Without getting into too much/any math, I just wanna talk about intuitions if possible.

If atomic increase, the attraction increase, and consequently the potential energy increase. More energy leads to higher frequency, different radiations.

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I guess the reason if this diagram is universal is that the gap between 1 and 2 is too large too that the difference in energy cannot simply be compensated by adding more protons.

  • $\begingroup$ Sorry, but it is very hard to understand what you would like to say. Can you be a little be more detailed about what is your question, what atoms you are talking about, how does protons come into the picture, and in general, more specific? Maybe giving an example? atomic what? Potential energy of what? Frequency of what? In what state? Why do you think that diagram is universal? Those energies are only for hydrogen... $\endgroup$ – Greg Jun 11 '15 at 1:19

Absolutely, the differences in potential energies will be different for every element. This is because the potential energies will be different for every element. Notice that the potential energy for each energy level can be determined from the ground state. Let $E_g$ be the potential energy of the ground state (with the sign). Then for energy level $n$ the potential energy is $\frac{E_g}{n^2}$. Thus Differences between energy levels $n$ and $m$ can be written as $E_g\left|\frac{1}{n^2}-\frac{1}{m^2}\right|$.

Clearly it will be different for a different ground state energy which is precisely what the number of protons affects.

Note that the only important part of the mathematics is that all differences rely on the ground state potentials.

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