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I have to say I have never really understood the half reaction method. What bothers me is that it seems to be too artificial. Too many little rules to remember! Why don't we use the mass conservation law to justify the process and use sistematically a set of linear equations to do the balances?

Thanks in advance!

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The concept of the half reaction has nothing to do with balancing redox reactions; how one arrives at a balanced reaction is not really important. The interesting thing about redox reactions is that electrons do not necessarily have to be transferred directly between the oxidized and reduced species. In many types of electrochemical cells, the anode and cathode are isolated from one another by a salt bridge and the species in each half-cell never approach one another. In techniques such as potentiometry, often one cares only about one half of the reaction, e.g. in a combination pH electrode, the response from the half reaction that contains hydronium ions produces the analytical signal and the the other half is a reference electrode reaction whose job is to simply produce a constant potential. In things like voltammetry, one is often only concerned about the half reaction occurring at the working electrode and the half reaction that balances it at the counter electrode isn't analytically significant (and may in fact be more than one different reaction).

Because the two half-reactions need not necessarily occur at the same place and one half reaction can generally be substituted without affecting the other, it makes sense to think of them separately. This lets us do things like think about different lithium batteries. There are many types and they all use the $\ce{Li -> Li+ + e-}$ oxidation half-reaction, but it's coupled with different reduction half-reactions based on cobalt, iron, etc.

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