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Im a french student in geochemistry.

My question might be silly, but I became really too confused to answer it myself. Does gravity affect the diffusion of ions in water ?

Lets imagine a vertical volume of water. There is no temperature gradient. If some NaCl salt is placed (without disturbing the water) at the bottom, it will be dissolved quickely. But then ? : -Na+ and Cl- will start to diffuse as to get their concentrations homogeneous everywhere. There will be an upward flux . - Does gravity want them to remain at the bottom because both of them are heavier than water molecule ? Then it would be a downward flux.. (?)

If these two process occur together, how could I evaluate which is dominant, or what would be the equilibrium ?

I thought I could use something like a Boltzmann distribution ? Like Concentration = $\mathrm{exp}(mgz/kT)$ ? Im not sure though of how to consider an ion (Cl or Na) sinking between water molecules because of gravity

Thanks for help

Sean
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Tom Giun
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1 Answers1

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The drift velocity of a particle of mass $m$ under gravity in a fluid of viscosity $\xi$ is $mg/\xi$, from which it follows that the relevant diffusion equation is

$$ \frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial z^2}+\frac{mg}{\xi}\frac{\partial C}{\partial z} $$

The steady-state ($\partial C/\partial t=0$) solution is of the form

$$ C(z)=\alpha \frac{D\xi}{mg}e^{-mgz/\xi D}+\beta $$

where $\alpha,\beta$ can be solved by imposing appropriate boundary conditions.

So the answer to your question is yes, the ions do sink. However, as $mg/D\xi$ becomes very small, the $C(z)$ distribution becomes almost uniform. And if you plug in the appropriate numbers for sodium chloride, you'll find that the gravitational gradient is negligible. Indeed, I do a lot of simulation studies of diffusion and we completely disregard effects of gravity.

lemon
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  • Thank you for this quick answer !

    What about millions years old water, and a relatively high height ? Is there some caracteristic time or length that I could calculate ?

    Also, I lied, Im not really interested in Na and Cl, but in the two isotopes of Cl (35 and 37)

     – Tom Giun Apr 02 '15 at 17:38
  • The solution I give is for the steady-state which is precisely the distribution you would expect in million-year-old water. Finding the time it takes to reach steady-state, however, is difficult (and often impossible) analytically, but it is straightforward using a computer by numerically integrating the first equation that I give. – lemon Apr 02 '15 at 22:04
  • I see.

    But from the first equation, it turns out that if I start from a solution absolutely homogeneous in concentration with z (let say, the solution was first mixed up), there would never be a "sedimentation of ion" because there is no concentration gradient with depth ?

     – Tom Giun Apr 03 '15 at 09:57
  • "the equilibrium concentration goes up exponentially with depth, by a factor of e for each 10 km or so" – Keith McClary Jul 23 '19 at 02:05
  • There is a complication, which is that temperature and pressure can affect the solubility of NaCl in water, and solubilities in general. I'm not sure how much that would matter if you're only interested in isotopic ratios, though, since the solubilities of 35Cl- and 37Cl- may be similar enough that the effect of the difference is much smaller than that of gravitational settling, but it also might not be. Also, in the Earth's ocean, I think convection currents prevent this kind of equilibrium from being reached, and it would likely be similar in many other "oceans". – Mr. Nichan Oct 11 '20 at 06:57
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    These equations make it look like gravity could matter for very deep "oceans", such as the interiors of gas giants with dissolved cores. That's why I'm glad I found this answer. Of course, things like convection, solubility changes, and differences in interaction between different pairs of molecules might matter, although I read that solutions of giant planet core materials in hot metallic hydrogen may be very much like ideal solutions. – Mr. Nichan Oct 11 '20 at 08:54