 We begin this module 3, devoted to electrostatic phenomena, an introduction to which has been made in the last part of the last lecture. So, we will have a quick recap of the beginning. We had noted that in an effort to understand what happens at the membranes of living cells, experiments had been conducted by these two schools of Butner and Bauer, wherein for a long period they have been, they had been conducting experiments with polar oils such as salicylulidate placed between aqueous phases. Their typical external setup was of this sort where you have an aqueous phase connected to another aqueous phase through a salt bridge and then there is a oil phase, which would be expected to mimic a membrane, although this would be quite thick. Then there is a last aqueous phase here. These two aqueous phases at the ends are connected through Ag-Ag cell electrodes for measuring any potential development across the system. And then you would introduce a salt here in this aqueous phase, which would dissociate and then any change in the potential would be recorded and basically the question was how does this potential get developed. It was anticipated that the potential across this interface would be the one which would get altered and the two schools differed in the interpretation or understanding of how that interfacial potential developed. Whereas Butner believed that it was because of unequal distribution of anions and cations between the organic and aqueous phases, the potential developed bars belief was that it is the adsorption at this interface, which lead to the potential which would be recorded. And it turns out that it is the distribution which would contribute to the development of potential in the equilibrium system. So, that potential whatever is developed is called delta phi and the contribution arising out of the unequal distribution would give you a psi potential. If there is any contribution from adsorption that would be the contact potential delta v, delta psi and delta v added together would contribute to delta phi. And this was the picture behind the rationalization that Butner offered. There would be concentration of anions, chloride ions here for example, in the aqueous phase and accumulation of cations NMe4 plus predominantly in the oil phase leading to different distances from the interface for the mean distance of charges given by Debye Huckel length. And that we have seen that in case of oil that is much larger than the Debye Huckel length in water. So, 4 to 5 orders of magnitude higher mean distances of separation would be achieved in oil. So, the inner or galvanic potential is given by some of the outer potential psi and the contact potential v. We admitted there might be transient contributions arising out of adsorption, but that would decay rapidly and equilibrium will have only contribution from unequal distribution. We need to understand this theoretically that is the objective today to be able to make a prediction of the interfacial potential. And to be able to let us say distinguish between the contributions from differences in concentration and from adsorption by suitable choice accidentally of the non aqueous phase. To begin with the concept is one of electrochemical potential which we define by this later mu bar that would be sum of the chemical potential mu plus the product of Faraday unit and the interfacial potential phi. So, that is the basic starting point for us mu bar is mu plus phi f. Now ionic distribution is expected to automatically readjust such that the total potential drop remains unaltered by the presence of monolayer. Now you could choose for non aqueous phase either an oil or air. By choosing air we will ensure that they would not be distribution of ions possible. So, only contribution which can come will be from the contact potential. Whereas, if you use an oil however non polar it may be they would be distribution potential coming there. Although by choosing a non polarity of non polar nature of the oil we might be able to minimize the distribution contribution, but still the potential will be distribution potential. So, at an interface between two liquids we will talk of interfacial potential at an interface between say water and air we will talk of surface potential. So, those are the differences between the different potentials that you can have depending on choice of your non aqueous phase. So, we could achieve changes in this distribution potential psi by adding salts to the aqueous phase or to both phases and among these you could even include surface active agents. Surface active agents by their very nature will tend to adsorb at the interface and one would then be able to measure the contact potential between water and air or the interfacial potential between non polar oil when you have this material which can spread or adsorb at the interface giving you interfacial potential. Changes in V are interfacial potentials when the non aqueous phase is any oil like paraffinic oil and I explained the surface potential already. Now, the changes