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A Russian chemist, Dmitri Mendeleyev, constructed the periodic table by which he calculated the atomic weights of the different elements?
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He sure did , and the way he could measure the weights without modern teqniuqes was by knowing that the weights of elements involved in chemical reactions always have to be in certain proportions. e.g hydrogen and oxygen combine to form water, H2O. If you don't have any leftover hydrogen or oxygen, then the weight of oxygen you start with has to be eight times the weight of the hydrogen.
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