Sunday, 25 September 2011

The Weierstrass Snare: On a fundamental flaw in solution chemistry

The Weierstrass Snare: How The Role of
  Surface Excess in Solution Chemistry Has

  Been Missed.

Federico I. Talens-Alesson © 2011
To the memory of my parents, Almudena and Federico.

Some G, g, and d symbols in the text must be read as Gamma (uppercase), gamma (lowercase) and delta (lowercase). While the blod editor supports many features, it des not seem to support Symbol font characters. The author.

The surface excess of chemicals in solution may be shaping several properties of solutes, like the solubility equilibria of sparingly soluble salts and the optical properties of solutions with significant surface excesses. Activity coefficients can be regarded as a ratio between the actual surface excess and the theoretical surface excess if the ratio excess to bulk concentration was maintained. A model is presented, based on the theoretical proposition by Petersen and Saykally that cationic and anionic components of Ray-Jones salts must form separate subsurface excesses. It regards such surface excesses as charged planes and considers that solubility products and other equilibrium conditions are based of a Bjerrum-like critical distance. If the surface charge density between the charge planes of the subsurface excesses is large enough, electrostatic forces cause the ions to approach and react. The model is consistent with the mathematical form of deviations from ideal behavior in colorimetric methods. It is suggested that non-zero order kinetic equations are also linked to both plane concentrations and surface renewal due to agitation. It is shown how the equations of this model have the same aggregate shape than conventional action mass models and their adjustments for non-ideal behavior, showing why misconceptions on the fundamental chemistry of many processes in solution have been covered by the fact that sufficiently effective predictive models were being developed based on wrong postulates.

1. Introduction
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.
Trivially, the result of altering the original function by incorporating another continuous function (e.g. a model refinement) would still be described by another polynomial function. Eventually, a highly refined model could be produced, and it would be possible to replace it by a polynomial series. After all, all physicochemical problems are investigated with the help of computers, and this is how complex equations are ultimately fed to computers.
As a consequence, if the initial function was chosen by making the wrong starting hypothesis, and the corrections are based on a misinterpretation of the problem, this will be disguised by the validity of the results. This may lead to complacency and the belief that a solid scientific knowledge has been attained. We can call this the Weierstrass Snare.
However, the polynomial approximation of the aggregate equation or function obtained will be also be that of the aggregate form of the real model for the system. Therefore, if a new model is proposed that deals with behaviours previously regarded as anomalous, as predictable and mainstream cases of the new model and the overall form of the new model is consistent with either the original or polynomial forms of the “numerically efficient” existing model, then there will be strong evidence to support the idea that the new model deals better with the description of reality.
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 [3]. 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[1]. 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.
Ionic surfactants are central to this study because their surface tension-modifying properties makes it easier to quantify their surface excesses. Their precipitation from diluted solutions takes place with the surfactant present as a monomer with a “normal” solubility product like, for example [2,3]
[Ca2+][DDBS-]2 = 2 10-11
[Al3+][DDBS-]3 = 4 10-19
[Ag+][DS-] = 6.3 10-5
[Cu2+][DS-]2 = 7.9 10-19
[La3+][DS-]3 = 6.3 10-14

