Burning hydrogen with oxygen basically just reverses the electrolysis you did to produce the hydrogen. One splits water into hydrogen and oxygen, and the other puts them back together again. Even if all of the steps were 100% efficient (i.e. no energy losses), by burning hydrogen at most you’d recover the energy you had to put in to do the electrolysis, and so you’d be back where you started from.
However, *fusing* hydrogen is a different beast altogether. You see, if you fuse hydrogen atoms into a helium atom, the resulting helium atom is actually a bit lower in mass than the sum of the masses of the hydrogens that went in. The difference in mass is released as energy, according to Einstein’s famous formula E=mc^(2), where c is the speed of light in m/s, i.e. (approximately) 300,000,000, and thus c^(2)=90,000,000,000,000,000. In other words, if you convert just 1 gram of mass to energy, you’ll get 90 **trillion** Joules of energy, i.e. enough to power New York City for 2 whole **years**. In hydrogen fusion, about 0.71% of the original hydrogen mass is converted into energy, so 1 kg of hydrogen in results in about **half a trillion** Joules of energy out. For comparison, when you burn 1 kg of hydrogen, this releases only about 100 million Joules of energy, i.e. about 5,000 times less energy.
So yeah, electrolysis requires energy, and if you just burned the hydrogen again, you’d *at best* get that same energy back. But fusion allows you to unlock a far greater source of energy that is totally worth the little bit of energy required for the electrolysis.
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