The Weierstrass Snare: How The Role of
Surface Excess in Solution Chemistry Has
1.1.The Weierstrass Snare.
The Weierstrass Theorem states in one of its forms that any continuous function over an interval can be described by a polynomial function.
1.2. On the idea that some of the solution chemistry is controlled by the surface excesses of solutes.
At 25oC it is possible to have 70.1 kg of AgNO3 mixed with 29.9 kg of H2O . The average distance between two ions of opposite charge on the basis of each occupying a cubic cell is 2.9 nm. The amount of water required to have Ag(H2O)2 hydrates in that mixture would account for half of the water. This indicates that such system's solubility is defined in the bulk phase. But only about 50 gram of AgCl can be dissolved in the same 29.9 kg of water. Evidence in favor of the idea that this is the consequence of formation at a singularity (like the surface of the liquid) of a stable, insoluble product will be presented, based on the idea that the behavior of ionic surfactants can be related to that of other ionic solutes.
where DDBS- and DS- are shorthand for anionic surfactants dodecylbenzene sulfonate and dodecyl sulfate. Precipitation may also occur in solutions where supramolecular aggregates known as micelles exist.
Figure 1. (left) Density profiles  for the de-coupling of cations and anions near the air water interface of a Ray-Jones electrolyte in water. (right) Surface excess of an ionic surfactant and opposing counter ion layer. Although dispersed as colloids, surfactant micelles are also shrouded by counterions.
1.2. Basics of a planar model for chemical reactions in a single solvent.
Figure 2. In a solution of two counterions one of the species may become pushed towards the boundaries of the solution by its sinability to solvate. The other may then be drawn by the first through electrostatic attraction but is also scattered due to turbulence in the solution, hence its more spread distribution (fig 1). Transfer to the bulk of the solution may result in decomposition of products creating the appearance of no reaction. Absence of stirring causes the process to be extremely slow.
Figure 3. Some simple calculations around the volume of a cube. Although the volume occupied by the dilute bulk concentrations more than compensates for the lower reaction rates, can species far apart really react in the bulk of the solution? Or is it just happening in the surface and subsurface excess regions within the peripheral region around the volume?
1.3. Refutation of a preliminary objection: irrelevance due to the low molar ratio surface excess to bulk concentration.
The proportion of chemicals involved in the surface reaction would almost always be small. However, in chemical technology and engineering mixing is used when preparing samples and in many reactors (“stirred tank” type reactors and “plug flow” reactors under turbulent mixing would be two typical cases ) and other devices. This renews the contents of the liquid surface , and we will show that it can be the surface of the liquid facing the reactor wall, not only the air water interface. In fact, the surface excess is likely to exist against any surface (e.g. a probe dipped into the solution or the submersed outside of an injection or sampling device).
2. Model and Discussion
2.1. Activity coefficients as a ratio between actual and expected surface excesses.
For the discussion in this section it is irrelevant whether the ideal behavior of a solute is linear proportionality between surface excess and bulk concentration or adherence to an ideal isotherm like Langmuir's. In section 2.2 it is suggested that linear proportionality should be regarded as the ideal behavior. We are going to illustrate the idea that non ideality is linked to the discrepancy between the actual surface excess and the projected value based on the bulk phase concentration. The expression presented here allows to estimate the evolution of the surface excess G with the increase of the bulk phase concentration C of a species. Its origin is described in detail in Supplementary Materials. The expression relates the surface excess to the total concentration and a threshold concentration at which a transition in the G-C relationship occurs:
where a, b are constants expressing the G-C transition and kC is the theoretical G. The activity coefficient, according to the model presented here, would be:
Figure 4. Mean activity coefficients for AgNO3 and CaCl2,  and a prediction based on an estimate of the ratio between effective surface excess and theoretical surface excess.
2.2. Beer's Law and surface excess across all the liquid surfaces.
We are going to use colorimetric results to propose that the surface excesses appear and must be accounted for against all limiting surfaces including walls, and not the air-water interface or the surface of an adsorbent. It is irrelevant if the surface excesses are identical or not in all cases.
where S is the surface and V the volume of the cell filled with a sample of concentration C. The bulk concentration would relate to the total concentration following:
where the first right hand term refers to the classically accepted contribution by the bulk of the solution and the second right hand term corresponds to the effect of the surface excess (monomolecular, hence the lack of a thickness) at the entry and exit planes of the cell we can see the relative drop in absorbance as shown in figure 5.
