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A structured approach to this is to use at least five mixtures of the solvents and saturate with caffeine, stirring for at least four hours. The five mixtures are not picked arbitrarily but can be set at idealised levels. If you use 100% ethanol and water as the extremes, then you want 50:50 and two inbetween on either side. These can them be combined ...

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It's not smoke, but Copper(II) oxide (cupric oxide, CuO), which is a black solid. It formed on the surface of your copper electrode and is falling off. Even after falling off, copper oxide could continue to form on the copper surface.

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Limescale consists mainly of calcium carbonate but can also contain other compounds depending on water composition. These include iron hydroxides which while a white solid gains a greenish tinge in the presence of oxygen. That may explain a green limescale in your kettle.

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If you don't want to have chlorine gas as byproduct in your electrolysis add sodium or potassium hydroxide - you would get O2 and H2. Water electrolysis is even possible with distilled or deionized water, because due to the CO2 in the air, there are always some HCO3 ions in solution that will aid electrolysis (given the voltage is high enough).

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If you pass current through salt water (sodium chloride solution), you will produce chlorine preferentially to oxygen. In practice, this chlorine will attack your anode if it's made of copper, or steel, or most any other metal. Since the linked demonstration uses pencils (graphite), that doesn't happen. Instead, the anode produces free chlorine, which ...

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Your ambitious project is not that trivial. Soaps are of several types. a) If you are talking about salts of fatty acids, the classical soaps, then one can precipitate them out with calcium salts. Many other cations form insoluble salts with soaps. Now if you are trying to purify laundry water, dishwasher water, shampoos, liquid soaps, nobody can remove ...

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You are right: 1 cubic meter of water contains $\pu{1E-4}$ moles of $\ce{H+}$ ions. But one mole is $\pu{6E23}$ atoms or ions. So, $\pu{1E-4 mol}$ contains $\pu{1E-4}\times\pu{6E23} = \pu{6E19 ions}.$ Here we are.

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You may pour them down the sink. NaOH will be soon neutralized and destroyed by the CO2 from the atmosphere, or by the bicarbonate ions present in all drinkable calcareous waters.

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