# Have I drunk the same water twice?August 28, 2009 11:09 AM   Subscribe

What’s the probability that I have drunk the same molecule of water more than once in my life?

All water on earth goes through the same evaporate and rain cycle over and over again, right? How likely do you think it is that I have drunk the same molecule twice in my life?

On a similar note: How likely is it that I have drunk a molecule that was once ingested by Shakespeare or Paul the Apostle?
posted by crapples to Science & Nature (17 answers total) 22 users marked this as a favorite

Similar question courtesy of Enrico Fermi.
posted by Pollomacho at 11:20 AM on August 28, 2009 [2 favorites]

wow. cool question. I have no idea. But I was told once by a professor (no actual evidence here) that its pretty likely, given the rapid exchange of molecules between our bodies and 'not our bodies' that its pretty likely that could have, say, a molecule that was once in Shakespeare's body.
posted by Lutoslawski at 11:21 AM on August 28, 2009

Have you never tasted your own sweat? Q1: Infintesimally close to 100% if not actually 100%
posted by biffa at 11:24 AM on August 28, 2009 [5 favorites]

Even if you never knowingly tasted it, you have touched your food with your fingers and have therefore most likely deposited some of your sweat on said food.
posted by soelo at 11:37 AM on August 28, 2009

Don't forget the water in your own saliva.
posted by Jorus at 11:39 AM on August 28, 2009 [1 favorite]

There are about 1.4 × 1021 liters of water on Earth. If you lived to the ripe old age of 100 and drank your 8 glasses a day, you could consume 69083 liters in your lifetime.

There are 55.5 moles of water in a liter and 6.02 x 1023 molecules in a mole. So there are about 4.68 x 1046 molecules of water on Earth, and you will consume 2.3 x 1030 in your hypothetical lifetime.

If we assume a completely even probability distribution and that each molecule is picked independently of any other, we can imagine you getting 2.3 x 1030 chances to pick molecules out of the Earth's water supply. What is the chance that any two will be the same? It's basically the birthday paradox except with gigantic numbers.

Solving this exactly is impractical, but we can use an approximation to get close enough. The probability is roughly equal to 1 - e-n2/(2m), where n is the number of water molecules you drink and m is the number of water molecules on Earth. The answer to that is as close to 1 as makes no difference. So, yes, you probably consume at least one molecule twice, especially given that a lot of molecules are 'off-limits' to you in that they never leave the ocean or ice during your lifetime.
posted by jedicus at 11:42 AM on August 28, 2009 [26 favorites]

Actually, it's probably pretty close to zero unless you refocus the question on atoms, not molecules. Atoms, as far as we can tell, are pretty permanent, nuclear reactions aside. But molecules have a tendency to dissociate and reform in a most promiscuous manner. Animals and plants regularly split water molecules as part of our metabolic processes, and what with dissolution and various large-scale, ongoing natural chemical reactions, the odds of a particular water molecule surviving with the same component atoms for more than a few years--outside ice packs--has to be pretty small.

Again, the idea that you could somehow ingest an atom that was previously somewhere interesting are preobably pretty high, but molecules tend to be a bit too ephemeral for that to be much of a possibility.
posted by valkyryn at 11:43 AM on August 28, 2009 [4 favorites]

Q2: Possible in the case of Shakespeare, but only if 1) you live in a village in England previously occupied by him and 2) the system of wells and septic systems in this village was concurrent with Shakespeare and survives to this day. Excreted water will eventually filter to the aquifer and make itself available for drinking by several generations, conceivably. But it would be impossible to verify whether this molecule was consumed by Shakespeare via ingestion, by respiration, or by the breakdown of chemical compounds by the body.
posted by Gordion Knott at 11:45 AM on August 28, 2009

Gordion Knott: Not so fast. Shakespeare would have exhaled lots of water that he had drank, as well as lost a fair amount through the evaporation of sweat. There's no need to limit it to water that remained liquid, shall we say. A great deal of water that passed through Shakespeare would have entered the larger hydrologic cycle.

valkyryn: Your point is well taken. I suspect the probability is still pretty high even if one phrases it as "what is the chance that the same two hydrogen atoms and one oxygen atom that once entered my body as a molecule of liquid water left my body then returned to my body as a molecule of liquid water, given a sufficient amount of time in the meanwhile so as to make pedantic nitpicking like tasting my own sweat not count?"
posted by jedicus at 11:54 AM on August 28, 2009

To make jedicus's point in a slightly simpler way...

