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 configuration is \(d^4\) in a strong ligand field.

Step 1: Formula for Spin-Only Magnetic Moment

The formula for calculating the spin-only magnetic moment \( \mu \) is given by \( \mu = \sqrt{n(n+2)} \, \mathrm{BM} \), where \( n \) is the number of unpaired electrons.

Step 2: Calculate for \(d^4\) in Strong Ligand Field

For a \(d^4\) configuration in a strong ligand field, the electrons pair up due to the splitting of orbitals (low spin), resulting in only 2 unpaired electrons. \(n = 2\). Plugging into the formula, we get \[ \mu = \sqrt{2(2+2)} = 2.83 \, \mathrm{BM} \].

Step 3: Calculate for \(d^4\) in Weak Ligand Field

In the weak ligand field, a \(d^4\) configuration results in 4 unpaired electrons. \(n = 4\). Plugging into the formula, we get \[ \mu = \sqrt{4(4+2)} = 4.90 \, \mathrm{BM} \].

Step 4: Evaluate \(d^3\) Configuration

For both weak and strong fields, a \(d^3\) configuration has 3 unpaired electrons (as there is no pairing irrespective of the field strength). \(n = 3\). Plugging into the formula, we get \[ \mu = \sqrt{3(3+2)} = 3.87 \, \mathrm{BM} \].

Step 5: Calculate for \(d^5\) in Strong Ligand Field

In a strong ligand field, a \(d^5\) configuration results in one unpaired electron (high pairing). \(n = 1\). Plugging into the formula, we get \[ \mu = \sqrt{1(1+2)} = 1.73 \, \mathrm{BM} \].

Step 6: Identify the Correct Configuration

The calculated magnetic moment closest to \(2.84 \, \mathrm{BM} \) is from Step 2: \(d^4\) in a strong ligand field with \(2.83 \, \mathrm{BM} \).

Key Concepts

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom's orbitals. These orbitals are organized by increasing energy, typically in the order of 1s, 2s, 2p, and so on. Understanding electron configurations helps us determine how electrons occupy these orbitals, influencing the chemical properties of the element, as well as its magnetic characteristics.

At the core of this concept, we are interested in valence electrons, which are the outermost electrons that play a pivotal role in chemical bonding and reactions. For transition metals, their electron configurations become especially important as they involve d-orbitals. For instance, in our example, the configurations are variations of d-orbitals like \(d^3\), \(d^4\), and \(d^5\). Each configuration will determine the number of unpaired electrons, impacting the spin-only magnetic moment.

The more unpaired electrons a compound has, the larger the magnetic moment tends to be. Thus, by knowing the electron configuration, we can predict magnetic properties among other chemical characteristics.

Ligand Field Theory

Ligand Field Theory (LFT) extends from Crystal Field Theory and helps us understand how ligands affect the energies of d-orbitals in transition metal complexes. LFT provides an essential framework for predicting and explaining the differences in d-orbital splitting in coordination complexes.

In strong ligand fields, ligands interact intensely with the d-orbitals, leading to large splitting and possibly causing electron pairing in lower energy orbitals. This is related to low-spin configurations. On the other hand, in weak ligand fields, the splitting is smaller, and electrons may prefer to occupy higher energy levels without pairing, resulting in high-spin configurations.
This directly affects the magnetic properties we measure, such as the spin-only magnetic moment. In the original problem, configurations in both strong and weak ligand fields demonstrate how these fields alter the number of unpaired electrons, thus impacting the Balzarine-Magnetic (BM) moment.

d-Orbitals

The shape and orientation of d-orbitals play a critical role in determining the electron configuration of transition metals. There are five d-orbitals: dxy, dxz, dyz, dx²-y², and dz². Each of these orbitals holds unique geometric orientations that cause them to have different energies when interacting with ligands.

When ligands present themselves near a transition metal ion, the five d-orbitals split into different energy levels. This phenomenon is crucial to understanding the electronic and magnetic properties of coordination complexes and is often referred to as d-orbital splitting.
In our problem, d-orbital configurations like \(d^4\) experience different extents of splitting based on whether the field is strong or weak, influencing the overall electronic ground state. This knowledge helps us calculate the spin-only magnetic moments accurately, as the number of unpaired electrons varies depending on the filling of these split d-orbitals.

Unpaired Electrons

Unpaired electrons have a significant impact on the magnetic properties of a substance. When electrons occupy an orbital singly without a partner with an opposite spin, they are regarded as unpaired.

The presence of these unpaired electrons accounts for the paramagnetic properties of a material, as each unpaired electron contributes a magnetic moment. The sum of these moments translates to what we observe as the spin-only magnetic moment.
In the context of our exercise, determining the number of unpaired electrons in configurations like \(d^3\), \(d^4\), and \(d^5\) across varying ligand fields illustrates how these electrons influence the calculated magnetic moments. Specifically, employing the equation \(\mu = \sqrt{n(n+2)}\) enables us to link unpaired electrons to the estimated magnetic moment in Bohr Magneton (BM) units, enabling predictions about a compound's magnetic behavior.