Figure 5. Relative absorbances from Yu et al, and from the model. The basic assumption of a contribution due to surface excess is enough to predict what would otherwise seem a drop in the extinction coefficient of the substance. The prediction is enhanced assuming saturation has been reached at the surface and the only increase in absorbance is due to the bulk phase.
where p,q,..,z are constants. Said equations for the deviation from ideal behavior are similar to Taylor series for a Langmuir or similar adsorption isotherm . This is further evidence of the connection between ideality as a consequence of linear dependence of surface excess and concentration.
2.3. Mechanism for precipitation of sparingly soluble salts with formula AB.
Let us consider that sufficiently high surface charge densities sA+ and sB- should generate enough electrostatic attraction to draw together counter ions from their planes and form a precipitate AB. Let us assume that the condition is similar to Bjerrum’s distance for ionic pair association , and dependent on the charge densities of the charged planes:
where the bulk phase concentrations [A+] and [B-] would be proportional to the surface excesses GA+ and GB-. Such expression is consistent with the form of a solubility product. The plane structure is shown in Figure 6 (bottom, left). It becomes evident that any deviation from linearity, for example the inability of G to increase with bulk concentration due to surface/subsurface saturation would reflect in activity coefficients different from 1.
Figure 6. The schematics show that it is possible to arrange a reaction path as a sequence of steps caused by migration from the starting planes into a prime intermediate plane where the first intermediate forms. Whether the first intermediate migrates to subprime intermediate planes or reacts with other species in the prime intermediate plane is debatable, but the scheme explains the reaction sequence.
2.4. Precipitation of sparingly soluble salts AxBy intermediate species planes.
For a more complex molecule AB2 the precipitation could be explained by an initial drawing together of A2+ and B- to form AB+ in a “prime intermediate” plane. The new plane (Figure 3, center) would be involved in the drawing together of AB+ + B- to form AB2. AB+ will be present in this plane at a concentration (comparable to a surface excess) dependent on the equilibrium GAB+= KeqG* AG*B, where G* denotes the “plane excess” in A2+ and B- in the AB+ “plane”.As a consequence the precipitation step to produce the insoluble salt would require to meet the condition:
2.5. Consistency between the model and the concept of mean activity coefficient.
If the meaning of the activity coefficient is the deviation between the actual value of surface excess of a species and an ideal value of surface excess linked to the actual bulk phase concentration and following the equations presented in the previous sections we can deduce the form of the mean activity coefficient of a substance as a function of ion activity coefficients 
If we consider the activity coefficients of species A+ and B- as
and equal to one for as long as Gamma = k [X], but different of 1 if Gamma becomes constant (for example) then we can see that equation 8 becomes
For equation 13 the inclusion of non-ideality creates the correction factor gA+(gB-) which can be made equal to (gmean)2. It is easy to see the mechanistic justification for the expression of the mean activity coefficient.
It is important to stress the novelty of the concept of surface plane mechanism and how it justifies the exponents for the concentrations. Although action mass models have been used for long time, they do not explain the exponents in the equations, but simply acknowledge them. In fact, if thought is given to the idea of bulk phase reaction, the form of a solubility condition should be a relationship involving concentrations with exponent one, such that the critical value indicated the minimal amount of either of them required to electrostatically atract the other in order to reach the final stage of the reaction sequence, implicitly accounting for availability for any previous intermediate reaction stages.
2.6 Other equilibria.
If we revisit Section 2.2, where we looked at the ”plane excesses“ at the “prime intermediate” plane, the proposition was that the intermediate plane excess was:
for any individual species forming in the “prime intermediate” plane. In the case of precipitation, the end result is a neutral species which does not significantly affect the structure of the plane system from an electrostatic point of view.
Figure 7. Scheme of the various intermediate planes for a sequence where all the intermediates and products remain soluble in their planes.
and then rearrange equation 15 we find that for a given salt with 1:1 stoichiometry, we obtain a dissociation equilibrium constant:
From here, in the case of a mixture of salts with common electrolytes, the left hand side of the equation provides the relationship between the common electrolytes, whereas the second term remains a constant that gives the restriction value of KAB. It becomes trivial that a subsequent stage leading from species AB to species AB2 involving distances d0/-, d0/1 and d1/- and characteristic parameters a0/1 and a1/- would result in:
which is easily combined with equation 16 to yield:
the form of the various equilibria and the connection between exponents and stoichiometric factors is therefore justified. If we go back to equation 16 we can see that we can rearrange the equation to have the general form:
where QS is the concentration-based apparent solubility product, KS is the acitivity based solubility product, a and b are constants, B(T) is a function of temperature, G(T) a series in T to different exponents, and I the ionic strength of the solution. Although equation 18 relates the “isotherm” form to the logarithms of the solubility products, the Taylor series of (1-x)-1 and -log(1+x) are very similar, including consistence in the signs for the terms in X to a given power. This means that non-ideality in the solubility product still will reflect the form of an adsorption isotherm. Nortier et al. , further developing on previous models on precipitation in complex systems, stress the fact that the underlying chemical mechanisms are not well known and cause the need for elaborate methods to account for non-ideality.
