 We will resume from where we left last time and this third lecture is devoted to discussion further on surface tension free energies and adsorption phenomena. To have a quick recap of a couple of considerations which lead to a certain relation between surface tension and free energy, we have this slide showing you. For the differential change in the Helmholtz free energy, the df is equal to minus sdt minus pdv plus gamma 0 da plus mu dm. Here gamma 0 is supposed to be the force per centimeter tending to contract a liquid surface. It could be even solid surface in general. S is the entropy t the absolute temperature, capital P is the pressure, capital V is volume and a is the surface area. Mu is chemical potential and n represents the number of molecules in the system. So, we have this fundamental equation from thermodynamics, df equal to minus sdt minus pdv plus gamma 0 da plus mu dm and this capital F is total Helmholtz free energy of the system. Now under conditions of constant temperature volume and for a given number of moles of the system, we see this equation reduce to df equal to gamma 0 da or that can be rewritten stating the conditions that have been now presumed gamma 0 is equal to dou f by dou a at constant t v and n. Now if Helmholtz free energy per unit area is fs, we may want to know a relation between fs and surface tension in a liquid system. We will later put certain restrictions on solids because certain precision is lost in the meaning of surface tension while talking about solids and there are other preferable terms in which we should be addressing this particular notion, but under the presumed conditions d times a fs is equal to df where fs is the Helmholtz free energy per unit area and a is the total area. And we just worked out gamma 0 equal to dou f by dou a at constant t v and n. Now if we were to attempt to combine these two equations together, what result could we obtain? We could see that by combining these two equations, we find gamma 0 is equal to dou by dou a of a fs f is a fs and then after performing this differentiation, we get gamma 0 equal to fs plus a times dou fs by dou a at constant t v and n. It is here that we inspect the second term, we have here a times dou fs by dou a at constant t v and n. When we ask a question, would fs depend on area or for a single stable liquid, would it depend only on the configuration of molecules? So, once we ask this question, we realize that fs for a single component system should only depend on configuration of molecules and therefore, dou fs by dou a can be identified to be 0. Therefore, dropping the second term, we find gamma 0 is equal to fs. Here is the relationship that we are looking for, we have the surface tension gamma 0 is equal to the Helmholtz free energy per area fs. We could likewise have argued and arrived at a result for conditions of constant temperature and pressure, in which case we would have obtained gamma 0 is equal to G s, where G s is the excess Gibbs free energy per unit area. One could take these last 2 results together, gamma 0 equal to fs under conditions of constant volume and gamma 0 equal to G s under conditions of constant pressure and next think about what kind of changes are likely to accompany changes in surface area. In practice, the changes of pressure or volume accompanying surface changes are small and therefore, G s is nearly equal to fs. Thus, we could say the surface tension and the Helmholtz free energy per unit area of surface are equal. That could be taken as a general result true for a single component or pure liquid. In practice, we measure surface tensions and express them in dines per centimeter. Helmholtz free energy on the other hand per area is expressed in ergs per centimeter square. Now, a little reflection will tell us that these 2 quantities are dimensionally identical dines per centimeter or ergs per dines per centimeter or ergs per centimeter square. So, this particular dimensional identity will keep in mind and many of our arguments later would revolve around this implicit equivalence of the dimensions of the surface tension and the Helmholtz free energy. Now, it was in 1950 that shuttle shuttle berth showed that this equation gamma 0 equal to fs tends to break down if our consideration is for liquids where very high viscosities are involved. The reason is under such conditions, the rearrangement of molecules in the surface occurs more slowly than the relaxation of shear stress in the interior of the liquid or solid. So, high viscosity conditions will lead to departure from this general result. We move one step closer now to realistic situation. What if we have a second component added to this pure liquid that we have begun with? The tendency of the surface to decrease spontaneously will be changed by addition of a second component. We can think of a simple example like addition of low molecular weight alcohol. For example, butanol. When we add butanol to pure water in small quantities it would dissolve and the hydroxyl group in the butanol would get hydrated. But what happens to the hydrocarbon chain? The hydrocarbon chain the hydrophobic part of butanol would be partially dislocating the structure of water. You all know that water is a highly associated liquid. There is a very strong hydrogen bonding and one may be right in saying that a