Problem 29 - A Complete Resource Book in Chemistry for JEE Main

Question

\(l=3\), then the values of magnetic quantum numbers are \(\AA\) (a) \(\pm 1, \pm 2, \pm 3\) (b) \(0, \pm 1, \pm 2, \pm 3\) (c) \(-1,-2,-3\) (d) \(0,+1,+2,+3\)

Step-by-Step Solution

Answer

Option (b) is correct: 0, ±1, ±2, ±3.

Step 1: Understanding the Magnetic Quantum Number

The magnetic quantum number, denoted as \( m \), determines the orientation of the orbital in space. It can take on integer values ranging from \( -l \) to \( +l \), where \( l \) is the azimuthal quantum number.

Step 2: Determine the Range of Magnetic Quantum Numbers

Given \( l = 3 \), the magnetic quantum numbers \( m \) range from \( -3 \) to \( +3 \). This includes all integers from -3 through +3.

Step 3: Identify and List Possible Values

The possible values for \( m \) when \( l = 3 \) are: \(-3, -2, -1, 0, +1, +2, +3\).

Step 4: Choose the Correct Option

From the listed options, the correct set of values for \( m \) is \( 0, \pm 1, \pm 2, \pm 3 \), which corresponds to option (b).

Key Concepts

Quantum Numbers

Quantum numbers are essential in understanding the behavior and arrangement of electrons within an atom. They are like addresses that pinpoint where an electron resides within an atom's electron cloud.

• The principal quantum number, denoted by n, indicates the primary energy level occupied by an electron. Larger values of n correspond to higher energy levels and larger orbitals.
• The azimuthal quantum number, or angular momentum quantum number, l, relates to the shape of the electron's orbital. Its values start from 0 and go up to n-1.
• The magnetic quantum number, m, determines the orientation of the orbital in space and can have values from -l to l.
• The spin quantum number describes the electron's spin direction and can be either +1/2 or -1/2.

Together, these quantum numbers provide a full description of an electron's position and motion within an atom, making them a cornerstone of quantum mechanics.

Azimuthal Quantum Number

The azimuthal quantum number, represented by l, is pivotal in defining the shape of an atomic orbital. Each value of l corresponds to a different type of orbital shape:

• s-orbital (for l = 0): Spherical in shape. Electrons within this orbital have no angular momentum around the nucleus.
• p-orbital (for l = 1): Dumbbell-shaped and can align in three orientations in a 3D space: along the x, y, or z axis.
• d-orbital (for l = 2): Generally, these have cloverleaf shapes and can align in various orientations, making them more complex.
• f-orbital (for l = 3): Even more complex in shape and orientation.

Each increase in l adds more complexity to the shape and orientation of orbitals, which in turn affects how electrons form bonds and interact with other atoms. Understanding the azimuthal quantum number is crucial for predicting molecular geometry and chemical reactions.

Electron Orbitals

Electron orbitals are fascinating regions around the nucleus where electrons are likely to be found. These regions are defined by the solution of the Schrödinger equation, a fundamental equation in quantum mechanics. Electron orbitals are not fixed paths but rather cloud-like regions representing probabilities. A key aspect of orbitals is their different shapes and orientations, determined by quantum numbers:

• The principal quantum number (n) sets the size and energy level of the orbital.
• The azimuthal quantum number (l) alters the shape of the orbital.
• The magnetic quantum number (m) influences the orientation or direction of the orbital in space.

Electron orbitals are responsible for the electron density that dictates the chemical bonding properties and reactivities of atoms. Visualizing these orbitals and understanding their quantum numbers enable scientists and students to predict how atoms will interact, form bonds, and create the molecules that make up the world around us.