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.
Micelles limit the precipitation of ionic surfactants by acting somehow as sequestering agents, and exist in solutions when there is enough surfactant to exceed its characteristic Critical Micellization. They are at the root of the unusual behavior of surfactants in solution, and it will be shown in Supplementary Materials that ionic surfactants are in fact a particular case of the general behavior of sparingly soluble salts by demonstrating that their solubility products can be linked to their surface excess.
It was recently proposed that the CMC is just the bulk phase concentration in equilibrium with a Critical Surface Excess of Micellization (CSEM) [4]. When reaching CSEM, the distance between surfactant ions at the air water interface the matches Bjerrum’s critical distance for ionic pair formation. That the trigger condition for micellization is a form of equilibrium reaction involving the surface excess of an ionic species is at the root of the model to be proposed here. For example [5] N,N′-Phthaloyl-bis(2-aminododecanoic acid) has a CMC of 1.8 10-6M, and a CSEM of 1.71 10-6 mol m-2, whereas NaDS [6] at 33oC has a CMC of 8.1 10-3M but its CSEM is still 3.33 10-6 mol m-2. Surface excess data at the CMC for many anionic, cationic and zwitterionic surfactants gave values consistent within 20% with Bjerrum’s distance, both above and below [4]. Therefore, in the case of surfactants an ionic pair-like assembly like micelles, found across the solution, is linked to a characteristic condition of its surface excess.
That the equilibrium leading to a reaction (like the formation of the complex or product “micelle”) is linked to a surface property and not a bulk phase property does not need to be restricted to surfactant. The solubility product of Fe(OH)3 is 2.79 10-39[1]. That means that precipitation will occur if there are more than 1.38 1014 molecules of Fe(OH)3 per litre of water. Or to put it in other form, if the volume per OH- ion is a cube of less than 1340 nm of edge, then precipitation will occur. It seems more reasonable to assume that the reaction takes place between far higher concentrations at the surface of the liquid and that the precipitate is too persistent to redissolve if it becomes dispersed across the solution.
When Petersen and Saykally [7] confirmed the Jones-Ray effect (the existence of a minimum in the surface tension of dilute solutions of some electrolytes around a concentration 1mM), in their modelling work they predicted the existence of separate anionic and cationic surface and subsurface excesses in a solution. The layers themselves are calculated to be several Å thick, with the cationic layer being thicker than the anionic. The 13 Jones-Ray electrolytes investigated include compounds like MgSO4, KClO3, NaCl, LaCl3 or K3FeCN6 The distribution can be regarded as two planes with average surface charge densities proportional to the surface and subsurface excesses of both the cationic and anionic layer (Figure 1). Such structure is similar to the structure of the surface excess of monofunctional ionic surfactants [8]. 
Figure 1. (left) Density profiles [1] 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.
Three ideas are central to the model proposed in this paper. First, that saturation of the surface surrounding the volume of liquid may limit the value of the surface excess G and cause a drop in chemical activity when the chemistry of the system depends strongly on G and not on the bulk concentration. Second, that free ions or reactant molecules, together with ionic pairs or product molecules forming at the surface are shuttled by mechanical agitation to and from the bulk phase, where they may remain stable or decompose due to solvation (Figure 2). Third, that products or intermediates shuttled back to the surface may experience ionic exchange reactions when they resume contact with the high concentration surface or subsurface planes.
The basis of the model is that, for the systems for which this model is relevant, irrespective of whether the surface excess is anionic [7] or vice versa [9], the counterion concentrations peak at a certain distance from each other. A likely explanation for the separate planes is that one of the species is only slightly soluble and its excess becomes pushed towards the boundaries of the solution by being excluded from solvent clusters. If the other species is more soluble, it may be dragged behind by the electrostatic attraction of the first. This must be countered by diffusional forces within the solution and its better solvation, which explains why it remains apart of its less soluble counterion.
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.
As the concentrations build up, the surface and subsurface excesses will raise until the electrostatic forces between planes drag together the ions and allow them to react. There are several possible outcomes of this condition. The salt will always be soluble if the reaction product fails to form due to the characteristics of the solvation spheres of the ions [10-11] or if it dissociates when transferred to the bulk of the solution due to strong tendency to solvate of at least one of the ions.
A salt will also be very soluble when the combined surface excesses of both counterions make it hard to meet the condition for electrostatic attraction. When both counterions have high surface excess to bulk phase concentration ratios and their salt is stable, they will be slightly soluble. Of course, while some salts may precipitate solely at the surface/subsurface of the liquid, more soluble salts may start precipitating there and then seed bulk phase precipitation. Any solid formed anywhere in the solution could become the source of new surface and therefore new surface excess, providing and an additional chance to react. That would be the interpretation of crystal seeding within this model.
A saturation surface excess value for ionic surfactants [7] would be in the order of magnitude of 3mmol m-2 contained in a layer of significantly less than 5 nm. The concentration would therefore be higher than 0.1M. A likely order of magnitude of bulk phase. e.g. surfactant concentrations in equilibrium with such values (e.g. a critical micellization concentration) may be 10-3M to 10-7M. Such surface excesses are very similar to values around 1.25 mmol m-2 found for cations of even highly soluble inorganic salts like NaClO4 as reported by Lopez-Perez et al [9]:. It is clear that reaction (e.g. micellar assembly) in the bulk phase is less likely to happen than reaction in the surface excess region.
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 [12]) and other devices. This renews the contents of the liquid surface [13], 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).
If the relative concentrations in the surface/subsurface layers are substantially higher than in the bulk phase, then the intermediate region between the two peak regions becomes the volume where reaction intermediaries and final products form, because the kinetics will be much faster.
Figure 3 suggests that the distances between species in the bulk phase may simply prevent any reaction to take place there. In such case, most kinetic reactions are likely to include within their frequency factor an element accounting for the rate of transfer between bulk phase and interface under ideal mixing conditions.
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:
Note that the value of k is irrelevant for the calculation of g. For AgNO3 in Figure 3 (top), the parameters for the prediction of the activity coefficient are:
C≤ 9 10-4M g=1
C 1.05 10-1M CThreshold = 9 10-4M a=1 b=0.9
C 2.53 10-1M CThreshold = 1.05 10-1M a=0.9 b=0.3
C> 2.53 10-1M g=[2.53 10-1M]/C
The values have been chosen manually, simply to demonstrate that a progressive decrease in the ability of the solute to deploy proportionally at the surface would explain the evolution of the activity coefficient. The visual aid lines show the changes in slope in the activity of AgNO3. The trend is consistent with a final situation where G cannot increase further but there is no change in the structural nature of the surface excess.
For the AgNO3 case, replacing the system above for a Langmuir isotherm yields a smooth functional variation (not plotted), but still gives a consistent trend for the evolution of the activity coefficient. For CaCl2 in Figure 3 (bottom), the parameters for the prediction of the activity coefficient are:
C≤ 9 10-4M g=1
C 1.25 10-1M CThreshold = 9 10-4M a=1 b=0.9
C 2.07 10-1M CThreshold = 1.25 10-1M a=0.9 b=0.3
C> 2.07 10-1M CThreshold = 2.07 10-1M a=0.3 b=50
Figure 4. Mean activity coefficients for AgNO3 and CaCl2, [10] and a prediction based on an estimate of the ratio between effective surface excess and theoretical surface excess.
In the case of CaCl2 the prediction is less accurate, which is to be expected because the model tries to adapt to a change in the nature of the surface of the liquid. The parameters propose a superactive surface excess by means of b = 50 above a certain threshold concentration. This may represent the development of a ionic liquid-like nanolayer or a condition where one of the species is replaced by an intermediate (e.g. an ionic pair) and becomes more reactive. Activity coefficients at concentration of 15 mol kg-1 like 33.8 for Pb(ClO4)2, 43.1 for CaCl2 or 323 for HClO4 [1] may be explained in this way.
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.
If ideal activity relates to linearity between surface excess across all the surfaces of the liquid and bulk concentration, then the transmittance/concentration plots must actually be describing the transition from linear to non-linear relationship between surface excess and bulk phase concentration, and the region of compliance of Beer's Law must match the region where the surface excess/bulk phase partition is linear. Otherwise the model would be inconsistent with actual observations. Within the region of compliance with Beer's Law [14] and linear proportionality between bulk concentration and surface excess, total and bulk concentrations and surface excess follow:


