Animated Solution for Chemistry - Coordination Compounds: Spin only magnetic moment in BM of $[Fe(CO{)}_4(C_2{O}_4){]}^{+}$ is

Imagine you are a molecular detective, and your first clue is the chemical formula of the complex: $[Fe(CO{)}_4(C_2{O}_4){]}^{+}$. Our ultimate goal is to find its spin-only magnetic moment, but we can't do that without knowing the state of our central suspect, the iron atom.

To find the oxidation state of iron, we need to set up a simple algebraic equation. We know that the sum of the oxidation states of all the ligands and the central metal atom must equal the net charge on the entire complex.

Let's assign the oxidation state of iron as $x$. Carbon monoxide ($CO$) is a neutral ligand, so it contributes a charge of $0$. Oxalate ($C_2{O}_4^{2-}$), on the other hand, is a bidentate ligand with a charge of $-2$. The overall charge of the complex is given as $+1$.

Setting up our equation, we get: $x+4(0)+(-2)=+1$

Solving for $x$, we find that $x=+3$. This tells us that our iron atom is in the $+3$ oxidation state, written as $F{e}^{3+}$.

Now that we know we are dealing with $F{e}^{3+}$, let's look at its electronic configuration. A neutral iron atom has the configuration $[Ar]3d^{6}4s^2$. When it loses three electrons to become $F{e}^{3+}$, it loses the two $4s$ electrons first, followed by one $3d$ electron, leaving us with a $3d^5$ configuration.

In an isolated ion, all five $d$-orbitals are degenerate, meaning they have the exact same energy. However, when ligands approach the central metal atom to form an octahedral complex, their electron clouds repel the electrons in the $d$-orbitals. This causes the $d$-orbitals to split into two distinct energy levels: a lower energy $t_{2g}$ set (consisting of three orbitals) and a higher energy $e_g$ set (consisting of two orbitals).

Here is where the nature of the ligands becomes crucial. Carbon monoxide is a strong field ligand. It creates a massive energy gap, known as the crystal field splitting energy (${\Delta }_o$), between the $t_{2g}$ and $e_g$ sets.

Because this gap is so large, it is actually more energetically favorable for the five electrons to pair up in the lower energy $t_{2g}$ orbitals rather than jumping across the gap to the $e_g$ orbitals.

Following Hund's rule and the Pauli exclusion principle within the $t_{2g}$ set, the first three electrons occupy one orbital each. The remaining two electrons then pair up with the first two. This gives us the configuration: $t_{2g}^{2,2,1},e_g^{0,0}$

Looking closely at our filled $t_{2g}$ orbitals, we can see that there is exactly one unpaired electron ($n=1$). This single unpaired electron is the source of the complex's magnetic properties.

To calculate the spin-only magnetic moment ($\mu$), we use the formula: $\mu =\sqrt{n(n+2)}$ BM

Substituting our value of $n=1$ into the equation: $\mu =\sqrt{1(1+2)}$ $\mu =\sqrt{3}$ BM

Evaluating the square root of 3, we arrive at our final answer: $\mu =1.73$ BM

The complex has a spin-only magnetic moment of $1.73$ BM, proving that even a single unpaired electron can have a measurable and significant impact on the physical properties of a molecule!