Why should an electron falling into the nucleus, according to the Maxwell's laws of electrodynamics, destroy the atom?

Question

It is often said in physics and chemistry classes and textbooks that atoms must be unstable when the electron continuously loses energy and finally fall into the nucleus according to classical physics. They often give an impression that the two when combined would annihilate into energy.

I am of the understanding that such a model of atom is unstable (with the electrons revolving around the nucleus in circular paths) and we need a better explanation.

Strictly within the ambit of classical physics, does it predict the annihilation of atom into energy? Protons and electrons are not matter and anti-matter they why should they get annihilated when the latter falls into the former?

Answer 1

They often give an impression that the two when combined would annihilate into energy. - No, that is not what is meant.

Classically because the electrons are charged and accelerating, the electrons would be losing energy by the emission of a continuous spectrum of electromagnetic radiation and thus spiral into the nucleus (cf satellite entering the atmosphere).

If the electrons spiral into the nucleus, what is the end result?
No longer an "atom" but a very small neutral mass.

Answer 2

It is often said in physics and chemistry classes and textbooks that atoms must be unstable when the electron continuously loses energy and finally fall into the nucleus according to classical physics.

The usual model of hydrogen atom in EM theory has some assumptions: the "electron" (lighter negative charge) orbits the "proton" (much heavier positive charge). The electron experiences force from the proton and from itself (self-force), but there are no other external forces; in particular, there isn't any background radiation acting on these charged particles. Also, it is assumed energy in the system flows according to the usual interpretation of the Poynting theorem.

In this model, the negative charge actually gains kinetic energy as it falls down to the nucleus. But during this fall, electric potential energy of the system (positive charge + negative charge) decreases more. So as a whole, energy of the system decreases, and this energy gets away with EM radiation. After some short time, it becomes zero and if the process continues, decreases further to negative values. Due to relation between energy and mass in relativistic theory $E=mc^2$, the system comes to a state where it has zero mass, and then negative mass.

The "atom must be unstable" means this electron fall to the nucleus happens in a very short time, around nanoseconds. We have never detected this with hydrogen atom.

They often give an impression that the two when combined would annihilate into energy. Strictly within the ambit of classical physics, does it predict the annihilation of atom into energy? Protons and electrons are not matter and anti-matter they why should they get annihilated when the latter falls into the former?

They do not annihilate in the sense of matter+antimatter particles disappearing and EM radiation appearing, like with quantum model of positronium. In the above classical model of hydrogen, the particles just get arbitrarily closer to each other, and radiate arbitrarily large amount of EM energy.

Answer 3

It is often said in physics and chemistry classes and textbooks that atoms must be unstable when the electron continuously loses energy and finally fall into the nucleus according to classical physics.

Not "...must be unstable when..." but "...would be unstable if..."

It was plainly obvious at the beginning of the 20th century that electrons do not spiral into their atomic nuclei because all of chemistry depends on the electronic structure of atoms and molecules. They understood that none of the physical substances of which our world is made could exist if the electronic "cloud" surrounding each nucleus was unstable.

I am of the understanding that such a model of atom is unstable (with the electrons revolving around the nucleus in circular paths) and we need a better explanation.

That's the heart of the problem. Maxwell's theories predict that an orbiting electron must radiate its energy away. Therefore, whatever the electron cloud actually is, it cannot be "electrons revolving around the nucleus in circular paths."

Strictly within the ambit of classical physics, does it predict the annihilation of atom into energy?

I have not heard that classical physics makes any prediction regarding the fate of electrons that spiral into nuclei. And why would they bother to predict the consequences of a thing that they knew does not happen?

If electrons spiraled into nuclei (which, they don't) then atoms would cease to be atoms anymore (which, they don't) and the Earth itself would never have existed (which, it does) and so it doesn't really matter what the atoms would have turned into if they turned into anything else at all (which, they don't.)

Answer 4

The protons and electrons do not annihilate. They combine to form neutrons. When a neutron star forms, the enormous pressure from a supernova explosion forces the electrons into the nucleus. The enormous force required to combine protons with electrons indicates this is not a spontaneous natural process. The electrons and protons combine to form neutrons and this is probably what would happen if the electron was to somehow fall into the nucleus. A nucleus composed purely of neutrons is almost certainly unstable, because nuclei composed entirely of neutrons are never observed in nature. Nuclei appear to require protons to hold the nucleus together and (as Learner has pointed out in the comments) neutrons are unstable in the absence of protons.