Saturday, 8 March 2014

Can surfactant micelles cheat solubility products?

©2014 Federico I. Talens-Alesson

What we are going to discuss here is whether a charged surface changes the solubility of a salt. If you have followed my blogs, you know that I postulate that solution chemistry is (mostly) the chemistry between of surface excesses of chemicals. A normal boundary to a solution is either the air/water surface or a wall (the container's) or a probe. However, seed crystals and colloidal particles would also be surfaces against which a surface excess may develop. This is the underlying reason for all sorts of fouling and cryytal growth/nucleation phenomena.

In the particular case of a micellar solution of an ionic surfactant (which is a type of colloidal system), it also demands saturation of the air/water interface with surfactant (which means that it also demands saturation of any other surface) forming a charged surface layer. At the time I was still publishing in scientific journals, and I went as far as to show that the critical conditions for micellisation is ionic pairing between the surface excess of an ionic surfactant and its counter ion's surface excess (Talens-Alesson, Langmuir 2010, 26(22), 16812–16817)

However, as the surface excess and concentration are related, I also showed that using Bjerrum's correlation it is possible to estimate the relative bindings of different polyvalent cations onto anionic surfactant micelles. Biding of cations onto anionic micelles is a particular case of ionic pairing, and ion pairing is key to precipitation and other reactions between electrolytes. It is through forming of ionic pairs that the building blocks for crystal assembly take place (Talens-Alesson, J. Phys. Chem. B 2009, 113, 9779–9785).

In short, the higher charge counter ion predominates and for as long as the half length of the hedge of a cube of volume (volume of solution/concentration of higher charge counter ion times Avogadro's Number) is less than Bjerrum's distance for ionic pairing, then this counter ion would bind onto the micelle. This works well for the binding of Al3+ and Zn2+ and subsequent flocculation of the surfactant SDS (sodium dodecylsulfate).

This required assigning an apparent charge of 5e- to the “counter ion adsorption patch” of an SDS micelle. This was required even though the binding of counter ions onto an SDS spherical micelle follows a ratio 1 counter ion (regardless of charge) per three SDS molecules in the micelle. This means that the binding attraction is more than it should be expected. And this brings us to the question of whether enhanced interaction of electrolytes near micelles is possible.

In practice, research on ionic surfactants focuses on how they help other chemicals (essentially organic ones) to dissolve through micellization. It also focuses on the solubility of ionic surfactants themselves in the presence of a range of counterions. The latter has to do with the presence of certain cations available (and therefore in soluble form) either in nature (soils, for remediation or oil recovery, for example) or in water in technological applications (detergency). Of course, how the surfactant may affect the solubility of something already precipitated seems a rather bizarre subject of research, particularly if one believes that chemical reactions in solution are bulk phase phenomena. Of course, I don't. And it should not be so difficult to realize that a broader investigation of how micelles affect chemical reactions should take place, considering that there is some evidence in its favor. An example of a micelle-enhanced Fenton reaction is given by Talens-Alesson (Chem, Eng Technol 2003, 26(6) 684-687).


Let us consider an aged data set, dating back to 1994, by a team from the IAST at the University of Oklahoma in Norman (Scamehorn et al., Separation Sci Technology, 1994, 29(7) 809-830). It describes binding of single cations and their mixtures onto micelles of anionic surfactant SDS.

In Figure 1 I plot my own calculation for experimental binding ratios for the cations investigated by the researchers at IAST, following:
total cation concentration minus final cation concentration in solution
one third the difference between total SDS concentration minus final SDS concentration in solution

together with my own predicted 'Bjerrum's” binding ratio as discussed above. There is a constant offset between the actual binding and the predicted one using Bjerrum's correlation.

If Bjerrum's correlation was correct, then binding would be lower than it actually is. But then, Bjerrum's correlation is derived on the assumption that solution chemistry is bulk phase concentration chemistry. Therefore, it is possible that there is a condition for diluted concentrations still matching the true critical ionic pairing condition at the surface excess regions. But this means that ionic pairing can occur well below Bjerrum's condition for ionic pairing: it would occur for the “surface excess” condition for ionic pairing, which would be masked by Bjerrum's use of bulk phase concentration.

This deviation is consistent with the fact that above saturation the surface excess is constant: above the saturation surface excess the value is independent of the concentration, and therefore the condition for ionic pairing appears to be a constant. The additional binding at a lower concentration would be a constant amount, related to the difference between the saturation surface excess and the critical surface excess for ionic pairing. Figure 2 shows this.

This does not imply that the linear relationship can be stretched indefinitely: Figure 3 shows some data by IAST researchers for 4mM counter ion concentrations, below a Bjerrum's concentration (would be 4.5 mM for a divalent cation against an SDS micelle). In these experiments the variation is in the SDS concentration: from 100 mM, to 200 mM and on to 400 mM. The concentration of counter ion is always 4 mM. As the concentration of surfactant increases, the binding drops which is consistent with the fact that there is more surfactant colloidal surface for the same amount of counter ion. As the ratio surface to volume increases, with the ratio surface excess to concentration being basically constant at low concentrations, then the total amount of solute is distributed over a lower surface excess matching a lower concentration. The binding ratio decreases but the total binding slightly increases. Binding of counterions onto ionic micelles may therefore exceed theoretical values and micelles are attractors of counter ions.


