Problem 19 - A Complete Resource Book in Chemistry for JEE Main

Question

Quantum numbers of an atom can be defined on the basis of (a) Aufbau's principle (b) Heisenberg's uncertainity principle (c) Hund's rule (d) Pauli's exclusion principle

Step-by-Step Solution

Answer

(d) Pauli's exclusion principle defines quantum numbers by specifying that no two electrons can have identical sets of quantum numbers.

Step 1: Understanding Quantum Numbers

Quantum numbers are sets of numerical values that provide solutions to the Schrodinger equation, allowing us to understand electron positions and energy levels in an atom. They are essential for describing the electrons' orbitals.

Step 2: Quantum Numbers and Principles

Each quantum number represents a different aspect of the electron's motion and the energy levels within the atom. These quantum numbers can be explained based on a fundamental principle that governs the arrangement and behavior of electrons.

Step 3: Analyzing the Options

Now, let's consider each option: (a) Aufbau's principle describes the order in which electrons fill atomic orbitals but does not define quantum numbers. (b) Heisenberg's uncertainty principle states that one cannot simultaneously know the exact position and speed of an electron, unrelated to defining quantum numbers. (c) Hund's rule is concerned with how electrons occupy orbitals of the same energy but does not define quantum numbers. (d) Pauli's exclusion principle states that no two electrons in an atom can have the same set of all four quantum numbers, explicitly defining quantum numbers.

Step 4: Selecting the Correct Answer

Based on the analysis, option (d) Pauli's exclusion principle is the correct answer because it directly defines and governs the uniqueness of quantum numbers in an atom by stating that no two electrons can share the same set of quantum numbers.

Key Concepts

Pauli's Exclusion Principle

The Pauli's exclusion principle is a fundamental concept in quantum mechanics, discovered by Wolfgang Pauli in 1925. It states that no two electrons in the same atom can have identical sets of quantum numbers. This principle is vital for the structure of the periodic table and explains why elements have unique properties.
To understand this better, remember that every electron in an atom is described by four quantum numbers:

• The principal quantum number \(n\)
• The azimuthal (angular momentum) quantum number \(l\)
• The magnetic quantum number \(m\)
• The spin quantum number \(s\)

These numbers provide a unique address for each electron. Thanks to the Pauli exclusion principle, each of the electrons in an atom can be distinguished distinctly by their quantum numbers, which helps in predicting chemical and physical behavior of elements.

Schrodinger Equation

The Schrodinger equation is a pivotal element in quantum mechanics that describes how quantum states evolve over time. Formulated by Erwin Schrödinger in 1926, it serves as a mathematical model to predict the behavior of electrons within an atom.
The equation is typically written as: \[\hat{H}\psi = E\psi\]where \(\hat{H}\) is the Hamiltonian operator, \(\psi\) represents the wave function, and \(E\) stands for the energy of the system.
It helps in understanding the probabilistic distribution of electrons in different orbitals and energy levels. By solving the Schrodinger equation, we can determine the possible energy states that an electron can occupy, and this leads to the derivation of quantum numbers essential for describing these states.

Electron Orbitals

Electron orbitals are regions around the nucleus where electrons are likely to be found. They are not fixed paths, but rather zones where the electron's probability of being located is highest.
Orbitals have different shapes and energy levels based on quantum numbers, primarily determined by the Schrodinger equation's solutions:

• \(s\)-orbitals are spherical
• \(p\)-orbitals are dumbbell-shaped
• \(d\)- and \(f\)-orbitals have more complex shapes

Each orbital can hold up to two electrons, provided they have opposite spins. Breaking it down:- Every electron in an orbital is uniquely defined by its quantum numbers.- This helps predict chemical behavior and bonding characteristics of atoms.

Electron Configuration

Electron configuration is a method of detailing the distribution of electrons in an atom's orbitals. It provides critical insight into an element's chemical properties and reactivity.
Understanding electron configuration involves:

• Filling orbitals with electrons in order of increasing energy levels, following the Aufbau principle
• Applying Pauli's exclusion principle, ensuring no two electrons within the same atom have identical quantum numbers
• Considering Hund's rule for distributing electrons into orbitals of the same energy

For instance, the electron configuration of oxygen is written as \(1s^2 2s^2 2p^4\), which indicates its electron distribution in different energy levels and orbitals. This hierarchical distribution not only organizes the periodic table but also explains the chemical behavior and interaction of elements.