How did chemists find how many electrons, neutrons, and protons each element had?
I mean I can understand the Mendeleev's table but without really knowing how you make one...
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One could do it in the following way.
The atomic number $Z$ can be determined by examining the x-ray spectrum of an atom. The frequency $f$ of the so-called K-alpha line is related to the atomic number $Z$ by Moseley's law $$ \sqrt f = k_1 \cdot \left(Z - k_2\right) \, . $$
The atom is electrically neutral, so the number of electrons is obviously calculated from the atomic number.
And the number of neutrons is also simply to choose: the only requirement at this stage would be the correct atomic weight for the element in question.