Problem 22 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The orbital angular momentum of an electron in \(2 s\) orbital is: (a) \(+\frac{1}{2} \cdot \frac{h}{2 \pi}\) (b) Zero (c) \(\frac{h}{2 \pi}\) (d) \(\sqrt{2} \cdot \frac{h}{2 \pi}\)

Step-by-Step Solution

Answer

The orbital angular momentum of an electron in a 2s orbital is zero.

Step 1: Understand the Quantum Numbers

The quantum numbers are used to describe the state of an electron in an atom. The relevant ones here are the principal quantum number (n) and the azimuthal (or angular momentum) quantum number (l). For the electron in a 2s orbital, n = 2 and l = 0.

Step 2: Recall the Formula for Orbital Angular Momentum

The formula for orbital angular momentum (L) of an electron is given by \[ L = \sqrt{l(l+1)} \cdot \frac{h}{2\pi} \]where \(h\) is Planck's constant, and \(l\) is the angular momentum quantum number.

Step 3: Calculate Angular Momentum for 2s Orbital

For the 2s orbital, the azimuthal quantum number \(l = 0\). Substitute \(l = 0\) into the formula from Step 2 to calculate the orbital angular momentum:\[ L = \sqrt{0(0+1)} \cdot \frac{h}{2\pi} = \sqrt{0} \cdot \frac{h}{2\pi} = 0 \].Thus, the orbital angular momentum is zero.

Step 4: Select the Correct Answer

From the given options, the correct answer for the orbital angular momentum value is (b) Zero.

Key Concepts

Quantum Numbers

Quantum numbers are a set of numerical values that describe the unique quantum state of an electron in an atom. They are essential for understanding the structure of atoms and the arrangement of electrons. Here's a simple breakdown of the relevant quantum numbers:

• Principal Quantum Number ( ) - This number indicates the main energy level of an electron. The larger the number, the higher the energy and the larger the orbital. For example, in a 2s orbital, the principal quantum number = 2.
• Azimuthal Quantum Number (l) - Sometimes referred to as the angular momentum quantum number, it defines the shape of the electron's orbital. Its value ranges from 0 to -1. For a 2s orbital, the azimuthal quantum number is l = 0, indicating a spherical shape.

Understanding these quantum numbers helps explain why electrons are arranged within orbitals and why certain orbitals have specific shapes and energy levels. These numbers work together to determine an electron's position and energy, providing a roadmap for how atoms bond and interact.

s Orbital

Among the different types of orbitals, the 's orbital' is the simplest yet fundamental one. When visualized, it resembles a sphere around the nucleus of an atom. Here are some vital points about the s orbital:

• The s orbital can hold up to two electrons, but they must have opposite spins due to the Pauli Exclusion Principle.
• Since its azimuthal quantum number (l) is 0, its shape is spherical. This shape allows it to be non-directional, meaning it has the same probability of containing an electron in any direction around the nucleus.
• s orbitals are found in every energy level, so you'll encounter a 1s, 2s, 3s, and so forth as you progress through higher energies.

The simplicity of the s orbital is crucial for understanding more complex orbital types and electron configurations. It serves as a foundational concept in quantum chemistry.

Azimuthal Quantum Number

The azimuthal quantum number, also known as the angular momentum quantum number, is a fundamental parameter in quantum mechanics. This number not only defines the shape of an electron's orbital but also plays a role in its energy calculations. Here's what you need to know:

• This quantum number is denoted by the letter l and can take values from 0 up to -1.
• Each value of l corresponds to a particular shape of the orbital: 0 for s (spherical), 1 for p (dumbbell-shaped), 2 for d, and 3 for f orbitals.
• The value of l also determines the orbital's angular momentum, which affects how electrons interact with magnetic fields and other electrons.

For a 2s orbital, the azimuthal quantum number is l = 0, meaning it has no angular momentum. As a result, the orbital angular momentum is zero, as calculated by the formula \( L = \sqrt{l(l+1)} \cdot \frac{h}{2\pi} \), where h is Planck’s constant. This specific quantum number is key to understanding electron distribution and chemical bonding in complex atoms.