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Is energy of electron of any atom at infinity zero?

As I know it's zero for hydrogen-like atom (ion) according to Bohr's model, but what about 2-electron atom or 3-electron or more? Will it also be zero at infinity?

And I know that the formula for energy in Bohr's model for hydrogen-like atom is $$ E=\frac{-13{,}6\,\mathrm{eV} \times Z^{2}}{n^{2}}, $$

but does it work for non-hydrogen-like atom?

And going back to why the energy level is zero at infinity in hydrogen-like atom is because if we plug infinity in the formula above we get zero, but will it work for non-hydrogen-like atom?

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  • $\begingroup$ We assume potential between two charged particles at infinite separation is zero as no electrical interaction exists in such a case, Bohr's model is a consequence of that result $\endgroup$
    – Draculin
    Commented Oct 30, 2021 at 14:01
  • $\begingroup$ Thanks a lot it helped! so potential energy of electron at infinity is zero for any and all atoms, but what about the formula above, does it work for all atoms? $\endgroup$
    – Elie Makdissi
    Commented Oct 30, 2021 at 17:32
  • $\begingroup$ It doesnt work for atoms or ions having more than 1 electron $\endgroup$
    – Jun Seo-He
    Commented Oct 30, 2021 at 18:07
  • $\begingroup$ see this hyperphysics.phy-astr.gsu.edu/hbase/hyde.html#c2 n is a quantum number , n going to infinity means the energy levels are densely packed, as in the illustration. Not in space infinity. $\endgroup$
    – anna v
    Commented Oct 30, 2021 at 20:46
  • $\begingroup$ "so potential energy of electron at infinity is zero for any and all atoms," electrons bound in atoms have zero quantum mechanical probability to be at infinity. Their location is bounded about the nucleus . en.wikipedia.org/wiki/Atomic_orbitalThe atoms are neutral $\endgroup$
    – anna v
    Commented Oct 30, 2021 at 20:48

2 Answers 2

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Yes, the energy of an electron for any atom is zero at $n \to \infty$.

Where does $E =0$ exist in an atom classically? Well past the atomic radius wouldn't it?

Why would an electron hang out at $n \to \infty$? Only if the atom's been ionized.

Logically this should be true for any atom of any element, multi-electron or not - when the atom's been ionized - the ionized electron has $E = 0$ with respect to the electric potential of the nucleus of its parent atom.

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    $\begingroup$ thanks a lot man! $\endgroup$
    – Elie Makdissi
    Commented Oct 30, 2021 at 21:08
  • $\begingroup$ @ElieMakdissi Love to help! $\endgroup$
    – Karthik
    Commented Oct 30, 2021 at 21:16
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For any hydrogen orbital there is a finite chance to detect the electron far away from the nucleus. In principle therefore the energy does not go to zero as r goes to infinity. Of course, in practice these probabilities decay exponentially. For a nearly ionised case this decay is much slower. For the ionised case the electron in general has positive, kinetic energy and can be anywhere so also at infinity.

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