Problem 165 - A Complete Resource Book in Chemistry for JEE Main

Question

In a multielectron atom, which of the following orbitals described by the three quantum numbers will have the same energy in the absence of magnetic field and electric fields? \(\quad\) [2005] (a) \(\mathrm{n}=1, l=0, \mathrm{~m}=0\) (b) \(\mathrm{n}=2, l=0, \mathrm{~m}=0\) (c) \(\mathrm{n}=3, l=1, \mathrm{~m}=1\) (d) \(\mathrm{n}=3, l=2, \mathrm{~m}=1\) (e) \(\mathrm{n}=3, l=2, \mathrm{~m}=0\) (a) \(\mathrm{b}\) and \(\mathrm{c}\) (b) \(\mathrm{d}\) and \(\mathrm{e}\) (c) \(\mathrm{c}\) and \(\mathrm{d}\) (d) a and \(\mathrm{b}\)

Step-by-Step Solution

Answer

(b) d and e

Step 1: Understanding Energy Levels in a Multielectron Atom

In multielectron atoms, the energy levels of orbitals are determined by the principal quantum number \( n \) and the azimuthal quantum number \( l \). Orbitals with different \( n \) usually have different energies. Within a given \( n \), the energy typically increases with increasing \( l \).

Step 2: Analyzing Given Quantum Numbers

Review the quantum numbers provided for each orbital: - (a) \( n=1, l=0, m=0 \)- (b) \( n=2, l=0, m=0 \)- (c) \( n=3, l=1, m=1 \)- (d) \( n=3, l=2, m=1 \)- (e) \( n=3, l=2, m=0 \)Within the same \( n \), orbitals with the same \( l \) but different magnetic quantum numbers \( m \) have the same energy in the absence of external fields.

Step 3: Identifying Orbitals with the Same Energy

Since \( m \) does not affect energy without external fields, focus on \( n \) and \( l \):- (d) \( n=3, l=2 \) - (e) \( n=3, l=2 \)Both (d) and (e) have the same \( n \) and \( l \), thus they have the same energy level.

Step 4: Choosing the Correct Answer

Since (d) and (e) have the same quantum numbers \( n \) and \( l \) \( (n=3, l=2) \), they are the orbitals that will have the same energy. Verify that (b) is correct, which corresponds to option (b) \( d \) and \( e \).

Key Concepts

Quantum Numbers

Quantum numbers are like an address system for electrons within an atom, helping us understand their positions and energy levels. Each electron in an atom is described by a unique set of quantum numbers, which are:

• Principal quantum number ( "): It indicates the main energy level or shell. Higher the value of , higher the energy level and the electron's probable distance from the nucleus.

The principal quantum number is fundamental in determining the overall size of an orbital, and it plays a crucial role in distinguishing between different shells.

• Azimuthal quantum number ( "): It defines the subshell or the type of the orbital within each principal level (e.g., s, p, d, f). Its values range from 0 to (n-1).

This number dictates the shape of the orbital and influences its energy, adding complexity to the electron's placement and behavior within the atom.

• Magnetic quantum number (m): This number describes the orientation of the orbital in space relative to the other orbitals. It takes values between -l and +l.

Even though the magnetic quantum number affects the orbital's spatial orientation, it doesn't impact the energy level without an external magnetic or electric field.

Energy Levels

Energy levels in a multielectron atom indicate the quantized energy conditions under which electrons reside. These levels are designated by the principal quantum number (n). Each integer value of represents a shell or energy level:

• Higher n values signify higher energy levels, which means that electrons are further from the nucleus on average.

The energy levels are not evenly spaced; as n increases, the energy difference between consecutive levels decreases.
In a multielectron atom, electrons repel each other due to their negative charges. This electron-electron interaction causes shifts in energy levels compared to a single electron atom (like hydrogen).
The presence of multiple electrons also causes a phenomenon known as "electron shielding." Electrons in inner shells partially shield outer shell electrons from the full attractive force of the nuclear charge, resulting in modified energy levels.
Despite the added complexity, the energy level framework remains a fundamental aspect for understanding atomic structure, chemical bonding, and spectral properties of elements.

Orbital Energy

Orbital energy refers to the energy associated with an electron residing in a specific orbital, taking into account its principal and azimuthal quantum numbers:

• Within the same principal energy level (same n), the azimuthal quantum number (l) becomes crucial in determining the orbital energy.

For any given , different az values lead to the formation of different subshells such as s, p, d, f, with increasing complexity and energy.
Within the same shell (same n), orbitals with a higher l value have higher energy. For example, in the n=3 shell, the energy order is s < p < d.
These subtle variations in orbital energy levels are significant in chemistry as they determine electron configurations, chemical reactivity, and bonding.
Understanding that these energy differences result from electron-electron interactions and other quantum mechanical properties helps in predicting the behavior of multielectron systems.