Ok, let us recapitulate our many-months correspondence and summarize it in one single text. And this text is given below.

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1) Let us connect a standard DC source to a standard conductor thus forming a circuit.

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2) We can write down the following two related equalities

V = I x R (1) <=> V x I x t = I x I x R x t (2),

where

V = voltage of the DC source

I = direct current, which flows through the conductor

R = Ohmic resistance of the conductor

t = time period, within which direct current flows through the conductor

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3) Equality (1) is the mathematical expression of the Ohm's law.

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4) Equality (2) is the mathematical expression of the Joule's law of heating.

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5) Equality (2) can be derived from equality (1) by multiplying both sides of (1) by (I x t).

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6) And vice versa, equality (1) can be derived from equality (2) by dividing both sides of (2) by (I x t).

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7) Equalities (1) and (2) are absolutely valid for any standard solid, liquid or gaseous conductor.

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BUT standard liquid and gaseous (and even "vacuum"(!)) conductors have some special features, which are as follows.

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8A) Liquid conductors. The minimum DC voltage, which is necessary for a standard DC water-splitting electrolysis to occur, is

equal to 1.23 V.

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8B) Gaseous conductors. The minimum DC voltage, which is necessary for a spark (or arc) to occur, is equal to 2 kV,

where 2 kV = 2,000 V.

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8C) "Vacuum" conductors. The minimum DC voltage, which is necessary for a direct current to flow through vacuum, is equal to 400 kV, where 400 kV = 400,000 V. This is the case of the so called cold-cathode emission.

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Note. As if some kind of approximate and (for the present) vague tendency is on its way to be shaped: the lesser the mass dencity and/or hardness of a conductor, the bigger the minimum DC voltage, which is necessary for a direct current to flow through this same conductor.

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9) A certain portion of hydrogen is generated while standard DC water-splitting electrolysis takes place. And if this portion of hydrogen is burned/exploded, then a certain portion of heat H is generated. Therefore we can write down the equality

H = Z x I x t x HHV (3),

where

H = heat, which is generated if the released hydrogen is burned/exploded

Z = electrochemical equivalent of hydrogen

I = direct current, which flows through the electrolyte while standard DC water-splitting electrolysis takes place

t = time period, within which standard DC water-splitting electrolysis takes place

HHV = higher heating value of hydrogen

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10) Let us add (3) to the right side of (2) thus forming the inequality

V x I x t < (I x I x R x t) + (Z x I x t x HHV) (4).

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11) As a further development of our basic concept let us consider the standard process of recharging of a standard car's battery. In this case in addition to the releasing of hydrogen we store electric energy E. And further, (1) if we disconnect the charger from the already fully charged battery and (2) if we connect the already fully charged battery to a standard copper wire load for example (thus forming a circuit), and (3) if we discharge the battery, then the stored electric energy E transforms entirely into a second additional portion of Joule's heat K. Therefore we can add K to the right side of (4) thus forming the inequality

V x I x t < (I x I x R x t) + (Z x I x t x HHV) + (K) (5).

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12) In one word, inequalities (4) and (5) unambiguously show that any standard DC water-splitting electrolysis process can be considered as a heater, whose efficiency is bigger than 1.

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Everything seems to be clear now, doesn't it?

Looking forward to your answers, comments, recommendations, questions.