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Water is composed of hydronium ions and hydroxide ions. Since ions do conduct electricity, why is distilled water such a poor conductor of electricity?

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You are mistaken.

Water is not composed of these ions. There is ongoing autodissociation equilibrium reaction$$\ce{2 H2O <<=> H3O+ + OH- }$$ shifted strongly toward left, with concentration of ions $\pu{1e-7 mol/L}$ at $\pu{25 ^{\circ}C}$ .

Therefore conductivity of water itself is very low.

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