Question

Find the magnetic moment of a divalent ion in aqueous solution if its atomic number is 25 : (a) \(6.9 \mathrm{~B} . \mathrm{M}\). (b) \(5.9 \mathrm{~B} . \mathrm{M}\). (c) 4.9 B.M. (d) \(3.0 \mathrm{~B} . \mathrm{M}\).

Short answer

The magnetic moment is approximately 5.9 B.M., option (b).

Step-by-step solution

Determine the electronic configuration of the neutral atom

The atomic number 25 corresponds to manganese (Mn). The electronic configuration of Mn is \([ ext{Ar} ] 3d^5 4s^2\).

Identify the electronic configuration of the divalent ion

A divalent ion means it has lost two electrons. Mn^2+ would lose its two 4s electrons, resulting in the configuration \([ ext{Ar} ] 3d^5\).

Calculate the number of unpaired electrons

In the 3d subshell, the configuration \(3d^5\) means that all five electrons are unpaired as each electron occupies a separate orbital. Thus, there are 5 unpaired electrons.

Use the formula for magnetic moment

The magnetic moment \(\mu\) can be calculated using the formula: \(\mu = \sqrt{n(n+2)}\) Bohr Magneton (B.M.), where \(n\) is the number of unpaired electrons. Thus, \(\mu = \sqrt{5(5+2)} = \sqrt{35}\).

Approximate the square root

Approximate \(\sqrt{35}\), which is approximately 5.92.

Compare with the given options

The calculated magnetic moment of approximately 5.9 B.M. closely matches option (b) \(5.9 \mathrm{~B} . \mathrm{M}\).

Key concepts

Divalent Ion

A divalent ion is a type of ion that has lost two electrons, resulting in a charge of +2. This occurs when an atom gives away its outermost electrons in order to achieve a more stable electronic configuration. In the exercise above, manganese (Mn) is a divalent ion when it becomes Mn²⁺ after losing two electrons. It typically loses its 4s electrons first because they are further from the nucleus and less tightly held compared to the 3d electrons.
This process of losing electrons is essential in forming ionic compounds and can significantly affect the properties of the ion, including its magnetic characteristics.

Electronic Configuration

Electronic configuration refers to the arrangement of electrons in an atom's orbitals. For any given atom, this configuration follows the principle of filling from lower to higher energy levels.
For manganese (Mn), which has an atomic number of 25, the electronic configuration can be represented as \([ \text{Ar} ] 3d^5 4s^2\). This means that Mn has five electrons in the 3d subshell and two in the 4s subshell initially. When it becomes a divalent ion (Mn²⁺), it loses its two 4s electrons, leaving the configuration as \([ \text{Ar} ] 3d^5\).
Understanding this configuration is crucial, as it directly influences the magnetic properties of the ion.

Unpaired Electrons

Unpaired electrons are those that do not have a corresponding electron with an opposite spin in the same orbital. They are instrumental in determining the magnetic properties of an atom or ion.
In the Mn²⁺ ion, after losing its 4s electrons, the resulting electron configuration is \([ \text{Ar} ] 3d^5\). In this case, each of the five electrons in the 3d subshell occupies its own orbital, resulting in five unpaired electrons:

• This is significant because the number of unpaired electrons directly affects the magnetic moment of the ion.
• The more unpaired electrons there are, the stronger the magnetic moment.

This concept is crucial while calculating the magnetic properties seen in the exercise solution.

Bohr Magneton

The Bohr Magneton (B.M.) is a physical constant that helps quantify the magnetic moment of an electron due to its angular momentum. It is used as a standard unit for expressing the magnetic moment of atoms and ions.
In this context, the formula used to calculate the magnetic moment \hspace{1mm} \mu = \sqrt{n(n+2)} \hspace{1mm} Bohr Magnetons (B.M.) \hspace{1mm} reflects how the number of unpaired electrons, \hspace{1mm} \(n\), impacts the magnetic moment. For Mn²⁺:

• Substituting \(n = 5\) (from five unpaired electrons) gives \ \(\mu = \sqrt{5(5+2)} = \sqrt{35}\).
• This approximation leads to a magnetic moment of about 5.92 Bohr Magnetons, which matches the pre-calculated option in the solution.

This shows how the Bohr Magneton is a pivotal component in determining and understanding magnetic activity.

Atomic Number 25

Atomic number is a unique identifier for each chemical element, representing the number of protons in the nucleus of an atom. For manganese (Mn), the atomic number is 25, which means it also has 25 electrons when neutral.
With 25 electrons, the complete electronic configuration can be established, \([ \text{Ar} ] 3d^5 4s^2\), providing key insights into the atom's chemical and physical behaviors.
The atomic number also accounts for delivery of detailed information regarding possible ionization states and serves as a foundation for calculating properties such as electron configurations and their associated magnetic moments, as seen in the exercise.