 So, there are other alpha forms and as I said today, if one takes a survey of literature, you will find that maybe there are 15 to 20 variants of sober and liquefied equation which are available and they have been fitted for variety of applications. Not only that these equations are applicable to the hydrocarbon systems for which they were originally developed, they also are applicable to some non-ideal systems. There is a specific sober and liquefied equation which has been developed which applies to ammonia water system. Ammonia water system is fairly a non-ideal system, but there is a modification available which works for that also. These are some of the references, but if you are interested in the basic description of the sober equation, this is the article which appeared in Cammin Science in 1972. The title was equilibrium constants for a modified Redlich-Kwong equation of state which turned out to be a turning point. Now as I mentioned that well, let us look at this diagram just to get an idea. This is a plot of K value with respect to pressure. The open circles here, they represent the experimental data and temperature in all the cases is 250 degrees Fahrenheit. So, it is not very high temperature, it is little above 100 degree Celsius. You have nitrogen, you have methane, carbon dioxide, ethane, hydrogen sulphide all the way up to normal decane. This is the heaviest component shown here. So, if you really see using the sober Redlich-Kwong equation, so these are the predicted values. The continuous lines are the predicted values and these are your experimental data. So, for variety of constituents, for variety of components, the fits are very good all the way from pressures of the order of about 200 psi up to about 2000 psi. So, the pressure effect is very nicely captured and when pressures are high, non idealities are high. So, K values are very nicely predicted. So, this just shows that applicability of sober Redlich-Kwong was quite widespread all the way from light gases to fairly heavy hydrocarbons. As I mentioned that the density prediction, the liquid density prediction of sober Redlich-Kwong remained erroneous and therefore, need was felt that something is not right about the equation of state form, the form of equation of state because from the Redlich-Kwong form we have now taken into account the effect of eccentricity and we have taken into account the effect of temperature and still the liquid density predictions are not right. And liquid density predictions means that you are in highly non ideal range because when we talk about liquids, the molecules are tightly packed, they are sitting next to each other. In vapor, the molecules are far apart. So, which means that the form of the equation needs to be played with and that was realized by these two researchers Peng and Robinson and in 1976, this was another turning point, another new equation of state with in which the form of the equation was slightly changed. It did not resemble the Redlich-Kwong, but otherwise the rest of the theory whatever I have described so far more or less remained the same. The same approach for calculation of pseudo critical pressures, temperatures, the same approach for calculation of the parameters at the critical point using the two derivative equations, the same approach of representing alpha as a function of eccentric factor and some sort of function of temperature, everything remained the same, but the form of equation was slightly changed. And it turned out I am not going to go into the details here because I have put only couple of slides on this, it turns out that the Peng Robinson form with the similar theory as was used by Sohwe Redlich-Kwong not only gave fairly good predictions of k values, in fact sometimes better than what Sohwe predicted, not only that the liquid and vapor enthalpies were predicted well, the vapor densities were predicted well anyway, the liquid densities showed tremendous improvement over what Sohwe Redlich-Kwong had. So, that particular deficiency was taken away and this method became much more reliable method for computations of thermodynamic quantities which are required in your simulation. And just like we had methods or variants for Sohwe Redlich-Kwong for variety of applications, here also people started looking for applications of Peng Robinson equation to other systems other than hydrocarbon systems, some non-ideal systems and you have maybe another 15, 20 variants floating in the literature where Peng Robinson equation form has been kept the same, but the internal omega and T forms have been played with and new equations have been developed. So, there is a family of Sohwe Redlich-Kwong equations available today and there is a family of Peng Robinson equations available today. So, if I were to compare and contrast Sohwe Redlich-Kwong with Peng Robinson, both the equations are considered equally well except for the part that in Peng Robinson the liquid density predictions are superior as compared to the Sohwe Redlich-Kwong equation. Superior does not mean here or I should not leave you with the message that Peng Robinson is excellent, it is better than Sohwe Redlich-Kwong, but it still does not come very close to the experimental values and therefore, for liquid density is even today we may not use Peng Robinson and we may go to better methods. But if you are not working with too tight accuracy, Peng Robinson becomes a very good equation of state to do complete spectrum of calculations all the way from your equilibrium constants to enthalpies to densities and I have not been mentioning about entropies, but entropy predictions are also very very good. Here is an example, this appears in a book by Sandler of how an equation of state, now this particular depiction is for Peng Robinson, but you can use similar plots by using any cubic equation of state. So, this is for oxygen as a species it is a pure component diagram using Peng Robinson as equation of state, how you can generate a PV diagram for the whole region of interest. As we know once we are in this zone, in this zone the characterization can be done using ideal gas law or may be small values of Z will be there which can be used. But the calculations become more and more difficult as you approach the critical point and they become very you can say tricky when you go beneath the critical point and this is the zone which is your two phase zone. And when we are working with two phase systems doing let us say distillation, we are always