 primarily about the Gibbs-Duhem equation this is actually applicable for any extensive property what we have seen is for any extensive property you will write everything in terms of specific property sorry where E is identically capped E by n by the total number of moles this is one equation the other equation says I will say example binary we will discuss binary mixtures and then I will extend it to multi-component mixtures E is equal to x1 then we showed that or this is one this is two since Gibbs-Duhem equation says two follows from three from differentiating one and using equation three. So we have the simple result that I can solve equations one and two simultaneously for E1 bar and E2 bar so I have E1 bar is equal to E plus x2 partial of E with respect to x1 and then E2 bar is E-x1 partial of E with respect to x1 just switch one and two but differentiation with respect to two is same as differentiation with respect to one with a minus sign so this is the set of the simplest way of solving for E1 E2 and it has the advantage that because there is one degree of freedom this has the advantage that this equations are symmetric you can simply model E and you do not have to model any one of the components. So if you have a model for E then you automatically have these are general results these are very important and in particular the most important in fact the three quantities that are important I will say E equal to G which is the most important thing GH and V these two are measurable and this is important from a theoretical point of view you know that G with respect to G by T with respect to T will give you – H by T square G with respect to P will give you V so the pressure dependence and the temperature dependence are covered through measurements of H and V because enthalpy and volume are also extensive properties these conditions are satisfied so if I can measure enthalpy change for example actually I should have written this I wrote I think everything as let me make a small change here I should actually have written everything in terms of ? E which is what I did last time right I will just repeat it here I will say more conveniently because what I have written there is correct we write the same equations we write equations like this ? E is equal to ? over XI when you get partial of ? E with respect to ? X I have to write it for specifically for binary I am just repeating the whole thing for you guys I do not know what to do about this chikung gania apparently some cases in the campus health officer and he said he was doing it okay double check it otherwise why do not you tell the wardens I will also tell the wardens we will make sure some spraying is done this thing is not fatal but apparently it causes near unbearable pain in the joints for about three four days okay we will come back to this I am rewriting all these equations so these are not new equations but these are equations written this will be one prime in terms of mixture properties so this is E1 bar – E1 – E2 bar – E2 and then we have E1 bar – E1 is equal to ? E plus X2 partial of ? E with respect to ? X1 and so on and then 1 and 2 can be interchanged what you are looking for is this property here E1 bar because if you are doing face equilibrium we have already shown in particular remember that G1 bar is the same as ? ? 1 so we have shown in face equilibrium that ? 1 ? is equal to ? 1 ? ? 2 ? is equal to ? 2 ? governs the face equilibrium so if I know the chemical potential as a function of composition then I can do the face equilibrium calculations so the purpose of this entire exercise is to get the composition dependence of the chemical potential since it is more general than that the theory actually applies to all partial molar properties I am deriving it in general for an arbitrary partial molar property I said three are of importance ? G because that will give you the chemical potential as a function of composition ? H ? V because they govern the temperature and the pressure dependence of the chemical potential and those are the measurable quantities these are not actually what is measurable is the change in enthalpy in the change in volume of course is explicitly measurable so we have these equations so what we have is in particular when E is equal to G E1 bar is equal to ? 1 and E is equal to H E1 bar is equal to I am sorry H1 bar for the others we do not have a name this is G1 bar which is identical with me one because E after mixing is XI EI bar we integrated the basic equations to show that you remember otherwise I can recall it for you here we did it for G the basic equation simply tell you this quantity holding TP NJ not equal to I is called EI bar that is the definition of partial molar property we integrated this at constant T and P but at arbitrary T and P the initial state was characterized by TP and some composition and we took this thought experiment in which we expanded the system by a factor of K into calculate the differences from that we got E is equal to sum over EI bar Ni so for property of a mixture it is not sufficient for you to know the specific property of pure I you have to know its value the permeable value in the mixture this is a function of composition this is also equal to E after mixing if I am considering a process is identical with the E before mixing is sum over EI Ni this EI is pure I I take N1 moles of 1 it is an its value of E is small E1 so I just add it all up because this is before mixing after mixing in the mixture the property changes from EI to AI bar so if I want to understand the mixing process I have to know how to calculate EI bar that is the central thing in mixture theory because it was not done this way actual science progresses by iterations by various things but since this is 175 years since all this was done in fact more I think 1927 when people started doing this process in when Gibbs introduced partial molar properties and so on so it is it is been refined over a period of time now we see it in perspective the way we look at it is simply to look at the process of mixing in which you can make measurements you can mix two components at constant T and P the quantities you can measure are actually ? H and ? V you cannot measure ? G but so happens that ? G is the most important quantity because it governs phase equilibria chemical equilibria through equality of chemical potentials in two phases so I have to look at ? G although it is not measurable ? H and ? V are measurable so I am looking at E before mixing E after mixing if I take the difference between these two you get Ni into AI bar – AI if I take per molar the mixture I get this equation I take X1 moles of 1 X2 moles of 2 and form one molar the mixture so per molar the mixture the change in E is simply given by XI into this and I am doing binaries because primarily we will be dealing in this course with binaries