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:
As, S/V decreases with an increase in the optical path length, for a given total concentration the bulk phase concentration will increase with the optical path length because less solute is required to provide the surface excess. As a consequence a sufficient increase of the optical path length would increase the surface excess of the liquid until saturation. This is observed in the work by Yu et al [15], where merely by enlarging the cell's optical path a non-Lambert Beer region is reached. If we consider the total contribution to the absorbance as made up by the contributions of the bulk phase and the surface.

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.
The “no saturation” plot combines equations 5 and 6, assuming CTOTAL =1 and K = 1. The extinction coefficient is irrelevant as the plot is for the ratio between the absorbance at a given optical path length and that at 0.5 mm. The “saturation” plot assumes that at 2.25 mm, the surface excess G=kCBULK becomes constant because the surface-volume ratio has already dropped so much that the surface excess becomes constant. The “Langmuir isotherm” line shows that using an isotherm instead of Equation 3 predicts an earlier drop in surface concentration and absorbance.
Within the scope of this paper, this would confirm that when an adsorption isotherm can be used to predict the dependence of surface excess with bulk concentration the system would already be non-ideal. We should keep in mind that the surface excess may be very dense. Petersen and Saykally [7] proposed in their model that the anions of the Ray-Jones electrolytes may constitute the whole of the solution in the layer where their surface excess peaks, even with the nominal concentration 0.001M relevant to the Ray-Jones behavior.
Therefore, while molecules in the solution may be randomly oriented when hit by a photon, part of the reason for high light absorption may be optimal orientation of the molecules in the surface excess to capture incoming photons, whereby there would be a bulk phase molar extinction coefficient eBULK and a surface excess extinction coefficient eSURFACE, with eBULK eSURFACE. This is illustrated by using equation 6 for a prediction, unadjusted for surface saturation, but for which it is assumed that the eSURFACE = 1.25 eBULK for the surface excesses found above the optical length 1.25 mm. That explains the change in slope at higher path lengths.
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.
However, within the region where Beer's Law holds, it will not be possible to discriminate the impact of the surface excess except through changes of the surface/volume ratio. This would be independent of whether the system's light absorption is controlled by the surface excess, by the bulk phase or a combination, because in all cases it would be a linear function of CBULK. The role of the chemical equilibrium in the solution is obviously important to establish the importance of the surface excess. Buijs and Maurice [16] provide absorbance values for solutions of Pu(IV) in the presence of different concentrations of H2SO4.
At their highest H2SO4 concentration (4M) , for cells of length 1, 2, 5 and 10 cm, the absorbance ratios relative to the 10 cm cell are 0.13, 0.25, 0.58 and 1, which are consistent with the asymptotic increase in S/V to when the optical path length tends to 0. This is consistent with the results by Yu et al. For other samples with lower concentrations of H2SO4, the absorbance ratio is more or less constant, suggesting that the contribution of the surface excess is directly proportional to the bulk phase concentration and that the model where the surface excess plays a role and the classical approach cannot be discriminated.
It should be mentioned that Buijs and Maurice describe some equations for the prediction of deviations from Lambert-Beer 's Law based on relationships between optical length, concentration and absorbance of the form: 

