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If an atom loses an electron, why doesn't its proton in the nucleus repel the charge and fall apart?
When an atom loses an electron, the positive charge on the resulting ion comes from the fact that there is now one more proton than electrons. There is no external positive charge for the proton in the nucleus to repel. The positive charge comes from the excess proton in the nucleus. Also, the strong nuclear force in the nucleus keeps the nucleus intact.
For example, a neutral sodium (Na) atom has 11 protons in its nucleus and 11 electrons in its electron cloud. If it loses one electron, it becomes a sodium ion with a charge of 1+, because there are now 10 negatively charged electrons, and 11 positively charged protons. The positive charge comes from the fact that there is now one more proton than electrons. So the positive charge is from the nucleus. There is no external positive charge for the protons in the nucleus to repel. Also, as stated earlier, the strong nuclear force holds the nucleus together.