Paton and Talens (Langmuir, 2002, 18 (22), pp 8295–8301) describe competitive binding of Al3+ and Zn2+ onto flocculating SDS micelles. The results indicate that the total charge of cations exceeds that of surfactant in the floc in some cases, with apparent charge ratios higher than 1 and sometimes up to 1.8 (Figure 4). The only explanation is ionic pairing between said cations and sulfate anions present in the solution. This leaves them as lower charge species and more of them are required to reach electroneutrality around the micelles.

This means that the flocculate is a mixture AlxZnyDSzSO4t. It is important to notice that the concentrations of all the chemicals are below their solubility limits. Aluminum sulfate and zinc sulfate are soluble at higher concentrations than the ones the flocculate forms. The excess chemicals are “opted in” into the flocs irrespective of their solubilites.

This means that a discontinuity in the solution, with the ability to cause enhanced adsorption of counter ions, may lead to enhanced capture of ionic pairs/salts. In a subsequent paper it was proposed that ionic pairs AlSO4+ and ZnSO40 may co-adsorb onto the micelles (Talens-Alesson, J. Phys. Chem. B 2009, 113, 9779–9785), explaining this apparent charge inversion. However, this explanation does not change the fact that well below their saturation conditions, fragments belonging to the Al2(SO4)3 and ZnSO4 salts are incorporated onto the flocs.

This is a peculiar experimental result, as it is reported as part of a fraudulent effort consisting in two papers in which staff at the University of Nottingham (one of them eventually moving to Oxford) plagiarized work of mine to pretend that they were active in the research of micellar flocculation.

In the second of them, published in Separation Purification Technol (2008) they claim flocculation of AL(DS)3 from solutions in which the residual Al and SDS concentrations they claim to have found (far below normal results in micellar flocculation of Al(DS)3) could only be explained by a strange charge ratio of around 4/1 between Al and DS (assuming the concentrations given where Al3+ and not aluminum sulfate, in which case it would be around 8/1). While in pH adjusted solutions it is possible to have rather high ratios Al3+/SDS (2 to 1), this only happens because in the adequate range of pH Al13 is present. The authors of this document do not state such pH adjustement, and the concentrations of surfactnat and SDS do not justify the charge inversion, as observed in previous work. A number of documents on the case of this story on plagiarism and fraud are available on the net.

However, there is a question: considering that various other statements make absolute no sense (stating that benzoic acid is a reagent for the analysis of anionic surfactants in two phase titration, that the formula of aluminum sulfate is AlSO4, or that phenol is an alkali, or that the Critical Micelle Concentration of a surfactant can ever be tolerable in a water targeted for processing for drinking purposes) even though the result seems unlikely there is no obvious reason why the authors, in their overall ignorance, should realize that the expected Al concentration would be unreasonably high considering they expect to remove a concentration of phenol which would be unrealistic (too low for an attempt to recycle/recovery). In principle this should be an irrelevant question if the experiment was a regular micellar flocculation one.

But there is a peculiarity in this experiment. Unlike previous work, it was carried out while air was bubbled through the solution as flocculation took place. This presents an interesting question. It is known that foam can be used to remove the foaming surfactant but also other chemicals present, adsorbed on the surfactant layer of the foam. There is for example work on the removal of moderate amounts of phenol in SDS foams.

What if the foam actually provides a barrier that may be crossed by ionic pairs or ionic molecules? What if on the air side they can remain for long enough for a shift in the effective equilibrium of solution, becoming an electrolyte sink? Figure 5 illustrates the idea.

On the grounds of previous information it is suggested that foams may assist to cause an “enhanced” insolubilistion of salts, which a variety of potential technological applications.
The way in which this may be brought to happen would be to use a micellar solution of surfactant, and cause air to bubble in the presence of the electrolytes targeted to produce the desired insoluble salt.

Saturday, 9 February 2013


Although the first section is mostly a repeat of a previous blog, I think it is better to include it. The proposition is that ignoring the dustortion of the surface excess in solution chemistry has an impact in creating perceived "non'idealities" and "experimental" and reproducibility errors. Such errors would not exist, but be the consequence of assuming bulk phase models to be true. This distortion plus precision errors and real human errors would be the sources of non'reproducibility and dispersion of data.


After showing in previous blogs that that the surface excess model of solubility lends itself to predict the existence of a constant solubility product while a bulk-phase concentration model would beg for a variable solubility depending on initial concentrations, we are going to show how observed variations in the absorbance of samples with length of optical path can be explained by a surface excess model (Talens, 2011). Let us assume two spectrophotometric cells, both 50 mm high and 10 mm wide, with a depth of 1mm and 10 mm. It is trivial that this can lead to a lower concentration in the narrower cell due to the higher “stress” caused on the resource (the number of molecules) by stretching the surface per unit of volume. It only takes k to be a significant number, which it often is.

