 just brought some data actual experimental data just put down the curves in the names of the systems these are yx diagrams I want to discuss vapor liquid equilibrium really the first one only looking at qualitative curves the kinds of curves you get this is tetrahydrofuran CCl4 okay the second system behaves like this this one is chloroform tetrahydrofuran it is sort of a convention that the more volatile component becomes component one why he has this third one because it does not illustrate anything different this is furan carbon tetrachloride is the last one looks like this one is ethanol toluene we will mark them by different you need to discuss only three this is one this is the second one in the plain line it will be the third one then this can be another one okay these are examples of xy diagrams in the corresponding T xy diagram these are happen to be data at one atmosphere if you T versus x,y I told you x is equal to 0 corresponds to pure less volatile component so it has a higher boiling point so normally the curves will look like this this is T versus x this is T versus y this is for example for system one call these A as he did and then B C and D so this is system A for example let us assume because this is at one atmosphere low pressure so gas phase is ideal and then you can assume that pointing correction is one so the phase equilibrium is simply governed by P1S x1 gamma 1 is equal to P y1 and P2S gamma 1 x1 gamma x2 gamma 2 is equal to P y2 follows of course at the total pressure where gamma 1 gamma 2 are known functions of composition these are functions of temperature so this is a function of temperature these two gamma 2 are functions of temperature and composition normally in curve fitting this is the equation you use in this case if P is one atmosphere you put it equal to one and find the parameters in this expression FT x1 has also has parameters in it for example if you have porters model this will simply be log gamma 1 will be A x2 squared that A is a parameter which is a function of both temperature and composition which is why actually from a theoretical point of view one prefers to do constant temperature data because prefers constant temperature data because this is a function of temperature alone automatically this becomes a function of temperature and x which is becomes then a function of composition alone otherwise you have a temperature dependence as well actually the way you deal with the gammas through GXS by RT is equal to some known function x1 and the temperature dependence is handled by RT squared so if you know the values at one temperature you should be able to calculate other temperatures using this so you have an integral of this equation that will then lead to from experimental data on HXS HXS remember is actually ? H of mixing because it is ? H of mixing – ? H ideal ? H ideal is 0 so from experimental data on HXS you can get the temperature dependence of the parameters here and that will have to go into this equation here for but if you take a look at this curve for example instantly these are compounds that are actually we did some measurements on these compounds on that are of direct importance in the chemical industry and what you do in vapor liquid equilibria is you do binary systems basically the question in thermodynamics is how do you stop having to measure experimental data for all systems so you have to classify systems make a measurement for one and guess the parameters for the next system if you do that at least if you measure all the binaries can you tackle the ternary that is the question so if I have GXS by RT for a binary for all the three binaries in a ternary system suppose I have a ternary system consisting of tetrahydro furan CCl4 and CHCl3 if I know the GXS by RT for each of the binaries and I know the parameters can I predict GXS by RT for the ternary system this assumes that basically all interactions are pair wise and they are independent of the third body if this is true you can make a prediction and what you do is if you are doing experimental measurements primarily what you do is look at systems for which the ternary data is already available and two binary data are available then you measure the third binary data and test out a theory that says how to predict the ternary data I will take that up very briefly because basically in an undergraduate course it is only more of the same we do not normally include ternary calculations but good for you to do one soil at least include that in the assignment but let us look at the azeotropic problem this is case B I have an azeotrope azeotrope is simply defined as the phenomenon in which a mixture when distilled gives you vapor of the same composition that means you cannot use ordinary distillation to enrich the mixture if X1 equal to Y1 then P1S ?1 is equal to PY1 I am sorry P so you have ?1 by ?2 is equal to P2S by P1S what you normally get from your models is log so log ?1 by ?2 is simply log of P2S by P1S in the example potas equations you have log ?1 is equal to AX2 squared and log ?2 is equal to AX1 squared so you get A into X2 squared – X1 squared which is the same as A x2 – X1 this is log ?1 – log ?2 which is the same as this is equal to log of P2S so if you measure the azeotropic composition this is known you measure this so A is simply equal to log of P2S by P1S by X2 – X1 at azeotrope and it is easy to measure the azeotropic composition so you can get the parameter A if actually in this in fitting data just take algebraic advantage of the form if I have for example one large equation then other model is often used in azeotropy especially for in this case for ethanol toluene alcohol water alcohol hydrocarbon systems the one large equation works very well you had an expression GXS by RT is equal to you just rewrite this is