 Today, in this lecture number 5, we will be discussing total surface energy. We briefly recall the equation for the total surface energy that we had derived from the equation 31 by plugging in for S s minus dou gamma 0 by dou t at constant N and V. This is considering that gamma 0 is equal to F s, the total Helmholtz free energy per area. So, by rearranging we had obtained U s as gamma 0 plus t times minus the partial derivative of surface tension with temperature at constant N and V. In physical terms, the total surface energy is equal to the total potential energy of molecules that form unit area of the surface of liquid in excess over what those molecules would possess in the interior of the liquid. And we can prove that this total surface energy U s unlike surface tension gamma 0 is practically independent of temperature. This can be done by differentiating the earlier expression for total surface energy U s with the temperature. So, we obtain dou U s by dou t equal to dou gamma 0 by dou t minus t times dou 2 gamma 0 by dou t square minus dou gamma 0 by dou t from where after cancelling dou gamma 0 by dou t we obtain dou U s by dou t equal to minus t times dou 2 gamma 0 by dou t square. In practice, the surface tension gamma 0 decreases linearly with temperature to a pretty good approximation. And therefore, we could actually show that if gamma 0 is written as C 2 minus C 1 times temperature where C 2 and C 1 are constant, then dou gamma 0 by dou t is equal to minus C 1. Therefore, dou 2 gamma 0 by dou t square is equal to 0. We return to the slides now and see that this is actually a pretty good approximation. And the second derivative indeed comes out to be extremely small leading to virtual independence of the total surface energy of temperature. For example, the values for the first partial derivative of the total surface energy with temperature for water is minus 0.00048 while for benzene it is 0.00012 bearing out the above expectation to be quite true. Next we deal with entropy of formation of a surface. The molecules in bulk of a liquid are surrounded by other molecules while those at the surface have a different environment on one side. At the surface, we may say that there is a new randomness possible when compared to the bulk liquid. As we will see in the next figure a molecule may occupy a position right in the surface or just below it. In this figure, we see in the top half the surface layer and the adjacent bulk liquid that is immediate bulk of liquid. In the bottom half we see all identical molecules in the bulk. So, looking at the position near the surface, we may be able to quantify this randomness. The equation by Boltzmann and Planck is handy in estimating the entropy for the surface. This equation relates to the entropy a notion of thermodynamic probability W. In this simple fashion S is equal to K ln W where W is the thermodynamic probability and it is worthwhile noting it that this is different from ordinary probability, but proportional to it. It could be considered as the total number of different ways in which a given system could be realized in a specified thermodynamic state. Here in the context of the surface layer, we would be able to argue that W is equal to 2 because this is the number of ways in which the surface could be realized in a given state. Hence, the standard entropy change associated with the formation of a surface would be given by S s equal to R ln 2 that is equal to 1.4 calories per degree centigrade or 1.4 entropy units. As we had shown earlier, the surface entropy was equal to the negative temperature coefficient of surface tension. S s the surface entropy is equal to negative temperature coefficient of the surface tension and therefore, the temperature coefficient of the surface tension of liquid must be negative because entropy surface entropy is positive. Here is a slightly different insight that one may be able to derive looking at this drop in surface tension with increase in temperature. For instance, if we regard the surface as a separate phase, it would be equivalent to saying that the surface phase becomes more miscible with the bulk phase just as the ordinary bulk phases of two liquids become more miscible on raising their temperature. From such a calculation, the entropy of formation of water surface is found to be equal to minus of minus 0.154 that is plus 0.154 Ergs per centimeter square per Kelvin or 1.6 entropy units. So, this estimate of the surface entropy of 1.6 entropy units is pretty close to 1.4 entropy units estimated from the Boltzmann-Planck equation. In case of mercury from the temperature coefficient of surface tension, one would find the entropy change to work out as two entropy units. Next we address molecular theories of surface energy. We begin here with a general observation that the cohesive forces among molecules of a liquid fall off very steeply with distance. The rapid decrease of interaction with distance leads us to concept of nearest neighbors, which means basically that we need not consider interaction beyond the nearest neighbors to a very good approximation. The next question which arises is regarding the relation between total surface energy and the molecular cohesion in liquids. And this question has been considered in the past by Langmuir in 1933 and Frankel in 1947. Here we briefly review their approaches to obtain an idea of simple ways in which this entity the total surface energy may be related to the molecular cohesion in liquids. We take these in turn. First the Langmuir's method. Let us say we assume that each individual molecule of the liquid possesses a spherical surface with the usual surface tension. Now this looks a bit preposterous as an assumption, particularly because the cohesive tendency and surface tension are a result of interaction of large number of molecules in the surface with those in the subjectant liquid. However, let us see what is the consequence of this