What is highest water pressure at which electrolysis can be performed to derive hydrogen and oxygen? Does the dielectric constant of water, which which lowers as pressure increases, have an effect on this?
[Physics] highest water pressure at which electrolysis can be performed
electrochemistrypressurewater
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I haven't done the calculations, but I doubt that this scheme would generate net energy. As was pointed out, electrolysis uses a lot of energy. However, after the H2 and O2 rises up the water column, you could get some of the energy back with a fuel cell that would convert the hydrogen and oxygen back to water and supply additional electric power, but inefficiencies at each stage would have to be made up from the energy gained by the rising gases in the column of water.
To show that there has to be a net overall loss of energy, consider the following modification to your problem. Imagine a long vertical pipe filled with water. Now instead of electrolysis, let's say you instead use an air pump to inflate a balloon near the bottom of the water filled pipe. This takes work which goes into lifting the column of water up such that the surface of the water rises enough to allow for the volume of the ballon. Once the ballon has risen through the water column the water level will go back to it's former position. That fall in water level is the source of the energy that the rising ballon could generate. So there is no free lunch or perpetual motion machine here - inefficiencies at each stage will insure that there is a net loss of energy.
By the way, I do not know this for a fact, but by this thought experiment, I would predict that electrolysis of water under high pressure would take more energy than under lower pressure. I say this because the electrolysis is effectively inflating a balloon against the pressure of the water - which will thus take more energy at high pressure.
Re your second question: have a look at http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/electrol.html. The 237kJ/mol is the Gibbs free energy required for electrolysis, while the 286kJ/mol normally quoted for combustion is the enthalpy. To compare the two you need to take into account heat exchange with the environment and even the work done by the escaping gas.
Re the first question: the energy density you quote is not a useful quantity in this context. To do thermodynamic calculations you normally only consider changes between initial and final states. The article I linked to above does the calculation in this way.
The efficiciency of electrolysis is always below 100% for various reasons. This can simply be resistive/heating losses, but a problem specific to electrolysis is overpotetials. These cause inefficiencies because if the overpotential is $V$ volts then $V$ electronvolts of energy are lost for each electron.
Best Answer
A high-pressure environment tends to favor liquids over gases, insofar as liquids have higher density. So the higher pressure it is, the higher voltage you need to apply to make the reaction H2O --> H2+O2 occur.
But, with a high enough voltage, it should always be possible. ANY electrolytic reaction will eventually occur if you crank up the voltage high enough. Electrolysis only becomes impossible (in practice) (impossible at any voltage), if competing reactions are favored. I'm not sure what the competing reactions would be for electrolyzing pure water in a high-pressure environment. For example, presumably, H2O --> H2 + O3 (ozone gas) would be favored over H2O --> H2 + O2 at high enough pressure.
It's impossible to be quantitative here, because the catalyst and reaction mechanism also make a huge difference to favor one reaction over another. It's not enough to JUST calculate free energies.