Problem 79 - Chemistry The Central Science In Si Units

Question

The following do not represent valid ground-state electron configurations for an atom either because they violate the Pauli exclusion principle or because orbitals are not filled in order of increasing energy. Indicate which of these two principles is violated in each example. (a) \([\mathrm{Ne}] 3 s^{2} 3 p^{6} 3 d^{5}\) (c) \(1 s^{2} 3 s^{1}\) (b) \([\mathrm{Xe}] 6 s^{3}\)

Step-by-Step Solution

Answer

(a) Violates the Aufbau principle; (b) Violates the Pauli exclusion principle; (c) Violates the Aufbau principle.

Step 1: Understanding the Pauli Exclusion Principle

The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. This means each orbital can hold a maximum of two electrons with opposite spins.

Step 2: Understanding Energy Order Filling

Electrons fill orbitals in order of increasing energy, known as the Aufbau principle. The typical order follows: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on. Exceptions can occur, but lower energy orbitals are filled first.

Step 3: Analyzing Example (a): [Ne] 3s^2 3p^6 3d^5

In this configuration, the 3d orbital is filled before the 4s orbital, which violates the Aufbau principle. Normally, after filling 3p, electrons should go to 4s before continuing to 3d.

Step 4: Analyzing Example (b): [Xe] 6s^3

The configuration includes '6s^3', which suggests three electrons in a single s orbital. This violates the Pauli exclusion principle, as an s orbital can only hold two electrons.

Step 5: Analyzing Example (c): 1s^2 3s^1

In this configuration, the 2s and 2p orbitals are skipped, which violates the Aufbau principle. After 1s, electrons should fill 2s and then 2p before moving to 3s.

Key Concepts

Pauli Exclusion Principle

The Pauli exclusion principle is a fundamental concept in quantum mechanics, vital for understanding electron configuration in atoms. According to this principle, no two electrons in a single atom can have an identical set of four quantum numbers. But what do these quantum numbers signify? They are crucial identifiers that describe the unique quantum state of an electron.

• The first quantum number refers to the principal energy level or shell of the electron, essentially its distance from the nucleus.
• The second quantum number relates to the shape of the orbital the electron occupies, known as the azimuthal quantum number.
• The third is the magnetic quantum number, which describes the orientation of the orbital in space.
• Finally, the fourth quantum number, the spin quantum number, defines the direction of the electron's spin, which can either be +1/2 or -1/2.

Thus, this principle ensures that each electron in an atom has a unique set of quantum numbers, resulting in a restriction on the number of electrons an orbital can accommodate. Specifically, each orbital can hold a maximum of two electrons, and these must have opposite spins. Violations of the Pauli exclusion principle occur when configurational errors result in an atomic orbital allegedly accommodating more than two electrons, as witnessed in example (b).

Aufbau Principle

The Aufbau principle, a German word meaning 'building up,' guides us in understanding how electrons are distributed across atomic orbitals in a ground-state atom. It suggests that electrons fill orbitals in order of rising energy levels, beginning with the lowest. This methodical approach follows a specific sequence: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so forth.

• Lower energy orbitals fill before higher energy ones.
• Exceptions may occur, especially beyond the 3p orbital level, due to close energy levels among suborbitals (like 4s and 3d).
• An example of this principle at work can be seen in the exceptions for transition metals.

Violations lead to incorrect electron configurations, as electrons may mistakenly fill higher energy orbitals before occupying lower ones. For instance, example (a) errs by filling the 3d orbital before the 4s orbital, disregarding the typical order upheld by the Aufbau principle. Understanding this principle helps in predicting the order in which electrons occupy orbitals and spotting errors in configurational depictions.

Quantum Numbers

Quantum numbers provide a comprehensive description of electrons in an atom, dictating not only their position but also energetics and occupancy. There are four quantum numbers, each playing a vital role in the electron's identity:

• The principal quantum number ( ) denotes the primary energy level. The higher the value, the further the orbital is from the nucleus.
• The azimuthal quantum number ( l) describes the shape of the orbital, such as spherically shaped s orbitals or dumbbell-shaped p orbitals.
• The magnetic quantum number (m) tells us the orientation of the orbital space and ranges from -l to +l.
• The spin quantum number ( s) identifies the direction of the electron's inherent spin, furnishing either a +1/2 or -1/2 value.

Collectively, these numbers ensure that each electron in an atom can be distinctly described and differentiated from others. This unique combination forms the foundations for understanding key principles like the Pauli exclusion principle and dictates electron configurations. Proper electron configuration not only relies on following the Pauli exclusion principle and Aufbau principle but also fundamentally understanding and assigning correct quantum numbers.