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Step-by-Step Solution
Step 1: Determine the Oxidation State and Electronic Configuration of Titanium
Titanium (Ti) in its neutral atom state has the electronic configuration [Ar] 3d2 4s2. In the complex [Ti(H2O)6]3+, the titanium ion is Ti3+. Losing three electrons means its electronic configuration becomes [Ar] 3d1.
Step 2: Understand the Coordination Environment
The complex [Ti(H2O)6]3+ is octahedral, where the six water molecules act as ligands around the titanium center.
Step 3: Apply Crystal Field Theory for an Octahedral Complex
In an octahedral crystal field, the five d-orbitals split into two sets:
t2g (lower energy), which has three orbitals
eg (higher energy), which has two orbitals
When there is one d-electron (as in Ti3+), it goes into one of the t2g orbitals. Hence, we have:
$n_{t_{2g}} = 1$ (one electron in the t2g level)
$n_{e_g} = 0$ (no electron in the eg level)
Pairing does not occur here (weak field ligand), so $n = 0$
Step 4: Express the CFSE (Crystal Field Stabilization Energy)
For an octahedral complex, the general formula for CFSE is:
$ \text{CFSE} = \bigl(-0.4 \Delta_0 \bigr) \times n_{t_{2g}} + \bigl(+0.6 \Delta_0 \bigr) \times n_{e_g} + nP $
Where:
$ \Delta_0$ is the octahedral crystal field splitting
$n_{t_{2g}}$ is the number of electrons occupying the t2g orbitals
$n_{e_g}$ is the number of electrons occupying the eg orbitals
$P$ is the pairing energy, and $n$ is the number of pairs
Substituting $n_{t_{2g}} = 1$, $n_{e_g} = 0$, and $n = 0$:
$ \text{CFSE} = \bigl(-0.4 \Delta_0 \bigr) \times 1 = -0.4 \Delta_0 $
Step 5: Use the Absorption Peak to Find $ \Delta_0 $
The single broad peak in the electronic spectrum of [Ti(H2O)6]3+ is at 20,300 cm-1. This value approximately corresponds to the energy required for the d–d transition, which we take as $ \Delta_0 $ for the octahedral complex:
$ \Delta_0 = 20{,}300 \text{ cm}^{-1} $
Step 6: Calculate the CFSE in cm-1
Substitute $ \Delta_0 = 20{,}300 \text{ cm}^{-1} $ into $-0.4 \Delta_0$:
$ \text{CFSE} = -0.4 \times 20{,}300 \text{ cm}^{-1} = -8{,}120 \text{ cm}^{-1}. $
The negative sign indicates stabilization.
Step 7: Convert the CFSE from cm-1 to kJ mol-1
1 kJ mol-1 ≈ 83.7 cm-1. Thus, converting:
$ \text{CFSE (in kJ mol}^{-1}\text{)} = \frac{8{,}120 \text{ cm}^{-1}}{83.7 \text{ cm}^{-1}\text{/}(\text{kJ mol}^{-1})} \approx 97 \text{ kJ mol}^{-1}. $
Final Answer
The crystal field stabilization energy (CFSE) of the complex [Ti(H2O)6]3+ is approximately 97 kJ mol-1.
