Homeopathy strength estimation

Finding the number of molecules in a medicinal pellet

Traditional medicine machinery

The number of molecules in 1 mol of any substance corresponds to $6,00253\times {10^{23}}$ (an Avogadro number)).

Summary: There was a lot of history for moles and atomic weights (John Dalton ) and Avogadro number Na (from the first measurement in Josef Loschmidt in 1865 to NA declaration as a fundamental constant in 2019). The idea is that it was found that substances are collections of molecules/atoms, and thus the ideal ratios of amounts of substances to react will always be proportional to integer numbers (each molecule can only decay for or acquire an integer number of other molecules). Correspondingly, it is convenient to measure amounts of substances to experiment with in units with a large fixed number of molecules (NA), which we can call a "mole." The weight of this unit (a mole) for all substances is different but proportional to the *sum of atomic weights of their components*. The atomic weights are small, $\approx 1.67⋅10^{−24}$g for a hydrogen atom, and it is more convenient to measure them relative to each other. The measurement unit was called Dalton; one Dalton is equal to 1⁄12 the mass of a carbon atom. Now we can use Daltons to take the relative amounts of substances in units the experimentalists like most (grams) and correspondingly define the mole as the amount of substance containing as many particles as there are atoms in 12 grams (12 is the carbon-12 atomic weight in Daltons) of carbon. Thus, one mole of carbon-12 will weigh exactly 12g, one mole of hydrogen approximately 1g (1Da is the approximate atomic weight of hydrogen atom), one mole of oxygen approximately 16g (16Da is the approximate atomic weight of oxygen), etc. The weights of 1 mole of any substance are called the molar weights with the dimension of grams per mol (g/mol) and are numerically equal to the sum of the atomic weights of their atoms.

The remaining task was to find the number of molecules in our 1 mol==12g of carbon, or in 1 mole-by-molecular-weigh of any substance - the Avogadro number, which was determined to be $\approx 6.022×10^{23}$ molecules/mole in 2019. However, experimentalists finally gave up on measuring it with better and better accuracy, and in 2019, they fixed NA with an exact value of $6.02214076\times 10^{23} mol^{-1}$

Homeopathy machinery

Like cures Like HMP uses "small" amount of substances that causes a symptom to treat the same symptom in illness

Homeopathic dilutions

"Here’s a quick guide to using homeopathic medicines: Low dilutions, such as 6X or 6C, will relieve local symptoms — a symptom you can point a finger at (e.g., an insect bite or bruise). Medium dilutions, such as 12X, 9C, or 12C, will relieve general symptoms — more than one symptom in more than one location (e.g., muscle aches and pains). High dilutions, such as 30X or 30C, will relieve general symptoms — more than one symptom in more than one location with possible behavioral or emotional symptoms (e.g., a very high fever and chills, accompanied by agitation or sleeplessness)."

"Diluting a substance by a factor of 100 at each stage. There is also a decimal dilution scale (notated as "X" or "D") in which the preparation is diluted by a factor of 10 at each stage. A 6C dilution repeats this process six times, ending up with the original material diluted by a factor of $100^{-6}=10^{-12}$"

There are several large homeopathic associations, and each of them has its own drug standards and requirements. See the examples below:

Homeopathy strength specification

HMP vendors (like HPUS, The Homeopathic Pharmacopoeia of the United States ) are required to specify the Latin name of an active homeopathic ingredient and the number of dilutions but do not specify the initial amount of substance or the exact chemical composition of a substance. Commonly, homeopathic vendors claim that the initial amounts are "small" (fractions of a mol). Correspondingly, we can set the upper limit of HMP per-mol dilution by assuming the initial amounts of substances are equal to 1 mol. Sometimes, vendors provide the amount of active ingredient in a pellet/capsule as done for approved medicine. In this case, we can estimate the actual per-mol dilution and compare it with the per-mol dilution of the approved medicine. The per-mol dilution can be converted to the number of molecules in a pellet Np by multiplying it on the Avogadro number NA - and thus is a natural metric for the pellet's capability to have some effect. Here, we will use ibuprofen 200 mg pellets as a reference for known and good-working medicine and also compare lethal doses of different toxins with corresponding traditional and homeopathic medicines.

