Question

Comment on the correctness of the following statement: The probability of finding two electrons with the same four quantum numbers in an atom is zero.

Short answer

The statement is correct due to the Pauli Exclusion Principle.

Step-by-step solution

Understanding Quantum Numbers

First, let's understand what quantum numbers are. Electrons within an atom are described by a set of four quantum numbers: \(n\) (the principal quantum number), \(l\) (the azimuthal quantum number), \(m\_l\) (the magnetic quantum number), and \(m\_s\) (the spin quantum number). These numbers dictate the energy, shape, orientation, and spin of an electron's orbital.

Reviewing the Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons in the same atom can have identical sets of quantum numbers. This is a fundamental principle of quantum mechanics and ensures the unique arrangement of electrons in orbitals.

Analyzing the Given Statement

The statement "The probability of finding two electrons with the same four quantum numbers in an atom is zero" is directly related to the Pauli Exclusion Principle. According to this principle, such a scenario where two electrons in an atom have identical quantum numbers is impossible.

Confirming the Correctness of the Statement

Given the Pauli Exclusion Principle, the statement is correct because having two electrons with exactly the same four quantum numbers contradicts the principle. This makes the probability of such an occurrence zero, confirming the statement's correctness.

Key concepts

Quantum Numbers

In quantum mechanics, quantum numbers are essential for describing the unique "address" of each electron in an atom. Think of them like coordinates that pinpoint an electron's location and energy state within an atom.

• The principal quantum number, denoted as \(n\), indicates the energy level of an electron and its average distance from the nucleus.
• The azimuthal quantum number, \(l\), describes the shape of the orbital where the electron is likely found (s, p, d, or f orbitals).
• The magnetic quantum number, \(m_l\), specifies the orientation of the orbital in space.
• The spin quantum number, \(m_s\), describes the direction of the electron's spin, which can be either +1/2 or -1/2.

Each set of quantum numbers uniquely identifies an electron, ensuring no two electrons in an atom can share the same set based on the Pauli Exclusion Principle. This foundational concept helps explain the electron configuration within atoms.

Electron Configuration

Electron configuration represents the arrangement of electrons in the orbitals of an atom. This configuration is essential for understanding chemical properties and behaviors in different elements. Electrons occupy shells and subshells in a specific order based on energy levels, starting with the lowest available energy state. The Aufbau principle guides this filling process where electrons fill orbitals in order of increasing energy.

• The Pauli Exclusion Principle ensures that each electron's configuration adheres to unique quantum numbers.
• Hund's Rule states that in orbitals of the same energy, electrons will fill empty orbitals first before pairing up.

These rules help define a stable electron configuration, which influences how atoms will interact chemically. For example, the electron configuration of oxygen explains its tendency to form bonds to fulfill its valence shell.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature on the scale of atoms and subatomic particles. The Pauli Exclusion Principle, part of quantum mechanics, asserts that no two electrons can have identical sets of quantum numbers within a single atom. This principle explains why electrons occupy different orbitals and spin states—actions that determine atomic stability and reactivity.

• Quantum mechanics introduces the concept of wave-particle duality, describing electrons as both particles and waves.
• The Heisenberg Uncertainty Principle, another key concept, indicates a fundamental limit to the precision with which pairs of physical properties like position and momentum can be known.

Understanding quantum mechanics enables us to grasp the complex behaviors of electrons, predicting how and why atoms interact with one another to form molecules and compounds. This scientific framework supports technological advancements such as semiconductors, lasers, and MRI machines.