in this zone, this is where our operations are. This being a cubic equation of state if I were to solve on let us say this particular critical isotherm T equal to minus 150 degrees Celsius and if I plot P versus V for the given temperature the equation of state will follow this particular line. It may not exactly coincide because it is a predicted value, this here of course it is shown as a predicted value, but there may be if I had experimental data there may be some error. So, it is not really predicting the experimental value, but what I am trying to say here is that it follows let us say this path. It would have gone like this as a smooth curve take a turn and then it would have gone up again it would have crossed this line again take a turn and it would have come like this. Do you agree with me? Because for the given temperature if this temperature is extended as a horizontal line for the given temperature this particular and given pressure here this particular curve would have come like this. Why? Because at that pressure it is and the given temperature it is supposed to give me three roots in volume. So, root number one which is the smallest for the liquid root number two would have been somewhere here which we would have discarded and the third root which is here which is for the vapor. What does this mean? This means that if I have the values of volume here this volume and this volume and if I have temperature fixed I should be able to calculate what could have been the pressure exerted by the constituent or by the species. Well, one way to do this is that I have this curve going down and then going up. So, if I draw a horizontal line with the assumption that the area under the curve is equal to area above the curve. So, I can say in one way that I am going to predict a pressure from this side. This would mean that an equation of state therefore, can be used to calculate that particular pressure which gives this kind of behavior and that I can call as the vapor pressure of the species. So, I am now going to state something different. I am going to state now that can I take an equation of state and at a given temperature once I have plotted this diagram I can use that particular prediction and back calculate the pressure of the system. If it is pure component the pressure of the system is nothing, but the pressure exerted by the constituent which is the species and that is the vapor pressure. So, the equation of state can become a good tool to calculate the vapor pressure of the system. Vapor pressure can be measured in the laboratory and you can compare the two and then you can find out how good is the equation of state alright because if the vapor pressure predictions are good there is no reason why the vapor liquid equilibrium will not be good. So, this is what has been done just to illustrate that Peng Robinson gives fairly good values. This particular example is for oxygen. So, this is PV diagram for oxygen calculated using PR EOS and what has been done here is that this pressure has been calculated. So, which means basically I am taking the vapor pressure prediction from the equation of state. Now just to illustrate another point that how important is the dependence of intermolecular forces on temperature and the eccentricity. Because we have been talking about that ever since we talked about the Sovereignty-Kwang equation. Again this open circles or squares they represent the experimental data. This is for the data is for normal butane and you have 1 by T scale here and this is the pressure. So, basically it is a pressure-temperature relationship vapor pressure essentially. Look here the line which is labeled as A this particular line this is Wendervoll equation. So, if you had Wendervoll equation and if you tried predicting the vapor pressure. So, this is how you will get whereas, the actual data is somewhere here. Line B is prediction of Peng Robinson, Peng Robinson has a slightly different form as compared to Sovereignty-Kwang. Sovereignty-Kwang was using the same form as of Wendervoll. So, if you use the Peng Robinson equation so, form is different from Wendervoll but alpha is said to 1 alpha has 2 effects in it alpha has temperature effect and alpha has a centric factor effect. So, both the effects have not been taken into account because alpha is numerically said to 1. So, if alpha is said to 1 we make very slight improvement in the prediction of vapor pressure as compared to Wendervolls. But if you took alpha as a function of temperature and this has the parameter m small m, m again is a function of a centric factor which I had shown you. So, if you took complete Peng Robinson equation the prediction is here. So, what does this mean? This simply means that the form of equation is not playing much role in the prediction that is only fine tuning. What is playing much role or a very significant role in the prediction of your vapor pressure? It is the eccentric factor which is something to do with the shape of the molecules and the temperature the intermolecular forces which are affected due to temperature. So, that is what makes all the difference. Now, we have discussed how to calculate Z I had shown you for Redley-Kwang and then I said similar thing applies for Sober-Redley-Kwang, similar thing applies for Peng Robinson. Just for completeness of subject some equations are shown here that once you have the Z dependence taken care of you can do this integration you can calculate the fugacity coefficients. This is the pure component fugacity coefficient, this is the fugacity coefficient of the i-th species and once you have the fugacity coefficients you can calculate other functions such as the enthalpy departures, the entropy departures, the density departures and all properties can be calculated. Now, the discussion will remain incomplete if I did not mention something about what are known as the binary interaction parameters or binary interaction coefficients. We had for mixtures mixing rules for AM and BM if you remember, A and B being the parameters if it is Sober-Redley-Kwang we were calling A and B as Sober-Redley-Kwang parameters if it is Peng Robinson we said A and B are Peng Robinson parameters for equation of state. On A we were using the geometric averaging rule and on B we were using the arithmetic average rule. Now, it turns out that in spite of going so much into details of use of eccentric factors, use of temperature dependence and other variables or other parameters such as the form of the