Reverse osmosis perpetual motion machine

This is a clarification of my earlier post.Assume that once set up there is a flow from right to left at the top.If there is a flow there will be a dilution of the salt water because water and salt ions will diffuse via the water bridge at the top of the loop.The membrane at the bottom may be able to stop the flow of salt ions but there is no membrane at the top.
Have you ever seen salt diffuse up a waterfall?

This is a very long thread, and I haven't read it all. But cesium frog has got it right.

Has anyone noticed that the drawing in post #1 is wrong? The salt water should be at a higher level than the fresh. I imagine that any supporting argument invloving osmotic pressure may have put the pressure on the wrong side of the equation, or droped a negative sign.
You are correct, if this would involve osmosis, but it involves reverse osmosis driven by the density of the salt water.

The compressibility of the (salt) water is irrelevant. The salt ions can still move freely in the water.

If we start with fresh water and then add salt, the additional pressure delta P as a function of height will be exactly that of the barometric height formula. Therefore the extra pressure at the bottom is due to the salt only, the partial pressure of the water at the bottom has not been changed at all. So, no water will flow through the membrane.
The change in partial perssure due to salt ions, and the change in barometric pressure due to the density change due to the salt ions are not exactly the same, which is the basis of the paradox.

The reason why there exists such a thing as osmotic pressure is simply because you are letting molecules of one type pass but not of the other type. This is the relevant physics, any formulas that are derived (theoretically or from experiments) can come with their small print with assumptions that may not be alway valid.

If you have a membrane that doesn't let the salt pass then you can add as much salt as you like one one side, it won't cause the water to move through. The more salt you add, the higher the osmotic pressure will become precisely because the membrane equalizes the partial pressure of the water and doesn't care about the salt.

Have you ever seen salt diffuse up a waterfall?
I can't see any reason why salt will not diffuse upwards, the thermal velocity of the salt ions will be high compared to the velocity of the falling water and each salt ion will follow a random walk path.The only difference I see is that diffusion against the flow will be slowed down.

madcowswe, Dadface. Well pshaw. I guess I'm it. Oh well, there always has to be some guy slower witted than the rest.

Dale
Mentor
I believe Einstein2nd has given a combination of density and osmotic pressure that is reasonably correct for some particular concentration of salt water. And if he hasn't, then his numbers are in any case consistent with some imaginary solution...all that needs to change is the atomic weight of the ions.
My concern is that he may be giving osmotic pressure for one solution and density difference for another solution. Such a combination of density difference and osmotic pressure may itself violate thermodynamics and thus be impossible.

My concern is that he may be giving osmotic pressure for one solution and density difference for another solution. Such a combination of density difference and osmotic pressure may itself violate thermodynamics and thus be impossible.
Hard to see how that could be. You can easily adjust the density while holding the osmotic pressure constant simply by varying the atomic weight of the solute particles. And even if you object that there is no actual positive ion with molecular weight 45 or whatever, I can't see a thermodynamic reason why there couldn't theoretically be such an ion.

To approch the problem obliquely, what would happen at both the top interface and the membrane interface if the solvent were water and the solute a gas such as carbon dioxide? One side is a column of water, and the other is a column of CO2.

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Bystander
Homework Helper
Gold Member
Water is not nearly as compressible as air, it is almost completely incompressible. Yet it's true that neither water nor salt-water is truly incompressible, but it doesn't really play a role in this problem since water and salt-water both increase their densities with pressure at the same relative rate.
"WORNG!" Salt water is around 10ppm/atm more compressible than fresh water.
Furthermore, the 2.5% difference in density is never countered, even if the compressiveness of the liquids were to prove to be very slightly different:
"The low compressibility of water means that even in the deep oceans at 4000 m depth, where pressures are 4×10^7 Pa, there is only a 1.8% decrease in volume.", from Wikipedia.
(snip)
Apply the Poynting correction to water activities in salt water and in fresh water at the pressure for whatever depth you're examining (use the different compressibilities for the two to calculate the partial molal volumes for water, fresh and saline, and the differences), and you will see a difference.

Mapes
Homework Helper
Gold Member
"WORNG!" Salt water is around 10ppm/atm more compressible than fresh water.
Bystander, could you give a reference for this please?

It is not clear yet that the RO system will stop when it releases some amount of potential energy and settles to equilibrium. (We're all sure it does, we're just not sure how or where.)
For a dilute solution, letting $r$ be the ratio between the column height and the fundamental length scale ${i_H k_B T}({\kappa g})^{-1}$ (where $\kappa$ is the difference in mass between a solute molecule and the solvent it displaces), reverse osmosis occurs iff the concentration at the semipermeable membrane (as a fraction of the mean concentration) is less than $r$. Thus the solvent will start cycling (provided $r$>1 and that the solution is initially mixed and not yet settled), however, at equilibrium (when the Archimedean force balances the osmotic potential gradient) the concentration increases exponentially with depth and the reverse osmosis cycle halts because the concentration at the membrane is $$\frac{r}{1-e^{-r}}$$.

Not sure yet how to generalise this to non-dilute solutions.

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