in interfacial distribution potentials are the distribution potentials. We are trying to have a uniform jargon, we have reserved interfacial for two liquids in contact and surface for a liquid in contact with a gas or air or its own vapor. So, in the same fashion changes in interfacial distribution potentials psi will be called distribution potentials. Now, we go into the theory of the distribution potentials. According to the classical theory postulated by Nernst in 1892, the distribution potential arise from differences in solubilities of the cations and anions in the two phases in the same manner that we have understood so far. And how do we quantify these ionic distributions? We could look into these two relations for cations. The distribution coefficient is B plus B subscript plus for anions it is B minus. So, B plus and B minus are the distribution coefficients for cations and anions and we could write RT ln B plus as W mu plus 0 minus O mu plus 0. There are number of subscripts and superscripts some of which we have covered already. So, in keeping with the normal conventions in thermodynamics this 0 represents the standard state. So, mu 0s are standard chemical potentials for cations and for anions in water and in oil. So, the standard chemical potential of cations in water minus the standard chemical potential of cations in oil will give you RT ln B plus. B plus is the distribution coefficient for cations. Similarly, we have the other equation RT ln B minus equal to W mu minus 0 minus O mu minus 0. Now, oil has a potential phi which would generally be determined by both psi and B. So, if you take a case of water in contact with a polar oil like amyl alcohol, nitrobenzene or salicyl aldehyde, if we add the tetramethyl ammonium chloride to this system, the organic cations would dissolve preferentially in oil. Chloride ions on the other hand would be left behind in water. So, if we have accumulation of these cations in the oil phase, oil phase must acquire a positive charge compared to water right. And in these systems as we will show later, they could be transient contribution arising out of adsorption that will be in the form of delta v which would rapidly decay to 0. If we are to measure delta phi, then we will be effectively measuring delta psi because delta v will not contribute anything. Now, this view is confirmed also by electro kinetic potential measurements on emulsion droplets. Now, when the potential change occurs, one may ask a question where does most of that change occur? It turns out that it is in the phase which has a lower dielectric constant that is where most of the potential change would occur. So, in the context of water and oil, oil representing something hydrophobic, normally the dielectric constant for the oil will be much lower. Dielectric constant for water is about 80 at 20 degree centigrade and for the organic phases, it could be perhaps as low as 1 to 2. So, it is in the oil phase that most of the potential change would occur. Now, let us address this task of relating the interfacial potential phi to the distribution cohesions that we defined for cations and anions B plus and B minus. Our objective is to arrive at a theoretical result to describe phi, the interfacial potential in terms of the distribution cohesions for the cations and anions. So, to this end, we have a number of relations. I would say that there are basically three categories of equations that we deal with and by viewing these equations together in different ways, we would be able to construct the overall picture. So, let us get begun with these initial equations. We have two phases, water and oil and we have two ions, cations and anions. So, electrochemical potentials for the cations and anions in the two phases will demand that we have these four electrochemical potentials defined in the first place. In water, it is W mu plus bar and W mu minus bar, electrochemical potentials for cations and anions in water. Similarly, in oil, O mu plus bar and O mu minus bar. So, those are the ones we need to start with. Let us see what the expressions are. W mu plus bar is equal to W mu plus 0, the standard chemical potential plus RT ln W c plus. This is the normal expression from thermodynamics. What we follow as convention in these systems, we attribute the absolute potential for bulk of water to be 0. We must understand this is a generic physics problem. We cannot measure potentials in their absolute values, we only can measure the differences. So, we have this convention that the bulk of water will have 0 potential and related to water, the oil will have positive potential. So, if that is the case, then this electrochemical potential is nothing but the chemical potential for water. W mu plus 0 plus RT ln W c plus. Consider what happens for cations. In cations in oil, we have the normal contribution here O mu plus 0 plus RT ln O c plus, but then the bulk of the oil phase is positive related to water and that potential is plus phi. So, we add to this mu phi F. So, we have here plus phi F for cations