Above is an image taken from a BBC webpage What is diffusion? – part of their Bitesize website, and aimed at 11 to 14 year olds*. So if you are from the BBC and you don’t like me using it, then just let me know and I will take it down, but the image on that page is fundamentally misleading, as I will explain. So arguably it should not appear on the BBC’s website either.
The image is used to illustrate coffee molecules (shown in brown) diffusing amongst the water molecules (shown in blue) in a cup of coffee. Point being that diffusion acts to even out concentration variations, so that if you add coffee (or milk, etc) to just one part of a mug of hot water, then diffusion will ensure that eventually the concentration of coffee (milk, etc) will be uniform.
This is true, but it is not what happens in your kitchen when you make coffee**. The key word in the previous paragraph is eventually. Even small molecules in water have diffusion constants no more than about D = 10-9 m2/s. The time taken to diffuse a distance x is given approximately by: time t = x²/D. So if the coffee mug is say 4 cm across, it takes about 10-3/10-9=106 s which is over a week.
No one can wait a week for their coffee to be ready. Apart from anything else, it will be cold. Molecules are not the only thing that diffuses, so does heat, in fact heat diffuses with diffusion constants much larger than D = 10-9 m2/s, both in water and in solids like the ceramic of a mug.
So, if you want a hot drink then relying on diffusion to mix coffee molecules will never work, unless you employ a very efficient thermos flask — to eliminate diffusion of heat out of the drink — and are very patient. Without a thermal flask, as heat diffuses so much faster than molecules do in water, then your drink will always be cold before it is mixed.
This is why, you, me and I assume BBC employees in their canteen, often stir their drinks. And even if you don’t stir then there will still be flow, for example if the drink cools a little at the top, then this sets up a temperature gradient — from hot at the bottom to the cold, and this will set up convection flows which will mix your drink for you.
Diffusion in water is slow over distances greater than around a millimetre, diffusing even 1 centimetre takes of order a day. So in practice, we almost never really see pure diffusion occurring in water, as it too slow for us, over all distances we are typically interested in, and can easily see.
I am not sure why we are telling impressionable 11 to 14 year olds that diffusion can redistribute coffee across a mug, when that is not how it happens in our kitchens, coffee shops, BBC canteens, etc. Possibly it is because diffusion is being taught, at least on this BBC webpage, without any numbers or equations. This is despite the only maths needed being squaring and division. I would say that the fact that BBC are apparently struggling to write an accurate webpage on diffusion is quite eloquent evidence of how important it is to be quantitative in science. We can’t even understand how to make a hot drink without at least some numbers.
* After some tweeting, and an earlier blog post, a different error on the same page was fixed. Thanks to them for doing that.
** Note to tea drinkers, the same arguments apply to making tea, infusing a mug via diffusion alone will never work.