Chemistry Question: Decomposition of Water by Electrolysis

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http://forums.anandtech.com – I understand that if we run an electrical current through water, it will decompose into its component parts (oxygen and hydrogen) according to the following chemical equation: 2 H2O -> 2 H2 + O2 I also know that this is an endothermic reaction (since the combustion of H2 is definitely exothermic). Because it is an endothermic reaction, theoretically, the temperature of the water that has a current running through it should go down. However, I was thinking that even though the chemical reaction noted above would cause the temperature of the solution to drop, wouldn't the current it (Hardware)