Q.Explain the violet colour of the complex $[Ti(H_2O)_6]^{3+}$ on the basis of crystal field theory.

The violet colour arises from a d-d electronic transition within the split $d$-orbitals of the $Ti^{3+}$ ion.

1. Electronic configuration: $Ti^{3+}$ has a $3d^1$ configuration. In the octahedral field of six water ligands, the single $d$-electron occupies the lower-energy $t_{2g}$ set.
2. Crystal field splitting: The energy gap between the $t_{2g}$ and $e_g$ sets is $\Delta_o$ (or $10 Dq$). …

Why Crystal Field Theory Explains Colour

Transition metal complexes appear coloured because electrons in partially filled d-orbitals can absorb visible light and jump to a higher energy level. The key is that in an octahedral field, the five degenerate d-orbitals split into two sets: the lower-energy $t_{2g}$ (three orbitals) and the higher-energy $e_g$ (two orbitals). The energy gap between them, denoted $\Delta_o$ or $10Dq$, determines which wavelength of light gets absorbed.

For $[Ti(H_2O)_6]^{3+}$, titanium is in the +3 oxidation state. Ti has atomic number 22, so Ti³⁺ has the electronic configuration $[Ar]\,3d^1$. This single d-electron is the entire story — there are no other d-electrons to complicate things with electron-electron repulsion or multiple transitions.

Step-by-Step Reasoning

1. Identify the metal ion and its d-electron count.

   Ti in $[Ti(H_2O)_6]^{3+}$ is Ti³⁺. The ground state configuration is $3d^1$. In an octahedral field, this single electron occupies the lower-energy $t_{2g}$ set. The $e_g$ orbitals are empty.

2. Determine the possible electronic transition.

   The only allowed d-d transition is from $t_{2g}$ to $e_g$: the electron absorbs a photon of energy exactly equal to $\Delta_o$ and jumps up. No other transitions are possible because there is only one electron.

3. Find the magnitude of $\Delta_o$ for this complex.

   Experimentally, the absorption spectrum of $[Ti(H_2O)_6]^{3+}$ shows a single broad peak centred at about 20,300 cm⁻¹ (which corresponds to a wavelength of roughly 490–500 nm). This is the energy of the $t_{2g} \rightarrow e_g$ transition.

   Tip

   The wavenumber $\tilde{\nu} = 20,300\ \text{cm}^{-1}$ corresponds to $\lambda = \frac{1}{\tilde{\nu}} \approx 4.93 \times 10^{-5}\ \text{cm} = 493\ \text{nm}$, which lies in the blue-green region of the visible spectrum.

4. Relate absorbed colour to observed colour. …

Method: Crystal Field Theory (CFT) – d-orbital splitting and d–d transition

Why this method?

Crystal Field Theory explains colour in transition metal complexes as arising from electronic transitions between split d-orbitals. The colour observed is complementary to the wavelength of light absorbed during this transition.

Steps

1. Identify the metal ion and its electron configuration

• In $[Ti(H_2O)_6]^{3+}$, titanium is in the +3 oxidation state.
• Ti atomic number = 22 → Ti$^{3+}$ has one d-electron (configuration: $3d^1$).

2. Determine the geometry and crystal field splitting

• The complex is octahedral (six water ligands).
• In an octahedral field, the five d-orbitals split into:
  • Higher energy: $e_g$ orbitals ($d_{x^2-y^2}$, $d_{z^2}$)
  • Lower energy: $t_{2g}$ orbitals ($d_{xy}$, $d_{xz}$, $d_{yz}$)
• The energy gap between these sets is denoted as $\Delta_o$ (or $10 Dq$).

3. Place the electron in the lower energy orbital

• The single d-electron occupies one of the $t_{2g}$ orbitals (lowest energy state).

4. Identify the electronic transition responsible for colour

• When visible light falls on the complex, the electron can absorb energy and jump from $t_{2g}$ to $e_g$ level.
• This is called a d–d transition.
• The energy of the absorbed photon equals $\Delta_o$. …

Here are the most common mistakes students make when explaining the violet colour of $[Ti(H_2O)_6]^{3+}$ using crystal field theory, along with how to avoid each.

Mistake 1: Forgetting the $d^1$ Configuration

• The Mistake: Students often try to apply logic for high-spin vs. low-spin configurations or start calculating pairing energies. They might incorrectly state that the $d$ orbital is completely filled or empty.
• Why it’s wrong: $Ti^{3+}$ has an electronic configuration of $[Ar] 3d^1$. There is only one electron in the $d$ orbitals. There is no possibility of pairing, so high-spin/low-spin is irrelevant.
• How to Avoid: Always start by finding the oxidation state of the metal and then its $d^n$ configuration. For $Ti^{3+}$, it is simply $d^1$. Write this down before discussing splitting.

Mistake 2: Describing the Wrong Geometry or Splitting Pattern

• The Mistake: Assuming the complex is tetrahedral or square planar, or drawing the wrong splitting diagram (e.g., $e_g$ above $t_{2g}$ in an octahedral field).
• Why it’s wrong: $[Ti(H_2O)_6]^{3+}$ is an octahedral complex. In an octahedral field, the $d$ orbitals split into the lower energy $t_{2g}$ set ($d_{xy}, d_{xz}, d_{yz}$) and the higher energy $e_g$ set ($d_{x^2-y^2}, d_{z^2}$).
• How to Avoid: Memorize the standard geometries and their splitting patterns. For a complex with six ligands, always draw the octahedral splitting diagram first. The single electron will occupy the lower energy $t_{2g}$ set.

Mistake 3: Confusing the Colour with the Absorbed Colour

• The Mistake: Stating that the complex is violet because it absorbs violet light.
• Why it’s wrong: The colour we see is the complement of the colour absorbed. $[Ti(H_2O)_6]^{3+}$ appears violet because it absorbs light in the blue-green region of the spectrum (around 490–500 nm, experimentally ~498 nm).
• How to Avoid: Use a colour wheel. If the complex appears violet, the absorbed colour is blue-green. Always state: "The complex appears violet because it absorbs blue-green light, and we see the transmitted complementary colour."

Mistake 4: Ignoring the $d-d$ Transition Mechanism

• The Mistake: Saying the electron jumps from the ligand to the metal (charge transfer) or that the colour is due to the ligand itself.
• Why it’s wrong: The colour arises from a $d-d$ transition. The single electron in the $t_{2g}$ orbital absorbs a photon of visible light and gets excited to the $e_g$ orbital.
• How to Avoid: Explicitly state the transition: "The electron undergoes a $t_{2g} \rightarrow e_g$ transition." This is the core of the crystal field explanation.

Mistake 5: Not Mentioning the Crystal Field Splitting Energy ($\Delta_o$)

• The Mistake: Describing the transition without linking it to the energy gap. …