Orbitals That Have The Same Energy Are Called

Orbitals That Have the Same Energy Are Called Degenerate Orbitals: A Deep Dive into Atomic Structure

Orbitals are regions of space around an atom's nucleus where there's a high probability of finding an electron. Understanding their properties, particularly their energy levels, is crucial to grasping the fundamentals of chemistry and atomic physics. A key concept in this understanding is degeneracy, which refers to orbitals possessing the same energy. This article will delve into the intricacies of degenerate orbitals, exploring their characteristics, the factors influencing their degeneracy, and the implications of degeneracy lifting.

What are Degenerate Orbitals?

Orbitals that have the same energy are called degenerate orbitals. This means that electrons occupying these orbitals possess the same energy level. In a simplified, isolated hydrogen atom, for instance, all orbitals within the same principal energy level (n) are degenerate. This means the 2s and 2p orbitals have the same energy. However, this degeneracy is often lifted in multi-electron atoms due to factors such as electron-electron interactions and shielding effects.

Principal Quantum Number (n) and Degeneracy

The principal quantum number (n) dictates the energy level of an electron. Higher values of 'n' correspond to higher energy levels and larger orbital sizes. In a hydrogen-like atom (one electron only), orbitals with the same 'n' value are degenerate. This is because there's only one electron experiencing the attraction of the nucleus without significant interactions from other electrons.

Azimuthal Quantum Number (l) and Degeneracy

The azimuthal quantum number (l) determines the shape of the orbital and its angular momentum. For a given 'n', 'l' can range from 0 to (n-1). Each value of 'l' corresponds to a subshell:

• l = 0: s subshell (spherical shape)
• l = 1: p subshell (dumbbell shape)
• l = 2: d subshell (complex shapes)
• l = 3: f subshell (even more complex shapes)

In a hydrogen-like atom, orbitals with the same 'n' but different 'l' values are still degenerate. This changes significantly in multi-electron atoms.

Magnetic Quantum Number (ml) and Degeneracy

The magnetic quantum number (ml) specifies the orientation of the orbital in space. For a given 'l', ml can range from -l to +l, including 0. Each value of ml represents a specific orbital within a subshell. For example, for l=1 (p subshell), ml can be -1, 0, +1, representing three p orbitals (px, py, pz). In a hydrogen-like atom, these orbitals are degenerate.

Degeneracy Lifting in Multi-Electron Atoms

The simple picture of degeneracy presented above changes dramatically when considering atoms with more than one electron. Electron-electron interactions and shielding effects play a crucial role in lifting this degeneracy.

Electron-Electron Repulsion

In multi-electron atoms, electrons repel each other. This repulsion alters the energy levels of the orbitals. Electrons in different orbitals experience different levels of shielding from the nucleus by other electrons. For example, a 2s electron penetrates closer to the nucleus than a 2p electron, resulting in a lower energy for the 2s orbital. Therefore, the 2s and 2p orbitals are no longer degenerate.

Shielding Effect

The inner electrons shield the outer electrons from the full positive charge of the nucleus. This reduces the effective nuclear charge experienced by the outer electrons. The extent of shielding varies depending on the orbital shape and penetration. s orbitals penetrate more effectively than p orbitals, which penetrate more effectively than d orbitals, and so on. Consequently, s orbitals experience a stronger effective nuclear charge and lower energy compared to p orbitals of the same principal quantum number.

Effective Nuclear Charge (Zeff)

The effective nuclear charge (Zeff) represents the net positive charge experienced by an electron, taking into account shielding by other electrons. Electrons in orbitals with greater penetration experience a higher Zeff, resulting in lower energy.

Penetration and Shielding

The penetration of an electron's orbital into the inner electron shells influences its energy level. Orbitals that penetrate more effectively experience a reduced shielding effect, leading to a higher effective nuclear charge and lower energy. This explains why, in multi-electron atoms, the s orbital always has lower energy than the p orbital within the same shell.

Consequences of Degeneracy and Degeneracy Lifting

The degeneracy (or lack thereof) of orbitals has significant implications for various aspects of atomic structure and chemical behavior.

Electron Configuration

The electron configuration of an atom describes how electrons are distributed among the orbitals. The Pauli Exclusion Principle and Hund's Rule guide this distribution, considering orbital degeneracy. When orbitals are degenerate, electrons will fill them individually before pairing up (Hund's rule). This maximizes the total spin, resulting in a more stable configuration.

Atomic Spectra

Atomic spectra arise from electron transitions between energy levels. The differences in energy levels between degenerate and non-degenerate orbitals directly affect the observed spectral lines. Degeneracy lifting leads to a more complex and fine-structured spectrum.

Chemical Bonding

The energy levels of orbitals are crucial in determining how atoms interact and form chemical bonds. The degeneracy of orbitals influences the bonding strength and the types of bonds that can be formed. For example, the difference in energy between the s and p orbitals influences hybridization and the geometry of molecules.

Spectroscopy and Quantum Mechanics

Degeneracy and degeneracy lifting are central concepts in various spectroscopic techniques. Understanding the energy levels of orbitals allows us to interpret spectral data and gain insights into atomic and molecular structure. Advanced quantum mechanical calculations are crucial for accurately predicting the energy levels of orbitals in complex systems and predicting the lifting of degeneracy.

Examples of Degenerate and Non-Degenerate Orbitals

Let's consider specific examples to illustrate the concepts discussed:

Hydrogen Atom (Z=1):

• n=1: 1s orbital (only one orbital) – no degeneracy
• n=2: 2s and three 2p orbitals (2px, 2py, 2pz) – degenerate in the hydrogen atom
• n=3: 3s, three 3p, and five 3d orbitals – degenerate in the hydrogen atom

Lithium Atom (Z=3):

• The 2s and 2p orbitals are not degenerate in lithium due to electron-electron repulsion and the shielding effect. The 2s orbital has lower energy than the 2p orbitals. The electrons will fill the lower energy 2s orbital first, before filling the 2p orbitals.

Other Multi-Electron Atoms:

The pattern of non-degeneracy continues for all multi-electron atoms. The s orbitals will always have lower energy than p orbitals within the same shell, followed by d and f orbitals. The energy gap between orbitals within the same shell increases with increasing atomic number.

Conclusion: The Significance of Degenerate Orbitals

The concept of degenerate orbitals, while seemingly simple, is fundamental to understanding the structure and behavior of atoms and molecules. While degeneracy is a characteristic of hydrogen-like atoms, the lifting of degeneracy in multi-electron atoms profoundly impacts electron configurations, chemical bonding, and atomic spectra. Appreciating the role of electron-electron repulsion, shielding, and effective nuclear charge in determining orbital energies is essential for a deeper understanding of quantum mechanics and its applications in chemistry and physics. The study of degenerate and non-degenerate orbitals provides a crucial bridge between simple atomic models and the complex reality of multi-electron systems, highlighting the importance of advanced quantum mechanical techniques for accurate predictions and interpretations.