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 [17]. This is further evidence of the connection between ideality as a consequence of linear dependence of surface excess and concentration.
Having shown that surface excess may have an influence on light absorption, that divergences in absorbance from ideal behaviour may be explained by saturation of the surface excess, and therefore that surface excess should be regarding as playing a major role in the chemistry of all surfaces of a liquid facing any other surface other than air, we are going to discuss the consistency of the proposed plane reaction model with existing equilibrium and kinetic equations.

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 [18], 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.
In this particular A + B = AB case, it makes sense to assume that a “prime intermediate” plane physically exist, where species A+ and B- migrate due to electrostatic attraction to reach. With this scheme, it makes sense that if any of the plane charge densities is initially too low to match the critical distance, precipitation will not occur. Precipitation will cease when both the A and B planes are depleted until their inter-plane electrostatic attraction is too low. The model also explains precipitation when two chemicals are present in non-stoichimetric concentrations: the force of the electrostatic attraction depends on the product of both surface charge densities.

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:



Rearranging 9 it can be seen that the product [A2+][B-]2 is equal to a constant value under some conditions: independence of Keq from surface excesses and linearity between surface excess and bulk phase concentration.
This migration between species planes and subsequent species plane generation would explain also any precipitates with more than one cation or anion and multiple precipitation options depending of the temperature and composition of the mixture, like alum (KAl(SO4)2) shown in Figure 3(right) [19]. Depending on the specific conditions of an experiment, either K2SO4, Al2(SO4)3 or alum may be the less soluble species, and precipitate. Although Figure 3 (right) does not show more detail, it is obvious that other competing “plane” paths include KSO4- + K+ to give K2SO4 and SO42- + AlSO4+ to give Al(SO4)2- followed possibly by AlSO4+ + Al(SO4)2- to give Al2(SO4)3.
A multiplanar arrangement (beyond three planes) would be a simplified description of something more complex. If an species of charge intermediate between the two original planes deploys between them, then it will partially shield them and allow them to be more separated from each other. That could allow further planes to exist. Such evolution would conform with the multi-plane idea. Alternatively, the intermediate plane may become the site for all the intermediate species and the locus for all the reactions between the initial species and the final product. The mathematical forms would be anyway equivalent.
Before extending the analysis to other types of reactions, with equilibria involving initial and final product concentrations, we will discuss how the definition of mean activity coefficients fit with this model, and also the way in which the nature of the surface/bulk partition would cause it to be masked by photometric methods is discussed below.
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 [18]


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.
However, for reactions like complexation, where the product may be a charged species, the equilibrium condition does not need to be depletion until the surface planes do not have enough charge to attract the species. The condition may be simply repulsion between same-sign adjacent planes (E.g. A+2 and AB+) cancelling transfer between them. This would suppress the ability to further reaction between incoming reactants from opposite sign planes and eventually terminate transfer. If we consider the total distance as dA/B = dA/0 + d0/B, and we define the three critical distances for equilibrium (Figure 7, top) in terms of their plane charges s as:


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:


This provides the equilibrium constant with a shape similar to an adsorption isotherm. This is a shape consistent with correction factors for non-ideality in solubility products derived from the Debye-Hűckel model [20]. The Debye-Huckel model does not consider surface effects, but nevertheless the isotherm equation form appears in a potential form in the corrective term for non-ideality [21]:

