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A gaseous mixture contains hydrogen atoms in the 4th excited state, $\ce{He^+}$ ions in 3d excited state, and $\ce{Li^{2+}}$ in 2nd excited state. Find the number of distinct spectral lines emitted when all these ions return to the ground state.

Number of lines emitted $=\displaystyle\frac{n(n-1)}{2}$, where $n$ is the number of the shell the electron is in.

If we simply add the number of lines, without considering repetitions, we get $10+6+3=19$ lines.

However since there are 3 atoms, there will be some cases of repetition. Now we can always find out the repetitive cases by listing down the energy of each transition, but that will be very long, and there is potential for error since there are 19 lines. Is there a better method to solve this question.

Energy of transition $=\displaystyle13.6z^2\left|\frac{1}{n_1^2}-\frac{1}{n_2^2}\right|\ \mathrm{eV}$

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    $\begingroup$ Involve algebraic evaluation. if/when for given z1, z2, n11, n12, n21 exists n22, matching equality, for given n constraints. $\endgroup$
    – Poutnik
    Sep 12, 2021 at 9:51
  • $\begingroup$ The best method is to write down each transition, just like you said. Is there a better method? Yes, but it is actually way worse, immensely long and heavily involved in math. $\endgroup$ Sep 12, 2021 at 9:55
  • $\begingroup$ Enumeration is definitely faster, especially if one involves a simple Excel table. $\endgroup$
    – Poutnik
    Sep 12, 2021 at 9:58
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    $\begingroup$ Also, repetitions are theoretically possible even with one atom (not in this case, though). $\endgroup$ Sep 12, 2021 at 9:59
  • $\begingroup$ Can such a mix even exist? an excited H and He+ would ionize at the energies needed to form Li++ ions. $\endgroup$
    – jimchmst
    Dec 4, 2023 at 0:32

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