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Chapter 5: Problem 48

Give the \(n, \ell,\) and \(m_{\ell}\) values for (a) Each atomic orbital in the \(6 f\) sublevel. (b) Each atomic orbital in the \(n=5\) level.

Short Answer

Expert verified
(a) 6f: (6,3,-3), ..., (6,3,3). (b) n=5 level: 5s, 5p, 5d, 5f, 5g orbitals.

Step by step solution

01

Understanding Quantum Numbers

Every atomic orbital is defined by three quantum numbers: the principal quantum number \(n\), the azimuthal or angular momentum quantum number \(\ell\), and the magnetic quantum number \(m_\ell\). \(n\) determines the energy level, \(\ell\) defines the shape (sublevel), and \(m_\ell\) specifies the orientation of the orbital.
02

Identifying Values for 6f Sublevel

In the 6f sublevel, \(n = 6\). The letter 'f' corresponds to \(\ell = 3\). The values of \(m_\ell\) range from \(-\ell\) to \(+\ell\). For \(\ell = 3\), \(m_\ell\) can be: \(-3, -2, -1, 0, 1, 2,\) and \(3\). So for each orbital, the quantum numbers \((n, \ell, m_\ell)\) are: \((6, 3, -3), (6, 3, -2), (6, 3, -1), (6, 3, 0), (6, 3, 1), (6, 3, 2), (6, 3, 3)\).
03

Identifying Sublevels for n=5 Level

For \(n = 5\), the possible \(\ell\) values are \(0\) (s sublevel), \(1\) (p sublevel), \(2\) (d sublevel), \(3\) (f sublevel), and \(4\) (g sublevel). Each sublevel contains a different number of \(m_\ell\) values.
04

Enumerating Orbitals in 5s Sublevel

For \(n = 5, \ell = 0\) (5s sublevel), there is only one value, \(m_\ell = 0\). So the orbital is \((5, 0, 0)\).
05

Enumerating Orbitals in 5p Sublevel

For the 5p sublevel with \(n = 5, \ell = 1\), \(m_\ell\) values can be \(-1, 0,\) and \(1\). So the orbitals are: \((5, 1, -1), (5, 1, 0), (5, 1, 1)\).
06

Enumerating Orbitals in 5d Sublevel

For the 5d sublevel, where \(n = 5, \ell = 2\), \(m_\ell\) values can be \(-2, -1, 0, 1,\) and \(2\). Therefore, the orbitals are: \((5, 2, -2), (5, 2, -1), (5, 2, 0), (5, 2, 1), (5, 2, 2)\).
07

Enumerating Orbitals in 5f Sublevel

For the 5f sublevel with \(n = 5, \ell = 3\), \(m_\ell\) values range from \(-3\) to \(3\). Hence, the orbitals are: \((5, 3, -3), (5, 3, -2), (5, 3, -1), (5, 3, 0), (5, 3, 1), (5, 3, 2), (5, 3, 3)\).
08

Enumerating Orbitals in 5g Sublevel

For the 5g sublevel, where \(n = 5, \ell = 4\), \(m_\ell\) values range from \(-4\) to \(4\). So the orbitals are: \((5, 4, -4), (5, 4, -3), (5, 4, -2), (5, 4, -1), (5, 4, 0), (5, 4, 1), (5, 4, 2), (5, 4, 3), (5, 4, 4)\).

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Key Concepts

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

Atomic Orbital
Atomic orbitals are regions around the nucleus of an atom where electrons are most likely to be found. These orbitals arise from the solutions of the Schr枚dinger equation and are defined by quantum numbers that spell out energy levels and shapes. Let's break down what each quantum number means:
  • The principal quantum number, denoted as \(n\), tells you about the energy level or shell of the electron. The higher the \(n\) value, the further out and higher in energy the orbital is located.
  • The angular momentum quantum number, \(\ell\), informs us of the shape of the orbital. This includes the familiar s, p, d, and f shapes. For example, \(\ell = 0\) corresponds to an s orbital, \(\ell = 1\) for p, \(\ell = 2\) for d, and \(\ell = 3\) for f.
  • The magnetic quantum number, \(m_\ell\), indicates the orientation of the orbital in space. Its values range from \(-\ell\) to \(+\ell\).
Understanding atomic orbitals is crucial when predicting how atoms will bond and interact with one another.
Electron Sublevel
Electron sublevels are divisions within energy levels (as indicated by \(n\)) and are specified by the angular momentum quantum number \(\ell\). Each sublevel can hold a certain type of orbital.
For example:
  • An \(s\) sublevel features a spherical shape and can contain only one orbital.
  • A \(p\) sublevel contains three orbitals, each shaped like a dumbbell, and oriented along different axes.
  • D sublevels have five orbitals with more complex shapes, while \(f\) sublevels contain seven orbitals with even more intricate structures.
The number of orbitals (and hence the number of electrons these orbitals can accommodate) increases with higher sublevels, following a pattern: \(s, p, d, f\).
This organization helps explain the periodic table's structure and the chemical behavior of different elements.
Angular Momentum Quantum Number
The angular momentum quantum number, \(\ell\), is one of the key players in understanding the behavior of electrons in atoms. It determines the shape of the atomic orbital, acting as a quantum mechanical version of rotating a spinning top.
Here's how it all plays out:
  • The possible values of \(\ell\) range from 0 to \(n-1\), where \(n\) is the principal quantum number. For example, if \(n=3\), the possible \(\ell\) values are 0, 1, and 2.
  • Each \(\ell\) value corresponds to a specific type of orbital: \(\ell = 0\) for s orbitals, \(\ell = 1\) for p orbitals, and so on, expanding into d, f, and theoretically beyond.
  • The angular momentum quantum number also influences the magnetic quantum number \(m_\ell\), which determines the number of ways the orbital is oriented in space. For example, for \(\ell = 2\), \(m_\ell\) can be \(-2, -1, 0, 1,\) or \(2\).
Each increase in \(\ell\) brings about more complex shapes for the orbitals, contributing to the intricacy of biological molecules and the richness of chemical reactions.

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