Animated Solution for Chemistry - Coordination Compounds: What is the spin-only magnetic moment value (BM) of a divalent metal ion with atomic number 25, in it's aqueous solution?

Calculating Spin-Only Magnetic Moment

Imagine you are a detective trying to uncover the magnetic secrets of a mysterious metal ion. The only clue you have is its atomic number: $25$. Let's embark on this thrilling journey to find its spin-only magnetic moment!

First things first, we need to identify our suspect. In the periodic table, the element with atomic number $Z=25$ is Manganese ($Mn$).

Now, let's look at its ground-state electronic configuration. Manganese is a transition metal in the 3d series, so its configuration is $[Ar]4s^{2}3d^5$. But the problem throws a curveball: it's a divalent metal ion in an aqueous solution. Divalent means it has a $+2$ charge, indicating it has lost two electrons.

Where do these electrons come from? They are always removed from the outermost shell first, which is the $4s$ orbital. So, the electronic configuration of our $M{n}^{2+}$ ion becomes simply $[Ar]3d^5$.

Now, let's dive into the $3d$ subshell. It consists of five degenerate orbitals. According to Hund's Rule of Maximum Multiplicity, electrons will fill these orbitals singly before any pairing occurs.

Since we have exactly five electrons to place in five orbitals, each orbital gets exactly one electron. This means all five electrons are unpaired! Therefore, our number of unpaired electrons, denoted by $n$, is exactly $5$.

To find the spin-only magnetic moment, we use a beautifully simple yet powerful formula:

Here, $\mu$ is the magnetic moment in Bohr Magnetons (BM), and $n$ is the number of unpaired electrons. This formula tells us that the magnetic moment is directly born from the spins of these lonely, unpaired electrons.

Let's substitute our value of $n=5$ into the master equation:

Now, we don't even need a calculator to estimate this. We know that $\sqrt{36}=6$. Since $35$ is just a tiny bit less than $36$, $\sqrt{35}$ must be slightly less than $6$. Looking at our options, $5.92$ is the perfect match!

And there you have it! By understanding electronic configurations and Hund's rule, we've successfully deduced that the spin-only magnetic moment of our divalent Manganese ion is $5.92\text{ BM}$.