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Chapter 30: Problem 26

It is known that the possible values for the magnetic quantum number \(m_{\ell}\) are \(-4,-3,-2,-1,0,+1,+2,+3,\) and \(+4 .\) Determine the orbital quantum number \(\ell\) and the smallest possible value of the principle quantum number \(n\).

Short Answer

Expert verified
\(\ell = 4\) and \(n = 5.\)

Step by step solution

01

Understanding the Magnetic Quantum Number

The magnetic quantum number, denoted as \(m_{\ell}\), can take integer values ranging from \(-\ell\) to \(+\ell\). Each value of \(m_{\ell}\) gives a specific orientation of an electron in space within a given subshell.
02

Determine the Orbital Quantum Number (\(\ell\))

Given that \(m_{\ell}\) takes values from \(-4\) to \(+4\), it implies that \(\ell = 4\) because \(m_{\ell}\) covers all integers from \(-\ell\) to \(+\ell\). Hence, if the largest positive value is \(+4\), \(\ell\) must be 4.
03

Determine the Smallest Possible Value of the Principal Quantum Number (\(n\))

The principal quantum number \(n\) must be greater than \(\ell\). Since \(\ell = 4\), the smallest possible value for \(n\) is \(n = 5\) because quantum numbers \(n\) and \(\ell\) must satisfy \(n > \ell\).

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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, symbolized as \(m_\ell\), is key in understanding electron orientation within atoms. It defines the specific orientation of an electron's orbital magnetic field in space. The values of \(m_\ell\) are integers that range from \(-\ell\) to \(+\ell\). This means if an orbital quantum number \(\ell\) is given, \(m_\ell\) can take any integer value within this range.
For example, if \(\ell = 4\), then \(m_\ell\) can be \(-4, -3, -2, -1, 0, +1, +2, +3,\) or \(+4\). These values describe different possible orientations that an electron can have, affecting how orbitals arrange around the nucleus. This concept is essential when determining how electrons are distributed in an atom.
Orbital Quantum Number
The orbital quantum number, denoted as \(\ell\), determines the shape and angular momentum of an electron's orbital. \(\ell\) can take integer values from 0 up to \(n-1\), where \(n\) is the principal quantum number.
The value of \(\ell\) also indicates the number of possible orientations, relating directly to the range of \(m_\ell\) values. For instance, in the given situation with \(m_\ell\) values ranging from \(-4\) to \(+4\), we find \(\ell = 4\), confirming that an \(f\) orbital is involved.
This correspondence is crucial when predicting the geometric structure of electron clouds within an atom, impacting chemical bonding and molecular geometry.
Principal Quantum Number
The principal quantum number, symbolized by \(n\), outlines the main energy level an electron occupies in an atom. It indicates the size of the orbital and the electron鈥檚 average distance from the nucleus. Values for \(n\) are positive integers starting at 1.
The principal quantum number must always be greater than the orbital quantum number \(\ell\). In the case where \(\ell = 4\), the smallest possible value for \(n\) is 5. This relationship is expressed as \(n > \ell\).
Understanding \(n\) helps determine the possible energy levels that electrons can occupy, influencing the atom's electron configuration and its overall energy state.

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Most popular questions from this chapter

Multiple-Concept Example 9 reviews the concepts that are important in this problem. An electron, traveling at a speed of \(6.00 \times 10^{7} \mathrm{m} / \mathrm{s},\) strikes the target of an X-ray tube. Upon impact, the electron decelerates to one-quarter of its original speed, an X-ray photon being emitted in the process. What is the wavelength of the photon?

An electron is in the \(n=5\) state. What is the smallest possible value for the angle between the \(z\) component of the orbital angular momentum and the orbital angular momentum?

A sodium atom \((Z=11)\) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to the outermost electron as an approximation, provided that we use an effective value \(Z_{\text {effective }}\) rather than 11 for the number of protons in the nucleus. (a) The ionization energy for the outermost electron in a sodium atom is \(5.1 \mathrm{eV}\). Use the Bohr model with \(Z=Z_{\text {effective }}\) to calculate a value for \(Z_{\text {effective }}\) (b) Using \(Z=11\) and \(Z=Z_{\text {effective }},\) determine the corresponding two values for the radius of the outermost Bohr orbit.

When an electron makes a transition between energy levels of an atom, there are no restrictions on the initial and final values of the principal quantum number \(n .\) According to quantum mechanics, however, there is a rule that restricts the initial and final values of the orbital quantum number \(\ell\). This rule is called a selection rule and states that \(\Delta \ell=\pm 1\). In other words, when an electron makes a transition between energy levels, the value of \(\ell\) can only increase or decrease by one. The value of \(\ell\) may not remain the same nor may it increase or decrease by more than one. According to this rule, which of the following energy level transitions are allowed? (a) \(2 \mathrm{s} \rightarrow 1 \mathrm{s}\) (b) \(2 p \rightarrow 1 s\) (c) \(4 p \rightarrow 2 p(d) 4 s \rightarrow 2 p\) (e) \(3 \mathrm{d} \rightarrow 3 \mathrm{s}\)

Find the energy (in joules) of the photon that is emitted when the electron in a hydrogen atom undergoes a transition from the \(n=7\) energy level to produce a line in the Paschen series.
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