Dissolving chalk and the Schmidt number

flowEight years ago a group at Penn State University published a paper I like a lot. What they did is simple: They just dropped a very small (about a hundreth of a millimetre across) piece of chalk (calcium carbonate) into water, and watched how small charged particles behaved near the dissolving piece of chalk. Chalk is not very soluble which is why if you stick a big piece in water only a small fraction will dissolve, but they were using tiny pieces which dissolve in water over, I think, about an hour or so.They found that the charged particles zipped along in trajectories that meant they passed near the dissolving piece of chalk and then headed off. Part of this motion must be due to the electric field around dissolving chalk. Chalk is made of positive calcium and negative carbonate ions which dissolve into the surrounding water. As the ions move out into the water they separate a little bit and this separation of charges creates an electric field. This field then pushes on the charged particles.

But those of the the diffusing calcium and carbonates ions that are near the surface the tiny piece of chalk is resting on, should do something else. They create flow of the water — a simple calculation of the flow pattern is shown above, arrows indicate direction of flow. In water almost all surfaces are charged and this charge interacts with the electric field of the calcium and carbonate ions, to make the water flow.

So the particles are moving both under the influence of the direct electric field produced by the dissolving chalk, and of the flowing water. Which effect is bigger?

I think the answer is the direct electric field. And the reason is due to a fundamental property of all liquids, including water. This is that momentum diffuses about a thousand times faster than ions and molecules do. So the motion of the water molecules that create the flow diffuse away from the piece of chalk a thousand time faster than do the calcium and carbonate ions.

This ratio of diffusion of momentum to diffusion of ions is so important it even has its own name: the Schmidt number, named after a 19th century German engineer. And it means that the effect of something on a surface, eg tiny piece of chalk, on the flow of the liquid, reaches out much further into the liquid, than do the ions produced by the chalk.

In turn this means that the more extended flow pattern due to the dissolving chalk is more dilute (because it is more spread out) and so has a weaker effect on nearby particles. As the reason it is more spread out is due to a fundamental property of all liquids, I think this should hold for anything dissolving any any liquid.

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