in oil. Likewise for anions in water is a similar expression with plus replace by minus. So, W mu minus bar is equal to W mu minus 0 plus RT ln W c minus. Again there is no contribution corresponding to the potential here. So, electrochemical potential and chemical potentials are equated. With respect to the anions in the oil phase, we have the chemical potential of anions in oil and because the oil phase is positive related to water, it is deficient in the anions. So, we have minus phi F here with respect to anions. So, those are our four fundamental expressions for electrochemical potentials for the two species in the two phases ok. What more can we see? So, these first four equations is the first category of equations that we need. Second is a statement about equilibrium. At equilibrium, the electrochemical potentials for the cations and anions in the two phases water and oil must be equal because they cannot be any net interchange at equilibrium. So, for cations we could say W mu plus bar is equal to O mu plus bar. So, cations have the same electrochemical potential in water and oil and similarly for anions. Electrochemical potential of anions in water is same as the electrochemical potential of anions in oil. This I will say is the second category of equations that we require a statement about equilibrium. When you think in these terms basic definitions of electrochemical potentials for 11 times in the two phases, then equilibrium relations. The last category is the third category is a statement about electro neutrality of the bulk phases. In the bulk of water as well as in the bulk of oil, the concentration of cations and anions must be equal. So, W c plus equal to W c minus and O c plus equal to O c minus. If we have to have bulk phases which are electro neutral. So, far away from interface the population of cations and anions will become equal in each of the phases. So, this is our third category a statement about electro neutrality. Now, we just play with these equations a bit. We substitute the first set of equations 1 to 4 into the second set which is 5 and 6. I would just suggest that you start following this. Put down on paper all the equations. Put these definitions 1 to 4 in equations 5 and 6 and see what you get. So, substituting equations 1 to 4 in equations 5 and 6, you would get these equations 9 and 10. W mu plus 0 plus RT ln W c plus equal to O mu plus 0 plus RT ln O c plus plus phi f. Similarly, W mu minus 0 plus RT ln W c minus equal to O mu minus 0 plus RT ln O c minus phi f. Now, we have two options here. One suggestion is you subtract equation 10 from 9 and recall what 7 and 8 are electro neutrality relations. See what you get. Sir, all these concentrations W c plus O c are nearly interposed. They can be at any location. So, depending on which location you are looking at, the corresponding value for phi will be different. So, you note here that because W c plus and W c minus are equal and O c plus and O c minus are equal, when we subtract 10 from 9, these second terms will be gone in subtraction process. And if you transpose the first term on the right hand side in 9 and 10 to the left hand side, then we can write W mu plus 0 minus O mu plus 0 minus this next bracket is W mu minus 0 minus O mu minus 0 and that will be equal to phi f minus minus phi f that is 2 phi f. But then you have the definitions of distribution coefficients. W mu plus 0 minus O mu plus 0 is nothing, but RT ln B plus. Same way W mu minus 0 minus O minus 0 O mu minus 0 is RT ln B minus. That is how the distribution coefficients were defined in the first instance. Look at this. So, we get that equation 12 which could be rewritten as follows. For phi, we can say RT by 2 f ln of B plus by B minus. That was our objective. We wanted to get the potential in terms of the distribution coefficients for the cations and anions. In this form, we would get the equilibrium value for interfacial potential that you actually measure across the interface between oil and water. Remember that the difference between this and the statement here is that now we will have to regard this as comparable to the measured value because this is the maximum potential that you can have. Now, there is no local concentration involved here. We got rid of the local concentrations. But as we will see later depending on the local concentration of cations and anions, the potential variation could be predicted also. So, this is now the basic result that we intended to get and we have it now. But we want to go little further and I will ask you to take a step backward here. What have we done here? We subtracted equation 10 from 9 and we achieved a relation between phi and B plus and B minus. It may be curious to see what happens, what result do we get if we instead choose to add these equations. Try doing that. If instead of subtracting 9 from 10, you add 10 and 9 and work out the consequence. When you add these equations plus phi f and minus phi f cancel and we get this RT ln B plus plus RT ln B minus is equal to minus RT ln W c plus W