Problem 97 - A Complete Resource Book in Chemistry for JEE Main

Question

For \(\mathrm{n}=2\) the correct set of \(\ell, \mathrm{m}\) are (a) \(\ell=2, \mathrm{~m}=-2,-1,0+1,+2\) (b) \(\ell=1 \mathrm{~m}=-2,-1,0+1,+2\) (c) \(\ell=1 \mathrm{~m}=-1,0,+1\) (d) \(\ell=0 \mathrm{~m}=-1,0,+1\)

Step-by-Step Solution

Answer

The correct set is (c) \( \ell = 1, m = -1, 0, +1 \).

Step 1: Review Quantum Numbers

Quantum numbers are used to describe the properties of electrons in atoms. For each value of the principal quantum number \( n \), the angular momentum quantum number \( \ell \) can take integer values from \( 0 \) to \( n-1 \). The magnetic quantum number \( m \) can take integer values from \(-\ell\) to \(+\ell\).

Step 2: Determine Possible Values of \( \ell \) for \( n=2 \)

Since \( n = 2 \), \( \ell \) can be \( 0 \) or \( 1 \). These are the only possible values for \( \ell \) when \( n = 2 \).

Step 3: Determine Possible Values of \( m \) for Each \( \ell \) Value

For \( \ell = 0 \), \( m \) can only be \( 0 \), because \( m \) ranges from \( -\ell \) to \( \ell \). For \( \ell = 1 \), \( m \) can be \( -1, 0, \) or \( +1 \). Therefore, acceptable \( m \) values depend on the corresponding \( \ell \) value.

Step 4: Evaluate Each Answer Choice

(a) \( \ell = 2 \) is not possible for \( n = 2 \), so this is incorrect. (b) \( \ell = 1 \) cannot have \( m = -2 \) or \( +2 \), so this is wrong. (c) \( \ell = 1 \) with \( m = -1, 0, +1 \) is correct. (d) \( \ell = 0 \) cannot have \( m = -1, +1 \), so this is incorrect.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), plays a crucial role in defining the size and energy level of an electron's orbit within an atom. This quantum number can take on any positive integer value (1, 2, 3, ...). Each increase in \( n \) represents a higher energy level and a larger orbit, as the electron can move farther from the nucleus.
For instance, when \( n = 1 \), the electron occupies the first energy level, closest to the nucleus. When \( n = 2 \), the electron is on the second level, slightly further out, and so on. In the context of our original exercise, for \( n = 2 \), electrons can occupy this energy level with specific configurations as determined by other quantum numbers.
This number also helps in determining the number of subshells or orbitals available at a certain energy level. For any given \( n \), there are \( n \) possible subshells, hence more room for electrons to occupy different spaces within an atom.

Angular Momentum Quantum Number

The angular momentum quantum number, represented as \( \ell \), is associated with the shape of the electron's orbit or subshell within an atom. Given a principal quantum number \( n \), \( \ell \) can take integer values ranging from 0 to \( n-1 \).
In simpler terms:

• When \( \ell = 0 \), it corresponds to an s-orbital.
• \( \ell = 1 \) aligns with a p-orbital.
• \( \ell = 2 \) corresponds to a d-orbital, and so forth.

These orbitals can further influence the properties of elements and their position on the periodic table. In the exercise, we focused on \( n = 2 \), meaning \( \ell \) can be 0 or 1. When \( \ell = 0 \), it indicates a spherically shaped s-orbital. For \( \ell = 1 \), a dumbbell-shaped p-orbital is implied.
The value of \( \ell \) is critical in identifying how electrons are arranged around an atom, impacting their chemical properties and bonding potential.

Magnetic Quantum Number

The magnetic quantum number, symbolized as \( m \), provides information on the orientation of the orbitals in space relative to an external magnetic field. For a given angular momentum quantum number \( \ell \), \( m \) can range from \(-\ell\) to \(+\ell\). This variety allows for the distinction and functioning of individual orbitals within a subshell.
To detail with examples:

• If \( \ell = 0 \), then \( m \) is definitively 0.
• For \( \ell = 1 \), \( m \) can be -1, 0, or +1.
• With \( \ell = 2 \), \( m \) could range from -2 to +2.

In this way, \( m \) actually denotes the various possible orientations an orbital can have, each offering unique paths in which electron probability clouds align. This feature lets electrons fill subshells in a manner that minimizes electron repulsions. Using our original exercise, when \( n = 2 \) and \( \ell = 1 \), \( m \) values of -1, 0, or +1 perfectly fit this electron configuration, indicating differing spatial alignments.
Ultimately, the magnetic quantum number assists in determining electron configurations and specified atomic orbital orientations.