If we reconsider the expression for the light absorbance of a solution, to include both surface excess (twice, as there are two surfaces involved in the absorbance test) and bulk phase concentration and take the 10 mm depth cell as reference

the distortion expected for the 1 mm depth cell is:

One thing explained by the 2 weight factor for the surface excess contribution to the absorbance is why often reducing the thickness by a factor does not lead to the same drop in absorbance. Of course, this has been explained away by entrance effects etc, but the point here is that certain levels of intrinsic human irrationality (adherence to magical explanations, gremlins, and some form of “original sin by proxy”, assuming man made artefacts are meant to be intrinsically flawed because, well, they are man made) may be interfering with our better judgment.

There is a potentially very important implication in this example about colorimetric reading distortion by surface excess. Do surface excesses distort readings in general? If a chemical reaction takes place at the surface of a solution primarily, and we are controlling the temperature with a thermometer which generates its own surface, is the local temperature different than the bulk temperature one? How many of our readings are surface property readings, and introduce a distortion when we assume they are bulk phase? Does this influence scale-up factors in the chemical industry, for example? Are our Platonic “Myth of the Cave” shadows on the surface of liquids?


The second aspect of this discussion is about reproducibility. If the end point of a tritation was really the condition where the content of a certain chemical has been depleted so that its surface excess is not high enough to keep reacting, then some of what we call experimental error may be just a device - related  dimensional adjustment factor. Equation Set 2 shows a set of equations. Equations a) and b) indicate the equations for the linear portion of the relationship between surface excess and concentration of titration target T and titration standard S at the end point  “ep”. The initial number of moles of the target T will be equal to the titrated (reacted) moles plus the residual moles at the end point (equation d). The residual moles will be equal to the surface excess time area of the volume plus bulk concentration in equilibrium times volume  (equation d). Equations e) and f) show that if the area changes and the volume remains constant the surface excess and bulk phase concentrations of the target change. This may seem strange because of the stated hypothesis that the surface excess dictates the end point and should be equal, until we remember that this area/volume change also affects the titration standard, and that the condition for end point is not a surface excess value but the product of two of them (equation g).

Equations h) and I) give trivially the final concentrations of the titration standard for a titration with 1:1 stoichiometry , which would be the titrated concentration of the target plus the end point concentration of the standard. Equations j) and k) show that, for a substance AnBm assumed to have a conventional equilibrium constant it is perfectly reasonable to expect a discrepancy in the experimentally calculated value derived from a dimensional setup distortion (equation l). With standard glassware nowadays it can be expected that such distortions will be smaller than in the past, and with chemicals where the surface excess is smaller the distortion will be less important, but nevertheless the prediction can be used to verify of this theory.

Plato, “Book VII” in: The Republic


© 2013 Federico I. Talens-Alesson

Solubility in the bulk phase?

For AlPO4, the solubility product Ksp equals 9.84 10-21. Apparently, it is not possible for around 10-10 mol of such salt to be dissolved in 55 mol of water. This means that up to 6 1013 molecules of salt can be dissolved in 3.3 1025 molecules of water. This means that a single molecule would break up and saturate the volume occupied by 0.5 1012 molecules of water. When the aluminum cation and the phosphate anion are each surrounded by a pack of one million molecules of water, then the surrounding 499,998 packs of one million molecules of water would be unable to hold a single aluminum or phosphate ions. It is hard to believe that a bulk phase model can explain this system. We can take it as an(other) exception to the general principles of chemistry, or we can assume that it tells us something is wrong.

On the other hand, let us consider a cube with a 10 cm edge. Yes, containing one liter of water and 6 1013 molecules of aluminum phosphate. The surface of the cube is 600 cm2, or 6 1018Å2. Let us assume that formation of ionic pairs Al:PO4 against the competition of solvation at the surface of the liquid decides whether precipitation occurs or not. Are the cations and anions within range of each other for ionic pairs to form? If we use Bjerrum correlation for the critical distance for ionic pair formation (it is a bulk phase equation, so it could still be biased, but lets go for it) we obtain a critical distance of 32.13Å at 25C (the distance would be 3.57 ∙│+3│∙│-3│). The surface controlled by every ion would be that value to the square, around 1000. And there would be two ions to the molecule, giving a total “area coverage” of 1000 ∙ 2 ∙ 6 1013 = 1.2 1017 Å2.

It still falls short by an area factor of 50 or a distance factor of 7, but it seems less of an act of faith to rationalize this discrepancy, particularly having considered the ions dimensionless. For example, if a shorter inter ion distance does not allow for solvation spheres to remain efficiently formed between phosphate and Al3+ ions, then this would explain why the total concentration required to reach the solubility product does not need to comply the Bjerrum correlation even if all of it is at the surface. Or maybe the model can still be wrong. But the feeling that the balance of probabilities might be in favor of something closer to the surface excess model than to the astronomical numbers of the bulk phase model is still there.