c minus by O c plus O c minus or when you combine B plus and B minus terms on the left hand side and cancel of RT, take this negative sign to invert the argument of ln. We realize that it is B plus into B minus is equal to O c plus by W c plus into O c minus by W c minus. Or that was the anticipated reflection on the distribution coefficient. But now if you argue that cations in the oil phase and cations in the aqueous phase would have come from corresponding salt. We could replace this O c plus by W c plus by concentration of salt in oil by concentration of salt in water. Similarly, O c minus by W c minus is again concentration of salt in oil by concentration of salt in water, which is nothing but the distribution coefficient for salt between oil and water. So, B square. Similarly, both O c plus and O c minus be equal because it also depends upon the stoichiometric. Now, see the O c plus and O c minus are equal in the bulk because of electroneutrality. If we take an electrolyte like MgCl, then in that case the concentration of negative ions will be twice that of positive ions. Yeah, but you have a corresponding stoichiometric factor coming in the salt concentration. When you derive the ions from salt, then the corresponding stoichiometric factor will come and which will cancel. So, we have now this result B plus into B minus equal to B square. Now, it may be possible to combine equation 15 and 13. We have presumed here that dissociation in water is of this sort Mx giving you M plus plus X minus that is the presumption. Depending on the kind of dissociation we have, we will have corresponding constants. Here, 1 cation and 1 anion is supposed to be derived from 1 molecule of Mx. That is the limitation here, but extension to any general situation is straightforward. So, before we go further, one more point should be mentioned here that according to this equation, if we have tetramethylammonium ions which are more soluble in oil phase than chloride ions are, B plus will be greater than B minus. Therefore, this ln of quantity greater than 1 will give you a positive quantity. So, phi is positive. So, that is on expected lines right. So, if we consider this B square equal to B plus into B minus, we can rewrite this equation by multiplying the numerator and denominator of the argument by B minus. So, this becomes B minus square. Here, we have B plus B minus and B plus into B minus is B square according to this. So, that means, our this equation can be written as RT by 2 F ln of B square by B minus square or you could think of this two getting cancelled with this. So, RT by F ln B by B minus. This equation cannot be used directly because we cannot measure absolute potential. So, one now needs to worry about how the potentials would be measured. We can certainly measure differences in potentials. So, if we take a system like nitrobenzene in contact with water, you could think of adding sodium iodide in one system and potassium iodide in another. So, we have nitrobenzene water in one case Na i is added in another case K i is added. We should be able to measure the differences in potentials between these two systems and we understand here that there is a common iodide ion here and according to this theory, the difference in phi should be independent of salt concentration. That is what is indicated here. Delta phi is equal to delta psi which is RT by F ln B Na i by B minus minus RT by F ln B K i by B minus. That is the potential for the Na i added case and that is one for K i added case. And since the iodide is common, we can write this as RT by F equal to ln B Na i by B K i. So, if we know the distribution coefficients for sodium iodide and potassium iodide, the difference in potential which can be measured can be related to these distribution coefficients of two salts. So, delta phi is indeed independent of the common ion and we can obtain delta phi equal to delta psi equal to RT by 2 F ln B Na plus by B K plus. The common ion is of no consequence in determining delta phi, clear? The next question will arise how good is this analysis? So, we have here summarized experimental results, measurements for different Mx kinds of salts KCl, N e t force Cl, tetraethyl ammonium chloride, sodium chloride and potassium iodide. So, you can think of using this variety of salts. If K plus is replaced by tetraethyl ammonium ion cation, the calculated delta phi is 124 millivolts, observed one is plus 126 millivolts, pretty good agreement. If you take, let us say KCl and NaCl, it is equivalent to replacing K plus by Na plus, calculated delta phi is 50 millivolts and observed is plus 53 millivolts. Again, a close agreement between the two, but think of this Cl minus replaced by I minus and the calculated value is minus 95 millivolts and observed value is about minus 102 millivolts, that is not close enough. So, one could conclude from here that extended data support the equations 18 and 19 well if the oil is polar like nitrobenzene. So, perhaps we can stop here for today and continue this discussion in the next lecture.