equation still there are gaps which remain in the predictions of these equations as compared to what you have from the experimental data. And to narrow it down what these authors did they brought in what are known as the binary interaction parameters or binary interaction coefficients which are thrown into the calculation of A, calculation of A. A was summation y i, y j and then a i, a j square root if you recall. So, multiplied by 1 minus k i j and if k i j is 0 the multiplier is 1 otherwise k i j will take some small value. So, what is that we are saying? We are saying that there are some unmodeled effects which still have not been taken care of and which we are not aware of ok. So, something more is happening which modeling has not taken care of. So, you are throwing that effect into k i j, but you still prefer to call them as binary interaction parameters because the interaction among the molecules is not yet properly modeled alright. So, 2 at a time you consider so you call them as binary interactions. So, here is a set of some sample values for binary interaction parameters. This is for soberedly Quang equation. So, if you see here I have hydrogen sulphide, carbon dioxide, nitrogen so on and so forth and then only 4 compounds on this side. So, i to i interaction of course does not make any sense, but if I say i to j so carbon dioxide molecule interacting with hydrogen sulphide. So, there is a correction factor of 0.102 which is required. Similarly, nitrogen to hydrogen sulphide it is 0.14. So, this is quite significant. So, sometimes the interactions are not modeled that well. So, you want to rectify by putting these parameters. Sometimes they are modeled very well for example, normal butane to hydrogen sulphide the value is only 0.06, some numbers are even smaller 0.03. Some numbers are negative also, sometimes these are over predictions, sometimes these are under predictions. So, basically what does k ij represent? The k ij represents that part of interaction, molecular interaction which has remained unmodeled which has not been modeled so far because when you have multi-component systems the intermolecular forces are very, very complex and there could be other effects also staying in the system. So, when we talk about Sovereign Lyquan, when we talk about Pangorobinsan or for that reason any other equation of state we do not get exactly what you get from the experiments and therefore, the residual effect is yet to be modeled and we do that through the binary interaction parameters. How do we get these binary interaction parameters? We get them from the experimental data, experimental data of binary mixtures, binary systems. So, if I have to predict the vapor pressure really right and if I make a claim that I will get from equation of state that statement may be an exaggeration because if that intermolecular forces effect has not been modeled to my satisfaction, it does not match with the experimental data then I need to bring in the binary interaction parameter first. After that I can do predictions the way I like, so that should be kept in mind. Remember I have said so far good part of that was devoted or dedicated to the cubic equation of state, but again for completeness of subject I must mention that there is another family of equations which are known as virial equations of state and they also have a very significant role to play in modeling and simulation. And one very popular equation which has worked fairly well for light hydrocarbon systems or hydrocarbon gases for quite some time people have used it is the BWR equation of state. The Benedict Webb Rubin equation as it is called originally when it was developed it was an 8 parameter equation essentially derived from the virial form. In virial form you write the pressure as you may say as an infinite series in terms of 1 by V. So, essentially that is the form we have been writing here 1 by V, 1 by V square, 1 by V cube like that, but somewhere you can terminate depending upon the degree of accuracy you want and then the number of parameters with which you want to play. To BWR there have been extensions done in the past again to improve and improve the applicability. So, one extension of Benedict Webb Rubin equation which was done by Sterling, so it became an 11 parameter equation. And then there are other extensions available due to Lee and Kessler and then there is a popular equation called Lee Kessler-Plocker which is considered probably one of the best equations in this family which allows you to calculate the liquid densities very accurately. So, that is another parallel development. So, in simulator when we talk about today not only that we have cubic equations of state we also have these family of virial equations, so that we should keep in mind. This is the Z form of the Lee Kessler equation which I mentioned to you it has finally, 11 parameters and then the expressions for the fugacity coefficient, the enthalpy departure, the entropy departure all these equations enable you to calculate these properties in totality. There is one interesting study which is available through this book of Sandler. If you write your equation cubic equations in the form P equal to RT over V minus B, this V underscore is the molar volume minus delta and then delta takes different mathematical forms as shown here and then you define these four parameters theta, eta, delta and epsilon, this delta is small delta. You can unify all the cubic equations of state. So, some examples are shown here that if theta is simply a small a which is the Van der Waal parameter a and eta is B, delta is 0, epsilon is 0, then this whole thing reduces to delta equal to a upon V square and this is nothing but Van der Waals equation. There is another equation there used to be it is not very popular the Clausius equation, then Berthelow equation, then Redley-Kwang. Redley-Kwang simply uses this form for theta, the eta is B, delta is B and epsilon is 0. So, this is the form which is given to delta. So, it becomes Redley-Kwang. Similarly, Sove, the form is slightly different, then you have Peng Robinson, then there is another equation called Patel-Teja, Liebber-Admister and the list goes on, this table is little old but one can go on generalizing for various cubic equations. So, what is the advantage? The advantage is that in the Z form in the compressibility factor form if you can write your computer program and if you have a nice routine which gives you the three roots of the volume, then depending