A plausible alternative: surface excess model for the solubility of a salt.

Let us assume that the first requirement for a salt to reach saturation is that, if present as a solid, the rate at which its ions will react with a solvent and becoming solvated and breaking away is balanced with the rate at which its surface excess will cause precipitation. Let us assume that it will precipitate wherever a sufficient surface excess happens anywhere in the solution, included the surface of any solid salt present. Notice this does not necessarily mean the air water surface: could mean any surface, like that of seeding particles or already formed crystals. Or a pipe wall or a membrane/filter. It is reasonable to assume that the surface excesses for each counter ion in a salt will be on average deployed at different distances of the surface as the solvation number will be different for both ions. Some research on the Ray-Jones effect ,a minimum in the variation of surface tension with the concentration of certain electrolytes at a concentration 1mM (Petersen and Saykally, 2005) suggests such offset would be part of the explanation of the effect.

In practice, it could be looked at as some sort of parallel surface excess planes containing each of the electrolytes (below) . Of course, the molecules and ions in this surface layer will be renewed by agitation, which is nearly omnipresent in chemical experimentation. This may also have an effect on which counterion is rejected faster by the solution by denying to it solvation water (or the relevant solvent in the case of a non-aqueous system).
Anions and cations circulate due to agitation between the bulk phase and the surface and back. If the associations are only stable in the high concentration surface region, they will dissociate, but why should they otherwise?

This model depends on the electrostatic attraction between the ions overcoming the separation between the planes containing the surface excesses (below) for reaction to occur.
The density of particles increases the electrostatic attraction: the difference between attraction (left) and complete separation (right).

When the surface excesses reach certain values and raise the surface plane “charge density” , then some of the ions may migrate towards each other and form ionic pairs of lower net charge. This may reduce the electrostatic attraction between the surface planes to the point that the migration does not happen any more and there is no reaction. This may have resulted in precipitation, or merely in the formation of an intermediate soluble species. The products and unreacted reagents migrate to the bulk phase due to surface renewal caused by agitation. There they would remain in whatever state they acquired at the surface until they re-enter the surface excess region. Because, if an ionic pair forms, why should it break up in a diluted solution. where the solvent could not prevent the pairing in the first place? And if it breaks up, because it was only the local high concentrations that drove it to form, why should it reform in the bulk of the liquid?

Of course, we know that some state of flux between species exists in solution. But this could be explained simply by a competition between surface excess components to exchange with a component of a particular ionic pair as it re-enters the surface excess region, and not a bulk-phase phenomenon (below).
”a” represents a kind of ions present in one of the surface excesses, “b” represent the counter ion present in the other. The figure shows how an already formed aggregate, present in the bulk phase, may re-enter the surface region and exchange part of its components.

How we would notice the difference? The surface excess would be up front the sensor, unlike the bulk phase concentration. If the condition for precipitation is that a given electrostatic attraction pulls together the ions on both concentration planes, this is actually independent of the individual superficial concentrations, and only depends on the product of the surface charges reaching a given value. This formally explains the solubility product of a salt: different values of surface charge density at both planes can yield the same electrostatic attraction, required for the reaction to take place. This can be seen as a product of concentrations, because the surface excess is a function of the bulk phase concentration (below). The detailed discussion, expanded to equilibrium constants and kinetic equations can be found in The Weierstrass Snare post.
If the attraction is electrostatic, it is irrelevant whether the charge density is contributed by one or the other of the ions suitable to cause the reaction.

Objection to an accepted truth: A bulk phase solubility condition without cheats.

The main difference between a bulk concentration and a surface excess model is the absence of a singularity: the condition must be met throughout the solution. We still should assume that ionic pairs must form as preliminary or be a trigger condition for precipitation. Therefore, that some form of critical distance between counterions must exist, even if it is not Bjerrum's. Extremely low solubility products, which can never be explained with final concentration links to ionic pair distances, will be dispenses with considering that their existence as precipitates is a consequence of the initial stages of mixing: after all Al(OH)3 may precipitate with different forms and very different solubility products, purely as a consequence of mixing conditions. Therefore, the condition may have been met during the mixing and almost no precipitable materials “survived” the mixing. This is already a concession, as in fact these extremely low solubility compounds would be against the bulk-phase mechanism because their justification would be a transient mixing condition.

There are a number of problems. The model assumes homogeneous mixing. Let us say that the target precipitate is AB2. A+2n is present in a given concentration. B-n is added. There is no real reason for any AB2 molecules to come apart in the bulk of the liquid if it has formed there. Remember, what we see may deceive us. We have already given a surface excess alternative to dynamic exchange or reconfiguration in solution. Just because high energy, fast moving molecules in a gas mixture may experience collisions between them and with the walls and change there (the equilibria in the synthesis of ammonia or sulfur trioxide, for example) does not mean that slow, shielded (solvated), low energy ions in solution should behave in the same way.

