Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

For a d-electron, the orbital angular momentum is (a) \(\sqrt{6 h}\) (b) \(\sqrt{2} \mathrm{~h}\) (c) \(\mathrm{h}\) (d) \(2 \mathrm{~h}\)

Step-by-Step Solution

Answer

(a) \(\sqrt{6} h\)

Step 1: Understand the Angular Momentum Formula

The orbital angular momentum \(L\) for an electron in a particular orbital (given quantum number \(l\)) is determined by the formula \(L = \sqrt{l(l+1)} \hbar\), where \(\hbar = \frac{h}{2\pi}\) is the reduced Planck's constant.

Step 2: Determine Quantum Number for a d-Electron

Identify the value of \(l\) for a d-electron. For d-electrons, the azimuthal quantum number \(l\) is equal to 2.

Step 3: Substitute and Calculate

Substitute \(l = 2\) into the angular momentum formula: \[ L = \sqrt{2(2+1)} \hbar = \sqrt{6} \hbar \]This simplifies to \(\sqrt{6} \hbar\). Since \(\hbar = \frac{h}{2\pi}\), the expression retains the form \(\sqrt{6} \hbar\) when not dividing by \(2\pi\).

Key Concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that explores the actions and interactions of particles at the atomic and subatomic levels. Unlike classic physics, which deals with macroscopic phenomena, quantum mechanics describes the peculiar behavior that particles exhibit when they are incredibly small. At this scale, particles like electrons do not follow the straightforward paths defined by classical physics laws.

Instead, quantum mechanics governs how particles behave in unpredictable or probabilistic ways. For instance, an electron can be in a state of spin or any orbital path that isn't easily defined at first glance. This branch of physics explains phenomena like superposition and entanglement, which seem counterintuitive but are essential to understanding how the universe functions at microscopic levels.

Some key characteristics of quantum mechanics include:

• Wave-Particle Duality: Particles like electrons exhibit both wave-like and particle-like properties.
• Uncertainty Principle: There is a fundamental limit to what we can know about a particle's position and momentum at the same time.
• Quantization: Certain properties of particles, like energy, take on distinct, discrete values.

By using quantum mechanics, scientists can predict the behavior of electrons in atoms, which is fundamental to chemistry and material science.

d-Electron

In the context of electronic configurations, electrons reside in various energy levels characterized by principal quantum numbers. Among these levels, d-electrons are those found in the third subshell, represented by the letter 'd'. These d-orbitals are complex and have shapes that require dependent orientations within an atom.

A typical atom with d-electrons usually has a principal quantum number (n) of 3, associated with the third energy level. Furthermore, d-orbitals can hold a maximum of 10 electrons, as there are five d-orbitals (each can hold 2 electrons due to their spin states). Locations in the periodic table where d-electrons significantly contribute to chemical properties include the transition metals.

Key points about d-electrons include:

• Shape: D-orbitals have distinctive clover or double dumbbell shapes.
• Magnetic Properties: The presence of unpaired d-electrons can influence an atom's magnetic properties, making some elements paramagnetic.
• Reactivity: Transition metals often exhibit multiple oxidation states due to the involvement of d-electrons in chemical bonding.

Understanding d-electrons is crucial for explaining the unique properties of metals, including conductivity, color, and catalytic behavior.

Azimuthal Quantum Number

The azimuthal quantum number, often denoted as \( l \), is a vital part of quantum mechanics since it defines the shape and types of orbitals in an atom. Each electron within an atom is represented not just by its energy level, but by this quantum number, which dictates its angular momentum and shape.

The azimuthal quantum number is an integer ranging from 0 to \( (n-1) \), where \( n \) is the principal quantum number corresponding to the electron's energy level. These numbers correlate to different sublevels:

• \( l = 0 \) corresponds to an s orbital.
• \( l = 1 \) corresponds to a p orbital.
• \( l = 2 \) corresponds to a d orbital.
• \( l = 3 \) corresponds to an f orbital.

The azimuthal quantum number is also linked to the atom's angular momentum, a measure of the rotational momentum of electrons around the nucleus. It is instrumental in determining the multi-faceted nature of electron configurations and their resulting chemical properties. For a d-electron, we know \( l = 2 \); hence the orbitals have more complexity and can influence an element’s reaction types and states.

Understanding this fundamental principle sheds light on why certain elements behave the way they do in bonding scenarios, ensuring we can predict molecular structures and interactions accurately.