Electrochemistry is a branch of chemistry. It is the science of the reactions that can take place at the interface of an electronic conductor (the electrode, which can be a metal or a semiconductor including graphite) and an ionic conductor (the electrolyte).
If a chemical reaction is caused by an external voltage, or if a voltage is caused by a chemical reaction, as in a Battery (electricity), it is an electrochemical reaction. In general, electrochemistry deals with situations where an oxidation and a reduction reaction is separated in space.