The possibility of AB2 to form before all A+2n as reacted to form AB+n could depend on the mixing conditions. As solubility products are constant and generally do not do this, we must assume that AB2 only forms when AB+n has formed in sufficient amount. We should remember that we should let the models predict dynamic equilibrium and the mathematical form of solubility products without our help. If a compound in a diluted solution forms in the bulk of the solution, there is no reason for it to decompose and reform continually. If we assume that any ion-ion interaction depends of a characteristic distance to happen (below) then we can develop a set of equation to describe the process.

If there is such characteristic distance for the interaction of A and B, then interaction will occur whenever the whole volume of the solution is encompassed by the two ions:

A + B must be equal to 1 for the solubility condition to be reached. If this is the case, then the trivial development for the precipitation equation is:

The relationship for a solubility condition clearly goes in the lines of one rises the other lowers, but it is not mathematically equivalent to a solubility product. If the sequence is not A+B equal AB, but A +2B equal AB2, then there would be a subsequent condition

between the intermediate species AB and B, with a relationship again of the form

It is hard to see how these equations have any connection with conventional solubility products. So, the fact is that following a rationalist approach, instead of an empiricist one allowing for a mixture of empirical facts and axioms, bulk phase solubilities do not look like the real thing, while surface tension-related solubilities do.

Petersen, P.B., Saykally, R.J., Adsorption of Ions to the Surface of Dilute Electrolyte Solutions: The Jones-Ray Effect Revisited J. AM. CHEM. SOC. 127(44), 15446-15452 (2005)
F. Talens-Alesson (2011)

Thursday, 7 February 2013

Surface Tension in Liquids: the structure and properties of the tension layer

© 2013 Federico I. Talens-Alesson

This manuscript discusses the idea that the boundary layer of liquid, where surface tension manifests itself, has an impact on a range of phenomena occurring in solution. It does it by being a region of singular viscosity and density compared with the bulk of the liquid. As such, it is the packing of molecules of solvent near the surface of the liquid influences the occurrence of surface excesses of solutes. Other phenomena are also described.

It is accepted that intermolecular forces in a liquid reach a singularity at the boundaries of the liquid with gas and solid phases. The molecules at the surface are pulled inwards in because there are no “outward” pulling forces. That creates the phenomenon of surface tension.

It is known that under certain conditions some liquids deposit layers on vertical or even upside down surfaces, continuous and well above the surface of their main surface, in a phenomenon linked to capillarity. Such liquids are called “super creepers”  and there is a maximum thickness to their layers coating surfaces higher than the surface of the liquid itself.

 There are many interesting questions about how this surface tension works. Amongst these, one is how surface tension modifiers work, and another is how surface tension plays a role in heat and mass transfer in multiphase flow.

Surface Tension and Surfactants.

Surface tension is seen as a force that “tenses” the surface of a liquid. The higher the surface tension the more difficult it is to ripple the surface. This is a relevant technological question, as ripples (or waves) on the surface of a liquid have an impact on phenomena like the ability of a gas-liquid absorption device to perform, or the ability of a heat exchange device to perform, because in some cases such devices consist in pipes with liquid  circulating along their walls “pushed” by a core of gas streaming through the center of the pipe. The ripples play a part on how much mechanical energy is transferred from the gas to the liquid to facilitate the latter's circulation. The “surface tensing” is also observed in the fact that, the higher the surface tension, the rounder will be the drops of liquid on a solid surface. Reducing the surface tension to flatten drops of liquids on solids and increase contact angles is part of wetting and detergent technologies, for example.

Surface tension modifiers, known as surfactants, have a number of characteristics, including the fact that they have a hydrophobic and hydrophilic fragment. Or several from one of the kinds, or from both. Surfactants require a minimum length of their hydrocarbon chain to be so. In aqueous solutions, they lower the surface tension. Sodium butyl sulfate is not a surfactant, sodium dodecyl sulfate is. A peculiarity of the drop in surface tension induced by surfactants is that it does not lead to an straightforward increase in “rippleness” in the liquid surface. A re-tensing effect called the Marangoni effect causes ripples to be flattened out more in a surfactant solution than in a pure liquid which had the same surface tension by its intrinsic properties. Surfactants are considered to insert their polar heads into aqueous solution, and keep the hydrophobic tails above, giving the solution the “hydrocarbon” surface tension.

Alternative View on Surfactants and Surface Tension.

If you have read any of my previous blogs, you will not be surprised to find that I disagree and have an alternative view about the thing works.

