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If I start with a small amount of a substance, and I dilute that substance with billions of gallons of water, what is the scientific explanation for there being no (actually zero) remaining molecules of that original substance?

This question is related to the practice of homeopathic dilutions. I was reading some articles which stated that homeopathic dilutions don't work because at most homeopathic levels of dilution, there are very few or no molecules or the original substance.

What I'm confused about is how there can no longer be any molecules of the original substance (or even fewer molecules) of an original substance when diluted.

If there are X number of molecules in a solution, and Y liters of water are poured into the solution for dilution, don't the X number of molecules remain? How can a molecule just "disappear"? How can even one molecule just "disappear"? It makes more sense if these molecules are undergoing a chemical reaction, but if two kind of molecules (the substance, and water) are not undergoing any chemical reaction, then how can one molecule of either kind simply vanish?

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You correctly point out that the number of molecules in a solution is finite and constant, however the volumetric concentration (that is, how many molecules per litre) changes upon dilution. If, for instance, you take one liter of a 1 mol/L solution of ethanol in water ($\approx{6.02\times{10}^{23}}$ ethanol molecules per liter) and add 9 liters of distilled water, you now have a 1 mole of ethanol in 10 liters, giving a concentration of 0.1 mol/L. If you were to draw out 1 liter of this and count the number of ethanol molecules in that liter, you would find approximately $\approx{6.02\times{10}^{22}}$ - a tenfold reduction. The rest of the ethanol is retained in the other 9 liters of solution.

Equivalently, if you were to measure out 100 mL of 1 mol/L ethanol solution and then make it up to 1 liter by adding distilled water, you would also get a 0.1 mol/L solution. This is the kind of thing taken to absurd extremes in homoeopathy, where tiny quantities of solution are repeatedly diluted, with some shaking. Obviously, they don't literally add billions of liters of water to their solutions, but rather pipette out a tiny amount of solution and discard the remainder, adding water to make up the desired volume. This process is repeated many times.

The finite number of molecules present in a solution leads to the concept of the dilution limit, which is where you have carried out sufficient dilutions that statistically speaking, your chances of finding even a single molecule of the dissolved ingredient become vanishingly small and that by extrapolation, you would have to drink approximately a zillion liters of the preparation to get a single molecule of it.

Where did all those other molecules go? They didn't simply disappear - they were poured down the drain during the dilution process, which curiously is the same place money goes when spent on homoeopathy.

Bonus: homoeopathic plutonium! (courtesy of Theodore Gray's periodic table)

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  • $\begingroup$ Any treatment has a placebo effect, which means it will cure about 1/3 of patients anyway. Homoeopathy is based on principles that are out of place here. $\endgroup$ – f p Mar 4 '13 at 12:43
  • $\begingroup$ Homeopathic plutonium? backs off $\endgroup$ – ManishEarth Mar 4 '13 at 14:30
  • $\begingroup$ Yeah I realized after I posted this question, that my mistake was in thinking a person simply adds more water to the molecules. This is false; homeopathic dilution works by taking a small amount of the original solution, in which case, I agree that there will be no molecules left after a certain number of dilutions. Thanks! $\endgroup$ – Jason Mar 5 '13 at 8:38
  • $\begingroup$ @fp - "placebo effect [...] will cure about 1/3 of patients anyway" - This is incorrect. If it were true, placebos would be the most important drug in medicine. $\endgroup$ – Richard Terrett Mar 7 '13 at 7:01
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    $\begingroup$ @Manishearth - As someone quipped, 'it doubles in strength every 88 years'. $\endgroup$ – Richard Terrett Mar 7 '13 at 7:04
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They don't vanish. It's just that if there are X molecules and Z bottles with Z > X, part of the bottles will not have even one molecule.

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A substance can be diluted to "no substance" because:

No substance present means we are sure there are no molecules of it in some finite volume.
Being sure means we assume it to be true to some finite level of probability.

We start with a finite number of molecules. And we can infinitely dilute the the solution.

We have only a finite amount of water available, because even in an infinite universe, we can reach only a finite amount of mass/energy because it expands.

But we can simply clean and reuse our water.

It's not even slow to do that: It is exponential, and limited only by how much water we can move: Imagine repeatedly diluting a drop in a Olympic size swimming pool. Filling the pool can be done in parallel to selecting a drop and mixing it in, and takes longer.
And there is industrial level experience available, ask your Civil Engineering Hydraulics expert. For off the shelf components, see hydro power stations.

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