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Do organic solvents such as pure ethanol and pure methanol conduct electricity? why is this so?

I have done some research and have not been able to find anything conclusive. I would think it would not conduct because I cannot see any medium for the passage of charge.

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Ethanol and methanol dissociate into ions, just as water does. For water:

$\ce{2H2O <=> H3O+ + OH-}$

For an alcohol $\ce{ROH}$,

$\ce{2ROH <=> ROH2+ +~ ^{-}OR}$

For water the self-ionization constant is $K_w \approx10^{-14}$. For ethanol or methanol, I was unable to find data on the self-ionization constant, which is essentially the same as the pKa of the alcohol in a solution of itself. The pKa of ethanol in water is slightly higher than the pKa of water in water. Also, the lower dielectric constant in alcohols will further disfavor self-ionization relative to water. So for methanol and ethanol the self-dissociation into ions will be weaker than in water.

These ions will still be present, though, and will conduct electricity even in pure ethanol and methanol solutions...but the conductivities will be very low. Very pure water has a resistivity of 18.2 $\text{M}\Omega\cdot \text{cm}$. If they self-ionize to a lesser extent than water does, the resistivities of methanol and ethanol will be even higher than water's.$%edit$

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