Children often play chase games, where one of them has to chase other children. In a variant, the chaser and the first chased join hands and then proceed to chase another child. As the chain grows longer, the centre of the line, which moves abreast to cover more field, is tensed with the children's arms stretched out. However, the children at both ends of the line are invariably pulled inwards the centre of the line. The actual way the mechanism causing surface tension should be looked at in this way: molecules vibrate, and when their vibration is away from a certain molecule they have an attractive force with, they yank it. If they move TOWARDS it, nothing happens but it is guaranteed that when it moves away it will yank the second molecule HARDER. This causes a chain reaction, which combines with other similar “pulls” by other molecules in all directions. The result should be that when there are molecules at long distances in all directions (well, from a molecular point of view), the matrix of the solution should be greatly expanded. When the phenomenon takes place closer to a surface, then the “pull” from that direction is reduced, until it decreases at the surface itself. As a consequence, molecules near a surface would be more tightly packed. By the way, this implies that the liquid near the surface is actually denser and, insofar as the viscosity of a solution or liquid is a consequence of the strength of the intermolecular interactions and the number of interactions per unit of volume (as a consequence of the density of molecules),  more viscid than in the core of the solution.

A possible explanation of super creeping is that the wall does not provide enough energy to “kick out” the molecules depositing on it. The fact that a critical thickness exists would be related to an increase of the intermolecular interactions and the “loosening” of the liquid layer, leading to molecules sliding down the surface of the film. This critical thickness layer hints to a surface tension layer. The surface tension layer would be the layer near the boundary of a liquid or solution where the molecules are more tightly packed and create the surface tension phenomenon. The super creeper liquid film would be a limiting case in which any surface, due to the low energy of the system, could contain a surface tension layer before any excess just slid down because the binding becomes too loose.

I propose that, instead of the usual view, the surfactant molecule is sunk into the liquid as a buoy, polar head inwards, with maybe the final fragment of hydrocarbon chain sticking out. The hydrocarbon chains of surfactants would interfere in a way similar to graphite rods in a nuclear reactor: instead of interfering with the “line of sight” of fissile nuclei “firing” neutrons at each other, they interfere the attractive interactions between the molecules. As their hydrocarbon fragments replace strong water-water interactions with weaker interactions with the surrounding liquid molecules, they “loosen” the surface tension layer, which leads to a loosened “wrapping” of the bulk of the liquid and increased surface to volume ratio. In three dimensions, it would not simply result in the hedges being pushes inwards, but in a more compact three-dimensional structure progressively loosening from the surface to a certain critical distance.

Marangoni Effect.

An explanation for the Marangoni effect follows easily from the above description: when a ripple breaks the surface and causes liquid from the isotropic bulk to surface, the balance of forces within the new surface must be re-established due to the new anisotropic conditions. That implies a densification of the surface liquid in the “gash” which somehow begins to “knit itself” together and as a consequence the hedges of the “gash” are pulled together. This approach is sufficiently similar to the regular approach to be indistinguishable, although it is the complete opposite: the “pull” is a consequence of the “bare bulk” liquid NOT being densified already and having to build up the surface tension. These build-ups are the ones causing the Marangoni effect. But it looks the same as assuming that the “bare bulk” region is a high surface tension area already, which is the conventional view.

Solubility Implications of the Existence of the Surface Tension Layer.

A densification of a polar solvent like water near the surface leads to further proximity between polar groups from solvent molecules. If we consider an electrolyte breaking up during dissolution in a polar solvent, with an anion recognized by current knowledge as having near zero hydration number, and a cation recognized as having 3,4 or whatever hydration number, what we know is that one of them has no tendency to associate with water molecules (for example) and the other does. But there is an issue there. If the anion really cannot hold onto solvent molecules, then it can migrate all the way to the surface and create a surface excess of itself. If the cation has an optimum number of “held” solvent molecules, it will have an optimum deployment place where the attractive forces of the anion surface excess and the polar solvent, and the repulsive charge of the transient bulk excess cation concentration balance. Therefore, the cation will have some form of subsurface excess, because it will be pulled away for the surface of the liquid by the solute-solvent interactions. This actually connects seamlessly with my reaction surface planets theory, and gives an explanation to why there would be separate surface excesses for anionic and cationic species.

In the case of a salt AnBm, which dissociates in A+m and B-n ions, the initial surface excesses would be one containing B-n likely to have near zero hydration number and be right at the surface of the liquid, by reason of nothing pulling it back into the solution. The cation A+m will have a tendency to be associated to solvent molecules, and it will create its surface excess at some distance of the surface. However, as the concentrations increase, the electrostatic pull between the two surface excesses causes some ions to meet at an intermediate point, forming ionic pairs AB(+m-n).

These pairs will have their own optimum solvent-interaction characteristics, and therefore their own hydration number and distance from the surface at which to build their own surface excess concentration. This surface excess shields the other two, because it is same charge than one and has a lower to interact with the other. Also, its appearance may have caused some depletion in the other surface excesses. If its distance to the opposite charge surface excess is too large to cause a subsequent step, then the reaction stops there. This explains why in an equilibrium with many species not all of them are present at all times. The reader is referred to the other document to see what the connection between this surface excess layer structure causes the know mathematical definitions for solubility, equilibrium and kinetic equations.

