# Ionization energy, how far does the electron go away?

The ionization is often defined as:

The energy needed to remove one or more electrons from a neutral atom to form a positively charged ion.

But what is meant by removed?

Suppose we have an infinitely large closed system with no content except a single atom (g). Now the ionisation energy can actually be as high as you like, at some point the electron will return to the atomic nucleus as it slowly slows down due to the constant (rapidly decreasing but always present) force of attraction, or not? So what does the energy quantity on ionisation energy refer to?

I mean, if I take the energy E = F * s with s equal to infinity and F decreases with 1/r^2, then I would have to integrate to infinity and even if F continues to approach the x axis asymptotically, it remains only an approach to zero and viewed at x (so r, the distance between electron and nuclei) infinity, E would also have to be infinite?

That is why I ask myself how a finite amount can be given in standard value tables for the ionisation energies of certain elements.

For example, I also don't understand why you can use infinity for n in the rydberg formula to get the ionisation energy, does that work more because of the definition of the rydberg constant? Infinity would imply that the Rydberg constant builds in some kind of finite absolute scale that even knows what potential energy an electron would have at infinite distance from a proton (wouldn't that be infinite again??). It is very difficult for me to understand that something can escape an everlasting attractive force (even if it continues to decrease with distance) with a finite amount of energy.

Your reasoning is wrong. The force of attraction between a nucleus and an electron is simply not strong enough to always pull the electron back. If you consider your isolated atom and give the electron enough energy, the electron will fly away from the nucleus and never come back, no matter how long you wait. The ionization energy can be defined roughly as the least amount of energy that you can give to the electron to achieve this effect.

The integral $$\int_{r_0}^RF\,dr\propto\int_{r_0}^R\frac1{r^2}\,dr=[-1/r]_{r_0}^R=1/r_0-1/R$$ describes (up to a factor) the potential energy difference between an electron at $$r=r_0$$ and at $$r=R.$$ You should expect to supply this much energy to lift an electron from distance $$r=r_0$$ up to $$r=R.$$ In particular, if you "kick" an electron by adding some kinetic energy to it when it starts at $$r=r_0,$$ by the time it reaches $$r=R,$$ it will have lost kinetic energy proportional to $$1/R-1/r.$$ But notice that, even if I want to lift the electron infinitely far away, the energy requirement is finite, since the integral remains finite as $$R\to\infty.$$ The force is never zero, fine, but it approaches zero fast enough as $$r$$ approaches infinity that it can only sap up to a finite amount of kinetic energy from a departing electron. So I will reiterate: an electron given enough kinetic energy will keep on moving away from the nucleus, forever.

• She assumed a completely isolated system, in which case there would be a finite neutralization time. That is, of course, another invalid assumption. Commented Apr 16, 2023 at 2:03
• @FlatterMann You're telling me that if I started with a single hydrogen in infinite empty space, then kicked the electron outwards at $0.99c,$ it would eventually find the nucleus again? I find that hard to believe, and the point of this answer is to show that that isn't the case.
– HTNW
Commented Apr 16, 2023 at 2:10
• I don't. She did in her OP which assumed that the atom was in a completely isolated box. I believe the question was edited since to make more sense. Commented Apr 16, 2023 at 2:16
• It is very difficult for me to understand that something can escape an everlasting attractive force (even if it continues to decrease with distance) with a finite amount of energy. But I will accept that for now, thank you.
– iwab
Commented Apr 16, 2023 at 9:38

Ionization energy doesn't refer to such a closed system. It refers to an open system in which the electron can be removed. It has to be that way because the effective ionization energy (as well as all other energy levels of an atom) will be slightly different in such a closed environment. By how much they differ from the energy levels of the free system will depend on the size and properties of the closed system.

The existence of "open systems" is, by the way, nothing particularly exciting. The entire universe is a giant open system. "Stuff" has been "escaping from us" since the beginning of the big bang. For all practical purposes my lab bench is a very good "open system". I have to buy very expensive insulation materials and vacuum systems to make "isolated systems" and none of it will ever be fully successful, no matter how much money I have to spend.

What's really bad is that the Health and Safety department is constantly in my hair to prevent that nasty things like gamma rays, neutrons or even mercury atoms are escaping. They are slightly less concerned about escaping electrons, but even that is pretty much energy dependent. Beyond a few keV they require x-ray shielding and above a few MeV they want us to put doors with electronic locks and extra thick concrete walls in... because secondary ionization of human tissue is not a nice experiment to conduct.