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. [21], 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.
A recent paper [22] on an accurate model for binding of cations onto anionic micelles by ionic pair formation showed that one obstacle for earlier recognition of the basic chemical process being a complexation-like reaction and not an adsorption is that the equilibrium constant and the adsorption isotherm can be found to have an analog underlying mathematical form. The fact that the approach presented here shows the same mathematical form than existing corrections for bulk-phase models suggests these are implicitly corrected to account for the surface excess factor.
2.7. Kinetics.
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.
If we consider the total volume of the solution V equal to ndA, where d is the thickness of the surface-subsurface layer, A the surface of the fluid phase and n the geometrical proportionality factor with the volume, we can define the bulk phase concentrations of the various species Ci once equilibrium has been reached as:

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:



We can relate the frequency factor in the kinetic equation to the renewal term A/(VtR) based on Higbie's surface renewal model [13], where tR would be the time required to replace the fraction of liquid within the surface layer and the k constants can be related to the activation energy in Arrhenius constant.
The overall rate of reaction may also depend on the rate of the reaction within the plane, if tR is shorter than the time required for the reaction to reach equilibrium. In such cases, there will be a progressive increase in the concentration of products as successive re-entries of chemicals take place. In all those cases, it is proposed that the products do not dissociate in the bulk of the liquid: dissociation in the bulk would be related to processes where the product is unstable and overall the process would be considered as no-reaction.
In chemical reactor engineering poor mixing areas are accounted for as “dead regions”[12] in compartmented reactor models, where part of the reactor volume is supposed to behave ideally, part of it is supposed to be stagnant (“dead”) and some is assumed to be a recirculating volume: reinterpreting “dead” regions as regions that are disconnected from surface renewal and thus denied access to a reactive surface region maintains consistency between the viewpoint of the model and experimental observation.

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.
Resistance of the surfactant to precipitation beings before micelles form. Figure 7 (bottom), closely inspired on work by Somasundaran et al [23], shows a generic logarithmic plot counterion concentration-surfactant concentration showing the precipitation phase boundary. On this figure, already below the CMC there is a region where the concentration of counterion required for precipitation increases from the linear precipitation region, and therefore resistance to precipitation is already increasing. The surface area of surfactants can be estimated from their surface excess at a saturated interface (CMC or CSEM).
Any transitions in aggregation below saturation would be unobservable for a pure surfactant, and in mixed surfactant systems the norm is to estimate an average surface area per surfactant head, as it is hard to elucidate interaction between surfactant molecules and ions. However, occasionally there are some systems which can provide more information. We will show that there is a change in activity linked to ionic surfactant pair formation at the surface of the liquid which can be linked with increased resistance to precipitation. The resulting expression is then used in the main text to justify that chemical activity is linked to surface excess generally, and not only in the case of surfactants.
Figure 9 shows two datasets by Gosh and Moulik [24], plus some visual aid lines and correlation results. A series designated as Tween/DS corresponds to a mixture of an anionic monovalent surfactant sodium dodecylsulfate (SDS) present in solution as DS (dodecylsulfate) ions and a nonionic surfactant molecule. The peculiarity of the data is that it agrees with a linear relationship:
0.58 + 3.91 XDS with R2 = 0.9594. (23)
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.
The value 0.58 is the area of the head of the non-ionic surfactant Tween, in nm2. 3.91 is the area of the head of ionic surfactant DS, which is odd because for pure DS in a saturated surface excess the same authors find the value 1.07. That is, DS is more compact when found at the condition where Bjerrum critical distance is reached (and therefore where it is found forming ionic pairs) than when it is very diluted in a nonionic surfactant which does not interact with it (hence the linear relationship).Figure 9 shows equation 21 as a dashed line and the data for Tween. It also shows a line broken in four segments corresponding to the calculation of average head surface for the Brij/SDS system. The leftmost is related to:
Molecular Surface area = 0.79XBrij+3.93XSDS (24)
for XSDS < 0.045

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.
The equation therefore describes the region where the surface area of ionic SDS per head of ion is constant. The small segment of the Brij/DS broken line, second from the right, that connects the two visual aid lines corresponds to:
Molecular Surface area = 00.79XBrij+1.9XSDS (25)
for 0.146<XSDS < 0.164