Summarizing, the surface tension layer creates a net which filters the various types of ions, creating a restriction to the relative deployment of the various surface and subsurface excesses versus the actual liquid surface. The solvent density at each subsurface excess plane is not the same, and this brings an interesting question: if the hydration number refers to an optimum value at the surface excess plane, there is no real reason for the value to be the same across the volume of the solution. Also, because even when we use probes we can only sense or measure at the surface of the solution, what we get is a distorted combination of the closer signal from the surface excess and the more distant signal from the bulk phase, which is nevertheless weaker and more diluted. Therefore, sensing indicating a given hydration number could refer mainly to surface excess hydration number, that nevertheless would be the relevant one, as reactions would take place at the surface/subsurface layer level.

Precipitation, Nucleation and Crystal Growth.

An immediate idea is that when precipitation takes place, then any crystals forming would contribute new surface against which surface and subsurface excesses can build up. In fact, the process by which new crystals form (nucleation) should also take place in the surface excess regions. Therefore any air-water interface (e.g bubbles), any liquid-liquid surfaces and any solid -liquid surfaces could cause nucleation (for example, by seeding). Even crystal growth would be a combination of direct assembly of ions onto the surface of a crystal (well formed crystal growth) or nucleation adjacent to the surface of an existing crystal (imperfect crystal growth).

Better structural matches between forming ionic groups and available surfaces would result in higher crystal growth rates and better quality crystalline structures. This is observed in surface coatings, where different chemical compositions of coating baths may lead to a range of amorphous or micro-crystalline coatings.

As a collateral finding, the existence of extended surface (dispersed particles of whatever kind) means that the proportion of liquid in surface tension layers increases and therefore the average density and viscosity of the liquid fraction of a dispersion/ suspension would already increase, irrespective of the impact of the intrinsic contribution of the dispersed fraction. Finely divided materials with higher surface/volume ratios should contribute more strongly to viscosity increases than more coarse particles to “static” viscosity.

While the increase in density is directly connected with the number of particles per unit of volume, the strength in the interactions between molecules causing viscosity will depend on a potential law of the distance and a function accounting for the number of interactions. The inverse square of the distance of interaction is:

(4π/3 ρ)^(2/3) = 1/r^2 α Fintermolecular interaction

where r = n/V is the particle density and the number of interactions between the n particles in a volume unit V is:

leading to a “energy per unit of volume and unit of time”(dimensions of dynamic viscosity) proportional to a function of the form:

This function shows that there is a link between increase in density and viscosity for a given liquid by the mere presence of an interface or boundary. The meaning would be the energy used up to generate the “pull around” required in a moving fluid: larger values for more viscid liquids, smaller values for less viscid liquids.

Let us go back to the question of suspensions of particles. Let us assume that the effective (average) viscosity has a proportionality to the viscosities and volume fractions of the bulk phase and tension layers. Let's assume that the average viscosity of the liquid in a dispersion can be considered an average of the bulk phase and tension layer viscosities. This is in conflict with the forms of viscosity equations for mixtures of substances, but it leads to an interesting result. Follow me.

where V is the volume of the liquid, and VB and VTL are the volumes of the bulk region and the tension layer:

The volume does not specifically includes the volume of the materials dispersed, but it takes into account their presence through VTL:

where AP is the area of the periphery of the solution, dTL is the thickness of the tension layer, and r and n the radius and number of particles contained in the volume of liquid V (which is not the same as the volume of the sample, which would be Vsample = V + Vparticles). Reorganising, the ratio between the effective viscosity and the „intrinsic“ viscosity  of the bulk phase looks similar to Einstein's equation. It also does not appear very different from Taylor's equation for fluid drops with a different viscosity to the bulk liquid:

It is worth stressing that the development proposed starts by assuming that the effective viscosity  is an average of bulk phase and tension layer viscosities, and yet produces the same kind of trend. A likely explanation is that, for the purpose of the example, there is no change in the nature of the inter particle forces. Therefore, an average of the internal “pull around” or cohesion tension (a way to look at viscosity) across the volume between different density regions makes sense for a given chemical composition. The role of surface excesses in fine-tuning the viscosities of the system still needs to be taken into account.

An interesting question is whether an increase in viscosity within the surface tension layer also increases the apparent viscosity of the bulk volume by creating some sort of molecular “bag” which holds the bulk volume together and restricts the mobility of the inside liquid. The response of this “bag” to the mechanical shear stress over a liquid may play a part in the occurrence of the various rheological behaviours found in liquids, suspensions and dispersions, like rheopexy or tixotropy. For example, if the increase in shear streamlines the tension layer molecules, but then as the shear stress decreases the molecules get entangled, this would result in a rheopectic behaviour.

Finally, there is the question of the dependence of (mu)TL with the nature of the surface and any substances adsorbed onto a dispersed phase. Molecules adsorbed onto particles (e.g. clay) may have a surfactant effect (disrupt the tension layer) or thickening (consolidates the tension layer). This viewpoint may be useful when considering the role of various additives in suspension/dispersion/ flocculation technology.

