Dissolving in a hurry

We have all added salt or sugar to water and seen it dissolve. Both salt and sugar dissolve rapidly, but exactly how fast do they dissolve? This is one of those seemingly innocent questions, that is a lot harder to answer than you might expect, or hope. For something highly soluble like salt we expect the sodium and chloride ions at the surface of the dissolving salt crystal, to very rapidly move into solution in the surrounding water. But  if the water is stationary, not stirred, then near the surface of the salt crystal, we quickly reach the point where the water is saturated with salt, and no more can dissolve, until the sodium and chloride ions move away.

So if there is no stirring or convection, or any other flow of the water, salt dissolution is excruciatingly slow. You need to stir, or find some other way to drive flow, for example if you are cooking in a pan then the heat from the bottom will drive flow via convection.

Then, near to the dissolving surface, diffusion moves the ions away from the dissolving surface, while farther away it is flow moving the ions. This is because diffusion is fast over short distances, but slow over large distances. The crossover distance is when ion motion is 50% diffusion and 50% flow, which occurs at a distance* of about l = D/u m – for a salt diffusion constant D and water flow speed u. Salt diffuses in water with a diffusion constant D ~ 10^-9 m²/s, so if you stir at a speed u ~ 1 cm/s, then the crossover from diffusion to stirring is at distance ~ 100 nm from the surface of the dissolving salt crystal. Effectively what happens is that salt ions leave the salt crystal, diffuse (i.e., move in a roughly random walk way) over about 100 nms, then are carried further away from the dissolving crystal by the flow of the water.

The flux of salt ions then leaving the dissolving crystal is then j = D (c_SAT/l), for c_SAT ~ 10²⁷/m³ the concentration of salt ions in saturated salt solution. This gives a flux of about 10 ²⁵ salt ions leaving the dissolving salt crystal, per unit area of surface of this salt crystal. As NaCl occupies about 0.1 nm³, this then causes the salt crystal surface to recede via dissolution at a speed of about 1 mm/s**. This speed varies linearly with the stirring speed, so if the water near the crystal is only moving at 1 mm/s, then the recession speed of the interface is only 0.1 mm/s.

So dissolving salt crystals is an example of a phenomenon where you can affect the rates of processes occurring over nm, at the surface of a crystal, by stirring with a spoon 10 cm long and creating swirls of motion also centimetres or bigger. This is very similar to evaporation, where blowing over the surface of hot coffee creates air currents centimetres across that speed up the process of sub-nanometre water molecules leaving the surface of the hot water.

* This distance l is when the Peclet number Pe = u*l / D = 1, as Pe is by definition the ratio of the flow transport rate to the diffusion transport rate. 

** Ions take about 10 μs to move 100 nm, so as long as salt ions take less than 10 μs to move from the salt crystal surface into the immediately surrounding water, then it will this diffusion-followed-by-flow process that controls how fast salt dissolves.