how high up a mountain do you need to be to make coffee

On mountains, molecules and coffee

A tea plantation in the mountains of the Cameron Highlands, Malaysia. But how high would you need to climb in order to boil water at the perfect temperature to prepare your brew?
A tea plantation in the mountains of the Cameron Highlands, Malaysia. But how high would you need to climb in order to boil water at the perfect temperature to prepare your brew?

Walking in the hills or, if you are lucky, the mountains, we can easily be reminded that atmospheric pressure decreases with height. We just have to look at the way that the plastic water bottles we may be carrying have been crushed, or open a yoghurt pot slightly too close to our face. We may remember that the boiling point of water decreases with decreasing atmospheric pressure and so that a kettle boils more quickly at the top of a mountain than at the bottom. But how high would we need to climb to make a perfect cup of coffee with just-boiled water? And what has this to do with the reality, or not, of molecules?

Although the effect appears obvious to us, it is not trivial to calculate exactly how the atmospheric pressure varies with height. To see why, we could think about what pressure is. The pressure exerted by a gas on an object is proportional to the number of gas molecules colliding with and recoiling from the object concerned. These collisions create a force on the object and pressure is just force ÷ area. So why would this change with height?

Small waves seen from Lindisfarne
Think about a layer of air with air pressure above and below it but further acted on by gravity pulling it down. What happens?

Think about a layer of air. Above it, the molecules in the gas are exerting a pressure, pushing down on the air. Below it, there are molecules pushing upwards and keeping it up. But there is one more force that we need to consider: gravity. In physics, we like to think of things in equilibrium, perfectly balanced. So when we think about our layer of air, the forces acting down on the layer of air (the pressure from above and the gravity of the earth) have to be perfectly compensated by the force acting up, i.e. the pressure from below. If this were not the case, the layer of air would sink. Perhaps it is starting to become clearer, why the density of the atmosphere decreases with height. The only thing that remains is to work out exactly how it does it.

And while we could do the calculation here, it has (fortunately) already been done for us by a remarkable physicist called Jean Perrin back in 1910. He was remarkable not just because of the detail of his experiments but also because of the connections he made.

“It appeared to me at first intuitively [that]…. Just as the air is more dense at sea-level than on a mountain top, so the granules of an emulsion, whatever may be their initial distribution, will attain a permanent state where the concentration will go on diminishing as a function of the height from the lower layers and the law of rarefaction will be the same as for the air.”

Jean Perrin, Brownian Movement and Molecular Reality, 1910
coffee at Watch House
In search of the perfect coffee. How far would you travel?

Perrin realised that to calculate the balance of forces acting on our imagined layer of air, one has to assume molecules exist, just as we have done above but something that was not obvious at the turn of the 20th century. But he also realised that this calculation would be the same for any fluid containing a suspension of particles whether that was the atmosphere or a drop of water colour paint. Assuming that the molecules exist allows us, and allowed Perrin, to make quantitative predictions for the variation of pressure with height or, in Perrin’s case, the variation of the number of granules in an emulsion with depth. Perrin considered a paint pigment suspended in water under the microscope, but his theory is also valid for the (non-soluble) matter in coffee. The fact that these quantitative predictions matched so extraordinarily well with the experimental observations of thousands of water droplets containing suspended paint pigment (the poor PhD students of Jean Perrin!) went a long way to proving the existence of molecules. Hence Perrin’s book “Molecular Reality” and the ceasefire in a philosophical disagreement about whether physics should seek to understand what was happening or merely describe phenomena such as pressure (but that’s another story).

Which takes us back to how to brew coffee properly. Calculating the variation of pressure with height is the first part of the problem. The second is calculating what that means for the boiling point of water, which actually is done by extrapolating from experimental data. But it does mean that we can calculate, for a small range of temperatures near 100C, the altitude at which you would need to boil a kettle for the boiling temperature to be identical with the optimum brewing temperature for your drink. Listed below are a few recommended mountains on which you can prepare your drink of choice. I will leave it to someone else to calculate the energy saving (and hence the saving in CO2 equivalent emissions) of boiling your kettle on top of a mountain rather than in your kitchen. We’ll assume that there’s electricity on top of Mont Blanc.
 

Drink – Recommended brew temperature – Equivalent Altitude – Suggested mountain

Coffee – 93.3 C* – 2000 m – Kebnekaise (Sweden),

Coffee – 96 C** – 1000 m – Any of the Scottish Munroes

Oolong tea – 87.8 – 93.3 C*** – upwards of 2000m – Mont Blanc (France) could be good

Pu’er tea – 93 – 100C| – why leave your living room?

*Coffee Detective

**The Kitchn/Blackbear coffee

***The Spruceeats

|The tea leaf journal