suggestion by Langmuir. Thus the volume of a single molecule of liquid could be written as the molecular weight divided by the density of the liquid divided by the Avogadro's number. So, M by rho n would give us volume of a single molecule of liquid. Now since we are going to regard this molecule as a sphere, we could equate this to pi d cube by 6 and therefore, we obtain d is equal to 6 M by rho n pi to the power 1 by 3. The surface area of the molecule is therefore, AM equal to pi d square and that is pi to the power 1 by 3 into 6 M by rho n to the power 2 by 3. The surface energy for unit volume of the liquid is next written as AM into US into rho n by M. AM is the area surface area of a single molecule, US is the surface energy per area and rho n by M gives you the number of molecules per volume. So, this entire quantity AM, US, rho n by M works out to be the surface energy for unit volume of the liquid. This could be equated to the latent heat of vaporization per volume that is L over V. By doing this, we can write US is equal to L by V into M by rho n AM. The assumption made here that the forces between two molecules in contact are mainly dependent on the surface properties is known as independent surface action. Each molecule behaves independent of other and interacts with other molecules as if it is surface where to exhibit properties of the surface of liquid itself that is independent surface action. In keeping with our theme of discussing water and looking at certain things related to water over and over again, we see here some of the pictures taken by Wilson Bentley in 1902 where ice crystals are shown to exhibit a beautiful symmetry and pattern or in a similar fashion an equally beautiful ice crystal is shown here. Surface energy minimization is the principle behind such symmetries found in nature. Here we see a diagram of super saturation versus temperature and we see that for different temperature ranges, different geometries for ice crystals are obtained like plates, columns, plates, columns again. We have below about minus 4 degree centigrade from 0 to minus 4 degree centigrade. We have then writes between minus 4 to about minus 10 degree centigrade at much lower temperatures. We have needles between minus 10 and about minus 22 degree centigrade. We get again then writes and sector plates and for very low temperatures below minus 22 degree centigrade, we have columns and plates and this is shown in context of a water saturation line. We return now to the second method that is Frankel's method for obtaining the total surface energy. In Frankel's method, the difference in the number of nearest neighbors for a molecule in the bulk of the liquid and that in the surface is first calculated. So, we basically make use of the rapid decay in interaction energy and focus our attention only on the nearest neighbors. So, if we take a molecule in the bulk and another in the surface, by seeing the difference in the number of bonds, we may be able to relate the mutual cohesive energy per bond and this difference in number of bonds to the total surface energy. The mutual cohesive energy per bond is used first with the difference in the number of bonds per molecule or the number of nearest neighbors and the number of molecules per area to calculate total surface excess energy and secondly, with the number of bonds per molecule, mutual cohesive energy per bond is used together with the number of bonds per molecule and the number of molecules per liquid volume and molar volume of the liquid to obtain the energy required to break away a molecule from the bulk of the liquid into vapor. This latter quantity is then equated to the latent heat of vaporization per mole of the liquid to estimate the mutual cohesive energy per bond. The total surface energy US could then be obtained by eliminating the mutual cohesive energy per bond or per two neighboring molecules to obtain US. The equations for this method are formulated as described below. Let us say Z is the number of nearest neighbors of a molecule in the interior of the liquid. Let me explain this with a diagram. If we have the surface let us say here and the surface molecules are seen to be interacting with the bulk. Deeper in the bulk, we have molecules in close proximity. So, there is a more complete attraction for molecules in the bulk. In the surface, there are fewer molecules interacting with a given molecule. So, we represent here by small z, the number of nearest neighbors of a molecule in the interior of the liquid. And by z prime, we represent the number of nearest neighbors of a molecule in the surface. Clearly, z is greater than z prime because a molecule in the interior has greater number of nearest neighbors whereas, one in the surface has no neighbor on the vapor side. And next we define nu, we represent by nu the number of molecules per volume of the liquid. By u 1, we represent mutual cohesive energy of two neighboring molecules or mutual cohesive energy per bond and by small b, the molar volume of the liquid. So, with z, z prime, nu, u 1 and b, we have the number of nearest neighbors in the interior, in the surface, number of molecules per volume, mutual cohesive energy and molar volume of the liquid. We should then be able to write in terms of these chosen notations, the surface excess energy per unit area. And that would work out to be u s equal to u 1 z minus z prime into nu to the power 2 by 3. Let me explain this. Supposing that we represent our molecules as arranged in a lattice and so on, the total number per volume of molecules is nu and if each side of this lattice is equal, then the number of molecules per side will be equal to nu to the power 1 by 3. Returning back to our slide, we then identify the number of molecules in any phase as nu to the power 2 by 3. So, u s is the mutual cohesive energy per bond u 1 into z minus z prime, there is a difference in the number of bonds into the total number of