Problem 152 - A Complete Resource Book in Chemistry for JEE Main

Question

An ion \(\mathrm{Mn}^{a+}\) has the magnetic moment equal to \(4.9\) B.M. Find the value of a

Step-by-Step Solution

Answer

The value of \( a \) is 3.

Step 1: Understanding Magnetic Moment Formula

The magnetic moment ( ( is expressed in terms of Bohr magneton (B.M.) for an ion and is related to the number of unpaired electrons ( () by the formula: \[ \mu = \sqrt{n(n+2)} \] where \( \mu \) is the magnetic moment in Bohr magnetons.

Step 2: Calculate Number of Unpaired Electrons

Given the magnetic moment is \( 4.9 \) B.M., we can equate it to the formula:\[ \sqrt{n(n+2)} = 4.9 \]Squaring both sides and solving for \( n \), we get:\[ n(n+2) = 4.9^2 \]\[ n(n+2) = 24.01 \] This implies \( n = 4 \) would satisfy the equation since \( n(n+2) = 4(4+2) = 24 \). Therefore, the number of unpaired electrons is \( n = 4 \).

Step 3: Determine Mn Electron Configuration

Manganese (Mn) in its neutral state has an atomic number of 25, meaning it has 25 electrons. The electron configuration is: \[ [ ext{Ar}] \, 3d^5 \, 4s^2 \]From the electron configuration, there are 5 electrons in the 3d subshell.

Step 4: Determine Charge of Mn Ion

The ion \( \text{Mn}^{a+} \) needs to have 4 unpaired electrons. The removal of electrons typically occurs from the outermost shells first. If \( a = 2 \), removing 2 electrons from the configuration \([\text{Ar}] \, 3d^5 \, 4s^2\) results in \([\text{Ar}] \, 3d^5\), which has 5 unpaired electrons. Since 5 unpaired isn't possible, we try \( a = 3 \), removing 3 electrons results in \([\text{Ar}] \, 3d^4\), which gives 4 unpaired electrons. Thus, \( a = 3 \).

Key Concepts

Bohr magneton

The Bohr magneton is a fundamental physical constant and the natural unit for expressing the magnetic moment of an electron. The magnetic moment is a measure of the strength and direction of a magnetic source. For electrons, it's particularly relevant because they spin and revolve around atomic nuclei, creating tiny magnetic fields.
The Bohr magneton (B.M.) is usually described by the formula: \[ \mu_B = \frac{e \hbar}{2m_e} \]Where:

• \( e \) is the elementary charge,
• \( \hbar \) is the reduced Planck’s constant,
• \( m_e \) is the electron mass.

In practice, when we talk about the magnetic moment of ions or atoms in units of Bohr magnetons, we are essentially comparing the strength of that magnetic source to this fundamental unit. In the context of the electron's magnetic properties, a Bohr magneton provides a baseline that facilitates the understanding of more complex magnetic behaviors.

unpaired electrons

Unpaired electrons are electrons within an atom's orbitals that are not partnered with another electron. These electrons are essential in determining the magnetic properties of atoms and ions.
In atoms, electrons populate orbitals starting from the lowest energy level and moving to higher ones, following principles such as the Aufbau process and Hund's Rule. When determining the number of unpaired electrons, one must consider how these rules affect electron distribution in a given element's orbitals.
Unpaired electrons are responsible for paramagnetism. Paramagnetic materials are attracted to magnetic fields because the unpaired electrons have unaligned spins, creating tiny magnetic fields that align with an external magnetic field. This leads to a net attraction.
For example, in the exercise, manganese ions have a specific number of unpaired electrons, which determines their magnetic moment. Understanding the way electrons are paired or unpaired within its orbitals is crucial in predicting Mn's magnetic behavior.

electron configuration of Mn

The electron configuration of an element describes the distribution of its electrons over different atomic orbitals. For manganese (Mn), which is element number 25 in the periodic table, the electron configuration in its neutral state is crucial as it becomes the basis for understanding its ions.
The neutral manganese has the electron configuration:\[ [\text{Ar}] \, 3d^5 \, 4s^2 \]This configuration means Mn has filled the argon core configuration and additional electrons in its 3d and 4s subshells. Specifically, the five electrons in the 3d subshell are each unpaired, perfectly adhering to Hund's Rule which states that every orbital in a subshell is singly occupied before any one orbital is doubly occupied.
In the context of Mn ions, each step of ionization involves removing electrons from the outermost shells first. For instance, the ion \( \text{Mn}^{2+} \) results from removing two electrons, usually from the 4s orbital, leading to the configuration \([\text{Ar}] \, 3d^5\). But, for establishing the correct ion to have exactly four unpaired electrons as per the exercise, Mn is ionized to \( \text{Mn}^{3+} \), resulting in the configuration \([\text{Ar}] \, 3d^4\). This will leave us with four unpaired electrons. Understanding these transitions helps explain how manganese's electron configurations dictate its magnetic properties.