and I will show you that the extension to multi-component system system real it is only algebraic complexity in practice though if you went to a refinery they will tell you you guys are being theoretical the actual refinery has 21 components but all you have to do is extend this and hit the idiot box on the head I mean the computer you just have to hit it on the head will give you sped out all the answers just simultaneous equations instead of solving two equations you will solve any equations in any component system they are all absolutely trivial in principle this is all there is to it but what we found from thermodynamics also is that the equations for the chemical potential or for AI bar for that matter for any AI bar you have N variables and only N – 1 equation so there is 1 degree of freedom thermodynamics cannot give you that degree of freedom there is uncertainty so that comes from experiment the only way to do it is try and measure use something experiment or empiricism you guess a model either modeling or experimentation for delta H delta V you can get it from experiment if you can get it from experiment you do not do modeling when you are uncertain you do modeling you start talking about wave mixtures occur classical thermodynamics does not provide you with any method for calculating this directly molecular thermodynamics gives you mechanisms because there are molecules you can say this molecule is replaced by this molecule for example if I want to talk about mixing I will imagine a box full of molecules of type 1 and I will bring a molecule far away from infinity molecule 2 to the middle of this mixture then I ask what is the configuration around this molecule 2 what was the configuration before what is the difference in the energy and what is the difference in the entropy I can do these calculations in a molecular theory but as far as classical thermodynamics is concerned I am simply guessing what delta ES in the case of G in the case of H and V so I will say for E equal to H or V use experimental data for delta H delta V for E equal to G models have to be developed we have exactly one model one ingenious model and variations of that this was done in 1898 as I told you by Van Laar I do not have the Dutch say use read V as F the Germans do I am not sure I have had only Van Laar here and in the US so I have to check with some Dutchman and I will same parenthetically Van der Waals and then any number of molecular models and then empirical models of course you will have to know the Van Laar model for one thing it works very well and it is also beautifully derived I will derive it here you do not need any extraordinary concepts with the existing concepts you can do the derivation it is a really clever one I mean after it is derived you can say what is the big deal I could have done it if you go back 1898 and look at the state of thermodynamics and ask if you could have guessed it and you said yes then you better be jolly good you better be in a well-priced quality but if you are honest most probably you will say no you could not have even thought of it molecular models there are many many of them the ones you have to know in this course are all of these are actually quack first like it is not then unique quack then there is a model called NRTL then I should have mentioned Wilson I think those are the only ones you have to know there are more but these are the tested ones and empirical models starts from what is called porters model holes models is a whole series of models I will add Van Laar again because Van Laar is actually derived of very good model it had only one parameter in it then people did not later who people who came along use these equations without realizing that there was only one parameter to there were two parameters related to one another they did not realize it so they made it a two parameter model and the two parameter model is so successful that it is fully established so we will use the empirical model of one large is the same model as what we derive here except that people forgot how it was derived this kind of thing happens all the time then finally there is a model by doles Alec no that should come under theoretical models I will say here Van Laar one classical model I suppose I should say I will also say plus one due to doles Alec this comes under slightly different it comes under what is called chemical theory of solutions all of these come under so called physical theories that means these theories for mixtures are derived without assuming any chemical reactions doles Alec said all mixtures are ideal except that you are looking at the wrong species so therefore it is a chemical theory it assumes it postulates chemical reactions that actually occur in mixtures as a result of which if you mix A and B you do not get a mixture of A and B but you get an ideal mixture of A to B and AB three if you like some combinations he postulates that there are reactions that take place in the solution in the final mixture if you looked at the right components is simply ideal but all others are physical theories actually there is some evidence for chemical theory chemical theory is now used in all cases where there is clear evidence of association salvation we have mixtures where the components in solution form dimers or for example hf in a mixture always seems to exist as h6 f6 if you take that into account then it does form ideal mixtures the other substances similarly you have your salvation where where a solvent solute molecule is surrounded by the solvent molecules because of high interaction so if you take that complex as one entity then you have an ideal mixture so these are very special cases other than that these are the models I will just say a few words about these acronyms this is actually called quasi chemical theory it is called quasi chemical theory because the form of the final result looks like the result you would have got had you assumed a chemical reaction it sort of recalls dole's alex theory but there is no chemical reaction postulated here the reaction postulated is of this form I mean postulated in the sense the final result looks like it in the sense that what is what this was done by Guggenheim that is the big name here there is a series of Guggenheim lectures Guggenheim this thing in Europe the students students students all combined to form these lectures and they are very prestigious anyway Guggenheim suggested that if you take a lattice model for the you do not have to go to molecular thermodynamics for this you simply imagine that one and one are neighbors and he says if you have a one and one and then out to two this is one one this is to two in the mixture you are replacing this one one bonds by one two and two one bonds the two two and the one one are you breaking up and you are forming one two and two one bonds simply did a counting