I am self studying MIT OCW chemistry 5.111 2014, one of the lecture questions states the following:

Consider a neutral atom with 8 distinct electron binding energies: −14 eV, −28 eV, −94 eV, −218 eV, −293 eV, −1730 eV, −1921 eV, and −14326 eV.
(a) Name all of the possible ground state atoms that could have these binding energies (without looking up any values).

It seems like the values are quite arbitrary without having a reference place to begin from i.e. electron configuration, quantum numbers, period and so on.

I thought maybe to use equation

$$E_{nl} = -\frac{Z_\mathrm{eff}^2R_\ce{H}}{n^2}$$

as it has an energy parameter, but it appears I have one equation and two unknowns.

Any ideas on how to go about this?

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    $\begingroup$ hint: consider the Aufbau principle, en.wikipedia.org/wiki/Aufbau_principle#/media/… $\endgroup$ – Buck Thorn Nov 21 '19 at 23:40
  • $\begingroup$ @BuckThorn Thank you for your response although I'm not sure I understand. Could you please help me reason through it? The Aufbau principle pertains to filling in sub-shells to create the electron configuration of an atom. How could I combine that principle with knowledge of only the binding energy to determine the identity of an unknown atom? $\endgroup$ – TheRedCamaro3.0 3.0 Nov 22 '19 at 2:07
  • $\begingroup$ Look at the binding energies of any known element. Look how do they grow when you are done with stripping one shell and are starting a new one. $\endgroup$ – Ivan Neretin Nov 22 '19 at 5:08
  • $\begingroup$ I think some context is missing in the question. They should have discussed the X-ray spectra in the lecture. Have they? Forget Bohr's equation for this question, it fails miserably after hydrogen. The answers are provided by the online answer key. See one possible answer and its X-ray spectra notation xdb.lbl.gov/Section1/Periodic_Table/Ga_Web_data.htm. If they did not discuss X-ray spectra then I don't know how to address it. $\endgroup$ – M. Farooq Nov 22 '19 at 5:22
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    $\begingroup$ @IvanNeretin Thank you for your response, it is my understanding that the binding energy is stronger the lower the energy level as they are closer to the nucleus so stripping electrons while maintaining the same amount of protons in the nucleus should result in a stronger binding energy. I am unsure how to quantify this in terms of energy though. $\endgroup$ – TheRedCamaro3.0 3.0 Nov 22 '19 at 7:59

There are a few clues on how to solve this question:

  • you are given binding energies. The energies in question are just enough to liberate electrons from their bound state.
  • The equation you've written $$E_{nl} = -\frac{Z_\mathrm{eff}^2R_\ce{H}}{n^2}$$ describes the binding energy. The important part is that it can be assumed to be only a function of n and l, that is, of the subshell to which an electron belongs ($Z_\mathrm{eff}$ is a function of n and l quantum numbers. Unfortunately this is probably a somewhat complicated function but that is not very important just now)

Finally consider the Aufbau principle, outlined in this libretext:

In essence, you can think of electrons filling in "n+l " order. If you have an equal "n + l" value for two orbitals, then electrons will occupy the orbital with the lower "n" value.

which is followed by a nice illustration (shown below). The number of distinct binding energies equals the number of subshells. The energy for emission from each subshell decreases in the same order in which the subshells are filled. All electrons in a subshell share a common binding energy. Following the Aufbau pattern you can identify the highest-energy occupied (valence) subshell, which corresponds to the smallest binding energy. I count 8 subshells and assign the minimum $E_{nl}$ to the 4p subshell (the electron configuration is $\ce{[Ar] 3d^{10} 4s^2 4p^?}$). This means there are six possible atoms that match the number of binding energies, namely Ga through Kr.

If we cheat and check a table of ionization energies we see that the lowest binding energy (-14 eV) matches that of Kr (other values in the table don't match those in the problem since the table lists the ionization energy for multiple electron emissions whereas the problem lists energies for all possible single electron emissions $\ce{A->A+ + e-}$, starting from the neutral atom, an emission from one subshell corresponding to one energy). You should note the pattern of decreasing binding energies (magnitudes) starting from the 1s electrons (−14326 eV) up to the valence shell (-14 eV). The following diagram adapted from the libretext quoted above illustrates how this works:

enter image description here

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  • $\begingroup$ Thank you so much for your very detailed answer, so simply by knowing the number of potential binding energies we are able to deduce the block on the periodic table that these binding energies must belong to. In this case it is not one atom with 8 binding energies at different levels but rather 8 different elements with differing ground state energies. But if it is the case that there are 8 differing elements with 8 differing ground states how could we know their respective energies? $\endgroup$ – TheRedCamaro3.0 3.0 Nov 24 '19 at 3:19
  • $\begingroup$ @TheRedCamaro3.03.0 I find there are 6 elements that match the emission pattern. I tried to explain more clearly in the answer. $\endgroup$ – Buck Thorn Nov 24 '19 at 10:57
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    $\begingroup$ Thank you so much truly for your detailed answer it makes it much more intuitive. I believe my confusion was in the wording of the question suggesting that we're looking for a single element with 8 ground states vs 8 elements with 8 distinct ground states when we were actually looking for potential elements with 8 potential sub-shells. $\endgroup$ – TheRedCamaro3.0 3.0 Nov 25 '19 at 1:11

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