Fun with homeopathy and math, again

After the last discussion on homeopathy, someone asked me to give a description of what a 200C homeopathic remedy would have to start out with in order to have at least one molecule at the end of the dilution. So let’s use the example of sugar (glucose) and see how much sugar we would need to get at least one molecule of sugar in a liter (1,000 mL) 200C remedy. We will use math for this, so hold on to your butts.

Remember that Avogadro’s constant states that there are 6.02×10^23 molecules of glucose in 180 grams of the stuff. So, if we add 180 grams of sugar to a liter of water, we will be adding 6.02×10^23 molecules of glucose into that liter. Remember that diluting that initial solution 1C (by one hundred) will leave us with 1.8 grams per liter or 6.02×10^21 molecules of glucose per liter. Finally, remember that we have to do this 200 times (to get to 200C), and that doign this makes us run out of molecules at around the 8th or 9th C dilution. After that, we are diluting water with water.

But what if we want to make sure there is at least one molecule of sugar at the end, at the 200th C dilution?

In that case, we work backwards with the assertion that there is one molecule per liter at 200C. To go to 199C, we would have to concentrate (the opposite of dilute) the solution by a factor of 100, leaving us with 100 molecules in the 199C dilution. Moving up to 198C, we have 10,000 molecules (100 multiplied by 100). Not quite Avogadro’s constant yet. Let’s go to 197C, and see that we have 1,000,000 molecules (10,000 multiplied by 100). Have you noticed the trend?

For every C concentration, we are adding two zeroes to the right of the 1 that we started with at 200C. So, after 200 concentrations, we will have 400 zeroes to the right of the 1. That’s an enormous number of molecules.

How enormous? Taking into consideration that Avogadro’s constant is 6.02 followed by 21 zeroes to make up just one mole, 1×10^400 molecules make up… well… a lot of moles.

Seriously, I don’t have a calculator with me with a display big enough for all those zeroes. I plugged in 1×10^400 into my mac’s calculator and it laughed at me! I tried to divide that number by Avogadro’s constant to get the number of moles, and the damn thing grew legs and walked away, cursing at me.

No, the computer didn’t do that. But if you believe homeopathy then the computer doing that doesn’t seem so far fetched.

If you multiply all those moles times 180 grams, you will have a lot of tons of sugar that you need to somehow cram into one liter of water in order to dilute that liter of water by 100 two-hundred times to get a final 200C homeopathic remedy that has at least one molecule per liter.

Did you catch that?

One molecule per liter. One. You have a 1 in 1,000 chance of catching that molecule if you only take one milliliter of the final solution – the remedy – at a time.

Even if you go from 30C to an original solution, you add 60 zeroes to the right of the 1, which is still a huge number, you will have 1.66×10^36 moles of sugar at your beginning solution. At 180 grams per mole, you’re looking at about 3×10^38 grams of sugar that will need to be put in one liter of water.

You’re going to need a lot of sugar. So…

Now do you see why homeopathy is more about magical thinking than any sort of science or medicine?