The atomic weight reported in the periodic table of elements for Carbon is the average weight of all the known isotopes of Carbon occurring naturally. There are 3: Carbon-12, Carbon-13 and Carbon-14. The numbers represent the mass numbers, or the sum of neutrons and protons in the nucleus. Each of the individual masses are multiplied with their individual percentage abundances. Then, when you add them up, it will yield the average atomic weight equal to 12.011 amu or grams/mole C.
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The periodic table is useful today for finding out all the elements that exists on Earth. The elements there can be used to write down chemical formulas, calculate things like molar mass/atomic mass of each element, atomic number for each, number of valence electrons each element has, the oxidation number for each, etc. Lastly, it can be used to predict the properties of elements yet to be discovered.
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