molecules in the surface nu to the power 2 by 3. This is an approximation, number of molecules per area is here calculated from the number per volume nu. Now, if L represents the heat of evaporation of 1 mole of liquid, it is a close approximation to the total surface to the total energy of vaporization of 1 mole of liquid. That is to say that we would ignore the difference that might arise from the difference in energies of the surface molecules and the bulk molecules for 1 mole of liquid. If equate the 2, we can then come to this conclusion L is equal to z by 2 u 1 into nu v, where z by 2 here is the net number of bonds broken. This could be explained with the figure on the next page here, next slide here. We find for 4 neighboring molecules represented as 1, 2, 3 and 4 in relation to the central molecule, we have 4 bonds. If this central molecule is taken away, 4 bonds would be broken, but 2 bonds will be reformed 1 and 2. So, if a molecule of liquid is removed from the bulk, we break z bonds, but z by 2 bonds reform. So, the net number of bonds broken will be z by 2. So, per molecule of liquid taken from the bulk, the energy required will be z by 2 into u 1 and that into the total number of molecules per volume nu will give you energy per volume and we multiply that by molar volume. So, we get the energy per mole that is the quantity L. Now, between these 2 expressions, we may be able to eliminate the mutual cohesive energy. In this equation 39, the first term is z by 2 u 1, which is the cohesive energy per molecule moved from the bulk that is the net number of bonds broken times the cohesive energy per bond. Upon multiplying by the second factor nu, we obtain the cohesive energy per volume of liquid and finally, by multiplying this quantity by the molar volume gives you total cohesive energy per mole of liquid. Now, we can substitute for the mutual cohesive energy of 2 neighboring molecules or mutual cohesive energy per bond from equation 39 into the expression for the surface excess energy per unit area that is equation 38. You could work it out in parallel or do the mental calculation substituting for u 1 as 2 L by z nu v. In this equation for total surface energy us, we obtain the required quantity total surface energy us as 2 L by p into z minus z prime by z into nu to the power minus 1 by 3. One could make use of this equation to calculate the total surface energy and then one could actually plug in relevant numerical quantities together with units to estimate the total surface energy from this model for different liquids and how good this simple model is could then be benchmarked by looking at the values from independent other advanced theories and experiments. So, we will look into some of these. We take a numerical example here on calculation of the total surface excess energy for water using Frankel's method. We can assume that z is equal to 6 that is for a molecule in the interior there are 6 nearest neighbors. Correspondingly in the surface the number of nearest neighbors will be z prime is equal to 5. The latent heat per volume L by v is 582 calories per centimeter cube per water or that could be translated as 2.4 into 10 raise to 10 ergs per centimeter cube. Then from the dimensions of water molecule we know that nu will be 33.3 into 10 to the power 21 molecules per centimeter cube. Substituting these quantities in equation 40 we get 2 times L by v is 2.4 into 10 raise to 10 into z minus z prime by z is 6 minus 5 by 6 into nu to the power minus 1 by 3 is 33.3 into 10 raise to 21 to the power minus 1 by 3 and when we evaluate this it works out to be 240 ergs per centimeter square. This number 240 ergs per centimeter square could be compared with the experimental figure of 118 ergs per centimeter square. At first thought it would appear that these numbers are very different but considering that this was such a simple theory an agreement to even within 50 percent may be considered to be satisfactory for such a simple model. You could take down this next numerical problem for your homework. Here we are asked to find out or calculate the total surface energy for normal octane and for mercury using Frankel's method. The experimental values for total surface energy are respectively 50.7 ergs per centimeter square for normal octane and 536 ergs per centimeter square for mercury. When we compare these numbers with other theories we will get an idea how good our simple theory is. In this table we see a comparison of experimental and theoretical total surface energies for different substances. Besides the experimental and Langmuir and Frankel's theories predictions we have two more columns one based on phase center cubic arrangement of molecules in liquid and the last one is for quantum mechanical predictions. We see here for FCC distribution or FCC packing the potential distribution could be evaluated and used for more accurate estimation of total surface energy and it gives a good agreement with experiment 32.9 against 35.3 whereas for mercury the best estimate is obtained from quantum mechanical calculation where we obtain a value of 490 against experimental 541 490 against 541. We then move on to another related concept that of interfacial tension. This is similar to the surface of a liquid exposed to air we may have an interface between two immiscible or partially miscible liquids which exhibits a contractile tendency. Unlike the surface tension the interfacial tension is the residual entity. We note here that the surface tension is a result of the asymmetry of the environment for molecules in the surface because the vapor in contact with liquid is so much rarer there is only the effect of the liquid molecules in the surface that is reflected in the surface tension. In context of interfacial tension we have a partially miscible or immiscible liquid in contact with a given liquid and because the vapor has been replaced by a condensed second phase that is second