While the above mathematical developments suggest that the plane model would explain the form of many equilibrium equations in solution, it seems possible to justify kinetic equations in the same way. Let us start by postulating that equilibrium in a solution is achieved because the surface behaves as a reactor while the bulk phase behaves as a storage area and feed tank. The equilibrium between the species would be achieved in the surface, and they would remain stable for as long as they remain in the bulk. Once they re-enter the surface-subsurface, high local concentrations may cause exchange of product fragments and free fragments. That would explain equilibria which are not completely displaced to products or reactants, as well as recombination when adding a further reagent or for example, an isotopic marker. Equilibrium in the context of this section refers to the final state of the reaction, whether a balance between direct and reverse reactions or full conversion.
The overall or macroscopically perceived rate of such reactions could depend on two factors. First, the rate of replacement of the liquid at the surface-subsurface layer may equal the rate of reaction if the actual reaction rate within the surface-subsurface layer is instantaneous. The rate of reaction would be affected by the stoichiometry of the equilibrium within the surface-subsurface layer in the same way that the equilibrium constant:
A fundamental flaw in science is the belief that succesive approximations to a numerically congruent solution/prediction to a problem are mechanistically approaching the real behaviour of the system. Succesive patches on an erroneous initial hypothesis, like bulk phase reaction vs surface reaction in a fully stirred solution, will correct the predicted value and will be pseudo-justified a posteriori by yield a final equation that provides an accurate numerical solution by amalgamanting a number of equations which, individually, do not predict the real behaviour of any of the steps or sub-processes of the system investigated. However, they will return the same numerical value than the correct and unkonw aggregate of functions. As a consequence, the real mechanims may be hidden until a sufficient range of contradicting experimental systems are found. However, the fundamental form of the real process may be found by careful analysis of the empirically aproximated forms, providing alternative sets of equations and asking what their physicochemical meaining could be. In the final analysis, this paper shows that a sustained preconception, even when it affects a system which can be easily controlled, may affect the understanding of a physical phenomenon. As a consequence, it hints at an even greater chance for error in the understanding of systems which are not controlled or contained, as those dealt with by high energy physicists or astrophysicists.
Supporting Materials. Precipitation of Ionic Surfactants: a surface reaction.
In this Supplement we are going to show that the decrease of the chemical activity of ionic surfactants as their surface excess increases can be linked to a progression of their aggregation from true monomers to two or three ion aggregates. This impacts on the solubility of the surfactants. The form of equations 1 and 2 is justified in this way.
Figure 8. Schematic views of solubility contours of ionic surfactants. (top) Residual surfactant concentration v total surfactant concentration for a given counterion concentration. (bottom) Solubility boundary.
where 0.79 nm2 is the area of a head of surfactant Brij, 3.93 nm2 is the estimate for truly monomeric SDS ions at the surface of the liquid when highly diluted in a non-interacting nonionic compound, and 0.045 is a value of XSDS for which the intersection with the lowest visual aid curve is found.
Figure 9. Experimentally estimated and calculated average head surface area of mixtures of ionic surfactant SDS and nonionic surfactants Tween and Brij. The two straight lines linking the series of two and three open square points are visual aids.
where 1.9 is a value attributed to an intermediate state, which is assumed to be M+n(DS)2, between truly monomeric DS and DS forming ionic pairs M+n(DS)3 or M+n(DS)4 as found in previous work on binding of counterions onto DS micelles.
The segments discussed can be explained away has being the consequence of a critical condition having been met which causes DS ions to be in a given state, and incidentally with a certain activity, which is uniform over a range of concentrations. However, the segments second from the left and rightmost require to be explained the respective expressions
Figure 10. Plotting surface excesses and solubilities of Ca(DS)2 against bulk phase concentration of DS. Particularly at low concentrations there is a surprising consistency between predicted transition points in ionic pairing and transitions in solubility. The vertical dotted line indicates the onset of micellization and of more complex colloidal interactions.
Figure 11.Model activity coefficients compared with activity coefficients estimated from the experimental solubility curves. The results confirm the model proposed in equations 1 and 2.
Keywords: ((surface excess · chemical activity · equilibrium constant · concentration plane · Lambert-Beer's Law))
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