What is the possibility: You’ve peed a molecule of Titanic’s iceberg i

AIUI, H2O disassociates fairly seldom, but when it does, it does so fast. And re-associates even faster, but not necessarily with the same individual atoms. As an indication of how fast that happens, it happens on average to every molecule of water every ten hours, but (from the definition of pH)

a quantity of 107 moles of pure (pH 7) water, or 180 metric tonnes (18×107 g), contains close to 18 g of dissociated hydrogen ions.

at every given moment. But this at an extremely fast turnover rate, those 18 g of H+ (and the corresponding 306 g of OH-) change all the time. It must be fast, or it couldn’t happen to every molecule every ten hours on average.
Now imagine that this is happening inside your body too, all the time.

I’m confused by this use of the word “pure”. Isn’t the author conceding that this sample has almost 2 parts impurity per billion? Maybe it’s not possible to get it any purer, but that doesn’t make it pure, does it?

I think so, hence my labeling it not a safe assumption. But I don’t even remember how to begin estimating time to some degree of homogeneity, assuming a quiescent (more poor assumptions), 3.6E8 km2 x 3.7 km ocean and a ~E-9 m2/s diffusion coefficient.

And I have a sneaking suspicion that exchange of surface water with the atmosphere (which I was ignoring) might be faster than self-exchange with the depths.

What is Avogadro’s reasoning here? I am not challenging this, I just don’t know how you reach this conclusion. How small a volume can you use as the reference volume and this still be true? An ice floe? An ice cube?

It seems by reductio ad absurdum that every time I pee, I am peeing out some of the pee of everyone who has ever lived, every animal who has ever lived, water from every river that has ever flowed on the face of the earth, the water that Jesus changed into wine, etc., etc. Which seems a stretch, but I know other things that are scientific facts that defy common sense.

Avogadro’s number says that in 18 g of water (1 Mol) there are 6.022 x 1023 molecules. That is a lot, and I can only guess that this amount and more sublimates from a decent sized iceberg every second, so in the meantime it is really well mixed in the atmosphere and has rained down all over the world (never mind the rest, that has melted, and a diffusion coefficient that seems far too low to me, and a Gulf current that still works).
But the discussion has shifted, I thought, to water molecules being stable or recombining all the time. That might be more relevant.

I see that you perhaps wanted a detailed calculation, so I’ll try. Avogadro’ number is big, but how much water is there on Earth and what would be the relation between both numbers?
The Earth has 361,132,000 km2 covered with water, the mean depth of the oceans is about 4 km, so multiplying both we get a volume of 1.45 x 109 km3 (rounded up). To convert km3 to gram you multiply by 1012, that is on Earth we have
1.45 x 1021 g of water.
Now if you divide Avogadro’s number 6.022 x 1023 by the amount of water on Earth, you get
415 molecules of the original Mol of water per gram.
But the iceberg did not have one Mol of water, it had tons and tons, a considerable amount of which did not melt, but sublimized: that is relevant for the mixing after 110 years.

That is not reductio ad absurdum, it is simple division. And if water molecules are stable, it is a fact. If not, then you apply this reasoning not to the molecules, but to the O and H atoms and you get the same result. Enough mixing is the only precondition to assume, and with enough time, it is given.

Assuming I did not mess up with all those zeroes.

The thing is, there a lot less water than water than you may think.

Bewdy, another opportunity to post one of the absolute coolest graphics on the internet from USGS.

Yes, I messed up with the zeroes, where I wrote

To convert km3 to gram you multiply by 1012, that is on Earth we have
1.45 x 1021 g of water.
Now if you divide Avogadro’s number 6.022 x 1023 by the amount of water on Earth, you get
415 molecules of the original Mol of water per gram.

I should have written multiply by 1015, so I was off by a factor of 1,000. Therefore you get 0.415 molecules of the original Mol of water per gram, not 415. Still, as the iceberg had a mass of several thousand tons, you have a lot (millions!) of the original molecules per gram of well mixed water anywhere in the world.

Now, I have been looking around to see if the dissotiated water molecules recombine with the original atoms (plausible, considering how fast they do so) or if they mix with other molecules (also plausible, as there are much more of them than the original molecules). I have found nothing. That got me thinking about how one would find out. That is how I would do it if I had a good lab:
Take some double-heavy water, D2O, where D stands for Deuterium. Then take the same amount of normal water, H2O, and mix both liquids. Wait a day or two or as long as you deem necessary and analyze the water. If you only find D2O and H2O, but no HDO (simple heavy water), the molecules stay stable. If you find HDO, the atoms of the water molecules have mixed.

Yes, but Socrates’ spit into the ocean would not have been in those bubbles, unless and until it evaporated and rained down someplace.

Water used in the manufacturing of cement does not re-enter the water cycle. Cement is made of hydrates, where the water becomes part of the molecular structure.