Question

Given the following sets of electron quantum numbers, indicate those that could not occur, and explain your answer. (a) \(1,0,0,-\frac{1}{2}\) (b) \(1,1,0,+\frac{1}{2}\) (c) \(3,2,-2,+\frac{1}{2}\) (d) \(2,1,2,+\frac{1}{2}\) (e) \(4,0,2,+\frac{1}{2}\)

Short answer

Answer: Set (a): \(1,0,0,-\frac{1}{2}\) - Valid Set (b): \(1,1,0,+\frac{1}{2}\) - Invalid Set (c): \(3,2,-2,+\frac{1}{2}\) - Valid Set (d): \(2,1,2,+\frac{1}{2}\) - Invalid Set (e): \(4,0,2,+\frac{1}{2}\) - Invalid

Step-by-step solution

Set (a): \(1,0,0,-\frac{1}{2}\)

For this set, the quantum numbers are n=1, l=0, m=0, and s=-1/2. This combination satisfies all the rules mentioned above. Therefore, this set of quantum numbers could occur.

Set (b): \(1,1,0,+\frac{1}{2}\)

Here, the quantum numbers are n=1, l=1, m=0, and s=+1/2. Notice that since n=1, the highest possible value for l is 0. Therefore, l=1 is invalid for this set of quantum numbers, and it could not occur.

Set (c): \(3,2,-2,+\frac{1}{2}\)

For this set, the quantum numbers are n=3, l=2, m=-2, and s=+1/2. Based on the rules, this set of quantum numbers satisfies all the requirements and could occur.

Set (d): \(2,1,2,+\frac{1}{2}\)

Here, the quantum numbers are n=2, l=1, m=2, and s=+1/2. Since l=1, the range of possible m values is -1, 0, and +1. As a result, m=2 is invalid, and this set of quantum numbers could not occur.

Set (e): \(4,0,2,+\frac{1}{2}\)

In this set, the quantum numbers are n=4, l=0, m=2, and s=+1/2. The value for l is valid but having m=2 when l=0 is not. The set is invalid because m should be in the range of -l to +l, which is only 0 in this case. Therefore, this set of quantum numbers could not occur.

Key concepts

Electron Configuration

Electron configuration is the arrangement of electrons in an atom's orbitals. Electrons fill available orbitals in a specific order based on increasing energy levels, often referred to as the Aufbau principle. The sequence follows energy sub-levels: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, and so on. Understanding electron configuration helps predict an atom's chemical behavior.
For example, hydrogen, with one electron, has a configuration of 1s¹. This indicates one electron resides in the first energy level's s orbital.

• Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers.
• Hund's Rule: Electrons occupy degenerate orbitals (orbitals at the same energy level) singly before pairing up.

These principles guide how electrons fill an atom's orbitals, ensuring stability and energy efficiency.

Quantum Mechanics

Quantum mechanics fundamentally describes how particles, like electrons, behave on an atomic scale. This field revolutionizes how we perceive matter and energy, contrasting classical physics with its probabilistic nature.
Key principles include:

• Wave-Particle Duality: Electrons exhibit both wave-like and particle-like properties.
• Uncertainty Principle: Proposed by Heisenberg, it states that the exact position and momentum of an electron cannot be simultaneously known.
• Quantum Numbers: Specify the state of an electron, including its energy level, orbital shape, orientation, and spin.

Understanding these concepts is crucial for explaining phenomena like chemical bonding and atomic structure. The rules governing quantum mechanics explain why only certain electron configurations are possible, as each electron configuration directly ties to quantum numbers.

Atomic Structure

Atomic structure refers to the composition of the atomic nucleus and the arrangement of electrons in orbitals around the nucleus. Atoms consist of protons, neutrons, and electrons, where protons and neutrons reside in the nucleus.
This nucleus accounts for most of an atom's mass, while electrons dictate many of its chemical properties.

• Protons are positively charged, electrons are negatively charged, and neutrons have no charge.
• The number of protons is the atomic number, defining an element's identity.
• Electrons in outer shells interact with other atoms, leading to chemical bonds.

Each electron's position and movement are determined by quantum mechanics, forming defined regions known as orbitals. This understanding is foundational to chemistry and explains the predictable behavior of elements and compounds.