Question

The maximum number of 3 d electrons having \(s=+\frac{1}{2}\) are: (a) 10 (b) 5 (c) 14 (d) 7

Short answer

The maximum number of 3 d electrons with spin +1/2 is 5.

Step-by-step solution

Understanding Electron Configuration

In an atom, electrons are arranged in orbitals following specific rules like the Aufbau principle, Hund's rule, and the Pauli exclusion principle. The 3 d orbitals can collectively hold up to 10 electrons since each orbital accommodates 2 electrons (5 orbitals × 2 electrons/orbital).

Identify Spin of Electrons

Electrons have a property called spin, which can be either +1/2 or -1/2. For the 3 d orbitals, we want to find out how many electrons can have a spin of +1/2.

Apply Hund's Rule

Hund's Rule states that every orbital in a subshell gets one electron before any orbital gets a second one, and all electrons in singly occupied orbitals have the same spin. For 3 d orbitals, initially fill each of the 5 orbitals with one electron each of spin +1/2.

Calculate Maximum Electrons with +1/2 Spin

Based on Hund’s rule, each of the 5 d orbitals can have one electron with spin +1/2 before any pairing occurs. Therefore, the maximum number of 3 d electrons with a spin of +1/2 is 5.

Key concepts

Electron Configuration

Electron configuration is like the blueprint of how electrons are distributed in an atom. Electrons occupy different energy levels or shells, which further split into subshells denoted as s, p, d, and f. The goal of electron configuration is to determine the most stable arrangement for electrons.

• Energy levels are like floors in a building, and each subshell is a room on that floor.
• Each subshell contains orbitals, and orbitals are spaces where there is a high chance of finding an electron.
• Within a given orbit, electrons fill the lowest available energy state first, aligning with the Aufbau principle."

It's essential to follow established rules to correctly assign electrons and ensure the atom remains stable. For instance, the 3d subshell can hold a maximum of 10 electrons as it has 5 orbitals, and each can house 2 electrons (one with spin +1/2 and the other with spin -1/2).

Hund's Rule

Hund's Rule is a guideline that helps us understand the arrangement of electrons within a subshell, specifically in the context of filling orbitals. Named after the German physicist Friedrich Hund, the rule simplifies the organization of electrons to minimize repulsion.
According to Hund’s Rule:

• Every orbital in a subshell receives one electron before any orbital gets a second electron.
• Electrons in orbitals within the same subshell stay unpaired as long as possible.
• All singly-filled orbitals will contain electrons with the same spin direction.

This way of filling orbitals reduces electron repulsion and increases the stability of the atom. When applying Hund’s Rule to the 3d level, we see that each of the five d orbitals will first receive one electron with the same spin, say +1/2, which maximizes their separation.

Quantum Numbers

Quantum numbers are like addresses for electrons; they provide a set of values that describe the unique quantum state of an electron. There are four quantum numbers to consider:

• The principal quantum number ( n ), which indicates the main energy level occupied by the electron.
• The azimuthal (or angular momentum) quantum number ( l ), denoting the subshell or shape of the orbital.
• The magnetic quantum number ( m _l ), which identifies the orientation of the orbital in space.
• The spin quantum number (s), which specifies the direction of the electron's spin, either +1/2 or -1/2.

Quantum numbers allow us to predict and describe the behaviors and arrangements of electrons in different atoms. Understanding these numbers is crucial for depicting how electrons are configured in their respective orbitals.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental principle in quantum mechanics introduced by Wolfgang Pauli. It states that no two electrons in a single atom can possess the same set of four quantum numbers.
Here's what it means practically:

• Every electron in an atom must be unique in its description by the quantum numbers.
• Electrons in the same orbital must have opposite spins to differ by the spin quantum number.
• This principle explains why each orbital, which holds up to two electrons, can only have one electron with a spin of +1/2 and another with -1/2.

By following the Pauli Exclusion Principle, we ensure that every electron in an atom is in its distinct state, leading to a stable and orderly filling of electron shells and subshells.