Problem 17 - 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) \(n=4, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=0\) (b) \(n=3, \ell=1, m_{\ell}=-3, m_{s}=-\frac{1}{2}\) (c) \(n=3, \ell=3, m_{\ell}=-1, m_{\mathrm{s}}=+\frac{1}{2}\)

Step-by-Step Solution

Answer

(a) Change \( m_s \) to \(+\frac{1}{2}\); (b) Change \( m_\ell \) to -1; (c) Change \( \ell \) to 2.

Step 1: Understanding Quantum Numbers

Quantum numbers describe the properties of electrons in atoms. The principal quantum number \( n \) can be any positive integer. The azimuthal quantum number \( \ell \) must be between 0 and \( n-1 \). The magnetic quantum number \( m_\ell \) ranges from \(-\ell\) to \(\ell\). The spin quantum number \( m_s \) can only be \( +\frac{1}{2} \) or \(-\frac{1}{2} \).

Step 2: Analyzing Set (a)

For the set \((n=4, \ell=2, m_\ell=0, m_s=0)\), the issue lies with \( m_s \), which cannot be zero. Change \( m_s \) to \(+\frac{1}{2}\) or \(-\frac{1}{2}\) to make it a valid set.

Step 3: Analyzing Set (b)

In the set \((n=3, \ell=1, m_\ell=-3, m_s=-\frac{1}{2})\), look at \( m_\ell \). For \( \ell = 1 \), \( m_\ell \) can only be -1, 0, or 1. Change \( m_\ell \) to -1 to make the set valid.

Step 4: Analyzing Set (c)

For the set \((n=3, \ell=3, m_\ell=-1, m_s=+\frac{1}{2})\), the error is with \( \ell \), which can only go up to \( n-1 = 2 \). Change \( \ell \) to 2 to correct the set.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by the symbol \( n \), plays a critical role in the atomic structure of an atom. This number can be any positive integer (1, 2, 3,...), and it essentially determines the size and energy level of an electron's orbit. The larger the value of \( n \), the higher the energy level and the larger the orbital.

Here's what you need to know about the principal quantum number:

• It's the first quantum number and signifies the main energy level of the electron.
• A higher \( n \) value means the electron is further from the nucleus.
• Each orbit holds a limited number of electrons: up to \( 2n^2 \).

Understanding \( n \) gives you the fundamental framework for where an electron resides in an atom.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \( \ell \), describes the shape of the electron's orbital. It is directly related to the principal quantum number as it takes on integer values ranging from 0 to \( n-1 \).

This number reveals the subshell and the type of orbital:

• \( \ell = 0 \) corresponds to an 's' orbital (spherical).
• \( \ell = 1 \) corresponds to a 'p' orbital (dumbbell-shaped).
• \( \ell = 2 \) corresponds to a 'd' orbital.
• \( \ell = 3 \) corresponds to an 'f' orbital.

Recognizing \( \ell \) is crucial when visualizing the shape and complexity of an electron's path around the nucleus.

Magnetic Quantum Number

The magnetic quantum number, \( m_\ell \), has the role of indicating the orientation of an orbital around the nucleus. The range of \( m_\ell \) values is dependent on the azimuthal quantum number \( \ell \), and it can take on values from \(-\ell\) to \( \ell \). This means that for each \( \ell \), \( m_\ell \) has \( 2\ell + 1 \) possible values.

Key points about \( m_\ell \) include:

• Determines how the orbital is oriented in space in a given magnetic field.
• Each value of \( m_\ell \) corresponds to a specific orbital within a subshell.
• Variety in \( m_\ell \) for multiple electrons leads to varied magnetic properties of substances.

Understanding \( m_\ell \) helps in predicting how diverse electron configurations can affect physical and chemical properties of atoms.

Electron Spin

The electron spin, denoted as \( m_s \), is a fundamental property of electrons, indicating their intrinsic angular momentum. Unlike the other quantum numbers, \( m_s \) is not derived from an equation related to the orbital, but is a property intrinsic to the electron itself.

Key points about electron spin are:

• \( m_s \) can only have values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \), often referred to as "spin up" and "spin down".
• Spin is essential for the Pauli Exclusion Principle, which states no two electrons in an atom can have identical sets of quantum numbers.
• The differences in spin lead to magnetic properties of substances, such as paramagnetism and diamagnetism.

Recognizing the concept of electron spin is crucial for understanding why electrons behave as they do within atoms.

Atomic Structure

Understanding atomic structure is fundamental to comprehending how atoms form matter and interact in chemical reactions. At its core, an atom consists of a nucleus surrounded by electrons, and these electrons are described by the quantum numbers discussed.

The structure includes:

• The nucleus, containing protons and neutrons, which determines the atom's mass and identity.
• Electrons, occupying discrete orbitals defined by quantum numbers, contribute to the atom's chemical properties.
• The interaction between electrons and the nucleus dictates the atom's stability and reactivity.

Mastering the atomic structure is fundamental for deeper insights into chemistry and physics, revealing the complexities of the smallest units of matter.