equal to AX1 X2 you just divide by B1 by a and so I will just write it as AX1 X2 by AX1 plus BX2 and B1 B2 are actually volumes liquid volumes it is it is equal to the liquid volume so these should be known but the way the equation has been used this is the empirical form of this is this where you forget the fact that B1 B2 are actually volumes of the liquid and you treat capital A capital B as independent parameters if you do that let me derive the expressions for log gamma 1 log gamma 2 all I have to do is to get D of GXS by RT by DX1 I need the derivative then I have AX1 plus BX2 the whole square numerator it is AX1 plus BX2 the differential of this is X2 minus X1 minus X1 X2 this is in the numerator I have X1 X2 into the differential of this which is A minus B so let us say for example this would be for system D alcohol hydrocarbon mixtures you have D of GXS by RT okay I have I will multiply this by AX1 plus BX2 the whole square this gives me AX1 X2 minus AX1 squared plus BX2 squared minus BX1 X2 then there is a minus A into A minus B minus AX1 X2 plus BX1 X2 so these cancel so I simply get D of GXS by RT by DX1 I am going to multiply this by X2 to get gamma 1 finally is simply AX1 squared minus AX1 squared X2 plus BX2 cube by AX1 plus BX2 the whole square so log gamma 1 as you know is simply GXS by RT plus X2 into D of GXS by RT so I have to add this X1 X2 so the numerator of this would be AX1 plus BX2 into X1 X2 minus AX1 squared X2 plus BX2 cube so this is AX1 squared X2 will get cancelled out so I get BX2 squared into X1 plus X2 so it is just BX2 squared so finally log gamma 1 is equal to BX2 squared by AX1 plus BX2 the whole square please that looks right okay it is the final expression for log gamma 1 and log gamma what you normally do is okay we will leave it as it is log gamma 2 is I just have to replace exchange 1 and 2 remember when you do replace 1 and 2 you also have to replace A and B when you do the interchanging they go together so you will get AX1 squared by AX1 plus BX2 the whole square you get log gamma 1 by log gamma 2 remember log gamma 1 is the same as in the case of azeotropes you get log or at azeotropic point point log gamma 1 is PY1 by P1S sorry P by P1S divided by log of P by P2S because P1S gamma 1 is equal to P and P2S gamma 2 is equal to P again if you take the ratio of these two and you put B by A equal to alpha you simply the denominator disappears so you get alpha into X2 squared by X1 squared so P is known what is measured as P temperature is known P is known X1 azeotrope is known so if you have an azeotropic point and you can measure these quantities and I have got this left hand side log P by P1S and log P by P2S therefore I can get alpha once I get alpha again the calculation of the other quantity a say for example you can rewrite this as if I divide numerator and denominator I will rewrite log gamma 1 divide numerator and denominator by B squared whatever I said I said B by A is alpha okay then I will divide this by A squared I will get B by A squared B squared is alpha by A sorry and this log gamma 1 is log of P by P1S so what you do is use this equation 1 calculate alpha then use equation 2 to calculate A therefore B so use to this is measured again 2 to calculate A means B is simply alpha times A so the azeotropic point becomes a very useful point all you need to is make one measurement and then you can predict the whole curve if I make one measurement namely if I make the measurements of T P and X1 of the azeotropic composition of the azeotropic mixture then I can calculate in two parameter equations I can get both quantities you may have to do some curve fitting if it is a different algebraic expression basically you can get both parameters if you get two parameters then you have the entire curve predicted because your equation simply is P by 1 is equal to P1S gamma 1 X1 you know gamma 1 so from one azeotropic point measurement you can get the entire VLE now the interest here is if you want to purify a mixture like this suppose you want to get pure ethanol out of it this is the one that we are working with this is the azeotropic point question is can you shift the azeotropic point this way or this way in a curve like this this is at constant pressure so you can change the pressure or you can do have a similar diagram at constant temperature can I shift the azeotrope by changing the temperature and you can obviously because the azeotropic point as you have seen here the calculation the azeotropic point depends upon the parameters A and B which are functions of temperature so the relative volatility the volatility one component versus the other will change with temperature if one of them changes faster then you should be able to shift this azeotrope this way that way so one way of shifting the azeotrope is simply changing the temperature of operation or changing the total pressure the pressure has a smaller effect than temperature so you normally change the temperature and try to shift the azeotrope and again it depends on this the temperature dependence of the parameters depends on the enthalpy of solution mixing okay and let me look at this is clear as far as vapor liquid equilibria is concerned the main problem is that once you get to this point you cannot purify the substance anymore this point it is the intersection with the 45 degree this point sorry this is the azeotropic point question is can you shift it this way or that way so you can do so by changing T or P another method of doing this is to add a third component if you for instance if I add salt to this mixture the salt addition of any substance we will see that also when you do solid liquid equilibrium addition of any substance to a solvent solute to a solvent will lower its vapor pressure in general if the lower the vapor pressure is