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Limits of the Classical Concept of Concentration Marco Maioli, Gyula Varadi, Robert Kurdi, Luciano Caglioti, and Gyula Palyi J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b02904 • Publication Date (Web): 06 Jul 2016 Downloaded from http://pubs.acs.org on July 7, 2016
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Limits of the Classical Concept of Concentration Marco Maiolia, Gyula Varadib, Róbert Kurdic, Luciano Cagliotid, Gyula Pályie,* a
Department of Mathematics, University of Modena and Reggio Emilia, Via Campi 213/B, I-
41125 Modena, Italy;
[email protected] b
Inpellis, Inc., 100 Cummings Center, Suite 243C, Beverly, MA 01915-6133, USA;
[email protected] c
Institute of Environmental Engineering, University of Pannonia, Egyetem u. 10, H-8200
Veszprém, Hungary;
[email protected] d
Department of Chemistry and Technology of Biologically Active Compounds, University "La
Sapienza"-Roma, P.le A. Moro 5, I-00185 Roma, Italy;
[email protected] e
Department of Life Sciences, University of Modena and Reggio Emilia, Via Campi 103, I-
41125 Modena, Italy;
[email protected] *Corresponding Author, e-mail:
[email protected]; Tel.: +39-349-1320637 Abstract Solutions of very low concentrations cannot be treated by the usual concept of concentration. Stochastic calculations are performed for the analysis of such solutions, containing one or a few molecule(s). It is concluded, that these systems escape the usual concentration parameters. Two “case histories” are also shown for demonstration of practical consequences of the theoretical analysis.
Introduction Chemical principles are generally based on the macroscopic behaviour of a very large number of particles (atoms, ions, molecules). Typically one mol means 6 .022 × 10 23 pieces of such particles.
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Homogeneous liquid phases of more than one chemical substance are usually characterized by "concentration": the relative amount of the minor component in the major one. A typical unit is molar concentration that is, the number of mol-s of the minor component in the major one, per volume unit of the solution. Apparently, molar concentration, e.g. 1 mol/Liter (1 M) can be divided continuously and infinitely. Most practically, this can be done by dilution, e.g. one takes 1 mL from 1 Liter of a 1 M solution, adds enough solvent to reach again 1 L final volume and thus a solution of 0.001 M (1 mM) concentration is obtained. A "technical" limit of such a dilution process is when 1 .66 × 10 − 24 M (1.66 yoctomol) concentration is reached, where one molecule is present in 1 L. A solution like this cannot be further diluted in the same manner, since 1 mL samples from this "solution" either contain the one molecule (hence diluting it again to 1 L will result in the same concentration) or are empty (resulting 0 mol/L concentration). At the point of the "last" molecule this result clearly shows the stochastic nature of the system (either - or), the same is true for the “first” molecule in a reaction mixture1-6. In this paper, we will show that stochastic principles must be taken into account even at much higher (by several orders of magnitude) concentrations. Recently, experimental efforts at visualizing7, reacting8-16or detecting17-21 chemical systems (mostly homogeneous liquid phases) with low numbers or even single molecule, have been studied very successfully in a considerable number of publications22. In these experiments, it is sometimes uncertain what the actual number is of the "interesting" particles present. In experimental efforts directed towards very low quantities and/or concentrations these problems emerged seriously (for an excellent review see23). Therefore, it appears to us that a systematic theoretical (mathematical) survey of the principles of probability to be regarded in such (current and future) experiments might be useful. In the present paper, we describe only our results regarding the stochastic (mathematical) side of the preparation and nature of very diluted solutions. Factors as solute-
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solvent, solute-solute, solute-vessel associations or temperature as well as time dependence will be regarded in a forthcoming publication.
Results and Discussion In the following calculations, we are dealing mostly with samples of small molecule numbers and therefore we shall use, with one exception, binomial formalism24 (see also ASI-1). 1 Dilution problem Let us dilute one mole 10
r
times: from 6 .022 × 10 23 particles per Liter to 6 .022 × 10 23 − r
particles per Liter. The number r can be viewed as a dilution rate. Let us take a sample of volume p Liter, having fixed 0
