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This should be a pretty quick answer but I can't figure it out.

I read somewhere that in an atom ionization energy and potential energy have the same absolute values, but it doesn't really make sense to me, how could they be equal if potential energy gets bigger as u move away from the atom while ionization energy gets smaller?

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  • $\begingroup$ The ionization energy is the energy required to move an electron from the atom to infinity. The potential energy that electron has at infinity, relative to its original orbital, is the ionization energy. $\endgroup$
    – Jon Custer
    Apr 6, 2023 at 15:36

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The specific qualm you have about this rule is resolved by the words "absolute value" - the potential energy becomes less negative (bigger), and the ionization energy gets less positive (smaller) when the outermost electron is farther from the rest of the atom.

Moreover a general concept is: in order to ionize an atom, you need to inject enough energy such that the system is in a quantum state consisting of one free electron and the rest of the electrons in a bound state. The free electron must have $>0$ energy so that it moves away from the rest of the atom and doesn't eventually turn around and come back. So the ionization energy is always exactly equal to the ground state energy of the ion you want to form minus the atom's current state's energy (assuming the freed electron has a very tiny amount of excess escape energy).

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