A live potato light? We have seen a potato clock. Could a LED run off a potato while the potato is in the ground and stay alive indefinitely?
 A: The actual energy source of a potato clock is not the potato, but the oxidation reaction of zinc in an electrochemical cell quite similar to the original voltaic pile invented by Alessandro Volta.
The battery of a typical potato clock consists of a zinc electrode (e.g. a galvanized nail) and a copper electrode (e.g. a copper wire), which are inserted into a potato. The potato separates the electrodes and provides an electrolyte solution with suitable electrolytic conductivity that also contains the counterions, which are required to maintain a neutral charge. Thus, the role of the potato may be compared to that of a salt bridge in typical electrochemical cells used in classroom demonstrations.
At the zinc electrode, zinc is oxidized:
$$\mathrm{Zn\rightleftharpoons Zn^{2+}+2e^-}$$
The standard electrode potential for this half reaction is $E^\circ=-0.7618\ \mathrm V$.
At the copper electrode, hydrogen ions are reduced to hydrogen:
$$\mathrm{2H^++2e^-\rightleftharpoons H_2}$$
By definition, the standard electrode potential for this half reaction is $E^\circ=0.0000\ \mathrm V$.
Since metallic zinc is dissolved at the surface of the zinc electrode during the operation of the potato clock, the lifetime of the battery is mainly limited by the available amount of metallic zinc in the galvanized nail that is used as zinc electrode. (The usability could also be limited by other factors, e.g. the accumulation of hydrogen bubbles at the copper electrode, the increase in concentration of zinc ions that counteract the oxidation of zinc, or the change in pH.)
How long the potato plant can survive this process probably depends on the toxicity of the released zinc ions in the potato tuber.
