Problem 85 - Chemistry: An Atoms-Focused Approach

Question

Which of the following combinations of quantum numbers are allowed? a. \(n=1, \ell=1, m_{\ell}=0, m_{s}=+\frac{1}{2}\) b. \(n=3, \ell=0, m_{\epsilon}=0, m_{s}=-\frac{1}{2}\) c. \(n=1, \ell=0, m_{\epsilon}=1, m_{s}=-\frac{1}{2}\) d. \(n=2, \ell=1, m_{\epsilon}=2, m_{\mathrm{s}}=+\frac{1}{2}\)

Step-by-Step Solution

Answer

a) \(n=1, \ell=1, m_{\ell}=0, m_{s}=+\frac{1}{2}\) b) \(n=3, \ell=0, m_{\ell}=0, m_{s}=-\frac{1}{2}\) c) \(n=1, \ell=0, m_{\ell}=1, m_{s}=-\frac{1}{2}\) d) \(n=2, \ell=1, m_{\ell}=2, m_{s}=+\frac{1}{2}\) Answer: b) \(n=3, \ell=0, m_{\ell}=0, m_{s}=-\frac{1}{2}\)

Step 1: Check Quantum Numbers for Option (a)

For option (a): \(n=1, \ell=1, m_{\ell}=0, m_{s}=+\frac{1}{2}\) 1. \(n=1\) (positive integer, allowed) 2. \(\ell=1\) (not in the range \(0\) to \(n-1=0\), not allowed) Since the value of \(\ell\) is not allowed, option (a) is not a valid combination of quantum numbers.

Step 2: Check Quantum Numbers for Option (b)

For option (b): \(n=3, \ell=0, m_{\epsilon}=0, m_{s}=-\frac{1}{2}\) 1. \(n=3\) (positive integer, allowed) 2. \(\ell=0\) (in the range \(0\) to \(n-1=2\), allowed) 3. \(m_{\epsilon}=0\) (in the range \(-\ell=0\) to \(+\ell=0\), allowed) 4. \(m_{s}=-\frac{1}{2}\) (allowed) Since all the quantum numbers are valid, option (b) is allowed.

Step 3: Check Quantum Numbers for Option (c)

For option (c): \(n=1, \ell=0, m_{\epsilon}=1, m_{s}=-\frac{1}{2}\) 1. \(n=1\) (positive integer, allowed) 2. \(\ell=0\) (in the range \(0\) to \(n-1=0\), allowed) 3. \(m_{\epsilon}=1\) (not in the range \(-\ell=0\) to \(+\ell=0\), not allowed) Due to the invalid value of \(m_{\epsilon}\), option (c) is not allowed.

Step 4: Check Quantum Numbers for Option (d)

For option (d): \(n=2, \ell=1, m_{\epsilon}=2, m_{s}=+\frac{1}{2}\) 1. \(n=2\) (positive integer, allowed) 2. \(\ell=1\) (in the range \(0\) to \(n-1=1\), allowed) 3. \(m_{\epsilon}=2\) (not in the range \(-\ell=-1\) to \(+\ell=1\), not allowed) As \(m_{\epsilon}\) is not valid, option (d) is also not allowed. So, the only allowed combination of quantum numbers is option (b).

Key Concepts

Electron Configuration

Electron configuration is a method used to describe the arrangement of electrons around the nucleus of an atom. Each electron occupies an atomic orbital, and its position is defined using four quantum numbers: principal ( ), azimuthal ( ), magnetic ( ), and spin ( ). Together, these quantum numbers provide a unique address for each electron in an atom.

• The principal quantum number ( ) indicates the energy level. It is a positive integer, which affects the size and the energy of the atomic orbital(s).
• The azimuthal quantum number ( ) determines the shape of the orbital and can take any value from 0 to (n-1).
• The magnetic quantum number ( ) indicates the orientation of the orbital in space and can range from - to + .
• Finally, the spin quantum number ( ) describes the intrinsic spin state of the electron, either +1/2 or -1/2.

These quantum numbers help in determining how electrons fill up different orbitals in an atom in accordance with the Aufbau principle, Hund’s rule, and Pauli's Exclusion Principle. Understanding electron configuration is essential for explaining many properties of elements, including their place in the periodic table.

Quantum Mechanics

Quantum mechanics is a fundamental branch of physics that provides a mathematical description of the wave-particle duality of atoms and particles. It is crucial for understanding the behavior of particles at the atomic and subatomic level, and explains how particles such as electrons can exist in different states of energy and location simultaneously.
Quantum mechanics radically changed the way scientists view atomic structure and electron behavior. Some of the key principles include:

• Wave-Particle Duality: Particles exhibit both wave-like and particle-like properties. Electrons, for example, can behave as both waves and particles.
• Uncertainty Principle: Formulated by Heisenberg, this principle states that the position and momentum of a particle cannot both be precisely known at the same time.
• Quantization of Energy: Electrons have quantized energy levels. They can "jump" between these levels, which is termed as quantum leaps.

Quantum mechanics provides the theoretical foundation for quantum numbers, which describe electron configurations and allow chemists to predict the chemical properties of elements.

Atomic Structure

Atomic structure refers to the arrangement of protons, neutrons, and electrons in an atom. The nucleus contains protons and neutrons, while electrons orbit around the nucleus in various energy levels. Understanding atomic structure is essential to grasp how elements interact with each other to form compounds. The role of quantum numbers in atomic structure is significant as they:

• Help define the electron cloud around the nucleus, giving insight into the atom's geometric shape.
• Determine electron orbitals, which influences chemical bonding and molecular geometry.
• Allow prediction of atomic and ionic sizes, ionization energies, and other related properties.

In summary, atomic structure and the theory of quantum mechanics combine to explain how atoms build up and interact with each other, resulting in the fascinating diversity of chemistry and material science we observe in our world today. Understanding these concepts lays the groundwork for many advanced topics in physics and chemistry.