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Chapter 40: Problem 8

In the subshell \(\ell=3\), (a) what is the greatest (most positive) \(m_{e}\) value, (b) how many states are available with the greatest \(m_{\ell}\) value, and (c) what is the total number of states available in the subshell?

Short Answer

Expert verified
(a) +3 (b) 2 states (c) 14 states total

Step by step solution

01

Understanding the Subshell

The given subshell has a value of \(\ell = 3\). This corresponds to the 'f' subshell as \(\ell\) values correspond to the following subshell designations: \(\ell = 0 \) for 's', \(\ell = 1 \) for 'p', \(\ell = 2 \) for 'd', and \(\ell = 3 \) for 'f'. The magnetic quantum number \(m_{\ell}\) can take on values from \(-\ell\) to \(+\ell\).
02

Determining the Greatest m_鈩� Value

Since \(m_{\ell}\) varies from \(-\ell\) to \(+\ell\), the greatest (most positive) value of \(m_{\ell}\) in the \(\ell = 3\) subshell is \(m_{\ell} = +3\).
03

Counting States with Greatest m_鈩� Value

Each \(m_{\ell}\) value can have two electrons with opposite spins (spin up, \(m_s = +\frac{1}{2}\), and spin down, \(m_s = -\frac{1}{2}\)). Thus, there are 2 quantum states available for the greatest \(m_{\ell}\) value of +3.
04

Calculating Total Number of States in the Subshell

For each \(m_{\ell}\) value from \(-3\) to \(+3\), there are 2 spin possibilities. Therefore, the total number of states in the subshell is \((2\ell + 1) \times 2 = (2 \times 3 + 1) \times 2 = 7 \times 2 = 14.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Quantum Number
The magnetic quantum number, represented by \(m_{\ell}\), is one of the quantum numbers used to describe the unique quantum state of an electron. This number is crucial in determining the orientation of the orbital in space. For a given subshell with angular momentum quantum number \(\ell\), \(m_{\ell}\) can vary from \(-\ell\) to \(+\ell\).
\(m_{\ell}\) specifies the projection of the angular momentum in a specified direction, usually the z-axis in quantum mechanics. It gives the number of orbitals and their orientation within a subshell.
For instance, in a subshell where \(\ell = 3\), which is the 'f' subshell, the magnetic quantum numbers range from -3 to +3, allowing for 7 different orbitals. Each of these orbitals can accommodate electrons with different spin quantum numbers, providing the orbital's magnetic properties.
Subshell Electron Configuration
Subshell electron configuration is an arrangement of electrons within the atomic orbitals. It describes how electrons are distributed among the various subshells and orbitals in an atom.
Each principal energy level (or shell) consists of one or more subshells, where the number of subshells is equal to the principal quantum number \(n\). Each subshell corresponds to a different shape of orbital and is characterized by the angular momentum quantum number \(\ell\).
For example, when \(\ell = 3\), the subshell designation is 'f'. Each subshell can hold a specific number of electrons. The general formula to determine the capacity of a subshell is \(2(2\ell + 1)\). Thus, the 'f' subshell can accommodate up to 14 electrons, providing space for many different electron configurations.
F Subshell
The 'f' subshell is characterized by the angular momentum quantum number \(\ell = 3\). This subshell can contain a maximum of 14 electrons distributed in 7 different orbitals, each capable of holding two electrons.
The orbitals in the 'f' subshell have complex shapes and are important in the chemistry of the lanthanides and actinides. They typically begin to fill in elements in the sixth period of the periodic table.
Understanding the 'f' subshell is crucial for explaining the chemistry of elements with higher atomic numbers which exhibit complex behaviors due to this sublevel of electron configuration.
Spin Quantum Number
The spin quantum number \(m_s\) describes the intrinsic angular momentum or spin of an electron within an orbital. It can take one of two values: \(+\frac{1}{2}\) (often referred to as 'spin up') or \(-\frac{1}{2}\) (referred to as 'spin down').
This quantum number is essential because it explains the magnetic properties of electrons and how they fill available orbitals. According to the Pauli exclusion principle, no two electrons in the same atom can have identical sets of quantum numbers. Hence, each orbital can hold two electrons, but they must have opposite spins.
This principle affects the arrangement and properties of electrons in atoms and lays the foundation for chemical bonding and molecular structure.

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Most popular questions from this chapter

Suppose two electrons in an atom have quantum numbers \(n=2\) and \(\ell=1 .\) (a) How many states are possible for those two electrons? (Keep in mind that the electrons are indistinguishable.) (b) If the Pauli exclusion principle did not apply to the electrons. how many states would be possible?

In an NMR experiment, the RF source oscillates at \(34 \mathrm{MHz}\) and magnetic resonance of the hydrogen atoms in the sample being investigated occurs when the external field \(\vec{B}_{\text {ext }}\) has magnitude \(0.78 \mathrm{~T}\). Assume that \(\vec{B}_{\text {int }}\) and \(\vec{B}_{\text {ext }}\) are in the same direction and take the proton magnetic moment component \(\mu_{z}\) to be \(1.41 \times 10^{-26} \mathrm{~J} / \mathrm{T}\). What is the magnitude of \(\vec{B}_{\text {int }} ?\)

An electron is in a state with \(n=3\). What are (a) the number of possible values of \(\ell,(\mathrm{b})\) the number of possible values of \(m_{e},(\mathrm{c})\) the number of possible values of \(m_{s},(\mathrm{~d})\) the number of states in the \(n=3\) shell, and (e) the number of subshells in the \(n=3\) shell?

(A correspondence principle problem.) Estimate (a) the quantum number \(\ell\) for the orbital motion of Earth around the Sun and (b) the number of allowed orientations of the plane of Earth's orbit. (c) Find \(\theta_{\min }\), the half-angle of the smallest cone that can be swept out by a perpendicular to Earth's orbit as Earth revolves around the Sun.

The active volume of a laser constructed of the semiconductor GaAlAs is only \(200 \mu \mathrm{m}^{3}\) (smaller than a grain of sand), and yet the laser can continuously deliver \(5.0 \mathrm{~mW}\) of power at \(a\) wavelength of \(0.80 \mu \mathrm{m}\). At what rate does it gencrate photons?
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