Problem 162 - A Complete Resource Book in Chemistry for JEE Main

Question

The value of 'spin only' magnetic moment for one of the following configurations is \(2.84 \mathrm{BM}\). The correct one is (a) \(\mathrm{d}^{4}\) (in strong ligand field) (b) \(\mathrm{d}^{4}\) (in weak ligand field) (c) \(\mathrm{d}^{3}\) (in weak as well as in strong fields) (d) \(\mathrm{d}^{5}\) (in strong ligand field)

Step-by-Step Solution

Answer

The correct configuration is (a) \( \text{d}^4 \) in strong ligand field.

Step 1: Understand the Magnetic Moment Formula

The spin-only magnetic moment \( \mu \) for a given electron configuration can be calculated using the formula: \( \mu = \sqrt{n(n+2)} \) Bohr Magneton (BM), where \( n \) is the number of unpaired electrons.

Step 2: Calculate for each configuration

Calculate the number of unpaired electrons for each configuration given the ligand field conditions:- (a) \( \text{d}^4 \) (strong ligand field): Strong field splitting may cause pairing, leading to 2 unpaired electrons.- (b) \( \text{d}^4 \) (weak ligand field): Weak field splitting leads to 4 unpaired electrons.- (c) \( \text{d}^3 \): Always results in 3 unpaired electrons regardless of the field strength.- (d) \( \text{d}^5 \) (strong ligand field): All electrons may pair up, leading to 0 unpaired electrons.

Step 3: Calculate Spin-Only Magnetic Moment

Apply the formula to find the magnetic moment for configurations:- (a) \( \mu = \sqrt{2(2+2)} = \sqrt{8} = 2.83 \) BM (approximation matches 2.84 BM)- (b) \( \mu = \sqrt{4(4+2)} = \sqrt{24} = 4.90 \) BM- (c) \( \mu = \sqrt{3(3+2)} = \sqrt{15} = 3.87 \) BM- (d) \( \mu = \sqrt{0(0+2)} = \sqrt{0} = 0 \) BM

Step 4: Identify the Correct Option

Compare calculated magnetic moments with the given value (2.84 BM). The configuration (a) \( \text{d}^4 \) in strong ligand field gives a moment of approximately 2.83 BM, which is closest to 2.84 BM.

Key Concepts

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom's orbitals. This concept is pivotal in understanding various properties of elements, as it affects how atoms interact with each other and with external fields. The electron configuration is typically described using the notation that specifies the energy levels, sublevels, and the number of electrons in each. For transition metals, such as those involved in this exercise, the d-orbitals play a significant role. When determining the electron configuration, each electron occupies the lowest energy state available, following the Aufbau principle, Hund's rule, and the Pauli exclusion principle.

For example, a \(\text{d}^4\) configuration indicates four electrons are present in the d-orbital. The precise distribution of these electrons across the five d-orbitals can vary based on external factors such as the strength of the ligand field, leading to variation in the magnetic properties of the compound. Understanding electron configuration is critical in the application of concepts such as ligand field theory, which explains how ligands affect the energy levels of d-orbitals.

Ligand Field Theory

Ligand Field Theory (LFT) extends beyond the earlier Crystal Field Theory by considering both the ionic and covalent nature of metal-ligand interactions. It provides insights into how ligands influence the splitting of the metal's d-orbitals, which in turn affects the properties of the complex, such as color and magnetism.

When ligands approach a metal center, they create an electrostatic field conducive to splitting the degenerate (equal energy) d-orbitals into two distinct energy levels. The size of this splitting, referred to as the crystal field splitting energy \(\Delta \), depends on the strength of the ligand field:

• Strong field ligands cause large splitting and often lead to the pairing of electrons within lower energy orbitals, known as the low-spin configuration.
• Weak field ligands result in smaller splitting, allowing electrons to occupy higher energy orbitals without pairing, known as the high-spin configuration.

This distinction influences the number of unpaired electrons and thereby determines the magnetic properties of the compound. For instance, in a strong field ligand situation with a \(\text{d}^4\) configuration, electrons tend to pair up leading to fewer unpaired electrons, while in a weak field, they remain unpaired.

Unpaired Electrons

Unpaired electrons are critical in determining the magnetic characteristics of an atom or ion. The presence of unpaired electrons in the d-orbitals contributes to paramagnetism, while completely paired electrons lead to diamagnetism.

Magnetic moment, which is a measure of the magnetism of a substance, depends significantly on the number of unpaired electrons and can be evaluated using the formula: \( \mu = \sqrt{n(n+2)} \) Bohr Magneton \(\text{(BM)} \). In this equation, \( n \) represents the number of unpaired electrons.

Let's explore the effect:

• For \(\text{d}^{4}\) with a strong ligand field, there may be 2 unpaired electrons, resulting in a calculated magnetic moment of about 2.84 BM, which matches the scenario in the exercise.
• However, in a \(\text{d}^{4}\) weak ligand field setup, there are 4 unpaired electrons, increasing the magnetic moment significantly.

Understanding the role of unpaired electrons in such calculations helps predict the behavior of complex systems when exposed to a magnetic field, thereby aiding in identifying the correct configuration in queries similar to the exercise provided.