In general, if in addition to the claimed dilution (6c, 12c, etc),

  • 1) we know the weight of the active component and the molecular formulae of a remedy, we can
    • calculate the initial amount of substance for the claimed dilution
    • calculate the dilution relative to the mole of the initial substance (== number of molecules in a homeopathic pellet)
    • compare the homeopathic mol-related dilution with dilutions common for approved medicine
  • 2) if we trust the claimed dilution, and we know the initial amount of substance used for dilution (in moles or grams), we

also can perform everything in 1)

  • 3) if we only know the claimed dilution and molar weight, but the weight of the active ingredient and the amount of the initial substance

is unknown - we assume that the initial amount was 1 mole and thus all dilutions higher than $10^{-4}$ (kids dose of one ibuprofen) are questionable, but the higher - the safer. $10^{-24}$ or 12C molar dilutions only have ~ 0.6 molecule per dose and dilutions above 12C (30C, 100C, 1000C!) have no active molecules at all.

The math

  • below (==) means "by definition", "=" means equal, "~" means approximately, "~=" means approximately equal, "~<=" meansapprocimately less
  • Molar weight (gram/mol) == the weight of one mole in gram, numerically equal to an atomic weight in Daltons [ ~= total number of protons and neutrons in a molecule ] ==> Dalton == gram/mol
  • Avogadro number NA ~= $6.02\times10^{23}$, the number of molecules in a mole

Examples:

  • the weight of 1 mole of hydrogen is ~1 gram
  • the weight of 1 mole of carbon is ~12 gram
  • the weight of mole of c2h5oh is ~46 gram
  • the weight of 1 mole of Ibuprofen | C13H18O2 is ~206 gram (and correspondingly one 200mg pellet contains $10^{-3}$ mol)
  • the weight of 1 mole of Caffeine | C8H10N4O2 is ~196 gram (and 1 teabag of 55mg contains $\approx 2.8x10^{-4}$ mol)
  • CyaniD CN - 26g (150mg is a lethal dose for 100 kg person) $\approx5.7 \times 10^{-3}$ mol
  • Lakhesis Muta Venom - popular homeopathy, molar mass 13.976 kDa == 13976 gram/mol, https://pubmed.ncbi.nlm.nih.gov/21308192/,

Follow-ups from above

  • 1) 1 dose of 6C solution of 200 mg ibuprofen $\approx 10^{-3} mol / 10^{-12} mol = 10^{9}$ == 1 trillion of standard tablets
  • 2) there are $6x10^{23}/10^{24}$ = 0.6 molecules in one 12C dilution, or 6 molecules/10 doses = 1 molecule / 1.6 dose
  • 3) one mole of almost any complex molecule > 100g; the initial amount of homeo 1g =~ 0.01 mole
  • 4) 0.5L of Russian Vodka at 40% = 200g ~ 4mole of pure spirit - no molecules in 13C dilutions
  • 5) to prepare $30C=10^{-60}$ homeopathic dilution that will have at least one molecule of a substance with a molecular weight of 100 $g/mol$,

one must have $10^{38}$g of such substance, which is equivalent to the mass of 10000 our Suns

The relative strength of popular substances.

We define the "strength" of a medicinal pellet as the number of moles of an active ingredient in a single pellet. A popular ibuprofen 200 mg pellet with a strength of $9.7\times10^{-4}$ mol of C13H18O2 is chosen as a reference. An example of per-mol dilution strength calculation for Boiron 0.443mg pellet of Lakhesis Muta Venom., Boiron 0.443 mg::

  • $ 0.443 \times 10^{-3} g /(13976 g/mol) \approx 0.443\times10^{-3}/(1.4\times10^{4}) \approx 3 \times 10^{-8}$ mol, with the relative strength to an ibuprofen pellet of $3\times10^{-8}/9.7\times10^{-4} = 3.2\times10^{-5}$.