Say you once drank a liter of water and returned it to be mixed in the environment. It now constitutes a fraction of 1 in 1.4 x 1021 in all the Earth's water (accepting jedicus's value).

Say you now drink just a mole of water (its weight in grams would be its water's atomic weight, which I think should be 18 grams? that'd be 18 cubic centimeters of water). In that mole, you'd have 6.02 x 1023 chances to recoup one of your lost molecules.

That's like entering 6 x 1023 lotteries, each of which has a chance of winning of 1 in 1021--very good odds of winning at least once! So in just a couple of sips, the chances are good that you'd reingest water from a liter you drank before.
posted by Schmucko at 12:19 PM on August 28, 2009

In some places it would be more likely than others. In my area, the source of water and dumping ground for treated wastewater is the same place in Lake Michigan within a few miles of each other. Obviously locations on the Mississippi would have a lesser chance because treated water is immediately sent downstream where it is unlikely to be seen again for a very long time.
posted by JJ86 at 12:26 PM on August 28, 2009

This might be interesting to you: An Estimate of the Number of Shakespeare's Atoms
in a Living Human Being

posted by pised at 12:35 PM on August 28, 2009

IANAS (I am not a statistician) so I'll ask, wouldn't the number be better then 1 molecule in all the earth's water since likely molecules and people tend to stay in the same general region?
posted by bitdamaged at 12:37 PM on August 28, 2009

Re: Shakespeare--he thought of it, too:
Alexander died, Alexander was buried,
Alexander returneth into dust; the dust is earth; of
earth we make loam; and why of that loam, whereto he
was converted, might they not stop a beer-barrel?
Imperious Caesar, dead and turn'd to clay,
Might stop a hole to keep the wind away:
O, that that earth, which kept the world in awe,
Should patch a wall to expel the winter flaw!
Hamlet, Act V, Scene 1. Right after the bit with the jester's skull.
posted by Halloween Jack at 12:57 PM on August 28, 2009 [7 favorites]

Does backwash in your drink count? Stranded in a desert and survived by drinking urine?
posted by brent at 1:09 PM on August 28, 2009

Do the molecules that you rinse out your mouth with count?
Do the molecules that you inhale and exhale on a humid day count?
How do you count the molecules that get synthesised into carbohydrates or proteins, and the original hydrogens and oxygens get separated?
Similarly, what if "your" molecules get ionised by strong acids or alkalis, and the original hydrogens and oxygens get separated?

I think the probability that you have drunk the same water twice is 100% if you count your own fingerprints, sweat, and blood.
My guess is the probability is still pretty high if you don't. JJ86 makes a good point, that some places recycle their drinking water locally faster than others.
You're going to get into a statistical nightmare if you consider that water molecules can break apart and recombine. What are the chances that the same hydrogens and oxygen will separate and recombine with each other? Do you count the ions of "your" molecules?

Other people have wrestled with this question, too. There is a traditional Yorkshire dialect song called "On Ilkla Moor Baht 'at" (On Ilkley Moor Without a Hat) in which the speaker chides the protagonist for going courting on the cold, windy moor without a hat, claiming, "You are bound to catch your death of cold / Then we will have to bury you / Then the worms will come and eat you up / Then the ducks will come and eat up the worms / Then we will go and eat up the ducks / Then we will have eaten you / That's where we get our own back".
posted by Multicellular Exothermic at 8:54 PM on August 28, 2009

When Caesar said "Et tu, Brute," in shock
his dying breath contained myriad things:
1024 atoms; a flock
of tiny birds on 1 million billion billion wings.
And in the time between his death and now
they have flown from Rome and into your mouth!
And I hear you wondering aloud, “How?”
They have circulated North, West, East, South—
Casting these oxygen and carbon seeds
across the world over land and ocean.
You likely inhale one or two of these
as your chest rises and falls; the motion
of every single quiet breath
brings the flavor of Caesar’s Death
posted by exlotuseater at 8:55 PM on August 28, 2009 [1 favorite]

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