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We all know that pure water can't conduct electricity. But during electrolysis, if add a small amount of $\ce{HCl}$ acid inside, water could be decomposed to hydrogen gas and oxygen gas, as it is reduced at the cathode and oxidised at the anode. So why can't pure water just conduct electricity by electrolysis?

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Indeed, water can conduct electricity by electrolysis. But this is hampered by the fact that water has very high resistance. From Electrolysis of water:

Electrolysis of pure water requires excess energy in the form of overpotential to overcome various activation barriers. Without the excess energy the electrolysis of pure water occurs very slowly or not at all. This is in part due to the limited self-ionization of water. Pure water has an electrical conductivity about one millionth that of seawater.

And from Self-ionization of water we can see how small it is:

Water molecules dissociate into equal amounts of $\ce{H3O+}$ and $\ce{OH-}$, so their concentrations are equal to 1.00×10−7 mol∙dm-3 at 25 °C

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Pure water has a very high resistivity, on the order of 18 MOhm/cm. This effectively means that any applied potential is going to be converted into thermal energy.

A side note, oxidation of an $\ce{HCl}$ (or any salt with chloride ions) solution will produce $\ce{Cl_2}$ gas, so electrolysis of these solutions should be performed in a well ventilated area.

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A solution conducts current when positive and negative ions are randomly dissolved in it. Pure water doesn't have enough ions but when you add $\ce{HCl}$ you add a strong electrolyte so, after the addition, solution can conduct electricity very well. However you can perform an electrolysis even without $\ce{HCl}$ but this is a very slow process because even if there are positive and negatives ions (and this could induce you to think that the pure water will conduct current!) the positive ions are near the negative electrode (the cathode) to be reduce on the other hand the negative ions are near the positive electrode (the anode) to be oxidise. So in fact you mustn't imagine a solution of positive and negative ions like in a solution of a true strong electrolyte but two bunch of ions, positive in the anode and negative near the cathode, so the current can't flow very well. This is why pure water can't conduct electricity even during the electrolysis when there are many ions in the solution.

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IN pure water there is no chance for electricity conduction as it has no free ions.It contains only $\ce{H2O}$ molecules. For electricity conductance, there should be ions present. In the case you are talking about some drops of $\ce{H2SO4}$ is added for the conduction of electricity.

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  1. Pure water cannot conduct electricity fundamentally because of lack of rapid ion transport inside bulk solution.
  2. In fact, pure water electrolysis has been achieved, by using nanogap electrochemical cells, where the distance between two electrodes less than Debye-length of water (less than 100nm!). But the fundamental mechanism has been changed significantly.

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  1. Lack of rapid ion transport in traditional electrolyzer.

We take cathode and H3O+ ions as an example. Initially near the cathode surface water molecules can be dissociated into H3O+ and OH- ions. H3O+ ions obtain electrons from cathode leading to hydrogen evolution; while newly-generated OH- ions can only transport very slowly through the bulk solution by slow diffusion or hopping process. These lead to local OH- ions accumulation (so that the solution near cathode turns alkaline) especially at the cathode surface, reducing the reaction rate of hydrogen evolution and thus water splitting. In other words, the reaction becomes very slow or even self-limited, showing a large equivalent resistance between the cathode and the anode. That is why pure water in macrosystem cannot be split efficiently.

  1. Pure water electrolysis in Nanogap cells

In pure water, when the counter-electrode is placed within the Debye-length, double layer regions of the cathode and the anode are overlapping with each other so that high electric field exists in the entire gap. Still at cathode, newly-generated OH- ions can be migrated rapidly from cathode towards anode due to large electric field in the entire gap. When the gap distance is small enough, initially the transport rate can be even higher than the electron-transfer rate. Once OH- ions are generated, they are immediately drawn from cathode to anode, leading to such OH- ions waiting for electron-transfer at the anode, rather than accumulated at the cathode. In this way, the whole reactions would continue even in pure water, but now are limited by electron-transfer.

https://pubs.acs.org/doi/abs/10.1021/acsnano.7b04038

enter image description here

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Pure water does not conduct electricity because there are no suspended particles of metals such as Iron and Copper. Pure water also has lack of zinc. Ions are even not present in pure water. We can make water conductive by adding drops of Sulphuric Acid.

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    $\begingroup$ Welcome to Chemistry.SE! Take the tour to get familiar with this site. Mathematical expressions and equations can be formatted using $\LaTeX$ syntax. For more information in general have a look at the help center. At the moment this reads more like a comment than an actual answer - could you elaborate a little more. With a bit more rep, you will be able to post comments on any question/answer. $\endgroup$ – Martin - マーチン Mar 3 '16 at 14:34
  • $\begingroup$ It is quite uncommon for water to contain suspended solid metallic particles. Metallic elements in a sample of water are almost always present in the form of dissolved ions. $\endgroup$ – Nicolau Saker Neto Mar 3 '16 at 21:54
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Pure Water can't conduct electricity because it has no free ions. But when water is contaminated, it increases the ionic level of it(then its no longer a pure water)-that's why you see waters conducting electric current and you got electrocuted by it.

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