glass full of water is like one single giant water molecule because there is so much of hydrogen bonding. Now, in this strongly hydrogen bonded water structure if you were to admit a hydrophobic chain like a hydrocarbon or C 4 H 9 here it would tend to disrupt or dislocate the structure of water. The hydrocarbon tail here will be viewed as an intruder it would not be acceptable in the otherwise equilibrium structure of water. The tendency therefore would be to dry away this hydrocarbon tail from the interior of water because energetically an interior location within water is not favorable for hydrocarbon chains. Now, the system may respond perhaps in this fashion if the alcohol molecule which we have added to the interior of water if it could reach the surface of water then there is a probability that some kind of rearrangement for this butanol molecule would make things better and the way the system would respond is the following. The hydroxyl group will be still anchored and immerse in water and the hydrocarbon tail would flip out into the vapor phase. This would mean now the system is energetically more welcome the chains will be more welcome in the vapor phase than in the water bulk. If this is the driving force then the molecules of alcohol would get accumulated in the surface in a roughly oriented mono layer at the interface between water and the vapor phase. Now, it is here that you might visualize what picture that you might have seen in many textbooks the palisade like structure strictly oriented alcohol or surfactant molecules at the surface. In reality the orientation is only a rough orientation. So, we should not take those kind of diagrams literally. Now, this particular phenomenon of molecules containing hydrophilic head groups and hydrophobic chains to pack into the surface is called adsorption. In a sense your understanding of adsorption is to be generalized here accommodating the packing of these amphiphilic molecules into the surface of liquid in a roughly oriented fashion. So, that is where the adsorption term arises in the context of a liquid surface. It is here that we need to now consider the contract contractile tendency of the surface of pure water. Adsorption has to be taken together with the tendency of the surface of pure water to contract. The film of adsorbed molecules will have a tendency to expand by spreading. The expansion of this film would exert a positive repulsive pressure in the surface we denote it by this later pi. And therefore, in the presence of this second component with the adsorbed film of the second component being present in the surface of water, the net tension of the liquid surface will be the contractile tension gamma 0 that is the surface tension for pure water minus this positive pressure arising out of the spreading tendency of the adsorbed film. So, gamma 0 is tending to contract the surface, pi is tending to expand the surface. And therefore, the net tension or net contractile tendency will be difference of gamma 0 and pi. Number of questions will arise here, we will address some of them at this stage and we will take up this equation many times again in later discussions. In general, this repulsive pressure pi is less than the contractile tension gamma 0. And therefore, gamma which is gamma 0 minus pi remains positive. There is this phase will tend to still contract into as little area as possible or we could say the two component system this solution will still be coherent. It will it left to itself it will tend to contract the surface. But the related question may arise what if the above inequality is reverse. In that case, the surface tension will become negative. If pi exceeds gamma 0, then the surface tension gamma will become negative. And therefore, the surface will tend to expand instead of contracting like it normally does. And if this is limited by the geometry of the container, then in a two liquid interface case it will lead to so called spontaneous emulsification in which one phase will split spontaneously into droplets and get dispersed in the other phase. So, spontaneous emulsification is something that you would be able to visualize in context of pi exceeding gamma 0. Now, it is here that I would like you to recall maybe some of your observations. There are certain oils or oil based formulations when added to water, they tend to emulsify fairly rapidly. Many of the agricultural products some of the insecticide for example, if added to water tend to form emulsions spontaneously. These products will be designed so that they have a negative surface tension or the interface in at least two component system will tend to expand instead of contracting as it normally would. And therefore, when the expansion of the surface is unlimited it will lead to buckling of the surface or spontaneous emulsification of one phase in other. In this case the oil based formulation will get emulsified in water. Now, we raise this question about the tension within a liquid on a different count. We may say that the free surface of a liquid behaves as if in it is in the state of tensile stress. Now, the contractile tendency of the surface you would understand from the positive free energy of the system. Can we understand the origin of a real stress in such a situation? Now, this was a question asked by Goodney and answered in