Problem 135 - A Complete Resource Book in Chemistry for JEE Main

Question

The substances which contain atoms with unpaired electrons in their orbitals behave as paramagnetic substances. Such substances are weakly attracted by the magnetic field. The paramagnetism is expressed in terms of magnetic moment. The magnetic moment is related to the number of unpaired electrons according to the following relation. Magnetic moment, \(\mu=\sqrt{n(n+2)} B M\) Where, \(\mathrm{n}\) - number of unpaired electrons. BM stands for Bohr Magneton, a unit of magnetic moment. An ion of a d-block element has magnetic moment 5.92 BM select the ion among the following: (a) \(\mathrm{Sc}^{3+}\) (b) \(\mathrm{Zn}^{2+}\) (c) \(\mathrm{Mn}^{2+}\) (d) \(\mathrm{Cr}^{3+}\)

Step-by-Step Solution

Answer

The ion is \( \text{Mn}^{2+} \) with 5 unpaired electrons.

Step 1: Understand the Problem

We need to determine which ion from the given options has a magnetic moment of 5.92 BM. This involves finding the number of unpaired electrons for each ion and confirming which ion matches the given magnetic moment.

Step 2: Recall the Formula for Magnetic Moment

The formula for magnetic moment is given by \( \mu = \sqrt{n(n+2)} \) BM, where \( n \) is the number of unpaired electrons. For a magnetic moment of 5.92 BM, we need to solve for \( n \) such that \( \mu = 5.92 \).

Step 3: Solve for Number of Unpaired Electrons

Setting \( \mu = \sqrt{n(n+2)} = 5.92 \), square both sides to get \( n(n+2) = (5.92)^2 \). Compute \((5.92)^2 = 35.0464 \). Therefore, we need to solve for \( n \) in the equation \( n(n+2) = 35.0464 \). This rounds to approximately \( n(n+2) = 35 \).

Step 4: Test for Possible Unpaired Electrons

We test integer values of \( n \):- If \( n = 4 \), \( n(n+2) = 4(4+2) = 24 \)- If \( n = 5 \), \( n(n+2) = 5(5+2) = 35 \)This means \( n = 5 \) fits because it matches the calculated value.

Step 5: Identify the Ion with 5 Unpaired Electrons

Determine the electronic configurations for each ion:- \( \text{Sc}^{3+} \) has no unpaired electrons.- \( \text{Zn}^{2+} \) has no unpaired electrons.- \( \text{Mn}^{2+} \) configuration: \([Ar] 3d^5\); thus, it has 5 unpaired electrons.- \( \text{Cr}^{3+} \) configuration: \([Ar] 3d^3\); thus, it has 3 unpaired electrons.

Step 6: Conclusion

\( \text{Mn}^{2+} \) has 5 unpaired electrons, and thus a magnetic moment of 5.92 BM. This matches the requirement set out by the problem.

Key Concepts

Magnetic Moment Calculation

Magnetic moment calculation is key in understanding paramagnetism. The formula used is \( \mu = \sqrt{n(n+2)} \) BM, where \( n \) represents the number of unpaired electrons. If a material contains unpaired electrons, it will generate a magnetic field when placed in an external magnetic field. This magnetic moment measurement helps identify how paramagnetic an element or ion is.

• The larger the value of \( n \), the stronger the magnetic moment.
• As magnetic moments increase, the strength of paramagnetism increases too.

This formula allows scientists to predict and confirm whether a substance displays paramagnetic properties, which is crucial for understanding the behavior of materials in magnetic fields.

Unpaired Electrons

Unpaired electrons significantly influence the magnetic properties of an element or ion. An unpaired electron is an electron in an orbital that does not have a corresponding electron with an opposite spin. These unpaired electrons are crucial because they act as tiny magnets themselves.The presence of unpaired electrons in atoms or ions causes them to be attracted to magnetic fields. In the context of the d-block of the periodic table, elements often have partially filled d-orbitals, meaning they can have unpaired electrons. In the case of Mn\(^{2+}\), for example, the electronic configuration is \([Ar] 3d^5\), indicating five unpaired electrons, contributing to its strong paramagnetic properties.

D-block Elements

D-block elements, often referred to as transition metals, play a pivotal role in the study of magnetism. These elements are characterized by their d-orbitals, which can host unpaired electrons:

• The presence of unpaired electrons in the d-orbitals often results in magnetic properties.
• Many ions of d-block elements display paramagnetism due to these unpaired electrons.

These elements are essential for various applications that utilize their magnetic properties. They are also a focal point in the study of chemical bonding and electron configurations. Understanding the behavior of d-block elements paves the way for harnessing their magnetic properties in industrial and technological applications.

Bohr Magneton

The Bohr Magneton (BM) is the fundamental unit for expressing the magnetic moment of an electron. It provides a convenient scale for measuring magnetic moments, based on quantum mechanics.

• A single Bohr Magneton corresponds to the magnetic moment of an electron due to its orbital or spin motion.
• In paramagnetism studies, magnetic moments are often expressed in Bohr Magnetons.

This unit is named after Niels Bohr, and it highlights the intrinsic relationship between quantum mechanics and magnetic phenomena. By using the Bohr Magneton, scientists can accurately describe and predict the magnetic behavior of atoms and molecules due to their electron configuration.