I was recently pointed towards the "10:23 Campaign", with their

YouTube video about "homeopathic vodka". They explain that a "30C" homeopathic solution, with 30 dilutions by a factor of 100 each, has almost no chance of even a single molecule of solute remaining. The overall dilution factor is 1:100

30, or 1:10

60.

This stirred a vague memory of having read that if one took a glass of water, poured it into the ocean and mixed thoroughly, then took back a glass of water, one would end up with some dozens of the original water molecules in the glass. I decided to try the calculation.

Per the

United States Geological Survey, the total amount of water on Earth is about 1.386 * 10

9 km

3. Most (96.5%) of this is in the oceans; for the sake of simplicity I'll ignore that factor. In more useful units, this is 1.386 * 10

21 L.

Assume a 250mL glass, i.e. about 8 oz., imperial. Mixing this into the volume of the ocean gives a dilution factor of 1:(1.386 * 10

21 L)/0.25 L or 1:5.544 * 10

21.

That 250 mL of water has a mass of about 250 g. Water is about 18 g/mol, and the conversion factor of moles to molecules is Avogadro's number, 6.02 * 10

23. So the number of molecules in the glass of water is (250 g) * (1 mol / 18 g) * (6.02 * 10

23 molecules/mol) = 8.36 * 10

24 molecules.

Applying the dilution factor, we end up with (8.36 * 10

24 molecules) / 5.544 * 10

21 = 1510 molecules. If you took that glass of water, mixed it thoroughly with all the water on Earth, and refilled that glass, you'd have about 1500 of those original molecules back. (If your starting volume was only a tablespoon, or 14.8 mL, and you did the mix-and-refill thing, you'd still have a statistical average of 5.3 molecules back in your spoon.)

In contrast, if you applied the "30C" homeopathic dilution to your glass, you'd have (statistically) 8.36 * 10

-36 molecules. That is, only 1 chance in 1.2 * 10

35 that you'd have even

*one* molecule left.

(And if you want to talk about the water having a "memory" of the original substance, consider all the other places that water has been, and should "remember" just as clearly.)