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There is a very specific order called a Periodic Table. All elements are arranged based on the number of protons. Hydrogen is number 1, it has 1 proton. Carbon is number 6, it has 6 protons. These values are called the Atomic Number.

The atomic number of the elements increases accross the Table from left to right.

In addition the table groups the elements into types (Inert Gases, metals etc and shows elements with similar properties in columns of increasing Atomic Number (top to bottom). As a bonus the Table generally contains information on Atomic Weights, completed orbitals, isotopes and similar.

The elements may also be listed alphabetically by name (Helium, Iron, Silver etc.) or symbol (He, Fe, Ag, etc.)

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Q: Is there any particuar order that the elements are shown?
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