upon whichever equation is there you can calculate these parameters from the respective formula here and the same procedure can be used to solve for volume for whole family of cubic equations of state. So, the programming becomes very compact and easy. These are some related definitions of capital A and capital B for the same equation. Here is some description of how the virial modified forms sometimes work better than the cubic equation. This study is available in Cedars book. The detailed reference is here, it is pretty old study done in 1973. For k values, collection of data 0.3510, enthalpy is 21 different compositions and liquid density is 700 compositions. Temperature range is given. So, minus 240 is fairly cold temperature in degrees Fahrenheit to 500 and this is enthalpy and this is liquid density range. Pressures are fairly high, these are in PSI. So, if you really see here, this is absolute average deviation in terms of percentage. So, where at Leuphan, the k value predictions average deviation is 13.6. So, it is not doing all that well, you know 13.6 is fairly large deviation. In enthalpy prediction 2.1, so it is doing fairly well and liquid density prediction is 9.78. Again, it is not doing all that well. Chow Cedar, which was the predecessor to Sauvier-Readley-Kwong, where Redley-Kwong was used with the solubility theory 15.5. So, as compared to Chow Cedar, Sauvier-Readley-Kwong is doing slightly better. Here dash means that Chow Cedar did not have provision for enthalpy prediction. As I mentioned to you, it was purely developed for k value prediction and similarly there was no provision for density prediction. But the Starling-Hann extension of BWR which is 11 parameter equation based on the virial concept, though in k value prediction it is doing worse than Sauvier-Readley-Kwong and even worse than Chow Cedar. In enthalpy is not all that bad, this was 2.1, 3.1, 3.1. But if you really focus attention on the liquid density, the liquid density predictions are extremely good. 1 percent average error is hardly any error in liquid density. So, what is the lesson? The lesson is that the cubic equations, they are doing as good as these involved multi-parameter virial equations as far as k value predictions and enthalpy predictions are concerned. Sometimes they are doing even better, but in liquid density the virial equations are certainly ahead of the cubic equations. So, that is why I had made this remark that even Peng Robinson many a times is not used for liquid density prediction and you go for either Starling-Hann extension of BWR or an equation which is called Lie Kessler-Plocker LKP that works fairly well, which again is in this family only. So, let us sum up now the equations of state. The focus has been on cubic equations. So, most of my remarks are on cubic equations. Well, the biggest advantage of using equations of state for our thermodynamic procedures is that they do not call for any standard states. PVT XY data are sufficient in principle, no phase equilibrium data are needed. So, even if I do not have XY data, PVT alone if I have, it is good enough. If you have PVT X, it is better because you can calculate the binary interaction parameters or you may have PVT Y data. So, this is to improve, but otherwise PVT alone if it is there that is good enough. Easily utilize theorem of corresponding states. I have taken you through the chronological history of the development through the corresponding state theory. They are applicable in the critical region because that is where the properties or the parameters are calculated. They allow prediction of k values, enthalpies, entropies and densities, but just as a warning that if it is cubic equation then liquid density predictions are not very reliable. The biggest advantage of course, is that the cubic equations are computationally fast. And when we do simulations of processes or simulation of, vigorous simulation of these columns which you are going to see through simulators. The fact remains that 60 to 70 percent of your computational time is in thermodynamics. So, if the thermodynamic support is computationally fast, your simulations will be done fast. If the thermodynamic support is slow, then the simulation will take much more time. And therefore, one should try to do all the improvements in the thermodynamic characterization and have efficient procedures to speed up the calculations. There are some disadvantages like I showed you that there are varieties of equations. So, no really good equation of state available for all densities. The one which predict densities nicely, they do not predict other properties nicely. And therefore, the thermodynamic consistency is lost. They are sensitive to mixing rules and then binary interaction parameters make the job even more difficult. And they are difficult to apply to, there is a spelling mistake here, it should read polar compounds which we have not talked about in this presentation. We have said predictions are good primarily when we have been talking about hydrocarbon systems or mildly non-ideal systems. But if you have polar systems, for example ethanol water system or oxygenated compounds present with water, aqua systems or large molecules or electrolyte systems where ions can be there, then these equations do not work well. So that is their limitation. So, as I just mentioned that we have this deficiency of polar systems not being modeled well by equations of state. So, we go to the non-symmetric or unsymmetric form of activity coefficient model. So, we go back to three basic things. I can come back to this slide. Let me start from here. We characterize three different types of solutions. The ideal solution, the regular solution and what are known as the polar non-ideal solutions. Ideal solutions are those solutions where the molecules are in random motion and by and large the molecules are of similar sizes. So, if this is let us say molecule A and this is molecule B species A and species B, B really does not care whether the next neighbor is A or the next neighbor is one of its own, which means the intermolecular forces are uniformly distributed. So, then the system remains in random motion and the system behaves like an ideal system. If this is the character of the system, then you can use Raoult's law. Now, notice here that I am talking about the neighborhood, which means I am talking about either the liquid phase, either the liquid phase because that is where the proximity is very important or one can