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
Best Answer
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.