We all know that when an electron is excited it moves into an excited energy level preforming a quantum jump between energy levels. If we take a H atom and excite it's electron so that it moves out one level would the potential energy of this electron in this level be equal to an electron in the S2 energy level when it is at ground state (say the electron in a Li atom). I have asked this question to several of my Chemistry professors and I can't seem to get a good answer out of them. Possibly because my professors are Asian and there is a cultural or language barrier between us. Anyone have any thoughts on this topic? I do not have sufficient math skills to formulate an equation that models this idea for myself.

Are you familiar with Balmer series and Rydberg formula?

http://en.wikipedia.org/wiki/Balmer_series

 

http://en.wikipedia.org/wiki/Rydberg_formula

In general, the levels of electrons in different atoms do not match up. As the Rydberg formula that Sensei linked to shows, the energy depends on Z² and n² for hydrogen-like atoms, and for the rest you have all the electrons interacting with each other. Any correspondence between levels will be accidental, not from some systematic pattern.

The energy of an electron on the 2s orbital of ground-state lithium and excited hydrogen match imprecisely:

http://www.webelements.com/hydrogen/atoms.html

1312kJ/mol for the 1s electron in hydrogen, hence 1312/2²=328kJ/mol as a 2s electron.

http://www.webelements.com/lithium/atoms.html

Compare with 520kJ/mol for lithium.

 

These are ionization energies, that is potential energy minus kinetic one, but the kinetic energy compensates half the potential, so you can compare the ionization energy just as well.

 

The energy difference tells that our representation of heavier atoms is an approximation. A lithium nucleus plus two 1s electrons would behave as a hydrogen nucleus if experienced from distance, but lithium's 2s electron is close to the nucleus and the 1s electrons, so all electrons interact and find a common behaviour that is more favourable, hence the bigger 520kJ/mol.