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Q5.25:
Explain the violet colour of the complex \([Ti(H_{2}O)_{6}]^{3+}\) on the basis of crystal field theory.

Solution :

Step 1: Electronic Configuration and Coordination Environment

The central metal ion in the complex $[Ti(H_{2}O)_{6}]^{3+}$ is Titanium in the $+3$ oxidation state. The ground state configuration of a neutral Titanium atom ($Z=22$) is $[Ar] 3d^2 4s^2$. Upon losing three electrons to form $Ti^{3+}$, the configuration becomes:

$Ti^{3+} = [Ar] 3d^1$

[Per the Aufbau Principle and Hund's Rule, the remaining electron resides in one of the five degenerate $d$-orbitals.]

Step 2: Crystal Field Splitting in Octahedral Geometry

As six $H_2O$ ligands approach the $Ti^{3+}$ ion in an octahedral arrangement, the degeneracy of the $d$-orbitals is lifted due to the electrostatic repulsion between the ligand lone pairs and the metal $d$-electrons. The $d$-orbitals split into two distinct energy levels:

  • $t_{2g}$ set: Lower energy ($d_{xy}, d_{yz}, d_{zx}$), containing 3 degenerate orbitals.
  • $e_g$ set: Higher energy ($d_{x^2-y^2}, d_{z^2}$), containing 2 degenerate orbitals.
Energy Free Ion e_g t_{2g} Δ_o

Step 3: d-d Transition Mechanism

The single electron occupies the lower energy $t_{2g}$ orbital. When the complex absorbs light from the white light spectrum, the energy of the photon corresponds to the energy gap ($\Delta_o$) between the $t_{2g}$ and $e_g$ levels [Per the Bohr Frequency Condition: $\Delta E = h\nu = \frac{hc}{\lambda}$].

The electron is promoted from the $t_{2g}$ level to the $e_g$ level. This process is known as a $d-d$ transition. For $[Ti(H_{2}O)_{6}]^{3+}$, the energy absorbed corresponds to the yellow-green portion of the visible spectrum.

Step 4: Complementary Color Observation

The color observed by the human eye is the complementary color of the light absorbed. According to the color wheel of visible light:

Absorbed ColorObserved Color
Yellow-GreenViolet/Purple

Since the complex absorbs the yellow-green light to facilitate the excitation of the $d$-electron, the transmitted or reflected light is perceived as violet.

Final Solution: The violet color of $[Ti(H_{2}O)_{6}]^{3+}$ arises from the excitation of the single $3d$ electron from the $t_{2g}$ level to the $e_g$ level ($d-d$ transition) upon the absorption of yellow-green light, resulting in the observation of the complementary violet color.


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