a 1947 nature paper wherein he offered the following arguments. Imagine a fresh surface of a liquid which is initially free from stress and it is just suddenly formed. So, we have fresh surface of liquid initially free from stress and suddenly formed. What would be the situation? Think of the chemical potential. The chemical potential of the surface molecules will be greater than that of the molecules in the bulk because of their asymmetrical environment right. We had seen this in the last lecture. Some of you might have another underlying explanation for why the chemical potential of the surface molecules will be greater than that of the molecules in the bulk. If the bulk is identical with the system before we created the fresh surface only difference can arise in the chemical potential of the surface molecules right. So, I will plant that particular thought in your mind and move ahead, but as a consequence of the higher chemical potential, how would the surface respond? The surface molecules will tend to rapidly sink into the bulk. This would be done in an effort to reduce the free energy of the system ok. So, molecules in the surface with higher chemical potential would tend to drop down into the bulk. And if they do so, they will leave voids behind in the surface. What will be the consequence of those voids? That is the next argument. It will increase the intermolecular spacing in the plane of the surface. And therefore, they should be an extra attractive tension coming into picture between the molecules in the surface phase right. The attractive tension will tend to reduce the tendency to escape of molecules in the surface. In other words, the attractive tension will tend to reduce the chemical potential of the surface molecules. So, we have now some kind of balancing effect coming in here. Net desorption from the surface would still continue until the tension has built to a sufficiently high magnitude to ensure that the chemical potential of molecules in the surface has now become equal to that of molecules in the bulk. So, the whole chain is the following. Higher chemical potential of molecules in the surface makes them sink into the bulk, thereby increases the intermolecular spacing. Therefore, increases the attractive tension, attractive tension reduces the chemical potential in the surface molecules. Therefore, at some stage, the chemical potential in the surface will become equal to that of molecules in the bulk. Once that balance is established, the number of molecules leaving the surface will be equal to the number of molecules entering the surface layer. And then onwards, there will be no net movement of liquid. This is to be understood in the sense of a dynamic equilibrium. Once the balancing of chemical potentials has taken place, we have no net movement of molecules from the surface to the bulk or from bulk to the surface of liquid. And under these conditions, then with this rationale, there should be no real reason to deny the existence of a physical stress or tension in the surface. This is one of the best explanations of surface tension that one can find around while dealing with liquid gas systems. There is another substantiation we can have to be a little more mathematical. In mathematical terms, we could say an increase in the surface tension will cause a decrease in chemical potential of the surface molecules. And the latter per molecule will be reduced roughly by the product of molecular area in the surface, A m times the change in surface tension, say dou gamma. The equation is dou mu by dou gamma at constant temperature is equal to minus A m. If you look at dou mu by dou gamma as a limiting quotient, then the reduction in chemical potential delta mu is equal to minus A m delta gamma at a constant temperature. So, this quantifies the change of chemical potential as it is linked to the change in surface tension. The otherwise higher chemical potential value for molecules in the surface is reduced by a positive tension stress gamma until the equality of chemical potentials of the surface and bulk is established. Having said this, I would make a general comment which is about scientific dispute, one of which we will be dealing with later. The best part of the scientific dispute is the high probability of over continuing with it and the above notion is really no exception. We still continue to discuss the surfaces. Let us now look at something different. You would all be able to visualize that human body is mostly water. Anywhere between 55 to 78 percent of water is present in a human body depending on the size. Then of course, we have most of our planet earth occupied by water. Continuing on visualization, but justifying our visualizations with quantitative terms, we take up kinetics of molecules in the surface for our discussion. Molecules of water may appear quiescent and smooth to the naked eye, but on a molecular scale it can be shown to be in a state of violent agitation. Let us see what I mean by this. The surface molecules are continuously replaced by other molecules. Kinetic theory of gases allows us to estimate the number of molecules striking a unit area of the surface per second and n dot, this number per second will be given by P by root of 2 pi m k t, where k t is in earths and m is the weight of a single molecule in grams. For saturated