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Chapter 7: Problem 74

Which of the following sets of quantum numbers are not allowed? For each incorrect set, state why it is incorrect. a. \(n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2}\) b. \(n=4, \ell=3, m_{\ell}=2, m_{s}=-\frac{1}{2}\) c. \(n=4, \ell=1, m_{\ell}=1, m_{s}=+\frac{1}{2}\) d. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-1\) e. \(n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2}\) f. \(n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2}\)

Short Answer

Expert verified
The sets of quantum numbers that are not allowed are: a. \(n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2}\) because \(\ell\) should not be equal to \(n\). d. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-1\) because \(m_{s}\) should not be equal to -1. e. \(n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2}\) because \(\ell\) should not be a negative value. f. \(n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2}\) because \(m_{\ell}\) should not be outside the range allowed for the given \(\ell\).

Step by step solution

01

Check Set a

For this set, we have \(n=3, \ell=3, m_{\ell}=0, m_{s}=-\frac{1}{2}\). Rules 1 and 4 are satisfied. However, there is a problem with the \(\ell\) value. According to rule 2, \(\ell\) should range from 0 to (n-1), so for \(n=3\), the valid \(\ell\) values are 0, 1, and 2. This set of quantum numbers is not allowed because \(\ell\) should not be equal to \(n\).
02

Check Set b

We have \(n=4, \ell=3, m_{\ell}=2, m_{s}=-\frac{1}{2}\). All the rules are satisfied: \(n\) is a positive integer, \(\ell\) ranges from 0 to (n-1), \(m_{\ell}\) ranges from -\(\ell\) to +\(\ell\), and \(m_{s}\) is either +\(\frac{1}{2}\) or -\(\frac{1}{2}\). This set of quantum numbers is allowed.
03

Check Set c

For this set, we have \(n=4, \ell=1, m_{\ell}=1, m_{s}=+\frac{1}{2}\). All the rules are satisfied. This set of quantum numbers is allowed.
04

Check Set d

We have \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-1\). Rules 1 through 3 are satisfied. However, there is a problem with the \(m_{s}\) value. According to rule 4, \(m_{s}\) can only be +\(\frac{1}{2}\) or -\(\frac{1}{2}\). This set of quantum numbers is not allowed because \(m_{s}\) should not be equal to -1.
05

Check Set e

For this set, we have \(n=5, \ell=-4, m_{\ell}=2, m_{s}=+\frac{1}{2}\). Rules 1 and 4 are satisfied. However, there is a problem with the \(\ell\) value. According to rule 2, \(\ell\) should range from 0 to (n-1) and should never be negative. This set of quantum numbers is not allowed because \(\ell\) should not be a negative value.
06

Check Set f

We have \(n=3, \ell=1, m_{\ell}=2, m_{s}=-\frac{1}{2}\). Rules 1, 2, and 4 are satisfied. However, there is a problem with the \(m_{\ell}\) value. According to rule 3, \(m_{\ell}\) should range from -\(\ell\) to +\(\ell\), so for \(\ell=1\), the valid \(m_{\ell}\) values are -1, 0, and 1. This set of quantum numbers is not allowed because \(m_{\ell}\) should not be outside the range allowed for the given \(\ell\).

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Key Concepts

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

Quantum Number Rules
Quantum numbers are essential in understanding the arrangement of electrons in atoms. They describe the electronic configuration and specify the unique quantum state of an electron. There are four key quantum numbers, each with specific rules:
  • The principal quantum number (\(n\)) is a positive integer that determines the energy level and size of an electron's orbit. It can have values like 1, 2, 3, etc.
  • The azimuthal quantum number (\(\ell\)), also known as the angular momentum quantum number, ranges from 0 to \((n-1)\) for each value of \(n\).
  • The magnetic quantum number (\(m_{\ell}\)) ranges from \(-\ell\) to \(+\ell\), including zero. It describes the orientation of the orbital in space.
  • The spin quantum number (\(m_s\)) can only be \(+\frac{1}{2}\) or \(-\frac{1}{2}\), corresponding to the two possible spin states of an electron.
Understanding these rules helps to predict the electron distribution and their allowed configurations within an atom.
Principal Quantum Number (n)
The principal quantum number, denoted as \(n\), is fundamental for determining the main energy level an electron occupies within an atom. It is indicative of the electron's distance from the nucleus. As \(n\) increases, the orbital becomes larger and the electron is further away from the nucleus.

