That's true. I think it's the first part of the process (the water → hydrogen) where you're missing something important. I'm not a teacher, and maybe my explanations are pretty bad. If so, sorry about that. I will try a couple new explanations below...
You said "it would be like using 10 J [...] and getting 80 in return". Yes, exactly! But you can't do that. I looked up real numbers so we can stop using 100 as an example. It takes 237 kJ per mole of water molecules, to separate water into hydrogen and oxygen. [1][2].
(I will call this energy, but technically it is the "Gibbs free energy" which is a value that takes into account both energy and entropy.)
An important point: this 237 kJ is not about the process used; it's the difference between the energy in the molecules in the start and end states. The water molecule, as you said, has a certain energy. The separated H
2 and O
2 also have energy,
higher than the water. To go from point A (low) to point B (high), you must put in energy. That's the law of physics that is in the way. You have to go "uphill" by 237 kJ/mol to pull the hydrogen out of the water.
Maybe a useful intuition: separating the hydrogen from water is literally "un-burning" the hydrogen. Think about all the energy released when you burn/explode hydrogen. You have to bottle that energy back up, to return to the original state with hydrogen as an un-burnt gas.
I think your intuition is off because you are imagining that hydrogen is right there in the water and can be gently separated with a little gentle nudging. That's not so. You are un-burning it, and it's a big deal and a lot of energy has to be put back into the system.
After separating the water, the 237 kJ/mol is the same energy you will get back by burning the H2 gas, because that's the exact reverse reaction.
The basic idea of a catalyst, like the cobalt phosphide you mention, is depicted in the diagram here:
en.wikipedia.org/wiki/Catalysis#Reaction_energeticsNotice the ΔG (our 237 kJ) does not change because of the catalyst. The catalyst lowers the "bump" in between the start and end state and makes it easier (faster) for the reaction to proceed.
I'm a little worried about talking off-topic so much, but this climate change thread was created for an off-topic discussion, so I think we're okay getting in the weeds some about alternative fuels.
[1]
www.wolframalpha.com/input/?i=gibbs+free+energy+of+water[2] A "mole" is a standard big number used in chemistry to talk about quantities of atoms or molecules. A mole is about 6x10
23.
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R.k.'s point is just simple accounting, nothing more, nothing less: a perpetual motion machine would defy the conservation of energy.