The emission spectrum of the He^+ion contains the Pickering series of spectral lines that is ana

The emission spectrum of the He^+ion contains the Pickering...

The emission spectrum of the $\mathrm{He}^{+}$ion contains the Pickering series of spectral lines that is analogous to the Lyman, Balmer and Paschen series in the spectrum of hydrogen. $$ \begin{array}{lllll} \hline \text { Balmer } i=1,2, \ldots & 0.456806 & 0.616682 & 0.690685 & 0.730884 \\ \text { Pickering } i=2,4, \ldots & 0.456987 & 0.616933 & 0.690967 & 0.731183 \\ \hline \end{array} $$ The table gives the frequencies (in $10^{15} \mathrm{~Hz}$ ) of the first four lines of the Balmer series and the first four even-numbered lines of the Pickering series. The frequencies of these lines in the Pickering series are almost coincident with the frequencies of lines of the Balmer series. Explain this finding. Provide a quantitative explanation of the small offset between these nearly coincident lines in terms of the reduced mass of the electron in the two systems. (In 1896 E.C. Pickering identified the odd-numbered lines in his series in the spectrum of the star $\zeta$ Puppis. Helium had yet to be discovered and he believed that the lines were being produced by hydrogen. Naturally he confused the even-numbered lines of his series with ordinary Balmer lines.)

The emission spectrum of the $\mathrm{He}^{+}$ion contains the Pickering series of spectral lines that is analogous to the Lyman, Balmer and Paschen series in the spectrum of hydrogen.
$$
\begin{array}{lllll}
\hline \text { Balmer } i=1,2, \ldots & 0.456806 & 0.616682 & 0.690685 & 0.730884 \\
\text { Pickering } i=2,4, \ldots & 0.456987 & 0.616933 & 0.690967 & 0.731183 \\
\hline
\end{array}
$$
The table gives the frequencies (in $10^{15} \mathrm{~Hz}$ ) of the first four lines of the Balmer series and the first four even-numbered lines of the Pickering series. The frequencies of these lines in the Pickering series are almost coincident with the frequencies of lines of the Balmer series. Explain this finding. Provide a quantitative explanation of the small offset between these nearly coincident lines in terms of the reduced mass of the electron in the two systems. (In 1896 E.C. Pickering identified the odd-numbered lines in his series in the spectrum of the star $\zeta$ Puppis. Helium had yet to be discovered and he believed that the lines were being produced by hydrogen. Naturally he confused the even-numbered lines of his series with ordinary Balmer lines.)

Key Concepts

Hydrogenic Energy Level Structure

The energy levels of one-electron systems (hydrogenic atoms) follow a 1/n² dependence, where n is the principal quantum number, and the transition energies—and hence the frequencies of the emitted or absorbed light—are determined by differences in these levels. Both hydrogen and He? share this structure, with the primary differences arising from the nuclear charge and the reduced mass. The reduced mass correction, though small, causes a quantifiable offset between the spectral lines (such as those in the Balmer and Pickering series) in the otherwise similar hydrogen-like spectra.

Reduced Mass Correction

In atomic physics, the electron does not orbit an infinitely massive nucleus; instead, the two-body problem is solved by using the reduced mass ? = (m_e M) / (m_e + M). This correction accounts for the fact that both the electron and the nucleus move about their common center of mass. In hydrogenic systems, the energies of the bound states are proportional to this reduced mass, leading to slight differences in the calculated energy levels (and hence the spectral frequencies) when comparing atoms (or ions) with different nuclear masses, such as hydrogen and He?.

Rydberg Constant Dependence

The Rydberg constant (R) governs the spectral line positions in hydrogen-like atoms and is given by an expression proportional to the reduced mass of the electron-nucleus system. When comparing an ion like He? to hydrogen, the Rydberg constant is modified by the factor ?/m_e, meaning that small changes in the reduced mass produce small shifts in the spectral line frequencies. This is the quantitative basis for explaining the observed slight offset between nearly coincident lines in the Pickering and Balmer series.

Spectral Series Overlap and Identification

The nearly coincident frequencies of specific transitions in the hydrogen Balmer series and the He? Pickering series are a result of the interplay between the Rydberg formula’s dependence on reduced mass and the similar structure of the hydrogenic energy levels. Although the transitions occur in different systems, the close correspondence in energies—modified only by a slight change in the effective Rydberg constant due to the reduced mass differences—leads to a near overlap in their spectral lines, with a small offset that can be quantitatively predicted.

Transcript

00:01 Singly ionized helium is a hydrogen -like atom, since it only has a single electron left in it, but it is an atom with a positive charge of two in its nucleus.

00:15 And people actually found helium first in the spectra of stars.

00:24 Hence the name helios derives the word helium helios being the greek word for sun so pickering an astronomer in the 1800s which was kind of the heyday of learning about spectra determined that in some of the hotter stars there was a series of spectral lines that again looked like the difference between the inverse of two integer squared and we can now understand where that's coming from by looking at the energy levels in a hydrogen -like atom, which all go as 1 over n squared.

01:12 But there is a z -square dependence as well.

01:16 If you do the model working with nuclei that have more than one proton in the nucleus.

01:23 So we can develop an expression for the energy difference in such an atom between two levels and minus n prime or however you want to write that.

01:42 So it would be 13 .6 over n squared, z squared, and then we should put that n squared in with a difference between a final and an state, and that is the energy of the photon coming off.

02:08 So if we kind of combine all these results, what we should see is that any hydrogen -like atom is going to create a series.

02:19 It may be hard to ionize the atom enough to make it into a hydrogen -like atom with a single electron orbiting the nucleus.

02:33 But what we would then predict is that the one -over -the -wavelength dependence, which relates in hydrogen to the rydberg constant, that we would have a new rydberg constant, we'll call rydberg -z, would be z squared times the rydberg constant of hydrogen.

02:57 So we would predict this constant in the pickering series with the helium to have four times the value of the rydberg constant for hydrogen.

03:13 And the rydberg constant for hydrogen is a well -known quantity.

03:20 It is 1 .0968.

03:24 I'll leave a number of sick figs in there times 10 to the minus 2 per nanometer.

03:30 I like to write that in terms of nanometers.

03:38 So what's unusual about helium is that there is a correspondence between lines in the pickering series and that of the bomber series.

03:49 So there are similar or corresponding, i should say, visible lines in helium...

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