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We all know that atomic number of an atom is "unique" just like fingerprint. And no two elements can have same atomic number. But how can we be So sure that no two elements can have same atomic number?

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    $\begingroup$ Well, if they do, they would be the same element by definition. $\endgroup$ – Ivan Neretin Sep 24 at 11:39
  • $\begingroup$ But what if we find two elements whose atomic number are same but their chemical properties are completely different from each other. So are we still gonna call them same elements. $\endgroup$ – Shekhar Sep 24 at 11:59
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    $\begingroup$ Well, chemical properties depend on the electron orbitals, which in turn depend on the nuclear charge, which is the same as atomic number. $\endgroup$ – Ivan Neretin Sep 24 at 12:03
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    $\begingroup$ Let's deal with it when we face it. When we face a double head unicorn, we will then deal how to treat it. $\endgroup$ – Poutnik Sep 24 at 12:11
  • $\begingroup$ It has nothing to do with any other properties - all nuclei containing only equal amount of protons and equal or different number of neutrons are nuclei of the same element. That's it. $\endgroup$ – Mithoron Sep 24 at 13:22
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Recall that atomic number is the number of protons in the nucleus of the element. Also recall that species having the same number of protons and different number of neutrons are termed as isotopes. Isotopes are all considered the same element. $\ce{^{12}C}$ and $\ce{^{13}C}$ are both very much carbon.

You may argue that its the definition that we chose that allows us to consider isotopes as the same element. You may wonder why we cannot consider isotopes are two different elements. However, we need to look into the practicality of how we organize the various entities that make up our world. Isotopes behave largely similar chemically, with only slight changes based on the increased nuclear mass. They only have different nuclear properties, but all chemical properties (which depend only on electronic interactions with protons) are virtually unchanged in most circumstances. It would be practically unwise to consider different isotopes of an element as two separate elements, when they have way too much in common.

tl;dr we chose atomic number as a fingerprint for elements because its the clearest way we can assign and record their properties and behaviors.

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The atomic number of an element equals to the number of electrons an atom of that element have in their neutral state. As you know the chemical properties of an element always depends on the number of electrons an atom of the element have. So two atoms which have same atomic number (which means they have same number of electrons) will always show same chemical properties which means they are two atoms in the same element. On the other hand if two atoms have different number of electrons they will have different chemical properties and will be atoms of two different elements.

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