SOLUBILITY CANNOT BE A BULK PHASE CONDITION

© 2013 Federico I. Talens-Alesson

Solubility in the bulk phase?

For AlPO₄, the solubility product K_sp 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 10¹⁸Ų. 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 Ų.

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.

References.

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) http://chemistry-f-talens-alesson.blogspot.co.uk/2011/09/some-g-g-and-d-symbols-in-text-must-be.html