Perpetual motion machines are a thermodynamic impossibility. However, I have had an idea for a perpetual motion machine for a long time that is highly impractical, but I can’t theoretically disprove it. The question is: why does this proposal violate thermodynamics?

Imagine you placed an electrolyzer on the ground, placed a fuel cell on top of a mountain, and connected the inputs and outputs via tubes. On the ground, you have a tank of water connected to the inlet of the electrolyzer. You provide the electrolyzer with a spark, creating hydrogen and oxygen. While the oxygen is denser than air, the hydrogen is not and thus rises to the fuel cell. Fortunately oxygen is abundant in the atmosphere; with the right oxygen membrane, you collect oxygen from the atmosphere at the top of the mountain. The hydrogen and oxygen then combine in the fuel cell to create water, and the electrical energy is fed back to the electrolyzer to split more water.

Of course, splitting and regenerating water is a useless exercise since each of these steps is inefficient; you can never generate more energy from the fuel cell than what you put into the electrolyzer. However, this setup has one key advantage: the water generated by the fuel cell at the top of the mountain can do work when it falls back to the electrolyzer on the ground. So, if you were to place a turbine at the outlet of the water pipe, you could capture additional energy. The idea is that at a sufficient height, this energy could compensate for the inefficiencies of the electrolyzer/fuel cell combo—and generate some extra on the side. In effect, we are exploiting the phase behavior of the hydrogen and the water.

To be clear, this proposal has all sorts of practical problems—low electrolysis efficiencies, terminal velocity of water, etc. Perhaps its biggest practical flaw is that the height of the “mountain” is truly massive. To get a sense for the absurdity of this idea in practice: water electrolysis requires 237 kJ/mol of electrical energy at a minimum, and fuel cells produce 237 kJ/mol of electrical energy at a maximum. Let’s very generously assume the electrolyzer, fuel cell, and turbine all have 95% energy efficiency (here let’s define turbine efficiency as the ratio of electrical energy generated from gravitational potential energy). This means for 1 mol of water:

• Electrolysis energy required = $237$ kJ / $0.95$ = $249$ kJ
• Fuel cell energy generated = $0.95$ * $237$ kJ = $225$ kJ
• Energy required by turbine to break even = (Electrolysis energy required) - (fuel cell energy generated) = $249$ kJ - $225$ kJ = $24$ kJ
• Effective force provided by water = (turbine efficiency) * (mass of water) * (acceleration due to gravity) = $0.95$ * $1$ mol * $18$ g/mol * $9.81$ m/s$^2$ = $0.177$ kg m / s$^2$ = $0.177$ N
• Height requirement of water to break even = (energy required by turbine to break even) / (effective force provided by water) = $24$ kJ / ($0.177$ N ) = $\textbf{143}$ km

Even with extremely generous assumptions, a truly ridiculous height is required to “break even”. This height is around 172 Burj Khalifas, 16 Mt. Everests, and well into outer space as defined by the Kármán line. So yes, a tad impractical. Long story short, chemistry $\gg$ gravity.

All that said, however, none of this answers the key question: why does this proposal violate thermodynamics? Why isn’t there a height where the gravitational potential energy of the water could, in theory, exceed the losses from the electrochemical system?

If you’ve made it this far and have the solution, email me and I’ll post the best answers in a new post.

EDIT (2022-03-13): Leonard Bezinge replied with the following message:

Your machine wouldn’t be perpentual for the following reason:

You are generating oxygen at altitude zero while consuming oxygen at 173 km altitude. The system is losing energy over time, and decreasing its total entropy. The energy loss corresponds to the work it would take to bring oxygen all the way up to your fuel cell (as you mentioned, oxygen is denser than air). In an isolated environment, your system will stop when the oxygen around your fuel cell is depleted.

EDIT (2023-01-05): Robert Murphy proposed a similar idea, but with the electrolyzer deep underwater and the fuel cell at sea level.