wavelength, followed by green, and blue has the shortest.

Mathematically, we can write:

Now, let's bring in Planck's quantum theory. The energy of a photon is inversely proportional to its wavelength, given by the famous equation:

This means that the light with the longest wavelength (red) carries the least energy, while the light with the shortest wavelength (blue) carries the most energy.

Crystal Field Splitting

How does this relate to our ligands? Enter Crystal Field Theory (CFT). When ligands approach a central metal ion, they cause the five degenerate d-orbitals to split into two sets of different energies: the lower $t_{2g}$ and the higher $e_g$ orbitals. The energy gap between these two sets is called the Crystal Field Splitting Energy, denoted by ${\Delta }_o$.

When an electron absorbs light and jumps across this gap, the energy of the absorbed light is exactly equal to ${\Delta }_o$:

Here is the crucial part: the magnitude of this splitting (${\Delta }_o$) depends directly on the strength of the ligand. A strong-field ligand pushes the orbitals further apart, creating a large ${\Delta }_o$, which requires high-energy (short wavelength) light to cross. Conversely, a weak-field ligand causes a small ${\Delta }_o$, requiring low-energy (long wavelength) light.

The Final Verdict

Let's synthesize our master equation:

Since ligand strength is inversely proportional to the absorbed wavelength, the order of ligand strength will be the exact reverse of the wavelength order.

We established that:

Therefore, the increasing order of ligand strength must be:

And there we have it! By simply looking at the colors absorbed by the complexes, we've successfully deduced the relative strengths of their invisible chemical bonds. The universe is truly elegant when you know how to read its equations.