4. Red herrings in electrochemical high dilutions
The analogy between the persistence of Nernstian response in high dilution electrochemistry and the results by Jacques Benveniste and his team on basophils degranulation () was pointed out by Henry Bauer in . In  the "unreasonable accuracy " of standard formulations of the solubility product and of complexation constants even at levels where no ions or molecules can be present is documented across different analytical techniques. Indeed concentration below the Avogadro threshold appear routinely in chemical calculations (see e.g. 7.4 in ). A remarkable instance of this phenomenon is the validity of the Nernst equation relating the electrodes potential to the the reactants concentration even at dilutions where the calculated probability of any reactants ion being localised in the sample is far below one, as reported by Dick Durst in , based on previous work ([7}, see also  and the references therein).
In the setting of of ion-selective membrane electrodes the Nernst equation relates the equilibrium electrode potential E to the concentration A of the free ions in the sample solution
E= E0+ 59.16/n ln A
where E0 is the standard potential and n is the ion's charge. The Nernst formula applies continuously to potential differences for which one of the two species is present in the samples in amounts far below a single atom. The results in  refer to ion-selective silver sulfide membrane electrodes where the silver electrode is immersed in a reference solution separated from the sample solution by a silver sulfide membrane, which is permeable only to silver ions. At equilibrium the potential across the membrane will prevent movement of silver ions , so that the measured potential corresponds to a specific amount of silver ions in the sample solution.
The experimental evidence for Nernstian response at concentrations below the Avogadro threshold appears to be well-established, while the theoretical issues behind it appear to be controversial (see ,,).
In Figure 9 in  , reproduced in  and sketched above , the response of a silver sulfide solid-state membrane electrode is plotted. The electrodes response turns out to be Nernstian down to molar concentrations of 10E-25 of silver ion on a sample volume of 5 microliters. The activity of silver ions are calculated from the solution's molar composition, which in the case associated with the lowest free silver concentration consists of 0.1 Na_S + 1 Na_OH. The silver ion concentrations in the table are obtained using the silver sulphide solubility product relating the concentration A of Ag+ (released by the membrane) and the known concentration B = 5.5*10E-2 of S-- (released by Na2_S) in the solution. The equation
yields the value A = 10E-24.9. This implies a probability below 10E-6 of a free silver ion being localised in the sample. An "ad hoc" explanation for the persistence of Nerstian response at this sub-single-ion concentration is hinted at in , but no concrete indication is provided. An alternative explanatory model is proposed in . In  the issue is discussed and contributions to the solution of the problem are again solicited.
The model and the experimental evidence
The model proposed in the first part of the paper to explain the results by Jacques Benveniste and his team can be applied to Nernstian response at very low concentrations, as well as to the whole issue of the efffectiveness of standard analytical formulations at very low concentrations. The usual objections based on ion's concentration being so low that in all likelihood no ions are present in the sample do not apply, since no "demolition" measurement (� la Braginsky) determining the position of the diluted ions is performed. Diluted ions may well subsist in the sample's wave-packet and induce physically observable effects also when their amplitude over the sample is less than one. The probability of a ion being present in the sample is just the squared modulus of the amplitude of the corresponding state in the sample's wave-packet. As long as no direct measurement of the ion's position is made, its non-null amplitude will induce a corresponding Nernstian response. Indeed it is unclear whether a position measurement of the free ions is possible in this setting, even in principle.
It may be recalled that the properties of various chemical compounds, e.g. benzene molecules, are routinely explained in terms of superpositions of different configurations with fractional amplitudes. Analogously, in the model proposed here, silver ions subsist as superpositions of free and bound states.
An intriguing further analogy between the biological and the electrochemical phenomena is provided by the experimental results reported a. o. by Srinivasan et al. in  , where Nernstian response at low concentrations is shown to be enhanced by stirring of the solution. According to the model expounded here shaking the solution enhances the spread of the diluted ion's wave-packet and therefore strengthens the induced response, as it appears to do both in Srinivasans's and in Benveniste�s experiments. It is not "a priori" clear how the explanatory model based on the theory ofcoherent domains ( ) could be applied to the electrochemical setting.
The Nernst equation corresponds to a theoretical model whose validity appears to persist at very low concentrations. While in the case of basophil's degranulation little is known in detail about the mechanisms triggering antibody detection, in this case we have a well-established model which appears to maintain its validity at concentrations below the Avogadro threshold. In other words the theoretical model underlying equation (1) provides us with a concrete instance of the amplifying device whose presence was conjectured in . Indeed the logarithm in Nernst's formula provides ion-selective electrodes with the the amplitude-detecting and amplitude-boosting property that was conjectured in the first part of this paper. Ion-selective electrodes for the detection of specific antibodies have already been developed and their sensitivity is being improved. They may yield some surprise.
A time-stamp: Message without reply
Here is an exchange with Prof. Jacek Tyczkowski.
1. D. Durst "Shades of Homeopathy (or "Where did all the ions go?")" SEAC Communications 12-2 (1995) available online at the Society for Electrochemical Chemistry website http://electroanalytical.org/communications.html
2. D. Durst "Electrochemical "Homeopathy" SEAC Communications 12-3 (1996).
3. H. Bauer "Physical Interpretation of Very Small Concentrations" Journal of Scientific Exploration , 4-1 (1990), pp. 49-51.
4. K. Srinivasan and G.A. Rechnitz "Activity Measurements with a Fluoride-Selective Membrane Electrode" Analytical Chemistry 40-3 (1968), pp 509-512.
5. K.L. Cheng "Capacitor Theory for Nonfaradaic Potentiometry" Microchemical Journal 42 (1990) pp. 5-24.
6. Davenas et al. "Human basophil degranulation triggrered by very dilute antiserum against IgE" Nature 333 (1988) pp 816-818.
7. D. Durst "Analyitical Techniques and Applications of Ion-Selective Electrodes" in Ion-Selective Electrodes, Ch. 11, D.Durst ed. NBS(1969).
8. I. Vecchi "On High Dilution Experiments" Frontier Perspectives 8-2 (1999).
9. G. Preparata., QED Coherence in Matter, Chapter 10: Dynamics and Thermodynamics of Water, Singapore, World Scientific, 1995.
10. J.W. Ross "Solid-State and Liquid Membrane Ion-Selective Electrodes " in Ion-Selective Electrodes, Ch. 11, D.Durst ed. NBS(1969).
11. D.C. Harris "Quantitative Chemical Analysis " New York, Freeman & Co, 1991.