liquid we would have less asymmetry in this case of two liquid interfaces as compared to the surface of a liquid in contact with vapor and the interface will still exhibit a diminished contractile tension. This residual entity or interfacial tension is denoted by this symbol gamma subscript i and once again is expressed in dines per centimeter. One might visualize this as follows that when you have two liquids in contact with each other each having a contractile tendency differ in their contractile tendencies somewhat and therefore effectively the interface is will have a net contractile tendency which results from the effect of these different contractile tendencies taken together. If the second liquid were identical to the first liquid obviously there would be no asymmetry and therefore interface between a liquid with its own type would obviously be zero. We will be dwelling upon the interrelations between interfacial tension and the miscibility in later lectures. For the time being we could visualize a simple experiment in which we consider an interface between butanol and water. The butanol molecules concentrate at the interface that is because butanol with its structure C 4 H 9 OH would have the hydrophobic C 4 H 9 tail and the hydrophobic C 4 H 9 tail and the hydrophilic OH head group. By concentrating at the interface the butanol molecules would get packed in an oriented manner with the hydroxylic head group submerged in water and the hydrophobic C 4 H 9 tail preferring the hydrophilic C butanol side we would have a situation where these packed butanol molecules would experience a certain extent of repulsion. A certain dipole moment is associated with the hydroxylic head groups which would cause the molecules in the surface to repel each other and this repulsion would partly offset the usual contractile tendency of water. We have seen why the packing would occur but the orientation of these C 4 H 9 OH or butanol molecules would actually lead to a state of low standard free energy and therefore is the favored state. The surface tension of butanol is 24 dimes per centimeter whereas its interfacial tension with water is just about 1.8 dimes per centimeter. One sees such low magnitude of interfacial tension for liquids containing polar groups. Low interfacial tensions are characteristic of polar liquids. A considerable orientation of dipolar molecules occurs at the interface as also seen from the interfacial tension of nitrobenzene with water which is the value 25.1 dimes per centimeter when compared against the surface tension of the dipole. For hydrocarbon oils the interfacial tensions are usually high typically of the order of 50 dimes per centimeter. The next table shows the standard interfacial tensions between pure liquids and water. For normal hexane at 20 degree centigrade the interfacial tension is 51 dimes per centimeter. For normal octane at 20 degree centigrade the interfacial tension is 50.8 dimes per centimeter. For carbon disulphide it is 48 dimes per centimeter and for carbon tetrachloride we have 45.1 dimes per centimeter. For bromobenzene then we have to use interfacial tension with water is 38.1 for benzene above 35 at 20 degree centigrade and then we have the alcohols like octanol and hexanol. Interfacial tensions are low 8.5, 6.8. For aniline it is 5.85. Once again for pentanol a low value of 4.4. For butanols it is the interfacial tensions are clustered around 2. For mercury in contact with water the interfacial tension is 375 and so on. Olig acid with its double bond and the acidic group has a low relatively low interfacial tension against water of 15.59 dimes per centimeter and for oligoil it is about 22.9 dimes per centimeter. Now it has been seen that interfacial tensions could be correlated to the mutual insolubility of hydrocarbon derivatives with respect to water. Several empirical correlations exist in literature and one particular result expressed by Bickerman shows that the interfacial tension decreases with the relative miscibility of various organic liquids with water. With a sharp drop from about 50 dimes per centimeter to about 3 dimes per centimeter occurring over an increase in relative miscibility from 0 to 10 and with very little change over a further increase up to 30. Let us have look at Bickerman's plot. We see here the interfacial tension plotted against the relative miscibility. So up to about 10 there is a sharp decrease in interfacial tension there onwards it is nearly constant. In three component systems same type of relationship holds. For example butanol dissolved in benzene adsorbs at the interface against water. The alcohol molecules packing into the interface form an oriented monolayer. The hydroxyl head groups get immersed in water and the repulsion of the hydroxyl groups in the monolayer once again reduce the contractile tendency of the interface. We see here the butanol molecules accumulating at the interface and this repulsion would reduce the interfacial tension between benzene and water. So like earlier we have gamma as equal to as equal to gamma i minus pi where gamma i is the interfacial tension of the system where there are only two liquids and pi is the mutual repulsion which arises out of adsorption of the monolayer of the third component. This resembles the equation 7 that we had seen earlier in the context of surface tension with gamma i replacing gamma 0. Greater the miscibility of two liquids lower is the interfacial tension. For the liquid pair of water and isopentanol the interfacial tension is 4.4 dynes per centimeter. If we add ethanol gradually the interfacial tension progressively reduces and becomes 0 when ethanol reaches 25 percent weight by weight. The whole system then becomes miscible and forms a single phase. So we see here that interfacial tension becomes 0 at a point where the two phases become miscible. We would stop this lecture today over here and comments from here next time. Thank you.