Question

Explain why each of the following electron configurations for a \(p\) sublevel is disallowed. (a) \(\downarrow \downarrow \downarrow \downarrow \downarrow \downarrow\) (b) \(\downarrow \uparrow \downarrow \downarrow \square \square\)

Short answer

Configuration (a) is disallowed because it contains six electrons with the same spin direction in the 'p' sublevel, violating the Pauli exclusion principle. Configuration (b) is disallowed as it violates Hund's rule: electrons must fill all unoccupied orbitals before pairing up in the same orbital, which is not the case here.

Step-by-step solution

- Understanding Electron Spin

In accordance with the Pauli exclusion principle, an atomic orbital can contain a maximum of two electrons with opposite spins. This principle is a quantum mechanical property that prohibits electrons within the same orbital from having identical quantum numbers, including their spin quantum number. The electrons are represented by arrows: an upward arrow (\(\uparrow\)) signifies one spin direction, while a downward arrow (\(\downarrow\)) represents the other.

- Analyzing Configuration (a)

We inspect the given electron configuration for the (a) case: six electrons with the same spin direction in a single 'p' sublevel. This violates the Pauli exclusion principle because we can only have at most two electrons in each 'p' orbital and they must have opposite spins, not the same. Moreover, a 'p' sublevel contains only three orbitals, which can accommodate a total of six electrons, three pairs with opposite spins.

- Analyzing Configuration (b)

We examine the given electron configuration for the (b) case: four electrons with one pair showing opposite spins and the other two unpaired with the same spin direction, and two empty spaces (\(\square\)) in the 'p' sublevel. This violates Hund's rule, which states that electrons will fill an unoccupied orbital before they pair up in the same orbital. In this configuration, instead of populating each of the available orbitals with a single electron first, the configuration has a paired electron while leaving two orbitals empty.

Key concepts

Pauli Exclusion Principle

The Pauli exclusion principle is a foundational concept in quantum mechanics that has critical implications for the structure of atoms and molecules. It simply states that no two electrons in an atom can have the same set of four quantum numbers. In other words, it's a rule that ensures each electron has a unique 'address' within an atom.

Imagine living in an apartment building where each unit has a unique identifier, consisting of a block number, floor number, unit number, and room number. Similarly, in the atomic 'building', quantum numbers serve as the address system, and the Pauli exclusion principle ensures that no two electrons 'live' at the exact same address.

Significance in Electron Configuration

Let's relate this principle to the example provided in the exercise. Configuration (a), with six downward-pointing arrows, suggests that all electrons are trying to occupy the same 'room' with the same 'room number' (spin quantum number), which is not allowed by the Pauli exclusion principle. This rule is critical for knowing how to correctly populate the orbitals of an atom with electrons and influences the chemical properties of elements.

Hund's Rule

Hund's rule guides us further in the art of arranging electrons within sublevels of an atom. Think of it as a strategy for filling seats in a theater to maximize comfort: most people will choose to sit alone before they start sharing space on the same bench. In the context of electron configurations, Hund's rule suggests that every orbital in a given sublevel gets one electron before any orbital gets a second one, and all single electrons in singly occupied orbitals must have the same spin (to maximize total spin).

Considering configuration (b) from the exercise, where we have a pair of electrons sharing one orbital while there are still vacant orbitals available, we see a clear breach of Hund's rule. It's as if two people chose to share a seat when there are other empty seats available. Each electron should first occupy empty space before pairing up, leading to a more stable configuration. Hund's rule is paramount in understanding the most stable arrangements of electrons in atoms and consequently influences the magnetic and chemical properties of an element.

Quantum Numbers

Quantum numbers are like the coordinates that give us the exact location of an electron within an atom's electron cloud. There are four types of quantum numbers:

• The principal quantum number ()) indicates the main energy level or shell of an electron.
• The angular momentum quantum number ()) denotes the sublevel (such as 's', 'p', 'd', or 'f') and the shape of the orbital.
• The magnetic quantum number ()) pinpoints the specific orbital within a sublevel where an electron is likely to be found.
• The spin quantum number ()) represents the spin direction of the electron (either +1/2 or -1/2).

When we talk about the electron configurations being 'disallowed' in the exercise, we are referring to situations where these quantum number rules are not being followed.

For example, having six electrons all with down spins in configuration (a) would mean more than two electrons have the same set of quantum numbers for a single orbital, which is not possible. Each electron must have a unique set of these numbers — almost like having a distinct ID card — to coexist within the same atom without violating the Pauli exclusion principle.