even think about this system, this need not be liquid, this could be a vapor phase under very high pressure and low temperatures. So, when temperatures are low, volumes are small, pressures are also high. So, that time the vapor molecules also are sitting next to each other. So, that behavior can be similar to liquid behavior. So, that has to be kept in mind. The other type of solutions where the molecules are of different sizes, the way it is shown in here and we expect that the intermolecular forces will be different and randomness will be lost, but it really does not happen and there is reason for that which will come in the next slide. So, molecules are still randomly located. Hydrocarbon systems of the homologous series, they fall in this category. So, if you have a system containing propane, butane, pentane, hexane, heptane like that, the molecular sizes are different, but they behave in this way. So, they are not ideal, but they are mildly non-ideal systems, mildly non-ideal systems. They are non-polar, but they are mildly non-ideal and these systems can be characterized by the regular solution theory which is due to Hildebrand. This is the one which Chauh cedar had used to calculate the activity coefficients from the pure component properties. The third category which I mentioned as a deficiency for the equations of state where we said the system is non-ideal, system non-ideal and this is normally observed when one of the constituents or one of the species is a polar substance. Take for example, aqueous systems, water is highly polar compound. If it is present with any other oxygenated compound, let us the most popular example is your ethanol water system. So, ethanol is polar, water is also polar. So, what happens? It is observed that the molecules are affected by the proximity of different type. For example, here this particular molecule is surrounded by the other molecules. So, it can feel the presence of the other molecule. So, there is a cluster formation here. Similarly, there is a cluster formation of the other type. What is the effect of this? The effect of this is that molecules then are segregated by type which means that if I have a vessel in which I withdraw the sample from different locations, I will find that local concentrations are different from the bulk concentrations. That does not happen with hydrocarbon systems. You can withdraw the sample from any place in the vessel and you will find that the average concentration and local concentrations are the same. Whereas, here the concentrations are location dependent which means there are gradients within the vessel. So, local concentrations are different from the bulk concentrations. Now, so these are the things like intermolecular forces between various pairs of molecules, ideal almost identical, regular moderately different and polar may be drastically different, sizes almost identical may be moderately different may be drastically different, local concentrations almost identical to average concentrations almost identical again and there this is the reason that the thermal energy overcomes the intermolecular force differences and polar non-ideal drastically different concentrations. And these are some of the examples like benzene toluene system, isopendene, pentane systems or your xylene systems, artha, meta, para xylene etcetera, they all form close to ideal system. So, Raoult's law is applied and fairly good calculations can be done. As I said the homologous hydrocarbon systems, ethane, propane, normal butane is a good example for regular solutions. Polar non-ideal, one of the constituents normally is an oxygenated compound or water is present. So, ethanol benzene, it is a polar system, acetone chloroform, acetone normal hexane, ethanol water. So, these are some examples of these systems. Now, let us go back and go to the first slide. There are three properties we talk about, number one that if A combines with B, so two species I have A and B, if A combines with B, so there is a volume change, volume change of mixing as it is called. So, if I have 100, let us say cubic centimeter or 100 milliliter of A and 100 milliliter of B and if I combine A with B and if the net volume remains 200, 100 plus 100 is 200, so there is no volume change of mixing which means the intermolecular forces are adjusted so nicely that there is no net change in the volume of the system. So, delta V of mixing is 0 as it is called, volume change of mixing is 0. There are systems which fall in this category. The other property is the where the heat effects are negligible. So, I take the enthalpy of 100 cc of A and enthalpy of 100 cc of B and then I do the mixing adiabatically and then measure the temperature. So, I find that if the mixing is done, there is no net change in the enthalpy, there is no net change in the temperature. So, delta H of mixing is 0. So, if delta H of mixing is 0 and delta V of mixing is also 0, then we say that the solution is an ideal solution, then only the solution is an ideal solution. Examples could be ethanol water system. The delta H of mixing is negligible, it is very small. Delta V of mixing is fairly large. The volume change is very large. If you mix ethanol with water, the net volume is lower than the two individual volumes. So, molecules they reorganize themselves due to intermolecular forces and the net volume is lower. You do not get 100 plus 100 equal to 200, you get lesser than that. Water ammonia is an interesting system where the delta V of mixing also is there, but not very significant, but when you mix water with ammonia, the system heats up depending upon the concentration range in which you are. So, there is a heat effect, there is a thermal effect. So, delta H of mixing is non-zero, fairly large, but delta V of mixing is approximately 0. So, I am giving you two examples. Ammonia water system where delta H is very large and ethanol water where delta V is very large. So, both cannot be characterized as ideal systems, but if we had ideal system, there is a third property now we want to look into and that is the delta G of mixing, the change in GIFs free energy, the change in GIFs free energy. Now, GIFs free energy comes in calculation and it has some physical significance. The physical significance we have in mind is the entropy, something to do with the randomness. So, if I have two species, species A species B and to start with if I have pure components, let us say I have benzene and if I have toluene and if I mix the two benzene toluene