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 \([\mathbf{2 0 0 5}]\) (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 \( \mathrm{d}^{4} \) in a strong ligand field (option a).

Step 1: Understand Spin Only Magnetic Moment

The spin-only magnetic moment (_{ ext{spin}}) is given by the formula, \( \mu_S = \sqrt{n(n + 2)} \), where \( n \) is the number of unpaired electrons. We are looking for a configuration with a magnetic moment of \(2.84 \mathrm{BM}\).

Step 2: Calculate Spin Only Magnetic Moment for Each Option

- For option (a) \( \mathrm{d}^{4} \) strong field: A strong ligand field pairs electrons, resulting in 2 unpaired electrons. \( \mu_S = \sqrt{2(2 + 2)} = \sqrt{8} = 2.83 \mathrm{BM}\).- For option (b) \( \mathrm{d}^{4} \) weak field: Weak field typically results in 4 unpaired electrons. \( \mu_S = \sqrt{4(4 + 2)} = \sqrt{24} = 4.90 \mathrm{BM}\).- For option (c) \( \mathrm{d}^{3} \): Both strong and weak fields lead to 3 unpaired electrons. \( \mu_S = \sqrt{3(3 + 2)} = \sqrt{15} = 3.87 \mathrm{BM}\).- For option (d) \( \mathrm{d}^{5} \) strong field: A strong field pairs electrons, resulting in 1 unpaired electron. \( \mu_S = \sqrt{1(1 + 2)} = \sqrt{3} = 1.73 \mathrm{BM}\).

Step 3: Compare With Given Magnetic Moment

Match the calculated magnetic moments with the given value of \(2.84 \mathrm{BM}\). The magnetic moment closest to this value is from option (a), which is approximately \(2.83 \mathrm{BM}\).

Step 4: Determine the Correct Configuration

Since option (a) \( \mathrm{d}^{4} \) in a strong ligand field yields a magnetic moment closely matching \(2.84 \mathrm{BM}\), this is the correct answer.

Key Concepts

Unpaired Electrons

When considering the spin-only magnetic moment, the key idea is the number of unpaired electrons present in the configuration of a compound. Unpaired electrons are those electrons in an atomic or molecular configuration that do not have a corresponding electron with an opposite spin in the same orbital. These unpaired electrons contribute significantly to the magnetic properties of an atom or molecule.
A basic understanding of these concepts is essential because:

• Unpaired electrons generate a magnetic moment due to their spin, resulting in paramagnetism.
• Paired electrons tend to cancel each other’s magnetic effects out, leading to diamagnetism.

Knowing the number of unpaired electrons allows us to utilize the formula for the spin-only magnetic moment, \( \mu_S = \sqrt{n(n + 2)} \), where \( n \) is the number of unpaired electrons. This calculation provides insight into the potential magnetic behavior of a substance.

Ligand Field Theory

Ligand field theory (LFT) is an important concept used to describe the effect of ligands on the distribution of electrons in transition metal complexes. It provides a more accurate depiction of the energy differences among d-orbitals and helps predict the properties of complexes.
Ligand field varies in strength:

• Strong Field Ligands: They exert a strong effect on the metal d-orbitals, often leading to pairing of electrons, reducing the number of unpaired electrons. This drastically affects the magnetic properties.
• Weak Field Ligands: They produce a relatively weak effect, often retaining more unpaired electrons, which increases magnetic moments.

LFT allows chemists to understand how altering the nature of ligands in a complex affects both the geometry and the electronic properties, crucial for predicting the behavior of different electron configurations.

d-Orbital Configuration

The d-orbital configuration refers to how electrons are distributed in the five d-orbitals of a transition metal. This configuration significantly influences the magnetic and other chemical properties of the substance.
When transition metals form complexes, their d-orbital configurations can be modified substantially by the presence of ligands. Depending on the field strength of these ligands:

• In a strong ligand field, electrons may be forced to pair up in lower-energy orbitals due to energy splitting, reducing the number of unpaired electrons.
• Conversely, in a weak ligand field, electrons typically remain unpaired to avoid pairing energy costs, resulting in a higher count of unpaired electrons and a larger magnetic moment.

Understanding the d-orbital configuration is essential for determining not just magnetic behaviors but also the color, stability, and reactivity of complexes.