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Chapter 6: Problem 93

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: $$ \begin{array}{l}{\text { (a) } n=4, l=1, m_{l}=2, m_{s}=-\frac{1}{2}} \\\ {\text { (b) } n=4, l=3, m_{l}=-3, m_{s}=+\frac{1}{2}}\\\\{\text { (c) } n=3, l=2, m_{l}=-1, m_{s}=+\frac{1}{2}} \\ {\text { (d) } n=5, l=0, m_{l}=0, m_{s}=0} \\ {\text { (e) } n=2, l=2, m_{l}=1, m_{s}=+\frac{1}{2}}\end{array} $$

Short Answer

Expert verified
The result is: (a) Invalid, \(m_l\) is not valid. (b) Valid. (c) Valid. (d) Invalid, \(m_s\) is not valid. (e) Invalid, \(l\) is not valid.

Step by step solution

01

Analyze Set (a)

Here we have \(n=4\), \(l=1\), \(m_l=2\), and \(m_s=-\frac{1}{2}\). Principal quantum number (n): \(n = 4\) which is valid. Angular momentum quantum number (l): \(l = 1\), as \(0 \leq l \leq 3\), which is valid. Magnetic quantum number (m_l): \(m_l = 2\), but the restriction was \(-1 \leq m_l \leq 1\). Therefore, \(m_l = 2\) is invalid. Spin quantum number (m_s): \(m_s = -\frac{1}{2}\) which is valid.
02

Analyze Set (b)

Here we have \(n=4\), \(l=3\), \(m_l=-3\), and \(m_s=+\frac{1}{2}\). All quantum numbers in this set follow the rules and are valid.
03

Analyze Set (c)

Here we have \(n=3\), \(l=2\), \(m_l=-1\), and \(m_s=+\frac{1}{2}\). All quantum numbers in this set follow the rules and are valid.
04

Analyze Set (d)

Here we have \(n=5\), \(l=0\), \(m_l=0\), and \(m_s=0\). Principal quantum number (n): \(n = 5\) which is valid. Angular momentum quantum number (l): \(l = 0\), as \(0 \leq l \leq 4\), which is valid. Magnetic quantum number (m_l): \(m_l = 0\) which is valid. Spin quantum number (m_s): \(m_s = 0\), but the allowed values are \(-\frac{1}{2}\) or \(+\frac{1}{2}\). Therefore, \(m_s = 0\) is invalid.
05

Analyze Set (e)

Here we have \(n=2\), \(l=2\), \(m_l=1\), and \(m_s=+\frac{1}{2}\). Principal quantum number (n): \(n = 2\) which is valid. Angular momentum quantum number (l): \(l = 2\), but the restriction was \(0 \leq l \leq 1\). Therefore, \(l = 2\) is invalid. The result is: (a) Invalid, \(m_l\) is not valid. (b) Valid. (c) Valid. (d) Invalid, \(m_s\) is not valid. (e) Invalid, \(l\) is not valid.

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Most popular questions from this chapter

The discovery of hafnium, element number \(72,\) provided a controversial episode in chemistry. G. Urbain, a French chemist, claimed in 1911 to have isolated an element number 72 from a sample of rare earth (elements \(58-71 )\) compounds. However, Niels Bohr believed that hafnium was more likely to be found along with zirconium than with the rare earths. D. Coster and G. von Hevesy, working in Bohr's laboratory in Copenhagen, showed in 1922 that element 72 was present in a sample of Norwegian zircon, an ore of zirconium. (The name hafnum comes from the Latin name for Copenhagen, Hafnia).(a) How would you use electron configuration arguments to justify Bohr's prediction? (b) Zirconium, hafnium's neighbor in group 4 \(\mathrm{B}\) , can be produced as a metal by reduction of solid \(\mathrm{ZrCl}_{4}\) with molten sodium metal. Write a balanced chemical equation for the reaction. Is this an oxidation- reduction reaction? If yes, what is reduced and what is oxidized? (c) Solid zirconium dioxide, \(\mathrm{ZrO}_{2},\) reacts with chlorine gas in the presence of carbon. The products of the reaction are \(Z r \mathrm{Cl}_{4}\) and two gases, \(\mathrm{CO}_{2}\) and \(\mathrm{CO}\) in the ratio \(1 : 2 .\) Write a balanced chemical equation for the reaction. Starting with a \(55.4-\mathrm{g}\) sample of \(\mathrm{ZrO}_{2},\) calculate the mass of \(\mathrm{ZrCl}_{4}\) formed, assuming that \(Z r O_{2}\) is the limiting reagent and assuming 100\(\%\) yield. (d) Using their electron configurations, account for the fact that \(\mathrm{Zr}\) and \(\mathrm{Hf}\) form chlorides \(\mathrm{MCl}_{4}\) and oxides \(\mathrm{MO}_{2}\)

The series of emission lines of the hydrogen atom for which \(n_{1}=3\) is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed. (b) Calculate the wavelengths of the first three lines in the Paschen series - those for which \(n_{1}=4,5,\) and \(6 .\)

Consider a transition in which the electron of a hydrogen atom is excited from \(n=1\) to \(n=\infty\) . (a) What is the end result of this transition? (b) What is the wavelength of light that must be absorbed to accomplish this process? (c) What will occur if light with a shorter wavelength that in part (b) is used to excite the hydrogen atom? (d) How are the results of parts (b) and (c) related to the plot shown in Exercise 6.88\(?\)

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