forms an ideal system, the delta V of mixing is 0, the delta H of mixing is also 0, but the moment I have done the mixing, I have mixed benzene with toluene, the entropy of the system has increased, the randomness has increased. And therefore, delta G of mixing is not 0, delta G of mixing is not 0 even for the ideal system. So, we will not be measuring the delta G of the system, what we are interested in, what is the difference between the actual delta G and the delta G had the system been ideal because we know that delta G of mixture is not 0 and that will be measured in terms of the excess property. So, that is the excess property excess over and above the ideal mixing. So, what is shown here is exactly that. So, I have a mixture now and I am taking the Gibbs free energy, I am mixing that in certain proportion mole fractions are given, I write the expanded form of the Gibbs free energy in terms of fugacities, this is a little derivation here, you go through that and one can show that the G of a mixture is G ideal as I said that G cannot be 0, there is a delta for even for ideal plus an excess property. Now, if the system is ideal then it follows Raoult's law and Raoult's law as compared to modified Raoult's law when we had written in the beginning we said gamma equal to 1, when we say Raoult's law we say activity coefficient is 1, gamma is 1. So, that means that G ideal does not contribute to gamma, when contribute to gamma it is G e which contributes to gamma. So, that brings us to the definition of how excess Gibbs free energy is related to the activity coefficient. So, if it is a non ideal system and we need to calculate gamma which is the activity coefficient then we should have a mechanism to model the excess Gibbs free energy. If we can model the excess Gibbs free energy then we will be able to calculate gamma, once gamma is known we can calculate the fugacity in the liquid phase. And obviously, that for ideal solution the excess Gibbs free energy is 0 and therefore, gamma goes to 1. So, that is the background on which this activity coefficient models work, this is the Hildebrand model for regular solutions where the excess Gibbs free energy can be calculated through the pure component properties. This is the only model which is available where pure component properties can be used to calculate the excess Gibbs free energy. This works fairly well for hydrocarbon systems, lambda is the heat of vaporization. Now, if we take binary systems and if we plot activity coefficient now this is coming from the experimental data. So, if we plot the activity coefficient back calculated from the experimental data. So, this is activity coefficient with respect to the composition. So, this particular diagram if you really see this scale is logarithmic again this is not linear scale. So, it is log of gamma and this is a linear scale. So, log of gamma is plotted with respect to the composition. The system here is ethanol normal heptane. So, what we find that as the composition of ethanol in the system increases the activity coefficient of ethanol continues to go down and when you have close to pure ethanol the activity coefficient is expected to be 1. Isn't it? Because it starts its own it starts showing its own behavior. Whereas, as the concentration is increasing in terms of ethanol the normal heptane molecules are feeling the presence of ethanol molecule. So, they are not comfortable. So, their activity coefficient keeps on increasing. So, as they are getting into small numbers they have what is known as an escape tendency. Larger the value of gamma which means escape tendency is more. So, when gamma is greater than 1 it is called the positive deviation from Raoult's law. It is a positive deviation from Raoult's law. So, there is the escape tendency is increasing. So, these molecules are not comfortable. So, normal heptane is not comfortable when it is present in small amounts in ethanol it wants to get away which means its effective vapor pressure increases. Escape tendency is there because gamma is a multiplier on the effective vapor pressure. And the same similar argument can be given for ethanol then when ethanol is present in small amounts it is not comfortable with the bulk of normal heptane. So, ethanol molecules have an escape tendency. In the limiting case these are known as infinite dilution activity coefficients. So, when ethanol concentration approaches 0 I have a value close to about 18 here. So, the activity coefficient of ethanol is 18 which means they are the most uncomfortable at that time when they are present in small amount. Similarly, for normal heptane the value is again close to the same number may be slightly lower and this is the infinite dilution activity coefficient of normal heptane this is the infinite dilution activity coefficient of ethanol. This scenario is very interesting scenario because both the curves are parabolic and they look like the mirror images of each other. They both have convex behavior from beneath if you see the curve is convex this curve is also convex and the one curve is the mirror image of the other. And what does this activity coefficient represent? Activity coefficient represents nothing but modeling of excess Gibbs free energy G E essentially. So, this was the basis that Margules wanted to come out with some expression. So, that this gamma could be modeled as a function of concentration. So, Margules now log of gamma because it is a semi logarithmic scale. So, log gamma if you treat as one variable y and this is equal to some constant multiplied by x square. So, y is equal to a x square y a x square because the curve is parabolic curve is parabolic. So, it is a parabolic equation. So, y is equal to x square while we are measuring y for gamma 1 for liquid 1 the concentration is of 2 mainly because when x 2 goes to 0 gamma 1 should go to 0 gamma 1. I said that when ethanol molecules are increasing in size in increasing in number the haptane molecules they are most uncomfortable. So, the gamma for that is becoming large its own gamma should go to 1 is this clear. So, you have the same form for species 1 and species 2 because the 2 curves are symmetric and they look like mirror image. So, the same parameter can apply on both. So, this is one parameter Margules equation basically modeling the variation of gamma as a function of composition which in turn is modeling the excess Gibbs free energy. Let us go back look at this particular function I am now modeling acetone versus