Wednesday, 17 October 2012

Are Molecular Orbital Hamiltonians actually black-box neural networks?

Regarding quantum physics The Copenhagen interpretation was recently summarized as “Shut up and calculate!” That’s blunt, but not completely unfair. It is, in fact, the right injunction
for most physicists most of the time.” say Rosenblum and Kuttner (2006).

A very simplified discussion on molecular orbitals excerpted from Levine (1977) follows. A Slater type orbital (for example, for a 1s orbital) has the form:
Molecular orbitals can be constructed with linear equations:
Following the Hellmann-Feynman theorem
it is possible then to calculate the molecular orbital energies. Levine(1977) cites Parr (1963) saying “We have at our reach precise descriptions of molecules”. That is, Parr's position is that advanced quantum chemistry provides a view of how molecules behave. However, we can see that the energy of an electron in an orbital is calculated by an operator which calculates a number of exponential functions f , whose addition multiplied by a set of parameters allows to calculate a set of functions φ, from which an integral is then calculated. That seems quite similar to a neural network of neurons in the first row, n neurons in the second row, and one neuron in the third row. An example of a similar neural network applied to the calculation of chemical activities (figure 1) can be found in Nami and Deyhimi (2011).

This brings the question of whether the calculation of energies for molecular orbitals is really mechanistically based, or just a black box model. We go back to Rosenblum and Kuttner (page 119): "Here’s something to ponder: Suppose the cat was placed in the box and the atom sent into the mirror system eight hours before you looked. The system evolves unobserved during those eight hours. If you find the cat alive, since it has gone eight hours without eating, you find a hungry cat. If you find a dead cat, an examination by a veterinary forensic pathologist would determine the cat to have died eight hours ago. Your observation not only creates a current reality, it also creates the history appropriate to that reality. You might consider all this absurd. Precisely Schrödinger’s point! He concocted his cat story to argue that, taken to its logical conclusion, quantum theory, at least its Copenhagen interpretation, was absurd. Therefore, he claimed, it must not be accepted as a description of what’s really going on."

A perverse hypothesis is proposed. Let us assume that electromagnetic fields, as a consequence of the emissions of subatomic particles, combine in an interference pattern instead of leading to homogenenous potential barriers. Such pattern would create regions of maximum and minimum field intensity, and regions of intermediate field intensity in between. The juxtaposition of the individual fields is represented pictorially in Figure 2(left) by means of cross sections of concentric spheres centered on field generating particles. The illustration does not intend to be rigorous (force lines of a magnetic field have nothing to do with an spherical distribution around the originating particle), but to show the complex patterns resulting of the juxtaposition of individual particle-centered patterns. The arrows try to show how a particle would try to negotiate passage through the field through favorable field intensity regions, which can obviously be regarded as a 3-D maze. Figure 2(right) represents even more schematically the now obvious consequence of the hypothesis: random motion through a potential barrier may result in the particle taking “the wrong turns” and getting out the way it entered. Although not illustrated, a potential well’s effect to reject particle may be the consequence of less interference between particle fields at the boundary of said field making it more “impermeable”. The conclusion would be that a wave function Ψ does not describe a quantum behaviour as we think of it, but is simply a function good at being fitted to give values between 0 and 1 within a certain interval, a probability generator.

A final element of this “heretic” quantum model would be the following: let us assume that considering electrons as basic subatomic particles is flawed because the fact they cannot be broken is due to their high mobility, which prevents the stress due to the impacting energy to break up the assembly of sub-particles constituting the electron, instead accelerating the electron and changing the configuration of the particle. If so, then the Hamiltonian of an electron:


Would be in fact
He-nΨe-n + ΣHse-ecΨse-ec = Ψe-nEe-n + ΣΨse-ecEse-ec

The subindex e-n would mean electron-nucleus interaction, and the subindex se-ec would refer to the interactions between the sub-electronic particles and the electronic virtual center. As a consequence, what we accept as wave function for an electron would be a combination:

Ψ = ( Ψe-nEe-n + ΣΨse-ecEse-ec)/E

No wonder the integral of the square of the wave function can assume very intricate forms! And no wonder trying to make sense of any steps of the calculation makes quantum theory “queerer than we can suppose”, as Haldane puts it or that Einstein found its implications “spooky” (Rosenblum and Kuttner, 2006, page 3). The fact is that it may be a black box tool with no significant meaning whatsoever: neural networks are precisely extremely good calculation tools with no pretense of giving us an explanation of why.


Levine, I..N., Quantum Chemistry, (Sp. trans of the first edition) Editorial AC (1977)
Nami, F., Deyhimi, F.; Prediction of activity coefficients at infinite dilution for organic solutes in ionic liquids by artificial neural network, J. Chem. Thermodynamics 43 22–27 (2011)

Rosemblum, B.; Kuttner, F.; Quantum Enigma, 112 Oxford University Press (2006)

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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