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I read the definition of element:
Element consists of only one kind of atom and cannot be broken down into a simpler type of matter by either physical or chemical means.
Then as per definition, $\ce{H2}$ is also an element as it is made of up one kind of atom but it can be broken down as in case of $\ce{H2 + Cl2 -> 2HCl}$. Here hydrogen get broken down in order to bond with chlorine.
Hence my question is, Is $\ce{H2}$ an element or not? If not, why?
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Instead of your definition, which can be misleading if you are a beginner, I would use the Wikipedia definition instead. You see that an element is better defined using the atomic number, i.e. the number of protons in the nucleus: we define elements as those entities which have the same number of protons in the nucleus. That's why we can distinguish between hydrogen and oxygen, or between carbon and iron. It is implied that you cannot break down an element into simpler terms, otherwise we would be doing nuclear physics, not chemistry, since we would be dividing it into neutrons, protons and electrons.
In the case of $\ce{H2}$, we see that we have two hydrogen atoms bonded together. Therefore, you see that $\ce{H2}$ can actually be broken down into simpler terms, i.e. into the two $\ce{H}$ atoms, because $\ce{H2}$ is a molecule (see the Wikipedia definitions for atom, and Wikipedia definitions for molecule). I hope this helps!
