Question

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

A half-filled subshell is particularly stable due to minimized electron repulsion, lower shielding effect, and increased exchange energy. When electrons occupy different orbitals within a subshell (half-filled), the repulsion is minimized, leading to a lower energy state. Additionally, the outer electrons experience a lower effective nuclear charge due to reduced shielding from inner electrons, resulting in increased stability. Lastly, in a half-filled subshell with maximum unpaired electrons, the exchange energy is maximized, contributing to the overall stability of the atom.

Step-by-step solution

Understand electron configuration

Each electron in an atom occupies a unique position known as its quantum state. This is described by a set of quantum numbers (n, l, m_l, and m_s). Electron configuration is the arrangement of electrons in the subshells of an atom, and it is determined by the Aufbau principle and Hund's rule.

Recall the Aufbau principle and Hund's rule

The Aufbau principle states that electrons fill the lower energy subshells first before filling the higher energy subshells, and the Pauli Exclusion Principle dictates that no two electrons in an atom can have the same set of quantum numbers. Hund's rule states that electrons occupy separate orbitals within the same subshell before pairing up, with parallel spins, which minimizes the repulsion between electrons.

Explain the electron repulsion

Electrons are negatively charged and repel each other. When electrons occupy the same orbital, the repulsion between them increases, which results in a higher energy state. On the other hand, when electrons occupy different orbitals within the same subshell (half-filled), the repulsion between them is minimized due to increased distance from each other, resulting in a lower energy state and increased stability.

Discuss the shielding effect

The shielding effect occurs when inner electrons shield the outer electrons from the full attractive force of the nucleus. In half-filled subshells, the outer electrons experience a lower effective nuclear charge due to the lower shielding effect. This results in increased stability as they are more strongly attracted to the nucleus.

Describe the exchange energy

Exchange energy is an additional stabilization factor in multi-electron atoms. In a half-filled subshell, there are a maximum number of unpaired electrons with the same spin. These electrons can swap or "exchange" their positions without violating the Pauli Exclusion Principle, increasing the exchange energy, which contributes to the overall stability of the atom. In conclusion, a half-filled subshell is particularly stable due to the combined effects of minimized electron repulsion, a lower shielding effect, and increased exchange energy. These factors together contribute to the overall stability of the atom.

Key concepts

Aufbau Principle

Electron configuration is an essential aspect of understanding atomic structure, and the Aufbau Principle plays a critical role in this process. According to the Aufbau Principle, electrons occupy the lowest energy subshells available before moving to higher energy subshells. This sequential filling ensures that electrons arrange themselves to maintain the lowest possible energy state, which is a fundamental goal in atomic organization.
To visualize how electrons are distributed in an atom, imagine a hierarchical set of energy levels or "shelves," with the lowest energy shelf being filled first. The order in which these subshells are filled is often remembered using the periodic table or specific mnemonics such as 1s, 2s, 2p, 3s, and so on.
The principle is named "Aufbau," a German word for "building up," reflecting the step-by-step nature of electron filling. This principle is supported by the Pauli Exclusion Principle, which states that no two electrons can have the same set of quantum numbers, highlighting the distinct nature of each electron's state.

Hund's Rule

When electrons occupy orbitals within the same subshell, they follow Hund's Rule, ensuring the lowest energy state is achieved. Hund's Rule specifies that electrons fill separate orbitals singly before pairing up. This arrangement reduces electron-electron repulsion by maximizing distance between electrons, naturally minimizing energy.
One can think of this rule as electrons "claiming their own space" before sharing it with another electron. Keeping spins parallel avoids additional repulsion, as electrons with opposite spins would generate increased energy interaction.
This behavior of electrons is likened to passengers on a bus preferring empty seats over sharing one with another person. Hund's Rule encourages better spatial distribution of electrons, promoting a stable and low-energy configuration.

Exchange Energy

A fascinating stabilization phenomenon in atoms is the concept of Exchange Energy. This concept becomes prominent when dealing with unpaired electrons in subshells. In a half-filled subshell, there are maximum unpaired electrons present, and electrons with parallel spins can swap positions in the orbital to enhance stability without conflicting with Pauli’s Exclusion Principle.
The swapping or "exchange" of these electrons leads to a lower energy state due to this increased "exchange energy." It's akin to a dance where dancers (electrons) swap places to optimize their performance (energy state) without colliding with one another.
This optimization contributes to the stability seen in half-filled subshells and is one of the reasons why configurations like those in chromium (Cr) and copper (Cu), which seem anomalous, actually represent a lower, more stable energy state for the element.

Shielding Effect

Shielding Effect refers to how inner electrons in an atom shield outer electrons from the nucleus's full positive charge. This effect notably impacts how tightly outer electrons are held. Electrons closer to the nucleus repel outer electrons, reducing the effective nuclear charge these outer electrons experience.
In half-filled subshells, the shielding is not as strong because the electrons are more evenly distributed across multiple orbitals, rather than being compacted into fewer orbitals. This allows the outer electrons to feel more of the nucleus’s pull due to reduced internal repulsive forces.
The decreased shielding effect in such configurations enhances stability, allowing electrons to maintain a low-energy configuration due to the stronger attraction to the nucleus. Therefore, understanding the Shielding Effect provides insight into why certain electron arrangements are particularly stable in atomic structure.