water vapor at 20 degree centigrade, the strike rate you could calculate will come out to be 8.5 into 10 raise to 21 molecules of water vapor per centimeter square per second. However, a small fraction of this just about 3.4 percent as the minimum estimate actually enters the liquid surface. At equilibrium, this would be the rate of evaporation of water. So, the lowest estimate of the number of water molecules leaving or condensing under such conditions works out to be 2.9 into 10 raise to 20 molecules per centimeter square per second. We now try to make calculations about some relevant time scales and by comparing them we may gain some insight into the kinetics of molecules in the surface. First to come to mind may be the mean residence time of the molecules in the surface. That would be T bar is equal to n divided by minus d n by d t for desorption, where the numerator n is the equilibrium number of molecules per centimeter square of the surface and minus d n by d t is the rate of desorption and at equilibrium it will be also the rate of adsorption at the surface. From dimensions of water molecules one can calculate that there would be about 10 to the power 15 molecules of water per centimeter square and minus d n by d t we just estimated is about 2.9 into 10 raise to 20 molecules per centimeter square per second. Thus the mean residence time can be calculated from this equation and it works out to be just about 3.4 micro seconds. That is a very short lifetime for a molecule in the surface of water. So, just for about 3.4 into 10 raise to minus 6 seconds a molecule of water remains in the surface and then evaporates. This would imply an extremely violent agitation in the surface layer. Think of the consequences and think also of the question as to what would happen to the surface and what might possibly be a role for surface tension. If we were to fire water surface with this molecular bullets at such a rapid rate with so short a time of precedence we should have practically disrupted the interface beyond recognition. However, there is a strong cohesion among molecules precisely the reason which lies in explanation of surface tension dating back to the first explanation by Young. It is because of the strong cohesion among molecules of water or in liquid the interface or surface of liquid would be still identifiable within a distance of few molecular diameters. There is a strong cohesion or the cohesive action which partly comes down the surface to be definite to within an extent of a few molecule diameters. And this kind of rapid exchange is valid only for molecular exchange at equilibrium. The actual or measured evaporation rates are much smaller than the one we calculated in this simple manner. I am sure those of you who are chemical engineers would have some idea as to why they should be so. There are two reasons for the lower observed rates and these are linked to first the presence of stagnant layers of paper in the vicinity of the interface which offer collisional barrier to the escaping or approaching molecules. Second is cooling of the water surface accompanying evaporation. So, the collisional barrier offered by stagnant layers of paper in the vicinity of the surface and the cooling of the surface actually tend to lower the measured evaporation or condensation rates to much smaller magnitudes. And therefore, the rate of mass transfer is retarded significantly in reality. One may be able to address this question alternatively from the modeling perspective. One could take the conventional modeling of desorption here. One could have expressed minus dn by dt the desorption rate as k times theta times exponential minus q by RT where theta is the fractional surface coverage and q is the heat of vaporization. Now, putting for simplicity theta equal to 1 and minus dn by dt equal to 2.9 into 10 raise to 20 molecules per centimeter square per second and q equal to 585 into 18 that is 10530 calories per mole. We find the desorption constant k equal to 1.2 into 10 to the power 28 per centimeter square per second 1.2 into 10 raise to 28 per centimeter square per second. We may want to compare this desorption constant with alternative predictions. This figure actually is an excellent agreement with the results for physical and chemical adsorption of simple molecules on solid surfaces and also with fundamental theory. For example, according to the Eyring theory k is equal to n kT by h and that works out to be 0.6 10 to the power 28 per centimeter square per second. Here h is the Planck's constant presuming that the molecule does not change the degrees of freedom during desorption. Yet another way would be to predict the desorption constant from quantum mechanics. A value given by quantum mechanics will be 0.14 into 10 raise to 28 per centimeter square per second. Yet another theory Polanyi Wigner and Langmuir theories predict a value of k of the order of 10 raise to 28 per centimeter square per second. The point I am trying to make here is that a simple model here is in agreement with the rigorous theories to well within an order of magnitude. The exchange of solvent molecules in the liquid surface with those in the immediate vicinity under the interface is something next one could address. As you could expect the exchange of solvent molecules in the surface with the