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Each time I'm too lazy too cook I think it'd be cool to be able to just plug myselt into an outlet. Yet I know it is not possible - I need amino acids and a lot of other stuff that electricity can't replace.

Can some simple organism live on electricity alone or is it just fundamentally impossible?

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No; the problem is, as you pointed out, that no organism will manage to multiply, grow or even sustain itself without absorbing matter to create new cells and fill metabolic losses.

Even photoautotrophic organisms which get energy from light (which is in fact an E-M wave, so pretty close to electricity) collect matter from the environment -- plants for instance seem to grow out of nothing, but in fact gather significant part of their mass from atmospheric CO_2.

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While true in practice there is nothing fundamental that prevents an organism from maintaining itself using a source of pure energy. Take the planet as a whole as an “organism”, it sustains itself with only an input of solar energy; all material is recycled. Of course it could not grow under such a situation –  Richard Tingle Mar 26 at 11:01
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The cells would have to produce glucose directly from electricity. If we figured that out we could also solve our energy crisis.

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I don't see how producing glucose from electricity solves the energy crisis; I would have thought that we want to produce electricity from glucose (efficiently) to do that. –  P Sellaz May 9 '12 at 13:12
    
I was mainly making a reference to the storage issue with clean energy generation. –  Gabriel Fair May 9 '12 at 20:13
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