formamide the infinite dilution activity coefficient of acetone is below 4 let us say 3.5 or something and here it is close to 6 or 5.5. Again the convexity is there, but we cannot say that there are mirror images. So, if there are not mirror images the parameters have to be more than 1 to model the same system. So, the moment we have situations where the infinite dilution activity coefficients are dissimilar for a binary system rest of the nature remains still the same convexity still remains the same we require minimum 2 parameters one cannot work. So, we go to Margules 2 constant equation 2 parameter. So, in the first diagram this equation will work, but this will fail in the second diagram. In the second diagram this will work, but if you use this in the first one no problem because one of the parameters will automatically go to 0. So, the whole theory is around how well you can model the ln gamma function as a function of composition that is all. Now, the complexity is tremendous I do not have time to go into too many details take for example, chloroform versus methanol. Now, chloroform has a very peculiar behavior here if the gamma value starts from slightly close to 2 and then if you can if you can see it is not very clear in the diagram it increases in the beginning it is exhibiting a maxima somewhere here and then it goes down. So, this form of the curve is concave and then this form of the curve is convex whereas, this curve is more or less similar to these type of curves it is little flat in the beginning, but still it is a type of parabola. Now, Margules 1 parameter and Margules 2 parameter both will not work here because they will not be able to capture this particular effect which means some other non-linear formula has to be brought in. Look here acetone chloroform both the curves are concave not only that the concave that here the curves rows above 0 above 1 here whereas, here they are starting from 1 and they are going beneath. So, both the compounds they are not showing higher escape tendency they are showing lesser escape tendency. So, this is a negative deviation from Raoult's law this is negative deviation from Raoult's law and of course, a very difficult case where based upon the composition you may have a scenario that the liquid may split into two distinct liquid phases. Benzene water is a very good example where small amount of benzene can stay with water small amount of water can stay with benzene, but if large amounts are present the benzene layer is separate and the water layer is separate. So, two liquid phases. So, when we talk about these non-ideal systems and this is typical of polar systems these are some of the scenarios and this is not complete this is just a representation these are the scenarios we have to live with. So, there are large number of equations which have been developed over a period of time they differ in complexity and the type of form they use for modeling the excess gifts free energy, but the fact remains that all of them are modeling the excess gifts free energy as compared to the ideal solution as a function of composition. And some popular methods are Van Laar two constant equation Wilson two constant model NRTL which is non-random two liquid model or Renan-Prausnitz equation it is called and then you have a equation called Uniquac which stands for universal quasi activity coefficient model. So, what is that they are all doing they are trying to model the excess gifts free energy as a function of temperature. All these models they have parameters sometimes one parameter sometimes two parameter sometimes three parameters. And because they have parameters it becomes essential that we have experimental data if we do not have experimental data we will not be able to back calculate these parameters. So, the liquid activity coefficient models if you have to use where the prediction of gamma becomes an important step we require first the experimental data to have these parameters in these equations. Now this is this was not the scenario in the equation of state case. In equations of state we said that PVT data is sufficient to do reasonable predictions. Here if you if your system is mildly non-ideal hydrocarbon type the same statement still applies you can use regular solution theory and you can do predictions using sketch hydrobrand method, but the movement it goes for polar systems highly non-ideal systems we require experimental data. So, that this is the deficiency, but the advantage is that once you have the data you can make very good predictions which will be much better than what you will get from the equation of state. Just to show you some examples of these parameters which are fitted and a huge compilation is available in this series some 14 volumes I think are available today when this was published it was in 10 parts these are called Dakima chemistry data series and for variety of binary systems these parameters have been fitted in the experimental data some examples are shown here Margules, Van Laar, Wilson so on and so forth. And as you can see looking at the parameters itself sometimes they are negative positive that they are fitted parameters they should not be assigned any physical significance. Some predictions with respect to experimental data and you can see that in this particular case gamma ethanol is being predicted these open circles are experimental data the firm line is Van Laar which is just a two parameter equation and Wilson also is two parameter, but it is much more complicated equation in its form and so when we are in the bulk of the composition here both are working very well, but when you are doing dilute solution distillations dilute solution calculations Wilson is working much better than the Van Laar. So if you were working in this zone and if you did your simulation using Van Laar you will have very erroneous results, but if you used Wilson you will have much better results. So infinite dilution range calculation is very important and it should be kept in mind. Again for the completeness of subject let me just mention that there is this concept of calculating activity coefficients through the functional group method and the most popular functional group method which is available in simulators is what is known as the Unifac group contribution method Unifac stands for Uniquac Functional Group Activity Coefficient. The basic concept here is that the gamma is broken into two parts one which is known as the combinatorial part and the second which