Animated Solution for Chemistry - Coordination Compounds: $[Ti(H_{2}O{)}_6{]}^{3+}$ absorbs light of wavelength $498\text{ nm}$ during a $d-d$ transition. The octahedral splitting energy for the above complex is ......... $\times 10^{-19}\text{ J}$ (Round off to the nearest integer). $[h=6.626\times 10^{-34}\text{ Js},c=3\times 10^8{\text{ ms}}^{-1}]$

Imagine you are looking at the heart of the $[Ti(H_{2}O{)}_6{]}^{3+}$ complex. Titanium, with an atomic number of 22, normally has the electronic configuration $[Ar]3d^{2}4s^2$. However, in this complex, it exists in a $+3$ oxidation state. Losing three electrons leaves it with just a single electron in its $3d$ subshell, making it a $d^1$ system.

When the six water molecules approach the central titanium ion to form an octahedral complex, their electron clouds repel the $d$-orbitals of titanium. Because of the spatial orientation of these orbitals, they don't all experience the same repulsion. The five degenerate $d$-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 lone electron naturally occupies the lowest available energy state, settling comfortably into one of the $t_{2g}$ orbitals.

Now, what happens when light shines on this complex? If a photon of light possesses exactly the right amount of energy, the electron in the $t_{2g}$ level can absorb it and make a quantum leap to the higher $e_g$ level. This phenomenon is known as a $d-d$ transition, and it is the primary reason why many transition metal complexes exhibit beautiful, vibrant colors!

The energy required for this transition is exactly equal to the energy gap between the $t_{2g}$ and $e_g$ levels. We call this gap the octahedral crystal field splitting energy, denoted by ${\Delta }_o$.

According to Planck's quantum theory, the energy of a photon is directly proportional to its frequency, or inversely proportional to its wavelength. The master equation connecting these concepts is:

Here, $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength of the absorbed light.

The problem provides us with all the necessary pieces of the puzzle. We are given the wavelength of the absorbed light, $\lambda =498\text{ nm}$. Before we plug this into our equation, we must ensure all our units are consistent. We convert nanometers to meters by multiplying by $10^{-9}$, giving us $\lambda =498\times 10^{-9}\text{ m}$.

We also have the standard constants: $h=6.626\times 10^{-34}\text{ Js}$ $c=3\times 10^8{\text{ ms}}^{-1}$

Substituting these values into our master equation, we get:

Let's break down the calculation to avoid any silly mistakes. First, we multiply the numerical coefficients in the numerator:

Next, we handle the powers of 10 in the numerator:

So, our expression simplifies to:

Now, we divide the numbers and subtract the exponent in the denominator from the exponent in the numerator:

To express this in standard scientific notation, we shift the decimal point two places to the right, which decreases the exponent by 2:

We have successfully calculated the octahedral splitting energy! The question asks us to find the value that fills in the blank for $\cdots \times 10^{-19}\text{ J}$, and to round it off to the nearest integer.

Looking at our result, $3.9915$ is incredibly close to $4$. Therefore, rounding off to the nearest integer gives us our final answer:

This problem beautifully illustrates how macroscopic properties like the color of a solution are intimately connected to the microscopic quantum world of electron transitions and orbital splitting!