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Chapter 8: Problem 58

Write an acceptable value for each of the missing quantum numbers. (a) \(n=3, \ell=?, m_{\ell}=2, m_{s}=+\frac{1}{2}\) (b) \(n=?, \ell=2, m_{\ell}=1, m_{s}=-\frac{1}{2}\) (c) \(n=4, \ell=2, m_{\ell}=0, m_{s}=?\) (d) \(n=?, \ell=0, m_{\ell}=?, m_{s}=?\)

Short Answer

Expert verified
(a) \(l=2\), (b) \(n=3, 4, ...etc.\), (c) \(m_{s}= +\frac{1}{2}, -\frac{1}{2}\), (d) \(n=1, 2, ...etc., m_{\ell}=0, m_{s}= +\frac{1}{2}, -\frac{1}{2}\)

Step by step solution

01

Solve for Missing Quantum Number in (a)

For (a) \(n=3, m_{\ell}=2, m_{s}=+\frac{1}{2}\), we need to find \(\ell\). As \(m_{\ell}\) ranges from \(-l\) to \(l\), \(\ell\) must be equal to or greater than \(m_{\ell}\). The only acceptable value that does not exceed \(n-1\) is \(l=2\).
02

Solve for Missing Quantum Number in (b)

For (b) \(\ell=2, m_{\ell}=1, m_{s}=-\frac{1}{2}\), we need to find \(n\). As \(\ell\) must be smaller than \(n\), the only acceptable value for \(n\) that is larger than \(\ell=2\) is \(n=3\), \(n=4\), and so on.
03

Solve for Missing Quantum Number in (c)

For (c) \(n=4, \ell=2, m_{\ell}=0\), we need to find \(m_{s}\). \(m_{s}\) only has two possible values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\). So either of them is an acceptable value.
04

Solve for Missing Quantum Number in (d)

For (d) \(\ell=0\), we need to find \(n, m_{\ell}, m_{s}\). As \(\ell\) should be less than \(n\), the only acceptable value for \(n\) that is larger than \(\ell=0\) is \(n=1\), \(n=2\), and so on. \(m_{\ell}\) ranges from \(-\ell\) to \(\ell\), so the only possible value for \(m_{\ell}\) here is \(m_{\ell}=0\). \(m_{s}\) only has two possible values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\). So either of them is an acceptable value.

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Most popular questions from this chapter

Without doing detailed calculations, arrange the following electromagnetic radiation sources in order of increasing frequency: (a) a red traffic light, (b) a \(91.9 \mathrm{MHz}\) radio transmitter, (c) light with a frequency of \(3.0 \times 10^{14} \mathrm{s}^{-1}\) (d) light with a wavelength of \(49 \mathrm{nm}\).

Concerning the electrons in the shells, subshells, and orbitals of an atom, how many can have (a) \(n=4, \ell=2, m_{\ell}=1,\) and \(m_{s}=+\frac{1}{2} ?\) (b) \(n=4, \ell=2,\) and \(m_{\ell}=1 ?\) (c) \(n=4\) and \(\ell=2 ?\) (d) \(n=4 ?\) (e) \(n=4, \ell=2,\) and \(m_{s}=+\frac{1}{2} ?\)

For the Bohr hydrogen atom determine (a) the radius of the orbit \(n=4\) (b) whether there is an orbit having a radius of \(4.00 \AA\) (c) the energy level corresponding to \(n=8\) (d) whether there is an energy level at \(-2.5 \times 10^{-17} \mathrm{J}\)

The recently discovered element 114 should most closely resemble Pb. (a) Write the electron configuration of \(\mathrm{Pb}\). (b) Propose a plausible electron configuration for \(8 \equiv\) element 114.

Use the Balmer equation (8.2) to determine (a) the frequency, in \(s^{-1}\), of the radiation corresponding to \(n=5\) (b) the wavelength, in nanometers, of the line in the Balmer series corresponding to \(n=7\) (c) the value of \(n\) corresponding to the Balmer series line at \(380 \mathrm{nm}\)
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