Problem 126 - A Complete Resource Book in Chemistry for JEE Main

Question

Four different set of quantum numbers for 4 electrons are given below: \(\mathrm{e}_{1}=4,0,0-1 / 2 ; \mathrm{c}_{2}=3,1,1-1 / 2\) \(\mathrm{e}_{3}=3,2,2+1 / 2 ; \mathrm{e}_{4}=3,0,0,+1 / 2\) Then which of the following order of energies of \(\mathrm{e}_{1}, \mathrm{e}_{2}\) \(e_{3}\) and \(e_{4}\) are against the quantum rules ? (a) \(\mathrm{e}_{1}>\mathrm{e}_{2}>\mathrm{e}_{1}>\mathrm{e}_{4}\) (b) \(e_{4}>\mathrm{e}_{3}>\mathrm{e}_{2}>\mathrm{e}_{1}\) (c) \(\mathrm{e}_{3}>\mathrm{e}_{1}>\mathrm{e}_{2}>\mathrm{e}_{4}\) (d) \(e_{2}>e_{3}>c_{4}>c_{1}\)

Step-by-Step Solution

Answer

Option (a) is against the quantum rules.

Step 1: Identify Electron Configuration

The given problem provides quantum numbers for four electrons - \( e_1 = 4,0,0,-1/2 \) , corresponding to \( n=4, \ell=0\) (4s orbital)- \( e_2 = 3,1,1,-1/2 \) , corresponding to \( n=3, \ell=1\) (3p orbital)- \( e_3 = 3,2,2,+1/2 \), corresponding to \( n=3, \ell=2\) (3d orbital)- \( e_4 = 3,0,0,+1/2 \), corresponding to \( n=3, \ell=0\) (3s orbital)

Step 2: Determine the Energy Levels

According to the Aufbau principle, energy levels are filled in the order of increasing \( n + \ell \) values. If \( n + \ell \) values are equal, then the level with the smaller \( n \) is lower in energy.Calculating \( n + \ell \) : - \( e_1: n + \ell = 4 + 0 = 4 \) - \( e_2: n + \ell = 3 + 1 = 4 \) - \( e_3: n + \ell = 3 + 2 = 5 \) - \( e_4: n + \ell = 3 + 0 = 3 \) So, in terms of increasing order of energy: \( e_4 < e_1 \) or \( e_2 < e_3 \)

Step 3: Analyze Given Options

Let's analyze each option based on calculated energy levels:- (a) \( e_1 > e_2 > e_1 > e_4 \): Contradiction in the sequence, and contradicts \( e_1 \) and \( e_2 \) equal- (b) \( e_4 > e_3 > e_2 > e_1 \): Correct in respecting both the \( n + \ell \) rule and \( n \) rules- (c) \( e_3 > e_1 > e_2 > e_4 \): This order seems plausible but is against correct earlier order- (d) \( e_2 > e_3 > c_4 > c_1 \): Incorrect, \( e_4 \) cannot come after \( e_3 \) Therefore, (a) is against quantum rules.

Key Concepts

Electron Configuration

Electron configuration is a way to represent the arrangement of electrons in atoms. Imagine electrons as occupants of an elaborate multi-story building, where each floor corresponds to a principal energy level. The rooms on each floor are different orbitals. To understand this, we use the quantum numbers:

• The Principal Quantum Number ( ) indicates the floor an electron is on, with higher numbers meaning higher floors.
• The Azimuthal Quantum Number ( ) identifies the type of room or orbital, such as s, p, d, or f.

For example, the electron configuration of hydrogen is written as 1s¹, showing one electron in the 1s orbital. Each electron in an atom has a unique set of quantum numbers that specifies its configuration. In the context of determining energies, knowing the configuration helps us understand which electron belongs to which shell and orbital.

Aufbau Principle

The Aufbau principle is a guideline for configuring electrons in atoms efficiently. It means "building up" in German, and it dictates that electrons fill orbitals starting from the lowest energy level, just like filling up seats in a classroom from the front row to the back. This ensures electrons are in their most stable form, minimizing the atom's energy.
Understanding this principle is key to correctly arranging electrons. You use their quantum numbers to sort them according to the increasing order of + , the total energy determination factor. For orbitals with the same + , the one with the lower precedes.
By applying the Aufbau principle, you can predict electron configurations and how electrons fill in accordance with their energy levels efficiently.

Energy Levels

Energy levels in an atom are like different floors of a building, where each floor represents a distinct shell. These levels are quantized, meaning electrons can only exist at specific energy values, much like standing on the particular floor of a building.
The principal quantum number ( ) determines the main energy level or shell, with higher numbers indicating more energy. However, within each main level, there are sublevels or rooms, identified by the azimuthal quantum number ( ). Each sublevel represents different orbitals like s, p, d, and f.

• Each type of orbital has a different shape and can hold a unique number of electrons. For example, an s orbital can hold up to 2 electrons.
• Arrangement in energy levels supports the atom's stability, where electrons closer to the nucleus generally have lower energy.

An understanding of energy levels helps clarify why electrons tend to occupy specific orbitals and how this affects their overall energies.

Quantum Rules

Quantum rules are fundamental principles that govern the behavior of electrons in an atom. These rules ensure that electrons are arranged, fill, and behave correctly in their respective orbitals.
Important quantum rules include:

• Pauli Exclusion Principle: No two electrons in an atom can have identical quantum numbers, meaning they cannot occupy the same state.
• Hund’s Rule: Electrons must occupy all orbitals singly before any orbital is doubly occupied, reducing repulsion between them.
• Heisenberg Uncertainty Principle: You cannot specify both the position and momentum of electrons precisely, an inherent property of their nature.

Together, these quantum rules help maintain order within atoms and dictate the permissible arrangements of electrons in a way that adheres to energy constraints, explains atomic interaction, and defines electron behavior more precisely.