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

The correct option is (b) \( 0, \pm 1, \pm 2, \pm 3 \).

Step 1: Understanding Magnetic Quantum Number

The magnetic quantum number, denoted as \( m \), can take on integer values ranging from \( -l \) to \( +l \). Here, \( l \) is the azimuthal quantum number or the angular momentum quantum number.

Step 2: Determine Range of m

In our case, \( l = 3 \). Therefore, the possible values of \( m \) will range from \( -3 \) to \(+3\), inclusive.

Step 3: List Possible Values

The comprehensive list of possible values for \( m \) are: \( -3, -2, -1, 0, +1, +2, +3 \).

Step 4: Match Values to Choices

Among the given options, choice (b) \( 0, \pm 1, \pm 2, \pm 3 \) best fits the list of calculated values for \( m \).

Key Concepts

Magnetic Quantum Number

The magnetic quantum number is symbolized by the letter \( m \). It's an essential part of quantum mechanics, helping us understand where electrons might be found around a nucleus. The value of \( m \) depends on another quantum number known as the azimuthal, or angular momentum quantum number (\( l \)). The range for \( m \) is determined by \( l \). Specifically, \( m \) can vary from \(-l\) to \(+l\), including all whole numbers in between. **Example of Values**- Suppose \( l = 3 \) - The possible values for \( m \) would be: \, \(-3, -2, -1, 0, +1, +2, +3\). These values tell us about the orientation of the electron's orbital in space, which is crucial for understanding the structure of atoms and how they will interact with each other.

Angular Momentum Quantum Number

The angular momentum quantum number is represented by \( l \). It is sometimes called the azimuthal quantum number. This number defines the shape of the electron's orbital. **Relationship with Orbitals**- Different values of \( l \) correspond to different types of orbitals: - \( l = 0 \): s-orbital (spherical shape) - \( l = 1 \): p-orbital (dumbbell shape) - \( l = 2 \): d-orbital (cloverleaf shape) - \( l = 3 \): f-orbital (complex shape)**Allowed Values**- The value of \( l \) depends on the principal quantum number \( n \) and must satisfy the condition \( 0 \leq l \leq n-1 \). This quantum number plays a pivotal role in capturing the energy levels of electrons within an atom, further helping to understand how they participate in chemical bonding.

Azimuthal Quantum Number

The azimuthal quantum number, also called the angular momentum quantum number, is denoted as \( l \). This number specifies the subshell or sub-level of an electron within a given principal energy level. **Importance in Quantum Mechanics**- It determines the shape of the electron cloud and is crucial for predicting chemical behavior and bonding characteristics.**Understanding Subshells**- Each principal energy level \( n \) can have multiple sub-levels, from \( l = 0 \) up to \( n-1 \). So for example: - If \( n = 3 \), then \( l \) can be \( 0 (s), 1 (p), 2 (d) \).The azimuthal quantum number is fundamental in quantum chemistry, helping chemists to identify and calculate possible electron configurations in complex atoms or molecules.