subjectant liquid is even more rapid. This could be anticipated from the smaller distances involved much smaller distances involved between the interface and bulk of liquid under method and the rapid molecular motion. One could estimate another time scale the diffusional time scale. One may define the diffusion cohesion as D equal to lambda square by T bar where D is the diffusion cohesion for water the self diffusion cohesion is about 2 into 10 raise to minus 5 centimeter square per second. Lambda is the distance between two successive equilibrium positions and T bar is the time taken by a molecule to move from one equilibrium position to another. It is possible that asymmetry may alter the diffusivity somewhat near the surface, but we might still do well in getting the feel for order of magnitude for T bar. For liquid water lambda is about 3.5 into 10 raise to minus 8 centimeters and with this T bar can be calculated to be equal to about 6 into 10 raise to minus 5 microseconds or about 60 picoseconds which is about 60,000 times smaller than the residence time for molecules being exchanged with the vapor phase. You get an idea now how much more rapid the exchange between the surface molecules or interface and the bulk of the liquid is compared to the exchange between the vapor and the interface. 60,000 times smaller time of residence is worked out for exchange between the surface and the adjacent liquid. One may want to contrast these times against another relevant time one comes across when dealing with surface tension systems, surface tension considerations. The time required for liquid surface to take up the equilibrium value is somewhat greater than the time scale that we just calculated, but it is still very small about 10 to the power minus 3 microseconds. That is about 1 nanosecond is required for the process of reorientation and rearrangement of molecules of a simple liquid subjected to exposure to a new surface. We could calculate the desorption constant yet again alternatively to describe this exchange desorption of molecules into the sub adjacent bulk liquid. Once again n is 10 raise to 15 molecules per centimeter square, T bar is 6 into 10 raise to minus 11 seconds and therefore, minus dn by dt which is n by T bar is 1.7 into 10 raise to 25 molecules per centimeter square per second. When we take theta equal to 1 and q to be the energy of activation for cell diffusion of water molecules which is about 5300 calories per mole, we would obtain k to be equal to minus dn by dt desorption by theta exponential minus q by RT that works out to be equal to 12 into 10 raise to 28 per centimeter square per second. Once again this estimate is in a good agreement with fundamental theory. The higher value here is attributable to the activated state of diffusion of water with an increase in entropy which is constable. So, we have now here certain bunch of time scales calculated and the desorption calculate constant calculated from a simple model turns out to be in fair agreement with the fundamental theory. It also brings to the fore the role of the surface tension in stabilizing the surface of a liquid in spite of having a very large number of collisions of the molecules from the surface and from the bulk. It is the high cohesive tension or because the liquid is highly coherent the surface is able to preserve itself to within a few molecular diameters. We gradually increase the complexity of the situations to come to grasp the reality. For an adsorbed monolayer of a surface active agent which would be in equilibrium with a solution we expect this time scale to be much greater than for the solvent. For example, the rate of desorption of acid from a monolayer and at equilibrium that will also be the rate of adsorption is lower than for water molecules by a factor which is exponential minus lambda by RT where lambda is the energy of desorption magnitude of which may be taken roughly as about 2300 calories per mole and therefore, that time comes out to be about 3 nanoseconds. Now, this equilibrium exchange rate for adsorption and desorption cannot be realized if we have a net flow and this can have practical consequences. Diffusion of solute along concentration gradients over several millimeters below the surface would greatly reduce the rate of adsorption and desorption and therefore, times ranging from a few milliseconds to several hours will be required before diffusion can bring the adsorbed film and bulk phase into equilibrium. So, we note now in passing that the adsorbed solute diffusion can actually make you come into realistic time scales which will be definitely measurable and as we would be seeing in the next lecture there are certain realistic issues which have to be considered related to this fact if we have to be accurate in our interpretations of experimental results. Not just that sometimes even to be right experimentally we have to be very clearly aware of the time scales involved in permitting a system to come to equilibrium and what goes on there is therefore of utmost importance. Realistic systems will have adsorbed monolayers of acids in a variety of organic chemicals and diffusion can play a very significant role there. There are other things also related to the equilibrium considerations will take them all in the next lecture maybe we can stop here for today.