Animated Solution for Chemistry - Coordination Compounds: The electronic spectrum of $[Ti(H_{2}O{)}_6{]}^{3+}$ shows a single broad peak with a maximum at $20,300{\text{ cm}}^{-1}$. The crystal field stabilisation energy (CFSE) of the complex ion, in ${\text{kJ mol}}^{-1}$, is ($1{\text{ kJ mol}}^{-1}=83.7{\text{ cm}}^{-1}$)

Decoding the Complex

Imagine you are looking at a vibrant solution of the hexaaquatitanium(III) complex, mathematically written as $[Ti(H_{2}O{)}_6{]}^{3+}$. Our first mission is to understand the state of the central metal ion.

Water ($H_{2}O$) is a neutral ligand, meaning it contributes zero charge to the overall complex. Therefore, the entire $+3$ charge belongs exclusively to the titanium ion, making it $T{i}^{3+}$.

Titanium, in its neutral ground state, has an atomic number of 22, giving it an electronic configuration of $[Ar]3d^{2}4s^2$. When it loses three electrons to become $T{i}^{3+}$, it loses the two $4s$ electrons and one $3d$ electron. This leaves us with a $3d^1$ configuration.

Crystal Field Theory in Action

Now, let's bring in the ligands. When six water molecules approach the central $T{i}^{3+}$ ion to form an octahedral complex, their electron clouds repel the electrons in the metal's d-orbitals.

Because of the spatial orientation of the d-orbitals, this repulsion is not uniform. The five previously degenerate (equal energy) d-orbitals split into two distinct energy levels: a lower energy set of three orbitals called $t_{2g}$, and a higher energy set of two orbitals called $e_g$.

Our single electron is lazy—it wants to be in the most stable, lowest energy state possible. Naturally, it drops into one of the $t_{2g}$ orbitals. Thus, the electronic configuration in the crystal field becomes $t_{2g}^1{e}_g^0$.

Understanding the Absorption Peak

The problem states that the electronic spectrum shows a single broad peak with a maximum at $20,300{\text{ cm}}^{-1}$. What does this mean physically?

When light shines on the complex, our single electron can absorb a photon and jump from the lower $t_{2g}$ level to the higher $e_g$ level. The energy required for this exact jump is the energy difference between the two levels, which is defined as the crystal field splitting energy, ${\Delta }_0$.

Therefore, the energy of the absorbed light directly gives us the splitting energy:

Calculating Stabilization Energy

By placing the electron in the lower $t_{2g}$ level instead of the hypothetical unsplit barycenter, the system has stabilized itself. We calculate this Crystal Field Stabilization Energy (CFSE) using the standard formula:

Here, $n_{t_{2g}}$ is the number of electrons in the $t_{2g}$ level, and $n_{e_g}$ is the number of electrons in the $e_g$ level. Substituting our values ($n_{t_{2g}}=1$ and $n_{e_g}=0$):

Now, we plug in the value of ${\Delta }_0$ that we found from the absorption peak:

Unit Conversion

We have the energy, but there is a catch! The options provided are in ${\text{kJ mol}}^{-1}$, not wavenumbers (${\text{cm}}^{-1}$).

Thankfully, the problem provides a conversion factor: $1{\text{ kJ mol}}^{-1}=83.7{\text{ cm}}^{-1}$. To convert our answer, we simply divide by this factor:

The negative sign simply indicates that the energy of the system has decreased, meaning it has become more stable. When asked for the value of CFSE in such multiple-choice questions, we look for the magnitude.