1. Find the electronic configuration of the metal ion

In $[Ti(H_2O)_6]^{3+}$, water is a neutral ligand, so the oxidation state of Ti is $+3$.

Titanium: $Z=22$

$\text{Ti} = [Ar]3d^24s^2$

So,

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

Thus, the complex is an octahedral $d^1$ system.

2. Relate the absorption peak to crystal field splitting

For an octahedral $d^1$ ion, the single electronic transition is:

$t_{2g}^1 \rightarrow e_g^1$

The energy of this transition is equal to $\Delta_o$.

Given:

$\Delta_o = 20{,}300\ \text{cm}^{-1}$

3. Write CFSE for an octahedral $d^1$ complex

For one electron in $t_{2g}$:

$\text{CFSE} = -0.4\Delta_o$

Magnitude of CFSE:

$0.4\Delta_o = 0.4 \times 20{,}300 = 8120\ \text{cm}^{-1}$

4. Convert from cm$^{-1}$ to kJ mol$^{-1}$

Use:

$1\ \text{cm}^{-1} = 0.01196\ \text{kJ mol}^{-1}$

Therefore,

$\text{CFSE} = 8120 \times 0.01196$

$= 97.12\ \text{kJ mol}^{-1}$

So,

$\boxed{\text{CFSE} \approx 97\ \text{kJ mol}^{-1}}$

5. Check options

• A: $83.7$ ❌
• B: $242.5$ ❌
• C: $145.5$ ❌
• D: $97$ ✅

Hence, the correct option is D.