# How long will it takes two miscible liquids to fully mix (homogenize) through diffusion?

This question occurred to me while I was pouring milk into my coffee and was too lazy to stir.

Assume you have some volume $$V_a$$ of liquid A directly on top of another volume $$V_b$$ of liquid B. The two liquids are fully miscible. With no stirring, how long until the solution becomes fully homogenous solely by diffusion?

• There's no 'formula' per se, without knowing some more details eg the mass of the liquids, the viscosity of the liquids etc. It's easier for gases, where you can apply Graham's Diffusion Law. For liquids, you should use a stopwatch, conduct the experiment and share the results with us :) – Aniruddha Deb Aug 30 '20 at 4:08
• @AniruddhaDeb Haha, I expected it to be a pretty tricky problem. If it helps, you can assume whatever else you like: the masses of the liquids, the temperature, force of gravity (or zero-g), air pressure, diffusion coefficients, etc. Even an answer to a much simpler problem (spherical cow) would be interesting. – gardenhead Aug 30 '20 at 4:21
• Diffusion is a slow process, see Ficks's Laws of diffusion and look up diffusion coefficients, typically $D=10^{-9}\mathrm{\,m^2\,/s}$ and distance diffused $\ell=\sqrt{6Dt}$ in time $t$. In most cased, as it your example, convection dominates over diffusion – porphyrin Aug 30 '20 at 17:12