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Chapter 7: Problem 27

Many times the claim is made that subshells half-filled with electrons are particularly stable. Can you suggest a possible physical basis for this claim?

Short Answer

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Electron configuration, which describes the distribution of electrons in orbitals, is related to the stability of subshells. Half-filled subshells are particularly stable due to three factors: minimization of electron-electron repulsion, symmetry, and exchange energy. In a half-filled subshell, electrons are spread out evenly, minimizing repulsive forces. Additionally, symmetry in electron configuration ensures even charge distribution and contributes to stability. Lastly, exchange energy, which arises due to electrons with the same spin occupying different orbitals, is at its maximum in half-filled subshells. An example of this stability can be seen in Chromium's electron configuration: 1s虏 2s虏 2p鈦� 3s虏 3p鈦� 3d鈦� 4s鹿, where the 3d subshell is half-filled, increasing its stability.

Step by step solution

01

Introduction to electron configuration

Electron configuration is the distribution of electrons of an atom or molecule in atomic or molecular orbitals. The arrangement is in ascending order of energy levels, with a maximum of two electrons in each orbital, thanks to the Pauli Exclusion Principle (each electron in an atom must have a unique set of quantum numbers).
02

Understanding subshells and their stability

Subshells are the subdivisions of electron shells, which are labeled as s, p, d, or f, with a specific number of orbitals and maximum electrons. For example, the s subshell has 1 orbital and can hold a maximum of 2 electrons, while the p subshell has 3 orbitals and can hold a maximum of 6 electrons. A half-filled subshell means having exactly half of the maximum number of electrons possible in a subshell. The stability of subshells is attributed to three factors: electron-electron repulsion minimization, symmetry, and exchange energy.
03

Describing physical basis for stability in half-filled subshells

1. Minimization of electron-electron repulsion: When a subshell is half-filled, the electrons in it have minimal repulsion, as they are spread out evenly among the available orbitals. This even distribution reduces the total repulsive force acting on the electrons, which increases the stability of the subshell. 2. Symmetry: Half-filled subshells are symmetrical in their electron configuration, which adds to their stability. Symmetry ensures an even distribution of charge and minimizes the energy associated with the system. 3. Exchange energy: Exchange energy arises from the requirement that electrons with the same spin must have different spatial quantum numbers in order to occupy the same orbital. In half-filled subshells, the exchange energy is at its maximum, which results in greater stability.
04

Example of a half-filled subshell

Let's consider the electron configuration of Chromium (Cr) with an atomic number of 24: Expected electronic configuration: 1s虏 2s虏 2p鈦� 3s虏 3p鈦� 3d鈦� 4s虏 Actual electronic configuration: 1s虏 2s虏 2p鈦� 3s虏 3p鈦� 3d鈦� 4s鹿 In the case of Chromium, it promotes one electron from the 4s orbital to the 3d orbital. This results in a half-filled 3d subshell (3d鈦�), which is more stable due to the reasons discussed above. This analysis demonstrates the physical basis behind the claim for stability in half-filled subshells and provides an example of an element that exhibits this stability.

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Most popular questions from this chapter

A carbon-oxygen double bond in a certain organic molecule absorbs radiation that has a frequency of \(6.0 \times 10^{13} \mathrm{~s}^{-1}\). a. What is the wavelength of this radiation? b. To what region of the spectrum does this radiation belong? c. What is the energy of this radiation per photon? per mole of photons? d. A carbon-oxygen bond in a different molecule absorbs radiation with frequency equal to \(5.4 \times 10^{13} \mathrm{~s}^{-1}\). Is this radiation more or less energetic?

Calculate the velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^{2} \mathrm{~nm}\) and \(1.0 \mathrm{~nm}\), respectively.

How many orbitals in an atom can have the designation \(5 p, 3 d_{z^{2}}\), \(4 d, n=5, n=4 ?\)

The elements \(\mathrm{Cu}, \mathrm{O}, \mathrm{La}, \mathrm{Y}, \mathrm{Ba}, \mathrm{Tl}\), and \(\mathrm{Bi}\) are all found in high- temperature ceramic superconductors. Write the expected electron configuration for these atoms.

Three elements have the electron configurations \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6}\), \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}\), and \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{1}\). The first ionization energies of these elements (not in the same order) are \(0.419,0.735\), and \(1.527 \mathrm{MJ} / \mathrm{mol}\). The atomic radii are \(1.60,0.98\), and \(2.35 \AA\). Identify the three elements, and match the appropriate values of ionization energy and atomic radius to each configuration.
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