lowered more for toluene than for ethanol then this will also shift the azeotrope so one way of breaking azeotropes is by adding a third component but you have to pay a penalty after you break the third component you will get a mixture of the salt and which you can have to separate again so it is simply a matter of economics how you work out this thing normally this is called salt effects in vapor liquid equilibria it is quite an important topic by itself depending on the kind of substance you are producing what happens is now in the pharmaceutical industry some of these substances have medicinal value and there is a market value automatically for such compounds then some of these methods that were considered uneconomical earlier have become very economical it is worth adding a third component to the whole thing to this and separating it out so what I will do is give you an assignment with vapor liquid equilibria large number of examples give you an azeotropic system and I also cover all the models that you have no the models that I expect you to know are these the GXS models do not if I listed them already because the one law for historical reasons and also for the fact that it is very good for alcohol containing mixtures in general alcohol water and alcohol hydrocarbon mixtures you have to know the simply Porter's equations third you have to know the Margules two suffix equation two parameter equation certainly Van der's model also what you do is write a and b as a12 and a21 that if you switch one and two the parameters get automatically switched for you need some of the rest of the equations are from molecular theory is a regular solution theory actually in this what you do is the square root of a11 by b1 that you got in Van der's theory you set it equal to what is called solubility factor derives independently but if you call that a parameter delta that is characteristic of substance one this is called solubility parameter you get essentially the regular solution theory the regular solution theory was divided derived by Hildebrand and Scott think I told you about Hildebrand he was at Berkeley and he passed away as a chemist he is the one who was a mountain here till 40 and then took to thermodynamics anyway and Scott derived it separately is then you need to know Wilson's model Wilson's model is the first local composition model at a molecular level he said that locally the composition could be different from the global composition you could have a 50-50 mixture of one and two but locally around every one you may have a 0.75 mole fraction of two and around every molecule of two you may have a 0.75 mole fraction of one since the immediate environment determines the work required to bring a molecule from infinity to here which is how these models are formulated there is a difference in the difference in the local composition makes a difference in the excess free energy calculation but this is basically a molecular model and following this you have this is this is molecular theory and so is this regular solution all these put a star here all these come under molecular theory under classical models exactly like the instead of the local composition model you can have Guggenheim's quasi-chemical theory what Guggenheim said was very simply this the quasi-chemical theory is a classical theory but it again depends on semi- molecular arguments what he said suggested was that if you have 1 1 pair in a pure substance and then you add a 2 2 2 you add 1 to 2 you have a 1 1 pair in the first container you have a 2 2 pair in the second container when you fix finish mixing them you get 2 1 2 pairs let me mark 2 as something different if you imagined that every 1 1 pair and the 2 2 pairs were replaced by 1 2 pairs in the mixture then you would write a reaction equilibrium constant K as the 1 2 whole square by 1 1 into 2 2 concentration of 1 2 divided by the concentration of 1 1 the number of 1 1 pairs into number of 2 2 pairs if you treat this K as a constant then you can calculate the number of pairs and therefore calculate finally arrive at an expression for again you have to use molecular theory I am not going to derive that here but you will essentially get an expression for excess free energy I will do that a little later after I will show you how you can start from molecular theory and do it do this but you essentially get what is called the quasi-chemical theory because it assumes that this K is a constant exactly like your treatment of reaction equilibrium this is not a reaction that actually occurs but you pretend that the reaction occurs and you get an expression so this quasi-chemical theory has then been generalized to universal quasi-chemical theory or universal quasi-chemical theory is a universalization of this this is Guggenheim 6 model 6 plus local composition concept this is due to prosnets and co-workers basically these this is good for alcohol good for alcohol containing mixtures these are for in some sense simple mixtures more or less hydrocarbons that are similar if you have mixtures of hydrocarbons that are similar then the single parameter this is sometimes referred to as the Margules 1 parameter equation this is Margules because all these come from Wohl's expansion the polynomial expansion for excess free energy the regular solution theory is like the Van Laar theory it is also good for mixtures it also has other predictions that are very useful as I told you one of the predictions that came from Van Laar theory and from regular solution theory is that substances that have the same critical pressure will mix ideally so the critical pressure differences are a measure of the non-ideality of mixing this is part of the regular solution one Wilson's model I must tell you in addition to all these being used in vapor liquid equilibrium I