Can an atom exist without any electron? If yes, then how?
In essence I'd say no, because an atom without any electrons (or the 'incorrect' number of electrons) wouldn't be a an atom but an ion, so technically the answer to your question is yes, an atom can leave behind it's electrons, but we wouldn't call it an atom anymore if it does (so it's really a matter of semantics).
But perhaps a few concrete examples of atoms leaving behind some or all of their electrons:
For example the hydrogen atom, which regularly leaves behind it's (only) electron in acid-base reactions and moves over to another molecule. However we generally don't refer to the result of this (a H+) as an atom but rather as a proton (given that hydrogen only has a single proton in it's core).
Additionally, metals when combined form a grid that consists of the cores of the metal atoms, while the electrons move relatively freely about the grid, you could say that the metal atoms therefore have no electrons that are fully their own....
Lastly the cohesion of atoms tends to fall apart under extreme circumstances, such as the surface of the sun. There we have a new state (along side the states of solid, liquid and gas) called plasma, where molecules and atoms completely fall apart and most atoms are essentially without their electrons (and therefore fully ionized).
Hope that helps.
Matter is built out of neutrons, protons(+), and electrons(-). Matter becomes stable only if it is electrically neutral. So atoms without electrons do exist and must have their own states (charged or uncharged) transferred back and forth in their environments.
An atom stripped out all of their electrons is called an ion and it's charged. No ions is uncharged.
What do you mean by this? Regardless of what you mean here, oscillatory motion of subatomic particles or a boundstate of them is by far only possible when driven by electric field, and that requires the driven object to be charged.
I didn't write much as this is a basic but tricky known question. And I would like to make it brief in a general sense. It is not called an atom anymore once after it is charged.
With regards to what I mentioned, I would have added more to explain reactions like this
[itex]H^+ + OH^- = H_2O = H^+ + OH^-[/itex]
and here is to explain the dihydrogen ion formation in nature. A neutral hydrogen atom is produced after the cation is reacted with hydrogen gas, which means electron-less atoms exist.
So nothing unusual happens to the lonely nucleus?
I still can't grasp what you intended to mean here. Is it because you are accustomed to calling bare ion an atom or you really wanted to mean something fancy?
Actually this isn't quite true for the reaction you've given. A proton never exists "on it's own" in water, or any aqueous acid - the proton is always solvated by one or more unionized water molecules: ##H_3 O^+ (H_2 O)_n##
A good example of a bare nucleus lacking electrons is an alpha particle. These will eventually pick up electrons from their environment to become helium atoms.
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