A glass of water - The calculation

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The calculation

The central point here is that in the example with sand the reason for the chance being close to 100% is that there are much more grains of sand in a cup than there are cups of sand in the bucket.

Similarly the chance is close to 100% in the example with the glass of water because there are much more molecules of water in a glass than there are glasses of water in the total amount of water on the planet.

The atomic mass of hydrogen is 1.00784 u and that of oxygen is 15.9994 u. Therefore, the molecular mass of water with formula H2O is (2  1.00784 u) + 15.9994 u = 18.01508 u. Therefore, one molecule of water weighs 18.01508 u, and one mole of water weighs 18.01508 grams.

A mole is defined as Avogadro's number of particles of any kind of substance (atoms, molecules, or ions). Avogadro's number is approximately 6.022  1023.

Thus 18.01508 grams of water contain 6.022  1023 water molecules.

A glass of water, or 200 grams of water, therefore contains 6.6855  1024 water molecules.

The total mass of the hydrosphere is about 1.4  1024 grams, and thus the total hydrosphere contains about 7  1021 glasses of water (with 200 grams of water in each).

Therefore, if the water molecules in a glass of water were spread evenly throughout the entire hyrdrosphere, you would find around 1000 of those molecules in any glass of water.

So the probability that you in any glass of water will find at least 1 molecule of water once drunk by Cleopatra is practically 100%.

Note that this argument presumes that the number of water molecules that disintegrate over a longer time is insignificant - a presumption which I am not sure one can make.

The argument also presumes that the water in the hydrosphere is spreading extensively over time.

If someone can give a rough calculation of how big part of the water in the hydrosphere disintegrate over a certain time, or an estimation of the spreading of the water over time, then please give me some feedback on this.