Question on Textbook problem G25

[Do Yeun Park]

Question G25: "Practitioners of the branch of alternative medicine known as homeopathy claim that very dilute solutions of substances can have an effect. Is the claim plausible? To explore this question, suppose that you prepare a solution of a supposedly active substance, X, with a molar concentration of
. Then you dilute 10. mL of that solution by doubling the volume, doubling it again, and so on, for 90 doublings in all. How many molecules of X will be present in 10. mL of the final solution? Comment on the possible health benefits of the solution."

I understand that there will be 0 molecules of X left in the solution. If this is the case, then what would be the health benefits?

[Matthew Vu 3C]

If there are no molecules left, then there should be no health benefits.

[KyleNagasawaDisc3C_Chem 14B2022W_]

Hey there Matthew,

I thought this was an interesting question to complete because the end numerical response correlates with a probabilistic interpretation. There aren't exactly 0 molecules in each 10mL aliquot. (After all, a dilution is a physical process, so the homeopathic substance doesn't "disappear".)

But first, to solve the problem:

Final Concentration (M): 0.1 M * (0.5)^90 = 8.07793... * 10^-29 M
-With each of the dilutions, the concentration of the substance decreases by half.
Total Molecules per Liter (molecule/L): 8.07794... * 10^-29 M * 6.022 * 10^23 molecule/mole = 0.000048... molecule/L

Molecules in 10mL: 0.01 L * 0.000048... molecule/L = 0.000000486... molecules



It would be easy to interpret this as there being no molecules in each aliquot or that there is a fragment of the molecule in each container. In actuality, the calculation implies that there is a mean probability of 0.0000486...% of finding a molecule in a jar. Some aliquots will have none of the homeopathic substances (likely scenario), while others will have 2 (statistically insignificant quantity).

[Naman Jain 3F]

Since there are an extremely minimal amount of molecules or no molecules left, the active substance is almost nonexistent. Therefore, the active substance X has very minimal health benefits and can be considered to basically have no health benefits.