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How do we know that atoms are not divisible?

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Atoms actually are divisible.  They are made up of subatomic particles including protons, neutrons, and electrons.  Even these particles are made up of smaller subatomic particles including quarks and bosons.  A proton is made up of two up quarks and one down quark.  So atoms are not the smallest form of matter. 

On the other hand, you may be thinking about atoms and elements.  Atoms are the smallest forms of basic, identifiable matter.  They are the smallest particles of an element.  The number of protons in the nucleus determine the identity of the element, the number of neutrons in the nucleus determine the particular isotope, and the number of electrons determine the net charge (or neutrality) of the atom.

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