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Why is there a max radius for an orbit of an electron around a nucleus ? I had a course in electromagnetism but I do not get this.

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  • $\begingroup$ Exact orbits literally can't be known even if they can be really called orbits. $\endgroup$ – Mithoron Dec 11 '16 at 0:22
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Perhaps you can clarify your question a bit, but if you are referring to the size of the radial distribution of electron density, this increases with increasing principal quantum number, up to the ionisation limit, at which point the spacing between energy levels trends towards 0 and the interaction energy trends towards zero. After this, the electron is for all intents and purposes unbound. This is illustrated for the hydrogen atom in the Lyman series (image borrowed from Wikipedia):

Lyman series, from wikipedia

The energy of the largest transition $n=\infty \rightarrow n=1$ is 13.6 eV, or half a Hartree (this is the energy of a photon with a wavelength of 912 Ångström). This is the binding energy of the hydrogen atom. Even the Lyman gamma line - representing a $n=4 \rightarrow n=1$ transition - is quite close to the ionisation limit. As n increases, from a strictly electrostatic point of view the orbital becomes far less stable, and this is reflected in the general negative trend of ionisation energies across the periodic table, however the complete picture is a bit more complex than that because electrons can interact with electrons as well as the nucleus.

The key point is that as n increases, the energy required lose an electron entirely becomes smaller and smaller, and this fixes a practical limit on where we can say the electron is actually bound. This has a caveat though - cold gasses can form unusually long-lived Rydberg states with ridiculous principal quantum numbers with correspondingly enormous orbital radii (up to micrometers!*) and semiclassical electron orbits. The reasons for the stability of these states is outside of my knowledge.

**To paraphrase, Alisa Bokulich points out in 'Re-examining the Quantum-classical Relation' that Rydberg atoms are of comparable size to grains of (fine) sand, which is pretty neat*

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