The atomic number Z defines what?

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The atomic number tells you how many protons are in the nucleus, and that proton count uniquely identifies the element. Each element has a specific number of protons, so Z fixes which element you have. The total number of nucleons (protons plus neutrons) is the mass number, not Z, so that describes something else. The average energy of electrons isn’t specified by Z, since electron energies come from the electron cloud and energy levels tied to the nuclear charge. The total number of electrons equals Z only in neutral atoms, but ions can have fewer or more electrons, so that isn’t what Z defines.