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





In equation 24 the contribution of the ionic surfactant in its true monomeric form with surface area 3.9 and of the first intermediate ionic pair with surface area 1.9 are weighed liearly over the range XDS 0.045 (the pair 1.9 appears) and 0.146 (the pair 1.9 is the only form).
In equation 25, we consider the range XDS to be from 0.164 to 0.3 and the surface 1.33. There are no further data to the right except for XDS = 1 where the surface area per head of DS is 1.07 nm2, but the XDS = 0.3 provides a reasonable fit with the surface area 1.33. The value 1.33 nm2 is picked as the area per head of a M+n(DS)3 because it seems a reasonable progression from about 4 nm2 for a truly free monomer to about 2 for each of the supposed 2 monomers more closely packed in a M+n(DS)2 group and on to about 1 (1.07 nm2) for the highest stoichiometry M+n(DS)4.
Having shown that the average surface per head of surfactant can be calculated on the basis of a constant contribution of the nonionic surfactant and a contribution of the ionic surfactant that varies as its partial surface excess raises, we plot the partial surface excess of the ionic surfactant (triangles) together with the solubility curve for its insoluble calcium salt [25] in Figure 9. The open triangles correspond to the partial surface excesses of DS at the transition points of its various surface areas. The plot has a similar shape to Figure 8 (bottom). Four arrows are depicted to show there is a change in slope in the solubility curve, which can be linked to the transition surface excesses. The surface excess which would correspond to the transition from 1.33 to 1.07 falls short, but it is an speculative value, and also at higher molar fractions of DS the difference between micellar molar fraction and surface excess molar fraction would increase. The conclusion is that the evolution of the resistance to precipitation and therefore of chemical acitivity is linked with a change in ionic structure at the surface.

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.
We will demonstrate now the physiochemical foundation for the equation system used in the main text. With the three leftmost points, we obtain an empirical correlation:
[Ca2+][DS]3.0604=2.0811 10-13 R2=1 (28)
The equation is peculiar because the insoluble compound is obviously CaDS2. The result has been carefully double checked having extracted the data from the original figure and used the pixel scale of a graphic package to obtain from the lengths of the X,Y scales and the positions of the points the actual values with high precision. A possible explanation is that the precipitation of CaDS2 requires to go through an intermediate ionic group like Na+[Ca2+(DS-)3]- which would form at the interface, where the local concentrations will always be higher, and decompose in the bulk phase due to the lower local ionic strength. The slight difference between the exponent and 3 can be just accumulated experimental error. Irrespective of the reason for the form of the correlation, if we attribute to Ca2+ a constant g = 1 through the range of concentrations and to DS a value of g = 1 only in this first series, we can estimate the activity coefficient for DS in the other points of the plot as:
gDS = (2.0811 10-13/[Ca2+])1/3.0605 (29)
For the three series considered non ideal (the three rightmost arrows in Figure 10), we proceed to calculate an estimated activity coefficient based on the following scheme:
  • gTB = 1 is chosen from the last ideal point as Threshold Base activity coefficient. The corresponding concentration is chosen as Threshold Concentration CThreshold.
  • A seed gGG value is chosen for the last point of a linear segment as Goal activity coefficient, with the concentration there becoming the Goal concentration CG. It will be iterated to find the actual value of gGG in the equation
  • gC = gTB Cthreshold/CG + gG (CG - Cthreshold)/CG, (30)
which will provide the values for all the concentrations between the Threshold and the Goal points, assuming that those two values are the activity coefficients of the only co-existing ionic states in the segment.
  • For the next segment gTB = gG, and the process re-starts.

Figure 11 shows the comparison between the estimates obtained from Equation 27 and the predictive calculations. The increasing divergency for lower activities reflects formation of surfactant micelles and further non-ideality inducing factors. Therefore, the concept of critical conditions causing a fundamental change in the characteristics of an ionic species (ionic pairs, ionic liquid layers) pre-existing condition and an emerging condition which will become predominant provides results that are consistent with observation.

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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  1. That looks really good.

    Is the MathCAD sheet available? It might be something worth posting on the MathCAD forum as I'm sure that many of the engineers who visit there would be interested.

    Philip Oakley

    1. I have to admit I am MathCAD useless. Well, in my time I was Mathematica useless, too. You can kick around the idea of doing something about it. (which is what I want, really). The idea is that solution chemistry happens at the interfaces. That is quirkier than it looks, because it also means that most of the heat release in exothermic reactions happens actually close to the boundaries of the
      device. One of the non-obvious consequences is that if you create a high surface reactor you need to watch out if your heat release is properly sized up.

      By the way, thanks for posting.