Problem 18 - Chemistry and Chemical Reactivity

Question

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. In each case, change the incorrect value (or values) to make the set valid. (a) \(n=2, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=+\frac{1}{2}\) (b) \(n=2, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=0\) (c) \(n=3, \ell=1, m_{\ell}=+2, m_{\mathrm{s}}=+\frac{1}{2}\)

Step-by-Step Solution

Answer

(a) Correct \(\ell\) to 1. (b) Correct \(m_{\mathrm{s}}\) to \(+\frac{1}{2}\) or \(-\frac{1}{2}\). (c) Correct \(m_{\ell}\) to 1.

Step 1: Understanding Quantum Number Rules

Quantum numbers describe the properties of an electron in an atom. The principal quantum number \(n\) is a positive integer. The azimuthal quantum number \(\ell\) ranges from 0 to \(n-1\). The magnetic quantum number \(m_{\ell}\) ranges from \(-\ell\) to \(+\ell\). The spin quantum number \(m_{\mathrm{s}}\) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Step 2: Analyze Set (a)

For the set \(n=2, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=+\frac{1}{2}\): The value of \(\ell\) cannot be equal to 2 when \(n=2\) because \(\ell\) must be between 0 and \(n-1\). Therefore, the correct set should be \(n=2, \ell=1, m_{\ell}=0, m_{\mathrm{s}}=+\frac{1}{2}\).

Step 3: Analyze Set (b)

For the set \(n=2, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=0\): The value of \(m_{\mathrm{s}}\) cannot be 0 because it must be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Therefore, the correct set should be \(n=2, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=+\frac{1}{2}\) or \(n=2, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=-\frac{1}{2}\).

Step 4: Analyze Set (c)

For the set \(n=3, \ell=1, m_{\ell}=+2, m_{\mathrm{s}}=+\frac{1}{2}\): The value of \(m_{\ell}\) cannot be 2 because it must be within the range of \(-\ell\) to \(+\ell\) or \(-1\) to \(+1\) when \(\ell=1\). Therefore, the correct set should be \(n=3, \ell=1, m_{\ell}=+1, m_{\mathrm{s}}=+\frac{1}{2}\).

Key Concepts

Electron Configuration

Electron configuration describes the arrangement of electrons in an atom's electron shells and subshells. This concept is essential for understanding the chemical properties of elements.
Electrons occupy orbitals in a specific order, usually filling the lowest energy levels first. This is known as the Aufbau principle.

• The first shell (closest to the nucleus) contains the 1s orbital, with a maximum of 2 electrons.
• The second shell has 2s and 2p orbitals, accommodating 2 and 6 electrons, respectively.
• Higher shells have even more subshells: 3s, 3p, 3d, and so on.

By understanding electron configuration, you can predict how elements will react based on their position in the periodic table. Electrons tend to arrange themselves to minimize energy, leading to elements having similar properties appearing in columns known as groups or families.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that explains the behavior of matter and energy on the atomic and subatomic level. It differs from classical physics in several ways, offering a more probabilistic understanding of how particles like electrons behave.
One of the key concepts is the idea of wave-particle duality, where particles like electrons act both as particles and as waves. This has significant implications for how we understand electron configurations because it introduces the concept of orbitals, which are probability distributions rather than fixed paths.

• Quantum mechanics provides the mathematical framework to interpret the position and momentum of electrons.
• It leads to the discovery of quantum numbers that help describe electrons' positions within atoms.

Though complex, grasping the basics of quantum mechanics is crucial for anyone interested in understanding the nature of matter and forces.

Principal Quantum Number

The principal quantum number, denoted by \(n\), is one of the primary quantum numbers used to describe an electron's state in an atom. It indicates the main energy level occupied by the electron.

• It is a positive integer \((n = 1, 2, 3, \ldots)\).
• Higher values of \(n\) correspond to higher energy levels.
• The value of \(n\) is related to the size of the orbital; larger \(n\) values mean larger orbitals.

This quantum number is foundational in understanding how electrons are categorized and how these categories influence an atom's properties and its place in the periodic table.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \(\ell\), defines the shape of an electron's orbital and partially determines its energy.

• It ranges from 0 to \(n-1\), where \(n\) is the principal quantum number.
• Each value of \(\ell\) corresponds to a specific orbital type: \(\ell = 0\) (s orbital), \(\ell = 1\) (p orbital), \(\ell = 2\) (d orbital), and \(\ell = 3\) (f orbital).
• The shape influences the electron's energy due to differences in distribution around the nucleus.

This quantum number is vital for understanding electron placement and orbital hybridization in chemical reactions.

Magnetic Quantum Number

The magnetic quantum number, denoted \(m_{\ell}\), provides information about the orientation of the orbital in space relative to an external magnetic field. It can be thought of as determining the number of orbitals and their orientation within a subshell.

• The value of \(m_{\ell}\) ranges from \( -\ell \) to \( +\ell \).
• For each type of orbital (s, p, d, f), there are multiple possible orientations that \(m_{\ell}\) describes.
• This quantum number is crucial for understanding how complex atoms interact with magnetic fields and plays a role in electron arrangements.

By determining the orientation of orbitals, \(m_{\ell}\) helps explain the spectrum of possible electron states within an atom.

Spin Quantum Number

The spin quantum number, symbolized by \(m_{\mathrm{s}}\), is a fundamental quantum number reflecting the intrinsic angular momentum of an electron.

• It can take only two values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\).
• This quantum number is not tied to the electron's motion through space but rather to its inherent "spin", a type of quantum characteristic unique to particles.
• Spin influences the electron's magnetic properties, playing a role in how electrons pair within an orbital.

Understanding the spin quantum number is crucial in explaining the behavior of electrons in external magnetic fields and in comprehending phenomena such as electron pairing in chemical bonds.