Shower thought

2024 April 03

When you are in the shower, watching the million billion molecules of water spray from the showerhead every nanosecond, take a moment to reflect on how, despite their innumerable multitude, each molecule has traveled a completely unique and remarkable journey for billions of years over and through the Earth to you.

Except, it’s not true. None of the water molecules in your showerhead will last long enough to reach you.

At room temperature, pure liquid water is ionized at a ratio of about 2.8 parts per billion: for every billion water molecules, there are 2.8 H+ ions and 2.8 OH- ions. Lone protons are never found in liquid water, but rather are associated with water molecules to make hydronium H3O+, and frequently bigger associations (H5O2+ “Zundel ion”, H9O4+, etc). The bonds in hydronium are very weak, not much stronger than hydrogen bonds, and have similarly short lifespan: hydronium typically lasts around a picosecond before one of the hydrogens jumps ship to sign on with a new water molecule. This is the Grotthuss process which causes protons to diffuse very rapidly in water:

Thus, for every H+ ion (by which we mean H3O+, etc, collectively) in a mass of water, there are about one trillion hydrogen swaps between nearby water molecules each second. This comes to around 400 swaps per water molecule: every water molecule in liquid water changes one of its hydrogen atoms on average 400 times per second.1This figure is super approximate, and could easily be more than an order of magnitude off.

If it takes, say, 140 ms for water in the showerhead to reach you, that is 100 times the half-life of a water molecule, and only 1 in 2^{100} water molecules (about 10000 liters) last that long.

(There is some nuance; liquid water forms large-scale networks or clusters of molecules connected by hydrogen bonds, which are constantly shifting and rearranging, but the timescale at which molecules turn and move is slightly longer than the timescale of the Grotthuss process. Thus when H+ moves from one water molecule to another, the most likely next place for it to go is back where it just came from, and the above calculation will modestly over-estimate the diffusive rate of H+.)

In the whole lifetime of the visible universe, no water molecule has ever persisted intact for a minute in liquid water at room temperature through chance alone.2It is possible that there is some stabilizing process that I don’t know about – water is complicated – but that would not be through chance.

Perhaps, one might think, although water molecules are rapidly intermixed, some water molecules might reform from the same constituents as at some point previously; if a billion H+ are shuffled between a billion water molecules, on average one of them will end up where it started. This would be reasonable if water molecules had one hydrogen, but with two hydrogens this would require both to return to the same starting molecule at the same time. The probability of this falls rapidly as the number of molecules involved increases; and as more time passes, there are more opportunities for the body of water to be mixed with other bodies, and the original constituents of a water molecule to be separated forever.

Let us relax our expectations a bit and ask whether any arrangement of an oxygen atom and two hydrogen atoms has ever recurred. Out of N water molecules there are N^3 possible such arrangements. By the birthday problem, you need to sample about \sqrt{N^3} = N^{3/2} random arrangements to have even odds of having sampled the same arrangement twice. 400 times a second, N new such arrangements are made.

Recall that deep in the mantle of the Earth, crystals are frequently found to have small pockets of water which may have been sealed for billions of years. If such an inclusion had a milligram of water, which is 10^{19} molecules, it would need 10^{9.5} / 400 seconds, or 3 months, for some water molecule configuration to have occurred more than once.3Or maybe not: “Recent research, however, has shown that water can sometimes be gained or lost from minerals at magmatic temperatures (>1000 C) in a matter of minutes. If this is true, then the fidelity of mantle xenoliths is questioned.” This remarkable coincidence will last for perhaps a millisecond (or, if a little unlucky, only a picosecond) before dissolving away.

(The preview image is an AI depiction of “shower thought about water”.)

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