Molecules and nanoparticles in extreme homeopathic dilutions: is Avogadro’s Constant a dogma?

We are all familiar with the simple iterative process which shows that more than a dozen 1:99 dilutions of a soluble material in a liquid solvent such as water results in a chemical system that theoretically should not contain any atom/molecule of the starting compound. In this view, the bogeyman of homeopathy is Avogadro’s Number: but what exactly is Avogadro’s Number?

Like many other chemists and pharmacologists worldwide, homeopaths have a rather naive view of chemical system in aqueous solution that is largely an inheritance of Dalton’s atomic concept. Dalton’s Law of Partial Pressures states that the total pressure exerted by a gaseous mixture is equal to the sum of the partial pressures of each individual component in a gas mixture. This empirical law was based on observations by John Dalton (1766–1844) published in 1801; it is related to ideal gas laws. Ten years later the Italian chemist Lorenzo Romano Amodeo Carlo Avogadro (1776–1856), on the basis of some contradictions between Dalton’s observations and Gay-Lussac’s Law of combining gases (Joseph Louis Gay-Lussac, 1778–1850), introduced the concept of molecules and hypothesized that two given samples of an ideal gas, at the same temperature, pressure and volume, contain the same number of molecules (in fact, this principle had been partly anticipated by Alessandro Volta in 1791). Thus, the number of molecules or atoms in a specific volume of gas is independent of their size or the molar mass of the gas.

This was stated as a universal chemical principle but it did not immediately give rise to a numerical value. The first quantitative assessment of its value was by Johann Josef Loschmidt, his work “Zur Grosse der Luftmolekule” (About the size of gas molecules) published in 1865 includes the first estimate of the size of the molecules of a gas.[ ¹ ] Based on change in the volume of a gas during condensation and evaporation, Loschmidt calculated that the number of particles in a unit volume of an ideal gas under standard conditions (T = 298 K, pressure = 1 atm) is 2.686763 × 10²⁵/m³. The French physicist Jean Perrin (1870–1942) calculated the currently accepted value of Avogadro’s Constant: 6.022 × 10²³ mol⁻¹. Perrin dedicated his discovery to Avogadro and termed it Avogadro’s Constant or Avogadro’s Number, it is known as Loschmidt’s Constant in the German-speaking world. Thus, Avogadro did not calculate the eponymous number; but the foundations of ponderal science were laid by Italian, English, French, and German scientists, more or less contemporaneously with the birth of homeopathy in Europe.

The value of Avogadro’s Constant can be calculated by many physical methods,[ ² ] and is related to fundamental physico-chemical concepts, including: the uniform molar mass of the solute as a multiple of 1/12 of the atomic mass of ¹²C, Brownian Motion and the homogeneous nature of the water solvent, described by Zachariesen’s continuous random network model.[ ³ ] The relationship with Brownian Motion, extensively studied by Albert Einstein and Marian Smoluchovski who published their findings in 1906, allows the description of solution chemical system very similar to a gas system (gas-like model of a water solution), in which statistics and thermodynamics play an important role. Einstein determined the mean square displacement of a particle subject to Brownian Motion in terms of Avogadro’s principle and Jean Perrin won the Nobel Prize in 1926 for determining Avogadro’s Constant by this method.[ ^4,5 ]

In recent years, research on material science and water has raised many criticism of this simple picture. Besides to the kT paradox[ ⁶ ] and the nano-heterogeneity of water structure,[ ³ ] the picture of the solvent as a blank backdrop against which biological phenomena are played out, is long outdated. In addition, criticisms of the real –log [M] concentration of a starting active principle undergoing a decimal or centesimal dilution have been raised elsewhere.[ ^7,8 ] The paper by Chikramane et al. recently published in Homeopathy,[ ⁹ ] points to the possible role of nanoparticles, as opposed to molecules, in carrying biological information in dilutions beyond the “Pillars of Hercules” of Avogadro’s Constant.

The chemistry of highly diluted substances deals with the behavior of nanoscale particles, which do not display the classical diffusive behavior of molecules and do not comply with laws of distribution in a solvent according to Dalton’s atomic principle and hence Avogadro’s Law, especially for their particular property due to their size[ ¹⁰ ] and electrodynamics.[ ¹¹ ] For example gold is usually shown as a shiny and yellow noble (non catalyst) metal, having a face centred cubic structure, being non-magnetic and melting at 1336 K. But gold nanoparticles (10 nm) absorb green light, thus appearing red, the melting temperature decreases dramatically with decreasing size, while 2–3 nm nanoparticles of gold are excellent catalysts and are magnetic. Gold nanoparticles have a different chemistry than atoms.[ ¹⁰ ] The quantum properties of nanoparticles do not correspond to classical chemistry,[ ¹⁰ ] as shown, for example, by Samal and Geckeler’s finding of unexpected aggregation of solutes following dilution with water, although Hallwass’s findings seem to disagree.[ ^12,13 ] Calculation of molar concentration, based on classical thermodynamics, for a given, entirely macroscopic, system can surprisingly result in small fractional numbers. Levis and Randall reported, for example, 4.0 μmol/L silver cyanide dissolved in 1 L of potassium cyanide 3 M, results in a concentration of silver ions about 1/10 of a single ion in 1 cm³, a small fraction of unity.[ ¹⁴ ] This estimate is impossible if one assumes that dilutions cannot result in fractions of atoms. It has been explained by hypothesising the existence of other “chemical” components, such as nanobubbles or nanoparticles, which have major effects on the classical molar calculation, for example by changing the nature and effective volume of the solvent.[ ^14,15 ] Das and Gupta-Bhaya have developed a statistical interpretation of such phenomena at concentrations of 10⁻²⁴ moles and less.[ ¹⁵ ] Certainly, these studies raise some questions about what happens to a fluid subjected to various regimes of dilution and different thermodynamic conditions and in relation to the type of solute and/or solvent. Spectroscopic studies may not be adequate to elucidate these issues.[ ¹⁶ ]

What exactly is the problem? Emil Roduner pointed out that nanoparticles behave totally differently from their massive relatives from a chemical–physical perspective.[ ¹⁰ ] Therefore we do not know, for example, if nanoscale particles in homeopathic potency respect the classical laws of chemical particles such as molecules. But there is another aspect to keep in mind. Electrolyte solutions or solutions of mixtures of non electrolytes and fluids generally have a characteristic microscopic structure due to system-specific interactions at the nanoscale. It is assumed that solutes are distributed evenly, but various data suggest that this is incorrect and that there may be inhomogeneous distribution of the solute, with non-uniform superstructures up to several hundred nanometers in diameter.[ ^17,18 ] Such supramolecular structures have been observed by dynamic light scattering (DLS) and static light scattering (SLS).[ ¹⁹ ]

The work of Chikramane et al. published recently in Homeopathy relates to this situation.[ ⁹ ] The question “is Avogadro’s Constant appropriate to the investigation of the “pharmacological” potential of an extreme high dilution into water?” is intriguing and calls for physical research on the physico-chemical nature of these systems. The question recently posed by Brandt, who asks if Avogadro’s Number is a dogma, is of great interest and a real provocation for basic science.[ ²⁰ ] We must take up the challenge.

• References

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