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

Option (c) is correct: \( \ell = 1, m = -1, 0, +1 \).

Step 1: Understand Quantum Numbers

For an electron in an atom, quantum numbers describe its state. For the principal quantum number \( n = 2 \), the possible values for the azimuthal quantum number \( \ell \) can be \( 0 \leq \ell < n \). Hence, \( \ell \) can be 0 or 1.

Step 2: Determine Magnetic Quantum Number \( m \)

The magnetic quantum number \( m \) is dependent on \( \ell \) and ranges from \( -\ell \) to \( +\ell \). Thus, for \( \ell = 0 \), \( m = 0 \). For \( \ell = 1 \), \( m \) can be \(-1, 0,\) or \( +1 \).

Step 3: Analyze Given Options

- Option (a): \( \ell = 2 \), but \( \ell \) cannot be 2 for \( n = 2 \), therefore (a) is incorrect.- Option (b): \( \ell = 1 \) but \( m = -2, -1, 0, +1, +2 \) is incorrect because \( m \) should be \(-1, 0, +1\).- Option (c): \( \ell = 1 \) and \( m = -1, 0, +1 \) which satisfies the conditions, so (c) is correct.- Option (d): \( \ell = 0 \) and \( m = -1, 0, +1 \). For \( \ell = 0 \), \( m \) should be only 0, so (d) is incorrect.

Key Concepts

Principal Quantum Number

The principal quantum number, often denoted by the letter \( n \), is a fundamental concept in quantum mechanics. It dictates the primary energy level of an electron in an atom. Understanding this is crucial for grasping the structure of atoms.

Some key points to note about the principal quantum number:

• It is always a positive integer, starting from 1, so \( n = 1, 2, 3, \) and so on.
• As \( n \) increases, the electron's energy level also increases and the electron is further away from the nucleus.
• The number of possible subshells (or orbitals) in a given energy level increases with \( n \).
• The total possible number of electrons in a shell is given by \( 2n^2 \).

For example, when \( n = 2 \), it indicates the second energy level of an atom, which can have different combinations of other quantum numbers. This guides us in determining the allowable combinations of azimuthal and magnetic quantum numbers.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( \ell \), defines the shape and angular momentum of an electron's orbital. This quantum number is essential for diving into the finer structure of electron orbitals and their properties.

Let's break down the azimuthal quantum number:

• The value of \( \ell \) depends directly on \( n \), where \( 0 \leq \ell < n \).
• When \( n = 2 \), \( \ell \) can be either 0 or 1.
• Each value of \( \ell \) corresponds to different subshell types:
  • \( \ell = 0 \) is called the 's' subshell (spherical shape).
  • \( \ell = 1 \) is known as the 'p' subshell (dumbbell-shaped).

Understanding \( \ell \) helps identify the spatial distribution and overlap between electron clouds, providing insight into chemical bonding and molecular structure.

Magnetic Quantum Number

The magnetic quantum number, noted as \( m \) or \( m_\ell \), defines the orientation of an electron's orbital around the nucleus. This quantum number is a pivotal factor in the magnetic properties of an atom.

Here's how the magnetic quantum number functions:

• The range of \( m \) is defined as \( -\ell \) to \( +\ell \), including zero.
• For instance, if \( \ell = 0 \), then \( m = 0 \). However, if \( \ell = 1 \), \( m \) can be -1, 0, or +1.
• These values signify different orientations of electron orbitals in space. This orientation affects how electrons interact with external magnetic fields.

In practical applications, understanding \( m \) allows us to determine the possible number of orientations for a given subshell, influencing the atom's spectral lines and chemical properties.