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Chapter 25: Problem 12

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

Short Answer

Expert verified
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

01

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.
02

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.
03

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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Most popular questions from this chapter

Explain the following: (a) The ionic radii of \(\mathrm{Ca}^{2+}, \mathrm{Mn}^{2+}\) and \(\mathrm{Zn}^{2+}\) decrease regularly. (b) The ionic radius of \(\mathrm{Ni}^{2+}\) is smaller than that of \(\mathrm{Cu}^{2+}\) in presence of octahedral crystal-field environment of halide ions. (c) \(\mathrm{Co}^{3+}\) preferably adopts octahedral geometry under the effect of strong field ligands. (d) \(\mathrm{Mn}_{2} \mathrm{O}_{4}\) exists in normal spinel structure while \(\mathrm{Fe}_{2} \mathrm{O}_{1}\) exists in inverse spinel structure.

Discuss the crystal-field effects on spinel structures.

Discuss the splitting of energy of degenerate \(d\) -orbitals in presence of tetrahedral crystal-field potential.

Discuss the crystal-field effects on lattice energy and hydration energies of transition metal ions.

Derive and expression for the tetrahedral and cubic crystal-field potential.
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