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Chapter 3: Problem 94

Determine the number of allowed values of the magnetic quantum number when \(n=5\) and \(l=3\).

Short Answer

Expert verified
There are 7 allowed values for the magnetic quantum number.

Step by step solution

01

Understanding Quantum Numbers

The principal quantum number \(n\) and the azimuthal (angular momentum) quantum number \(l\) are given. For \(n=5\), the \(l\) can be 0, 1, 2, 3, or 4. Here, \(l=3\) as provided in the task.
02

Magnetic Quantum Number Range

The magnetic quantum number \(m_l\) depends on \(l\) and ranges from \(-l\) to \(+l\). Thus, for \(l=3\), the possible values are \(-3, -2, -1, 0, 1, 2, 3\).
03

Counting Allowed Values

Count the values from the set \(-3, -2, -1, 0, 1, 2, 3\). There are 7 possible values for \(m_l\).

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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, often denoted as \(m_l\), is one of the four quantum numbers used to describe the unique quantum state of an electron. It specifies the orientation of the orbital's angular momentum relative to an external magnetic field. The range of the magnetic quantum number is determined by the azimuthal quantum number \(l\), which defines the shape of the orbital. The values that \(m_l\) can take are integers ranging from \(-l\) to \(+l\). Therefore, if \(l\) is 3, \(m_l\) can be \(-3, -2, -1, 0, 1, 2, 3\).
  • The magnetic quantum number gives more information about the orientation of an electron's orbit.
  • It influences the number of orbitals that can exist in a given subshell.
Principal Quantum Number
The principal quantum number, designated as \(n\), is a major component in the quantum mechanical model of the atom. It determines the energy level of an electron in an atom and is always a positive integer: \(n = 1, 2, 3,... \).
  • This number largely dictates the size and energy of an electron's orbit.
  • Higher values of \(n\) mean electrons are more energetic and further from the nucleus.
For the problem at hand with \(n=5\), it indicates the electron is in the fifth energy level. Moreover, it limits the possible values of the azimuthal quantum number \(l\), as \(l\) can range from 0 up to \(n-1\), which for \(n=5\) would be 0 through 4.
Azimuthal Quantum Number
The azimuthal quantum number, symbolized as \(l\), defines the shape of an electron's orbital within a given principal energy level. It is also known as the angular momentum quantum number. The value of \(l\) ranges from 0 up to \(n-1\), where \(n\) is the principal quantum number.
  • Each value of \(l\) corresponds to a specific type of subshell or orbital, such as \(s\), \(p\), \(d\), and \(f\).
  • In our example, \(l=3\) corresponds to a \(d\)-type orbital.
This quantum number not only determines the shape but also affects the number of possible orientations (given by \(m_l\)) an electron can have in a subshell. For \(l=3\), it directly influences the 7 orientations described by the magnetic quantum number, making this subshell very diverse in electron configuration possibilities.

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Most popular questions from this chapter

Identify the quantum number that specifies each of the following. (a) The size of the orbital (b) The shape of the orbital (c) The way the orbital is oriented in space

Look up the covalent radii for magnesium and sulfur atoms and the ionic radii of \(\mathrm{Mg}^{2+}\) and \(\mathrm{S}^{2-}\) ions in Appendix B.4. Explain why \(\mathrm{Mg}^{2+}\) ions are smaller than \(\mathrm{S}^{2-}\) ions even though magnesium atoms are larger than sulfur atoms.

Which of the following contains sets of atoms or ions that have equivalent electron configurations? (a) \(\mathrm{B}^{3+}, \mathrm{C}^{4+}, \mathrm{H}^{+}, \mathrm{He}\) (b) \(\mathrm{Na}^{+}, \mathrm{Ne}, \mathrm{N}^{3+}, \mathrm{O}^{2-}\) (c) \(\mathrm{Mg}^{2+}, \mathrm{F}^{-}, \mathrm{Na}^{+}, \mathrm{O}^{2-}\) (d) \(\mathrm{Ne}, \mathrm{Ar}, \mathrm{Xe}, \mathrm{Kr}\) (e) \(\mathrm{O}^{2-}, \mathrm{S}^{2-}, \mathrm{Se}^{2-}, \mathrm{Te}^{2-}\)

The following data are known for element \(X\). $$ \begin{array}{ccccc} \begin{array}{c} \text { Number } \\ \text { of Peaks } \\ \text { in the PES } \end{array} & \begin{array}{c} \text { Radius of } \\ \text { Neutral } \end{array} & \begin{array}{c} \text { First } \\ \text { Atom } \end{array} & I E & \text { AVEE } & \begin{array}{c} \text { Core } \\ \text { Charge } \end{array} \\ \hline 5 & 0.099 \mathrm{nm} & 1.251 \mathrm{MJ} / \mathrm{mol} & 1.59 & +7 \\\ \hline \end{array} $$ (a) Identify element \(X\). Explain your reasoning. (b) Give the complete electron configuration for element \(X\) (c) Give the electron configuration for \(X^{-1}\). How would the radius of \(X^{-1}\) compare to the radius for \(X ?\) Explain. (d) How would \(I E\) for \(X\) compare to the energy required to remove the most loosely held electron from \(X\) ? Explain. (e) How would the AVEE for \(X\) compare to the AVEE for the element with one more proton? Explain. (f) Qualitatively compare the covalent radius of \(X\) to that of the element with a core charge of +7 and a first \(I E\) of \(1.681 \mathrm{MJ} / \mathrm{mol}\). (g) How many valence electrons does \(X\) have?

Relate to the following data. An atom with an equal number of spin-up and spindown electrons is said to be diamagnetic because the atom is repelled by a magnetic field. In this case we say that all of the electrons are "paired." If this is not the case - if there are one or more unpaired electrons on an atom - the atom is attracted to a magnetic field, and it is said to be paramagnetic. The strength of the attraction is an experimentally measurable quantity known as the magnetic moment. The magnitude of the \(m a g-\) netic moment (measured in magnetons) is related to (but not proportional to) the number of unpaired electrons present. In other words, the larger the number of unpaired electrons, the larger the magnetic moment. Here are some experimental data collected by an investigator of this phenomenon. $$ \begin{array}{llc} {3}{c} {\text { Magnetic Moments of Several Elements }} \\ \hline & & \\ \text { Element } & \text { Type } & \begin{array}{c} \text { Magnetic Moments } \\ \text { (magnetons) } \end{array} \\ \hline \mathrm{H} & \text { Paramagnetic } & 1.7 \\ \mathrm{He} & \text { Diamagnetic } & 0 \\ \mathrm{~B} & \text { Paramagnetic } & 1.7 \\ \mathrm{C} & \text { Paramagnetic } & 2.8 \\ \mathrm{~N} & \text { Paramagnetic } & 3.9 \\ \mathrm{O} & \text { Paramagnetic } & 2.8 \\ \mathrm{Ne} & \text { Diamagnetic } & 0 \\ \hline \end{array} $$ An ion, \(X^{2+}\), is known to be from the first transition metal series. The ion is paramagnetic, with four unpaired electrons. What two possible elements could \(X\) be?
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