Problem 12 Discuss the concept of electroni... [FREE SOLUTION]

Chapter 25: Problem 12

Discuss the concept of electronically degenerate states with the help of suitable examples.

Short Answer

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Electronically degenerate states are those that have the same energy even though they are represented by different wave functions. An example in atomic orbitals would be the hydrogen atom's 2s and 2p, which have different wave functions but the same energy, hence they are degenerate. A molecular example could be the pi_2px and pi_2py molecular orbitals of Oxygen molecule (O2), which also have equal energy and hence are degenerate.

Step by step solution

Conceptual Understanding

To understand electronic degeneracy, it's crucial to understand that electrons in an atom are arranged into different energy levels. When multiple electrons possess the same energy level (not necessarily the same state), they are said to be degenerate. In quantum mechanics, an energy level is said to be degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, an energy level is said to be non-degenerate if it corresponds to exactly one state.

Example of Electron Degeneracy in an Atom

Consider a simplistic model of an atom, such as a hydrogen atom. In the hydrogen atom, the 2s and 2p orbitals are said to be degenerate because they have the same energy. The 2s orbital is spherical, while the 2p orbitals are dumbbell-shaped and oriented along the x, y, and z axes. Despite their different shapes (and hence different wave functions), their energies are the same, so they are degenerate.

Example of Electron Degeneracy in a Molecule

In a molecule like Oxygen (O2), the Highest Occupied Molecular Orbital (HOMO) is degenerate, comprising the pi_2px and pi_2py molecular orbitals. Both of these orbitals are filled with two electrons, and due to the symmetry of the molecule, they have equal energy. Hence, the pi_2px and pi_2py molecular orbitals of Oxygen molecule (O2) are considered degenerate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Levels

Energy levels are fundamental in understanding electronic degeneracy. Each atom has different levels where electrons reside.
These levels are quantified by the principal quantum number and determine the amount of energy an electron possesses. It's like steps on a ladder.

The lowest rung, or energy level, is closest to the nucleus and corresponds to the lowest energy state.
Higher rungs represent higher energy levels, meaning the electron is further from the nucleus and possesses more energy.

When multiple electrons share the same energy level, they form a group of degenerate states. In quantum systems, **degenerate** means these electrons can exist in different quantum states while having the same energy.

Quantum Mechanics

Quantum mechanics provides the framework that explains how electrons occupy different energy levels and why degeneracy occurs. At its core, quantum mechanics introduces us to the concept of quantization, where physical properties like energy are not continuous, but come in discrete amounts, known as quanta.
A pivotal element of quantum mechanics is the wave function, represented by the Greek letter psi (蠄). This function describes the probability of finding an electron in a specific location around the nucleus.

The Schr枚dinger equation is a foundational formula. It calculates the wave function and, consequently, energy states of an electron.
Electrons do not travel in predictable paths; instead, they exist in "clouds" of probability.

This view leads to multiple possible states for each energy level, some of which may be degenerate.

Atomic Orbitals

Atomic orbitals are regions around an atom's nucleus where there is a high probability of finding electrons. Each orbital corresponds to a specific energy level and shape, influencing electron configuration within an atom.
There are several types of orbitals:

s-orbitals: Spherical in shape and can hold up to two electrons.
p-orbitals: Dumbbell-shaped and come in sets of three (px, py, pz), each can hold two electrons.
d-orbitals: More complex shapes and appear in groups of five orbitals.
f-orbitals: Even more complex, with seven orbitals per set.

Degenerate states often arise with p, d, and f orbitals, where electron energies are equal due to their symmetric distribution in space.

Molecular Orbitals

Molecular orbitals are formed when atomic orbitals from two or more atoms combine, creating a new set of orbitals that spread over a molecule.
They play a critical role in understanding chemical bonding and molecular electron configuration.

Bonding orbitals result from in-phase interactions, which stabilize the molecule.
Antibonding orbitals result from out-of-phase interactions, decreasing stability.
Non-bonding orbitals don't participate significantly in bonding but are crucial for determining molecular properties.

In molecules, degeneracy can occur when orbitals like the pi_2px and pi_2py orbitals of oxygen have the same energy. This happens due to the molecular symmetry and spatial orientation, making molecular orbitals essential for understanding electronic structure and behavior in chemistry.

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Discuss the concept of electronically degenerate states with the help of suitable examples.

Short Answer

Step by step solution

Conceptual Understanding

Example of Electron Degeneracy in an Atom

Example of Electron Degeneracy in a Molecule

Key Concepts

Energy Levels

Quantum Mechanics

Atomic Orbitals

Molecular Orbitals

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