An unspecified amount of active ingredient, 30C dilution (no molecules at all if the initial amount of substance used was less than $10^5$ moles ($\approx 100000$ kg of venom):

The results are summarized in Table 1

Summary and conclusions

The presented examples illustrate the relative strength of popular substances, including pain relief medicine, toxins, and homeopathy, based on the estimated "per-mole" dilution. The estimated "per-mole" dilution differs from the claimed dilution for homeopathy. The upper strength limit is provided based on the claimed dilution strength if the amount of the active ingredient is not specified. The highest applicable dilution of $2.1\times10{-14}$ mole (14D or 7C in HMP notation) of scientifically proven medicine was found for the Botox family of substances. The LD50-LD100 amounts (doses that are lethal with probabilities of 50 and 100 percent) of snake venoms are in the $10^{-3} - 10^{-10}$ mole (3D-10D) range. Nivolumab anti-melanoma Immunotherapy is administered at $10^{-6}$ mole (6D) per day. Traditional pain relief medicines (Ibuprofen, Tylenol) have approximately $10^{-3}-10^{-2}$ moles (2D-3D) of active ingredient in one pellet. Homeophatic dilutions cover the wild range of $10^{-3}-10^{-4000}$ moles (3D-4000D).

From the above, we can classify homeopathic dilution strength in comparison with known toxins and approved medicine:

  • estra high dilutions, expensive placebo: $10^{-24}-10^{-4000}$ mole, or 24D- 400D == 12C-200C range - no molecules of active ingredient. Should be safe if obtained from a "trustable" HMP vendor (HPUS, Boiron, etc)
  • hight dilutions, $10^{-12}-10^{-24}$ mole, or 12D-24D==6C-12C range - at least a million times weaker than approved immunotherapy and a billion times weaker than popular pain relievers, but only 100 times weaker than some snake venoms and in the range of approved medications based on botulinum toxin. HMP medicine based on strong toxins can have negative effects near 6C concentrations, see FDA warns consumers about homeopathic teething products.
  • low dilutions, $10^{-3}-10^{-12}$ mole, or (3D-12D) are in the range of approved medicine. May have unpredictable effects on a patient's health if not based on known safe ingredients (like Potassium phosphate)>

It is essential to mention that HMP medicines often use 1 gram of active substance for dilution instead of one mole. Depending on the molar mass ( 100 - 200000 g/mol) for analysed substances, the estimated homeopathic dilution strength (and thus the number of molecules of active substance in HMP pellets) can be correspondingly 100-200000 times lower.

Supplemental materials

Homeopathic claims/methods/practices that do not comply with an elementary level of chemistry and physics:

  • dilutions with no molecules of active ingredient do have medical effects, and these effects still depend on the no-molecules-at-all dilution strength
  • dilutions of the same strength but different "potency" (== number of dilutions) like 6C and 12D are considered different medications even if they should have the same number of active molecules
  • the amount of active ingredients is often not shown on the labels, and when shown, it may not correspond to the claimed dilution (0.443mg of active ingredients in Boyron medications)
  • the same amount of active ingredient 0.443mg is claimed for different dilutions (6c, 30c, 200c), which is not physically possible
  • the same amount of active ingredient 0.443mg is claimed for the different dilutions of different substances (and this corresponds to a different number of molecules)
  • for dilutions of 12C and higher, it is instrumentally impossible to distinguish between different homeophatic medicines (no molecules of active ingredients)
  • when the amount of active ingredient does correspond to the claimed dilution, $10^{-12}$g for 6C dilution of "Argentum metallicum", it is taken relative to grams, not moles and thus the dilutions of the same strenght for different substances will have a different (proportional to the molecular mass of a substance) number of active molecules
  • More than a hundred complaints against Boiron
  • Homeopathy—where is the science?. EMBO Rep. 2019 Feb 14;20(3):e47761. doi: 10.15252/embr.201947761

Examples of homeopathy efficacy studies

Legal lawsuits and regulations

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