Running the numbers on homeopathic dilution

Why do we keep saying that homeopathy is not medicine? Why do we keep saying that there is no medicine in it? Surely when you dilute something there is still some of it there. So why do skeptics insist that there is nothing?

Well, I’ll explain.

In a typical homeopathic dilution, a tiny sample of something is dissolved in water, and then 1 mL of solution is transferred to 99 mL of water (making an even 100 mL). It is mixed (well, “succussed”), and then 1 mL is transferred to another 99 mL container, and so on. This may be done 20 or 30 times.

Sure, that’s a lot of dilution, but even if you put a few drops of something in a swimming pool, there are so many atoms in a drop that any sample should contain traces of the substance, and the homeopathic dilution is, after all, only a few litres.

Well, no. Let’s do the math on this, and see what the concentration comes out to be at the end. Let’s choose something like Lithium, as it is light, and it is used as a medication (I have no idea if it is used in homeopathy, but let’s just use it as an example). Lithium has a molar mass of 6.94g, and a density of 0.53 g/mL. Let’s begin with a 1% solution (v/v), and dilute it 20 times:

1 mL of Lithium contains 4.6×10^22 atoms. That’s 4600000000000000000000 atoms. That’s a lot. When dissolved, 1 mL of that solution should contain 1/100 of that, though still 46000000000000000000 atoms. But each time we dilute it, we decrease it by a factor of 100, so with serial dilutions, we have

1. 460000000000000000 atoms
2. 4600000000000000 atoms
3. 46000000000000 atoms
4. 460000000000 atoms
5. 4600000000 atoms
6. 46000000 atoms
7. 460000 atoms
8. 4600 atoms
9. 46 atoms
10. 0.46 atoms
11. 0.0046 atoms
12. 0.000046 atoms
13. 0.00000046 atoms
14. 0.0000000046 atoms
15. 0.000000000046 atoms
16. 0.00000000000046 atoms
17. 0.0000000000000046 atoms
18. 0.000000000000000046 atoms
19. 0.00000000000000000046 atoms
20. 0.0000000000000000000046 atoms

What this means is that after only 10 dilutions there is a roughly 50:50 chance of a single atom being present, and after 20, there is a chance of 1 in 217 billion billion that there will even be one atom present. Or, put another way, it would be the same as taking one mL of a substance and dissolving it in a volume of water equivalent to a sphere over two and a half billion kilometres across, or roughly the size of the orbit of Saturn correction: with 20 dilutions the volume would be about the volume of a sphere the size of earth’s orbit. But if you add two more dilutions, that sphere expands to the orbit of Uranus. And if I’m doing my math correctly, 30 dilutions would be equivalent to dilution in a sphere 130 light years across!*

So we know that there is none of the active ingredient in homeopathic medicine, if it is in fact prepared properly. None at all. But then, homeopaths don’t claim the original substance is there – they claim the water is influenced, and retains a “memory” of the substance, and that this memory is what is amplified by the dilution. This, too, is of course utter bullshit.  This idea is born of magical wish-thinking, and is not even remotely related to reality. Water is water, and has the properties of water, not a “memory” of what was once in it.

So that’s why, when we say “there’s nothing in it”, we really mean it.

*check my math:

• there are 10^15 cm^3 in a cubic km
• 30 dilutions by 100 is 10^60 mL
• which is 10^45 km^3
• which is a sphere with radius 6.2×10^14 km, or 65 light years.