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According to this:

$ E = \frac{-13.6 Z^2}{n^2} $

the energy of an electron is, well, higher the farer it is away from the core. I found this confusing as I need to put less energy to release an electron when it is on the 5th shell than when it is on the 1st shell. Am I right? Concluding, one would say "The more energy an electron has the less energy I need to release it". This wouldn't be the case if "they" didn't introduce the minus sign. It probably has a reason and I'm probably neither the first nor the last who asks about that. But, sorry, I still have to ask: Why did they do that? Why is a minus better than a plus?

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Why did they do that? Why is a minus better than a plus?

Firstly, understand that when the electron is 'infinitely' far from the nucleus its energy is $0$ (assuming it is also stationary).

But empirically we also know that when an electron 'falls' from a higher orbital ($n=n_1$) to a lower one ($n=n_2$ where $n_2<n_1$) energy is released in the form of a photon emission, so $\Delta E <0$. And indeed:

$$\Delta E=E_2-E_1=-13.6 Z^2\Big(\frac{1}{n_2^2}-\frac{1}{n_1^2}\Big)<0$$

So the choice of sign is correct.

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  • $\begingroup$ The higher orbital is $ n_1 $ and the lower is $ n_2 $ ?Despite from that I don't see an argument for the minus at that place? $\endgroup$
    – Ben
    Jun 17, 2020 at 13:48
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    $\begingroup$ That IS the argument: put a plus sign in and you'd get $\Delta E >0$ which is contrary to our experience. It's a convention to ensure that for a 'falling' electron, $\Delta E<0$. That's all there is to it. $\endgroup$
    – Gert
    Jun 17, 2020 at 14:05
  • $\begingroup$ Ah, because otherwise, it would gain energy? $\endgroup$
    – Ben
    Jun 17, 2020 at 14:34
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    $\begingroup$ The convention (universal, not just electrons) is that an energy loss is a negative $\Delta$ and a gain is a positive $\Delta$. $\endgroup$
    – Gert
    Jun 17, 2020 at 15:30
  • $\begingroup$ Well, that makes sense, indeed. Confusing but once that is known it makes sense :) $\endgroup$
    – Ben
    Jun 18, 2020 at 5:14

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