is called the residual part. So combinatorial part means that that part of gamma which you can calculate when the molecules combine when A combines with B. So based upon the size and shape of the molecule volume and surface parameters you should be able to calculate the activity coefficient. So that part is calculable. What cannot be calculated but is found in the experimental data that is left as the residual effect and that is where you require the experimental data. So rather than working with species in hydrocarbon systems particularly if you have homologous series the suggestions were made quite some time ago that why cannot we work with the functional groups because it is the functional groups which combine and give you larger molecules. So rather than having volume and area parameters for let us say butane why not I have parameters for CH3 as a functional group CH2 as a functional group and then I can combine the effects and find out the area and volume parameters for the molecule. So it is a building block approach. So molecule is being made out of small constituents, smaller constituents which are functional groups. So for large number of functional groups for organic substances these parameters have been tabulated and this enables you to do the calculations based on the data which is already there in the library. So basically it is group contribution method and you do not require the experimental data. Some example of for certain functional groups one of the parameters AMN it is called these values have already been given in the library. The very good reference for this is this vapor liquid equilibria using UNIFAC which was published in 1977. Again the area and volume parameters these are known as the volume and surface parameters for different types of functional groups and subgroups. What are the advantages of UNIFAC method? Well the advantage is that it uses the theory which was originally used in the UNIFAC equation that break the contribution in two parts the combinatorial part and the residual part. So it does have some theoretical basis. The parameters have been fitted by taking experimental data. So more or less they are independent of temperature. Size and binary interaction parameters are available for many functional groups. They have been calculated. Temperature ranges are fairly ok 275 to 425 it does not give you too large a temperature range but in the non-ideal systems normally are distilled in this range only. It turns out that most of the non-ideal systems are distilled in this range and therefore these temperatures are sufficient. They are not applicable beyond 10 atmosphere pressure and again the non-ideal systems are normally distilled below 10 atmosphere. And the predictions have been compared extensively with experimental data. The errors are there but they are not very large. So for preliminary analysis for preliminary design UNIFAC method works fairly well. Of course because it is predictive method the disadvantage remains that it cannot be as accurate as NRTL or Wilson or Uniquag because they use the experimental data. So here is an example. This supposed to be a gamma activity coefficient. These are the experimental values. You have predictions by Van Laar, Wilson Uniquag and UNIFAC. This is predictive. These three are experimental based on experimental parameters. So Van Laar does not predict the data in this form it rather predicts a phase split. Wilson gives you 2.35 which is very close to this value. So prediction is very good. There is slight under prediction for the second constituent, second species. Uniquag gives you slightly over prediction here 2.44 and 1.36 which is same as Wilson. And UNIFAC based on group contribution gives you 2.71 which again is further over prediction and slight over prediction here as well. So one thing to see is that if suppose data was not available if parameters were not available. So neither Wilson nor Uniquag or let us say if NRTL was there we could not have used any of those. Then should we go and resort to Raoult's law? The answer is no because if you had put Raoult's law you would have made this value, this value 1 and this value 1 whereas if you have UNIFAC you are getting 2.71. So 2.71 vis-a-vis 2.348 is much better option than 2.71 vis-a-vis 1. Similarly 1.52 over 1.43 is a good approximation rather than replacing 1.52 by 1. So do not use Raoult's law and use UNIFAC. But UNIFAC of course is over predicting and it is not giving results as good as other methods. Here are some computationally based comparisons, Van Laar, Wilson, NRTL, Margules, Uniquag, UNIFAC. This is the applicability part. As you can see Wilson is applicable to vapor liquid equilibrium but it is not applicable to liquid-liquid. All other methods are well applicable to all the calculations. So Wilson should not be used for extraction problems or problems where 2 liquid phases are expected to be in equilibrium. Accuracy wise Wilson is best, Van Laar is good, NRTL, Uniquag are considered very good. Simplicity wise Van Laar and Margules are simple because they were developed that way. All other are quite complex models and therefore computationally fast, slow, slow, fast, slow, slow. This is important again because 60 to 70 percent of your time in simulation will go in thermodynamics. So you should worry about which method is faster and which method is slower. And then there is another interesting table where some selected equations have been taken Redlich-Kister, Wilson etc. And depending upon the constituent you have or the species you have, you define the type of mixture and come back to this table and try to read whether a given method is very good, Vg means very good, E means excellent so on and so forth. G dash Vg means good to very good like that. Average deviation ranges are given for very good, good, fair and poor and this is the type of description for mixtures. So type one mixture when we talk about mixture of non-polar or slightly polar compounds and these are the examples. Then mixture of moderately polar compounds, there are total 12 classifications. So you locate the classification here. Let us say if this was of type 7, I am just giving an example. Then you go to this table and go for type 7 and then find out whether 2 parameter NRTL is very good or 3 parameter NRTL is very good or Uniquak is very good. You know so this is a guideline for choosing the right type of thermodynamic procedure for prediction of activity coefficient. So let me end this discussion here.