Finding the Magnetic Moment of a Divalent Ion (Atomic Number 25)

The question asks for the magnetic moment of a divalent ion in an aqueous solution, derived from an element with atomic number 25. To solve this, we first need to identify the element and its electronic configuration, then determine the configuration of its divalent ion, find the number of unpaired electrons, and finally calculate the magnetic moment using the spin-only formula.

Identifying the Element and its Electronic Configuration

The element with atomic number 25 is Manganese (Mn).

Its electronic configuration is determined by filling electrons in orbitals according to the Aufbau principle, Hund's rule, and the Pauli exclusion principle.

• Atomic number (Z) = 25
• Element: Manganese (Mn)
• Ground state electronic configuration of Mn: \([\text{Ar}] 3\text{d}^5 4\text{s}^2\)

Determining the Electronic Configuration of the Divalent Ion

A divalent ion means the atom has lost two electrons, resulting in a net charge of +2. When forming cations from transition metals like Manganese, electrons are removed first from the outermost shell (highest principal quantum number), which is the 4s shell in this case, before the 3d shell.

• Ion: \(\text{Mn}^{2+}\)
• To form \(\text{Mn}^{2+}\), Manganese loses the two electrons from its 4s orbital.
• Electronic configuration of \(\text{Mn}^{2+}\): \([\text{Ar}] 3\text{d}^5 4\text{s}^0 = [\text{Ar}] 3\text{d}^5\)

Finding the Number of Unpaired Electrons in the Divalent Ion

The magnetic properties of an ion, specifically paramagnetism and its magnetic moment, are largely determined by the number of unpaired electrons. For \(\text{Mn}^{2+}\), the configuration is \(3\text{d}^5\). The 3d subshell has five orbitals. According to Hund's rule, electrons will individually occupy each orbital within a subshell before pairing up.

• The \(3\text{d}^5\) configuration means there are 5 electrons in the 3d subshell.
• Filling the five 3d orbitals according to Hund's rule: Each of the five 3d orbitals receives one electron, all with parallel spins.
• Number of unpaired electrons (\(n\)) in \(\text{Mn}^{2+}\) is 5.

Calculating the Spin-Only Magnetic Moment

The spin-only magnetic moment (\(\mu_s\)) is calculated using the formula:

\[\mu_s = \sqrt{n(n+2)}\] where \(n\) is the number of unpaired electrons and \(\mu_s\) is measured in Bohr Magnetons (BM).

For \(\text{Mn}^{2+}\), we found \(n=5\).

Let's calculate the magnetic moment:

\[\mu_s = \sqrt{5(5+2)}\]

\[\mu_s = \sqrt{5(7)}\]

\[\mu_s = \sqrt{35}\]

Calculating the square root of 35:

\[\sqrt{35} \approx 5.916 \text{ BM}\]

Comparing with Options

Let's compare our calculated value with the given options:

• 1. 2.84 BM
• 2. 3.87 BM
• 3. 4.90 BM
• 4. 5.92 BM

The calculated value, approximately 5.916 BM, is closest to 5.92 BM.

Summary of \(\text{Mn}^{2+}\) Properties

Thus, the magnetic moment of the divalent ion (\(\text{Mn}^{2+}\)) in aqueous solution is approximately 5.92 BM.

Revision Table: Magnetic Moment Calculation Steps

Additional Information: Understanding Magnetic Properties

Bohr Magneton (BM): The unit of magnetic moment is the Bohr Magneton (BM). It is a fundamental constant representing the magnetic moment of an electron caused by its angular momentum.

Paramagnetism vs. Diamagnetism:

• Substances with unpaired electrons are generally paramagnetic. They are weakly attracted to a magnetic field. The magnetic moment value helps quantify this paramagnetism.
• Substances with all paired electrons are diamagnetic. They are weakly repelled by a magnetic field and have a magnetic moment of zero.

Hund's Rule: This rule states that for a given electron configuration, the lowest energy state is the one with the maximum number of unpaired electrons allowed by the Pauli exclusion principle within a subshell. This is why the five electrons in the \(3\text{d}^5\) configuration of \(\text{Mn}^{2+}\) spread out into five different orbitals before pairing.

In aqueous solution, transition metal ions are often surrounded by water molecules (aqua ligands). This forms a complex ion. While crystal field theory can sometimes influence the number of unpaired electrons (e.g., in strong field ligands causing pairing), for many common ions like \(\text{Mn}^{2+}\) with its stable \(d^5\) configuration, the high spin case (maximum unpaired electrons) is typical, especially with weak field ligands like water. The spin-only formula provides a good approximation for the magnetic moment in such cases.