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Chapter 36: Problem 17

Which of the following is not a possible value for the magnitude of the orbital angular momentum in hydrogen: (a) \(\sqrt{12} \hbar\) (b) \(\sqrt{20} \hbar ;\) (c) \(\sqrt{30} \hbar ;\) (d) \(\sqrt{40} \hbar ;\) (e) \(\sqrt{56} \hbar ?\)

Short Answer

Expert verified
The magnitude \(\sqrt{40} \hbar \) is not a possible value for the orbital angular momentum in hydrogen.

Step by step solution

01

Identify possible values of l

Write out the first few possible values for the orbital quantum number \( l \) (nonnegative integers), and calculate \( \sqrt{l(l+1)} \) for each.
02

Compare \(\sqrt{l(l+1)}\) to given values

Compare each calculated value of \( \sqrt{l(l+1)} \) to the given possible values of the orbital angular momentum. Remember that \(\sqrt{l(l+1)} \hbar \) values are equivalent to \(\sqrt{2, 6, 12, 20, 30, 42, 56, 72, 90, ..}\).
03

Identify the value that doesn't fit the rule

Identify any given values that could not have come from an integer \( l \). This is the value that is not a possible value for the magnitude of the orbital angular momentum in hydrogen.

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Most popular questions from this chapter

With sufficient energy, it's possible to eject an electron from an inner atomic orbital. A higher-energy electron will then drop into the unoccupied state, emitting a photon with energy equal to the difference between the two levels. For inner-shell electrons, photon energies are in the keV range, putting them in the X-ray region of the spectrum. These characteristic X rays are labeled with the letter indicating the shell to which the electron drops, followed by a Greek letter indicating the higher level from which it drops; thus \(K \alpha\) designates a transition from the \(L\) shell to the \(K\) shell. Characteristic X rays provide scientists and physicians with an important diagnostic tool. Environmental scientists bombard pollution samples with high- energy electrons, knocking out inner-shell electrons and thus producing X-ray spectra that help identify contaminants (Fig. \(36.20 a\) ). Geologists do the same with rocks. Medical radiologists reverse the process, exploiting the fact that X rays cause inner-shell transitions as well as complete ejection of inner-shell electrons. In particular, radiologists use the element barium in this way to produce high-contrast X-ray images of the intestinal tract \((\text { Fig. } 36.20 b)\)(GRAPH CANNOT COPY) (a) An \(\mathrm{X}\) -ray spectrum from air pollutants trapped on a filter. The labeled peaks show the presence of lead (Pb) and arsenic (As), as evidenced by \(K \alpha, K \beta, L \alpha,\) and \(L \beta\) characteristic X rays. (b) \(\mathrm{X}\) -ray of an intestinal tract, made by coating the intestinal wall with X-ray-opaque barium Elements \(A\) and \(B\) have atomic numbers \(Z_{A}\) and \(Z_{B}=2 Z_{A} .\) How do you expect element \(B^{\prime}\) s \(K \alpha\) X-ray energy to compare with that of element \(A\) ? a. \(B^{\prime}\) s \(K \alpha\) energy should be about one-fourth that of \(A\) b. \(B\) 's \(K \alpha\) energy should be about half that of \(A\) c. \(B^{\prime}\) s \(K \alpha\) energy should be about twice that of \(A\) d. \(B^{\prime}\) s \(K \alpha\) energy should be about four times that of \(A\)

Why is stimulated emission essential for laser action?

An electron in hydrogen is in the \(5 f\) state. What possible values, in units of \(h,\) could a measurement of the orbital angular momentum component on a given axis yield?

The \(4 p \rightarrow 3 s\) transition in sodium produces a double spectral line at 330.2 and \(330.3 \mathrm{nm} .\) What's the energy splitting of the \(4 p\) level?

Can the component of a quantized angular momentum measured on a given axis ever equal the magnitude of the angular momentum vector? Explain.
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