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The ion electron method is a.k.a the half reaction method.

How are we just allowed to add electrons and protons and water molecules? Why does it work anyway?

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    $\begingroup$ What are you asking really? Why stoichiometry and oxidation numbers work? $\endgroup$ – Mithoron Jun 4 '16 at 11:21
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Why can't we just add protons and water molecules? According to the law of conservation, we can neither create nor destroy matter. Hence, to balance the atoms of each respective element on either side, we add protons and water molecules. Also, I think the method is based on the fact the a certain amount of reduction would entail that an equal amount of oxidation would take place. If one atom of a P4 molecule underwent a change in oxidation state of +5 (say forming PCl5), then the total change would be +20 and hence you would need an equivalent amount of reduction (i.e. -20) of other species participating in the reaction.

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