Animated Solution for Chemistry - Coordination Compounds: The octahedral complex of a metal ion $M^{3+}$ with four monodentate ligands $L_1,L_2,L_3$ and $L_4$ absorb wavelengths in the region of red, green, yellow and blue, respectively. The increasing order of ligand strength of the four ligands is

Have you ever wondered why transition metal complexes display such a breathtaking array of colors? From the deep blue of copper sulfate to the vibrant purple of potassium permanganate, the secret lies hidden within the intricate dance of electrons and the invisible forces exerted by surrounding molecules called ligands. Let's embark on a journey to decode this colorful phenomenon by solving a classic problem from coordination chemistry.

When white light strikes a coordination complex, it doesn't just bounce off. The complex acts like a selective filter, absorbing specific wavelengths of light. The color we perceive with our eyes is actually the complementary color—the light that is transmitted or reflected after the specific wavelengths have been absorbed.

But why does the complex absorb light in the first place? It all comes down to energy. According to the Planck-Einstein relation, the energy of a photon ($E$) is inversely proportional to its wavelength ($\lambda$):

Where $h$ is Planck's constant and $c$ is the speed of light. This means that light with a longer wavelength (like Red) carries less energy, while light with a shorter wavelength (like Blue) carries more energy. We can easily remember this using the VIBGYOR spectrum, where wavelength increases from Violet to Red.

The Heart of the Matter

In an isolated transition metal ion, all five $d$-orbitals have the exact same energy (they are degenerate). However, when ligands approach the metal ion to form an octahedral complex, their negatively charged electron clouds repel the electrons in the metal's $d$-orbitals.

Because of the spatial orientation of the $d$-orbitals, they don't all experience the same repulsion. The orbitals pointing directly at the ligands ($e_g$ set) are pushed to a higher energy level, while those pointing between the ligands ($t_{2g}$ set) remain at a lower energy level. This energy gap is known as the Crystal Field Splitting Energy (${\Delta }_o$).

When the complex absorbs light, an electron jumps from the lower $t_{2g}$ level to the higher $e_g$ level. The energy of the absorbed photon must exactly match this energy gap:

Not all ligands are created equal. Some ligands push the $d$-orbitals apart very strongly, creating a large ${\Delta }_o$. These are called strong-field ligands. Others cause only a small splitting and are known as weak-field ligands.

In our problem, we are given four ligands ($L_1,L_2,L_3,L_4$) that absorb Red, Green, Yellow, and Blue light, respectively. Let's arrange these colors by their energy:

1. Red ($L_1$): Longest wavelength $⟹$ Lowest Energy 2. Yellow ($L_3$): Medium-long wavelength $⟹$ Low-Medium Energy 3. Green ($L_2$): Medium-short wavelength $⟹$ Medium-High Energy 4. Blue ($L_4$): Shortest wavelength $⟹$ Highest Energy

Since the absorbed energy is directly equal to the crystal field splitting (${\Delta }_o$), the ligand that absorbs the highest energy light must be causing the largest splitting. Therefore, it must be the strongest ligand!

Following our energy order, the increasing order of ligand strength is:

This perfectly matches option (b).

A Word of Caution: Always read the question carefully! This problem explicitly gave us the absorbed wavelengths. If a question gives you the observed color of the solution, you must first use the complementary color wheel to determine what color was actually absorbed before comparing energies. Stay sharp, and the colors will always guide you to the right answer!