Problem 21 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

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

Step-by-Step Solution

Answer

The orbital angular momentum is \(\sqrt{6}\frac{h}{2\pi} \), so the answer is (a).

Step 1: Understanding Orbital Angular Momentum

The formula for orbital angular momentum \( L \) is given by \( L = \sqrt{l(l + 1)} \frac{h}{2\pi} \), where \( l \) is the azimuthal quantum number (also known as the angular momentum quantum number) and \( h \) is Planck's constant.

Step 2: Identify the Quantum Number

For a \(d\)-electron, the azimuthal quantum number \( l \) is 2. This is because \(d\)-orbitals correspond to \( l = 2 \) in the quantum mechanical model of the atom.

Step 3: Plug into the Formula

Substitute \( l = 2 \) into the formula for \( L \):\[L = \sqrt{2(2 + 1)} \frac{h}{2\pi} = \sqrt{6} \frac{h}{2\pi}\]

Step 4: Compare with Options

Compare the calculated value \( \sqrt{6} \frac{h}{2\pi} \) with the given options. Option (a) matches the calculated value.

Key Concepts

Azimuthal Quantum Number

The azimuthal quantum number, often symbolized as \( l \), is crucial in the quantum mechanical model for determining the shape of an electron's orbital. This number is one of the four quantum numbers used to describe the unique quantum state of an electron. It directly influences the electron's angular momentum.
The azimuthal quantum number can take integer values ranging from 0 to \( n-1 \), where \( n \) is the principal quantum number. Each value of \( l \) corresponds to a specific orbital type:

• \( l = 0 \) 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.

Understanding \( l \) is key to predicting how electrons occupy orbitals and thus the electron configuration of an atom. For example, when dealing with a \( d \)-electron, \( l \) equals 2, designating the electron resides in a d-orbital. This is critical for calculating properties like orbital angular momentum.

d-electron

A \(d\)-electron is simply an electron that occupies a d-orbital, which is a specific type of atomic orbital characterized by the azimuthal quantum number \( l = 2 \). D-orbitals come into play starting from the third energy level (or shell) in an atom.
These orbitals have some of the most complex shapes known in quantum mechanics, with their unconventional clover-like structures. Because of this complexity, d-electrons are involved in intricate bonding and are significant in defining the chemistry of transition metals.

• D-orbitals can hold a total of 10 electrons.
• This makes them important in filling the electron shell of transition metals.
• They give rise to unique properties like variable oxidation states and magnetic behavior.

Understanding \(d\)-electrons is essential for grasping the electronic structure of many elements. Their presence and behavior explain a wide range of chemical properties and reactivity.

Quantum Mechanical Model

The quantum mechanical model of the atom is a modern description that explains the nature of electrons within an atom. It moves beyond the classic "solar system" model by introducing probability clouds, known as orbitals, where electrons are likely to be found.
This model interprets electron behavior using four quantum numbers:

• The principal quantum number \( n \), indicating the main energy level.
• The azimuthal quantum number \( l \), which specifies the orbital shape.
• The magnetic quantum number \( m_l \), identifying the orientation of the orbital.
• The spin quantum number \( m_s \), representing the electron's spin direction.

Under this model, particles like electrons exhibit both wave-like and particle-like properties, a concept central to quantum mechanics. This model allows chemists and physicists to understand and predict atomic and molecular structures and reactions with great precision.