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In an electrochemical cell, there are two reactions are occurring. One oxidation and one reduction reaction. In a reduction, electrons are gained by the system and in oxidation electrons are lost. By arranging these reactions in an electrochemical cell, with an external circuit, we can measure the amount of electron flow that occurs. This flow of electrons leads to the flow of current. The flowing current can be used for power. Electrolytes are there to be a source of ions. Either forming ions during oxidation or being consumed during reduction. The electrochemical cell has a salt bridge which helps carry ions from one cell to another to counter the flow of electrons. Not all reactions will be an oxidation-reduction reaction so we won't always see the flow of current for chemical reactions.
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