Problem 71 - Fundamentals of Physics

Question

An electron in a multielectron atom is known to have the quantum number \(\ell=3 .\) What are its possible \(n, m_{\ell}\), and \(m_{s}\) quantum numbers?

Step-by-Step Solution

Answer

Possible values: \( n \geq 4 \), \( m_{\ell} = -3, -2, -1, 0, +1, +2, +3 \), \( m_{s} = +\frac{1}{2}, -\frac{1}{2} \).

Step 1: Understanding Quantum Numbers

The quantum number \( \ell \) represents the angular momentum quantum number. It can take values from \( 0 \) to \( n-1 \), where \( n \) is the principal quantum number.

Step 2: Determine Possible Values of n

Since we have \( \ell = 3 \), the smallest possible value of \( n \) is \( 4 \) because \( n \) must be greater than \( \ell \). Thus, \( n \geq 4 \).

Step 3: Determine Possible Values of m_{\ell}

The magnetic quantum number \( m_{\ell} \) ranges from \( -\ell \) to \( +\ell \). Given \( \ell = 3 \), the possible values of \( m_{\ell} \) are: \(-3, -2, -1, 0, +1, +2, +3\).

Step 4: Possible Values of m_{s}

The spin quantum number \( m_{s} \) for an electron can be either \( +\frac{1}{2} \) or \( -\frac{1}{2} \). This is independent of other quantum numbers.

Key Concepts

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as \(\ell\), is a fundamental concept in quantum mechanics. It determines the shape of an electron's orbital within an atom. Specifically, the value of \(\ell\) can vary from 0 to \(n-1\), where \(n\) represents the principal quantum number of the electron. The orbital's shape is typified by different \(\ell\) values, commonly associated with the letters 's', 'p', 'd', and 'f'.
For instance, when \(\ell = 0\), it corresponds to an 's' orbital, which is spherical. When \(\ell = 1\), it corresponds to a 'p' orbital, which is dumbbell-shaped.
In our exercise, the given \(\ell = 3\) suggests that the electron is likely in an 'f' orbital, which has a more complex shape. Understanding \(\ell\) is crucial as it not only defines the shape but also contributes to the overall energy levels of the electrons in multi-electron atoms.

Magnetic Quantum Number

The magnetic quantum number \(m_\ell\) is closely tied to the angular momentum quantum number. It denotes the orientation of an electron's orbital within a given subshell. The values \(m_\ell\) can take are determined by the value of \(\ell\) and range from \(-\ell\) to \(+\ell\).
For example, if \(\ell = 3\), the possible values for \(m_\ell\) include \(-3, -2, -1, 0, +1, +2, +3\). This variety in values allows for electrons to occupy different spatial orientations within the same subshell. When dealing with multi-electron atoms, these orientations play a pivotal role in understanding both the electron configuration and magnetic properties of the atom.
Hence, recognizing and determining the magnetic quantum number is vital for comprehensive insight into the structure and behavior of electrons in an atom.

Spin Quantum Number

The spin quantum number, symbolized as \(m_s\), is a fundamental component of an electron's intrinsic angular momentum, a unique quantum property not directly related to an orbital's shape. Unlike other quantum numbers, \(m_s\) is limited to just two possible values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\). These values are often verbalized as "spin up" and "spin down," respectively.
Each electron in an atom needs a unique set of quantum numbers, making the spin number essential for ensuring no two electrons share identical quantum properties (as per the Pauli Exclusion Principle). The spin quantum number also influences the magnetic nature of atoms, contributing significantly to their chemical and physical behavior.
Understanding \(m_s\) is crucial, particularly when predicting how atoms bond or interact with magnetic fields.

Principal Quantum Number

The principal quantum number, denoted by \(n\), serves as an indicator of the energy level of an electron within an atom. It is always a positive integer (1, 2, 3, etc.) and defines the major electron shell in which an electron resides.
The principal quantum number is integral in indicating the overall size and energy of an electron's orbital. Higher values of \(n\) imply electrons are found in orbitals far from the nucleus, and usually, these electrons possess higher energy levels.
In the context of our exercise, if \(\ell = 3\), then \(n\) must be at least 4, since \(\ell\) can range from 0 up to \(n-1\). This ensures that the electron is housed in an appropriate orbital subshell corresponding to its energy state within the atom.
Understanding \(n\) is essential for grasping the electron distribution and the energy state of atoms, thus aiding in the comprehension of atomic properties and chemical behavior.