The possible values of \(n\) are positive integers (1, 2, 3, etc.). Higher energy levels can support more electrons. The number of electrons each level can hold is given by the formula \(2n^2\). For example, if \(n = 2\), it can hold up to 8 electrons.
Azimuthal Quantum Number (鈩�)
The azimuthal quantum number, \(\ell\), plays a crucial role in determining the shape of an electron's orbital. It is also referred to as the angular momentum quantum number and it defines the subshells or sublevels within each principal energy level.

The value of \(\ell\) ranges from 0 to \((n-1)\). For each value of \(\ell\), there is a corresponding subshell, typically represented by letters:
  • \(\ell = 0\): s subshell
  • \(\ell = 1\): p subshell
  • \(\ell = 2\): d subshell
  • \(\ell = 3\): f subshell
The properties associated with these subshells, like shape and orientation, are dictated by the value of \(\ell\).
Magnetic Quantum Number (m_鈩�)
The magnetic quantum number, \(m_{\ell}\), specifies the orientation of an electron's orbital around the nucleus. This quantum number is crucial for understanding how orbitals fit together within an atom.

The possible values of \(m_{\ell}\) range from \(-\ell\) to \(+\ell\), including zero, providing a spectrum of possible orientations. For instance, if \(\ell = 1\) (associated with a p subshell), then \(m_{\ell}\) can be -1, 0, or +1, which corresponds to the px, py, and pz orbitals.
Spin Quantum Number (m_s)
The spin quantum number, \(m_s\), describes the intrinsic spin of an electron, a fundamental property similar to the angular momentum. Electrons have a spin, which can only take one of two values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\). These values reflect the two possible spin orientations of the electron: "spin-up" and "spin-down."

This property is critical for the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of quantum numbers. Hence, even within the same orbital, electrons must have opposite spins, making \(m_s\) a vital part of electron configuration.

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Most popular questions from this chapter

The electron affinity for sulfur is more exothermic than that for oxygen. How do you account for this?

Using the element phosphorus as an example, write the equation for a process in which the energy change will correspond to the ionization energy and to the electron affinity. Explain why the first ionization energy tends to increase as one proceeds from left to right across a period. Why is the first ionization energy of aluminum lower than that of magnesium, and the first ionization energy of sulfur lower than that of phosphorus? Why do the successive ionization energies of an atom always increase? Note the successive ionization energies for silicon given in Table \(7.5 .\) Would you expect to see any large jumps between successive ionization energies of silicon as you removed all the electrons, one by one, beyond those shown in the table?

The bright yellow light emitted by a sodium vapor lamp consists of two emission lines at \(589.0\) and \(589.6 \mathrm{~nm}\). What are the frequency and the energy of a photon of light at each of these wavelengths? What are the energies in \(\mathrm{kJ} / \mathrm{mol}\) ?

Assume that we are in another universe with different physical laws. Electrons in this universe are described by four quantum numbers with meanings similar to those we use. We will call these quantum numbers \(p, q, r\), and \(s\). The rules for these quantum numbers are as follows: \(p=1,2,3,4,5, \ldots\) \(q\) takes on positive odd integers and \(q \leq p\). \(r\) takes on all even integer values from \(-q\) to \(+q\). (Zero is considered an even number.) \(s=+\frac{1}{2}\) or \(-\frac{1}{2}\) a. Sketch what the first four periods of the periodic table will look like in this universe. b. What are the atomic numbers of the first four elements you would expect to be least reactive? c. Give an example, using elements in the first four rows, of ionic compounds with the formulas XY, \(\mathrm{XY}_{2}, \mathrm{X}_{2} \mathrm{Y}, \mathrm{XY}_{3}\), and \(\mathrm{X}_{2} \mathrm{Y}_{3}\). d. How many electrons can have \(p=4, q=3 ?\) e. How many electrons can have \(p=3, q=0, r=0\) ? f. How many electrons can have \(p=6\) ?

What type of relationship (direct or inverse) exists between wavelength, frequency, and photon energy? What does a photon energy unit of a joule equal?
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