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Chapter 30: Problem 23

The principal quantum number for an electron in an atom is \(n=6,\) and the magnetic quantum number is \(m_{\ell}=2 .\) What possible values for the orbital quantum number \(\ell\) could this electron have?

Short Answer

Expert verified
The possible values for \( \ell \) are 2, 3, 4, and 5.

Step by step solution

01

Understand the Principal Quantum Number

The principal quantum number, denoted as \( n \), determines the energy level or shell of an electron in an atom. For this problem, \( n = 6 \), meaning the electron is in the 6th energy level.
02

Recall the Orbital Quantum Number

The orbital quantum number, denoted as \( \ell \), defines the shape of the electron's orbital. \( \ell \) can have integer values ranging from 0 to \( n-1 \). For \( n = 6 \), \( \ell \) can therefore be 0, 1, 2, 3, 4, or 5.
03

Understand the Magnetic Quantum Number

The magnetic quantum number, \( m_\ell \), gives the orientation of the orbital and can have values between \(-\ell\) and \(+\ell\), including zero. In this problem, \( m_\ell = 2 \), so \( \ell \) must be at least 2.
04

Determine Possible Values of \( \ell \)

Given that \( m_\ell = 2 \), \( \ell \) must be greater than or equal to 2. Simultaneously, \( \ell \) must be less than \( n \). Thus, the possible values for \( \ell \) are 2, 3, 4, and 5.

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Key Concepts

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

Principal Quantum Number
The principal quantum number, denoted as \( n \), plays a critical role in determining the energy level or shell where an electron is located within an atom. Think of it as the floor number of a building, where each floor represents a different energy level.
For instance, if an electron has a principal quantum number of \( n = 6 \), it resides in the sixth energy level.
This level indicates not only the specific energy possessed by the electron but also its average distance from the nucleus.
  • Higher principal quantum numbers mean the electron is further away from the nucleus.
  • The energy level increases as \( n \) gets larger, moving the electron to higher energy states.
Understanding this helps in determining other quantum numbers related to this electron.
Orbital Quantum Number
The orbital quantum number is symbolized by \( \ell \), and it is essential for describing the shape of an electron's orbital.
Essentially, it defines the pattern of regions in space where the electron is likely to be found around the nucleus.
The value of \( \ell \) is not random; it relies entirely on the principal quantum number. For any principal quantum number \( n \), \( \ell \) can take on integer values from 0 to \( n-1 \).
  • This range describes different orbital shapes: \( \ell = 0 \) corresponds to an s orbital, \( \ell = 1 \) to a p orbital, \( \ell = 2 \) to a d orbital, and so on.
  • In our case with \( n = 6 \), the possible \( \ell \) values are 0 through 5.
  • These values allow the electron orbitals to vary from spherical to increasingly complex shapes.
It's crucial to understand that the orbital quantum number limits the maximum complexity of the orbitals an electron can occupy at a given principal quantum number level.
Magnetic Quantum Number
The magnetic quantum number, denoted as \( m_\ell \), adds another layer of detail by revealing the orientation of an electron's orbital.
Consider this number as the compass for electron orbitals, ensuring that they are aligned in the three-dimensional space around the atom.
The value of \( m_\ell \) can range from \(-\ell\) to \(+\ell\), encompassing zero.
  • This means \( m_\ell \) gives us the different orientations the given orbital shape (determined by \( \ell \)) can take.
  • For example, if \( m_\ell = 2 \), the minimum \( \ell \) must be 2, since \( m_\ell \) cannot be larger than \( \ell \).
  • This rule ensures that for \( m_\ell = 2 \), \( \ell \) must be one of the following: 2, 3, 4, or 5.
The magnetic quantum number dictates how many different orientations, or "compass points," are available for an electron's orbital, offering a complete picture when combined with \( n \) and \( \ell \). This information is pivotal for understanding the electron's position, energy state, and chemical bonding capability.

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Most popular questions from this chapter

(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. Verify that your answers are consistent with your answers to the Concept Questions.

A certain species of ionized atoms produces an emission line spectrum according to the Bohr model, but the number of protons \(Z\) in the nucleus is unknown. A group of lines in the spectrum forms a series in which the shortest wavelength is \(22.79 \mathrm{nm}\) and the longest wavelength is \(41.02 \mathrm{nm} .\) Find the next-to-the-longest wavelength in the series of lines.

Fusion is the process by which the sun produces energy. One experimental technique for creating controlled fusion utilizes a solid-state laser that emits a wavelength of \(1060 \mathrm{~nm}\) and can produce a power of \(1.0 \times 10^{14} \mathrm{~W}\) for a pulse duration of \(1.1 \times 10^{-11} \mathrm{~s} . \mathrm{In}\) contrast, the helium/neon laser used at the checkout counter in a bar-code scanner emits a wavelength of \(633 \mathrm{~nm}\) and produces a power of about \(1.0 \times 10^{-3} \mathrm{~W} .\) How long (in days) would the helium/neon laser have to operate to produce the same number of photons that the solid-state laser produces in \(1.1 \times 10^{-11} \mathrm{~s}\) ?

Two of the three electrons in a lithium atom have quantum numbers of \(n=1, \ell=0\), \(m_{\ell}=0, m_{s}=+\frac{1}{2}\) and \(n=1, \ell=0, m_{\ell}=0, m_{s}=-\frac{1}{2} .\) What quantum numbers can the third electron have if the atom is in (a) its ground state and (b) its first excited state?

The \(K_{\beta}\) characteristic \(\mathrm{X}\) -ray line for tungsten has a wavelength of \(1.84 \times 10^{-11} \mathrm{~m} .\) What is the difference in energy between the two energy levels that give rise to this line? Express the answer in (a) joules and (b) electron volts.
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