can also predict immiscibility I will do that in a minute because all I have to do is thermodynamics tells me that the ? G of mixing should be a minimum G should be a minimum at equilibrium if I have constant temperature and pressure G should be a minimum so the question is G a minimum for the two substances that remain unmixed or for them in the mixed state so you have to look at G as a function of composition ask if there is a minimum at some point if there is if the curve actually produces a situation where the two separate substances have a lower free energy than the mixture they will remain a separate substance so I will show you that so Wilson's model is used where Wilson's model you can show cannot predict the only limitation of since product is it cannot predict immiscibility you can show this theoretically the quasi chemical theory is generally used in all these cases and the universal quasi chemical theories one of the most successful theories it has actually a fundamental flaw but it is so successful that the flaw is overlooked I will point out the mistake in it the mistake in it is actually that the quasi chemical the universal quasi chemical theory is much more successful than the quasi chemical theory so people thought it was a great this thing this came in 19 this came in 1950s and this was 1973 guy called Abrams is a graduate student with Prausnitz and Berkeley turned out that what they they made a mistake in this theory that the number of pairs you know the Guggenheim worked with number of pairs Prausnitz worked with from molecular theory with a local composition concept he had a very nice way of deriving it so you need one doesn't have to take away credit from Prausnitz but in the process of deriving it it turned out that the theory had one defect that the number of one two pairs counted was not the same as the number of two one pairs in Guggenheim's theory the one two pairs in the two one pairs number of one two pairs was counted as the same as number of two one pairs it was obvious in the way the theory was derived in this case it wasn't so there was a mistake and because of that mistake it had one parameter in a binary more than Guggenheim's theory and that parameters correlated data so well that even now Uniquac has accepted as it is with the flaw in pair counting so this model you should know simply because the industry uses it very widely and it has a very natural generalization to multi-component systems if you know binary calculations you can do ternary calculation so for multi-component systems I will say multi-component prediction from binary data so this is its main power then you have to know one more this is it is called NRTL very pompous names this is non-random two liquid theory so NRTL this is again due to Prowstnitz it is due to a guy called Renon this is Abrams in Prowstnitz the Hildebrand and Scott theory is sometimes referred to as the two liquid theory you can derive this on the basis of I can show you how it is done later but this Scots theory was called also two liquid theory so non-random two liquid theory was a generalization of this does not assume random mixing and this is extremely good for immiscibility immiscible mixtures this theory is even worse than Uniquack because Uniquack had a mistaken pair counting NRTL has a very fundamental flaw in its derivation which is now well known in fact if I derived it in class which I will try and do later after you finish the classical part I will preserve a few lectures for the molecular theory I have to actually ask you to close your eyes and change internal energy to free energy in order to derive that result it is now well known that it has that defect but again it is one of those cases where the experimental correlation of liquid-liquid equilibrium data that is if you have two liquid phases and you correlate the equilibrium it is one of the most successful theories in correlating the data so there must be some compensation whereas that makes it so good so as far as engineers are concerned this theory will continue to be used for quite long time but this question is why does it work so well is there a good derivation of it that will show that the assumption that it makes is actually valid as of now there is no justification for it so you should be aware of it and you should I mean although you use it as a tool you must know that there is no fundamental basis for it you can treat it all you can treat all of them as empirical models because the model classical thermodynamics does not give you an expression for the composition dependence so any model you write for the composition dependence that goes to 0 for the pure substance is a valid model so as far as classical thermodynamics is concerned that the derivation as a flaw makes no difference there is finally an expression an algebraic expression that correlates data extremely well that is all that is important but if you are looking at it from a theoretical point of view it is very unsatisfactory so the situation in molecular thermodynamics is very unsatisfactory as far as classical thermodynamics this course is concerned I will only give you the final expressions in the final expressions in these theories are at the moment the best ways of correlating data so this NrTl again this is the same thing applies to NrTl as well you can do multi component prediction from binary data so you have to know these are the models that you have to know I do not think you need anything else so you will need to know a total of 8 models my suggestion is that actually Wallace has a summary of these so you should have the expressions for gxs and for log ? if you start deriving it like I did now you just take a long time the whole idea of an open book exam is that you can use these so you must get a little familiar with these