on that and assumed that one one plus two two produces two times one two if you imagine the reaction of this kind where you go from a one one pair and a two two pair to one two pairs then you would write a chemical equilibrium equation which tells you the number of one two pairs whole square by the number of one one pairs times the number of two two pairs is equal to constant that is the equation that comes out of the theory and therefore the theory is called quasi chemical theory this is called the universal quasi chemical theory this is derived by Proust Nets so I will write the names down this is Guggenheim and then this is due to Proust Nets Proust Nets and co-workers Proust Nets is probably 75 and he is at Berkeley even now NRTL is again due to Proust Nets but the name associated with okay I will tell you the for Uniquac the name associated with that is Abrams for NRTL it is Proust Nets and Renon Wilson is due to Wilson so he generalized Proust Nets generalized this theory again made a mistake end up with two independent parameters that correlated data so well that you cannot argue against Uniquac I mean if you say Uniquac should not be used people laugh at you then further corrections have been made in fact we ourselves have made the correction in Uniquac there was a mistake you got to what is called the it is not at widely used as no point teaching it in an undergraduate course but it is called it is essentially a self consistent theory NRTL is actually again I do not want to get you guys to get very skeptical NRTL is derived by a mistake but the resulting equations correlate data so well that people have accepted it again it is called non-random two liquid theory there is a corresponding theory actually I must have mentioned that here apart from these there is another whole list of theories that go by the name of corresponding states I will discuss that separately there is a corresponding states theory due to Scott and due to Scott and what from that theory is NRTL theory was derived I mean that is a basis I am going to discuss all these in greater detail this is giving you general stories behind but anyway this is the this theory is a very successful theory for liquid liquid mixtures so we will use it Wilson is the guy who started the whole concept of liquid mixtures theory of liquid mixtures actually it has a very small paper in 1962 I think it is in if I remember correctly in Journal of JCP and it was a very small one page paper and Wilson simply said maybe I am jumping the gun but let me say this a bit suppose you have 1 1 1 1 1 here and then you introduce a molecule of 2 here what he suggested was for example this could be suppose you looked at a liquid mixture closely and locally you could take a look at let us say this is the lattice configuration I have 1 4 neighbors arranged like this around a central molecule actual lattice configurations are different but does not matter only illustrating a point if I have cells like this inside the liquid then I have a 50-50 mixture of 1 and 2 all around all it sees his neighbors are once in here it is 2 and I will show you the following physical result we have already seen this result you have seen that the work done in a process in an isothermal process at constant temperature and pressure the work done is equal to minus delta G we saw that for open systems correspondingly this number the work done in bringing a molecule to 1 mole of 2 into or 1 molecule of 2 into a mixture of 4 molecules of 1 and so on is determined completely by the chemical potential mu 2 I will show you that mu 2 is the change in mu 2 from the pure state gives you the work done in bringing this molecule to from infinity to here that is the physical interpretation so this the work done in this mixture in a 50-50 mixture is actually not dependent on the global composition but on the local composition. So what Wilson said was you should not be talking of x2 what you should be talking of x2 1 that means the mole fraction of 2 in the neighborhood of 1 and that is what determines the work done and that is what determines delta G that is started off a whole spate of theories called local composition theories and all these come under local composition theories this does not quite come under local computers derived before Wilson's theory with Uniquac and NRTL come under local composition theories all these so these three are called local composition theories I think that is all as far as these theories and what I will do now is derive the theory of one law one law basically said I am interested in a model for delta G you want the composition dependence of the free energy of mixing this is clearly equal to delta H you are doing mixing at constant T and P this is the process of interest in looking at delta G mix it is equal to delta H – T delta S is equal to delta U plus P delta V – T delta S constant T and P so at delta T delta P one coming no change in temperature and pressure of these this is an argument you can make the change but this is a valid argument for a large number of mixtures the change in delta V is so small that P delta V is very small compared to delta U so I am going to neglect this because it is small it is a reasonable assumption very rarely is delta V large enough for you to worry about it is an assumption that is made very often then he said I am going to have an ideal mixture I already understand an ideal mixture I showed you that one solution to the Gibbs Gamm equation is log x 1 but let us say I assume delta G ideal is equal to delta H ideal – so I will say approximately but I will say – T delta S ideal here I can write P delta V ideal also but in the ideal mixture there is no change in volume there is no change in enthalpy I do not have to worry about those in an ideal mixture you have molecules that do not whose interactions are not different in the mixture from in the pure state so there is no change in enthalpy there is no change so this difference delta G – delta G ideal is equal to T delta U and write – delta U ideal but this is actually 0 – T this is written identically as G excess similarly this will be written as U excess this is written as S excess the superscript is for excess it means over and above the ideal case all theories the molecular theory or the classical theory so far we have a problem with finding both they either assume U excess is 0 or S excess is 0 I must say one more model here in the molecular models is one more model which is due to flurry called the flurry against model it was polymer solutions where he said the big difference is not in energy but it is in arrangements so if you put in one polymer molecule here the polymer molecule is so large that has about 10,000 neighbors so your actual polymer concentration will be in parts per million but will have a tremendous influence on properties it is sort of deceptive if you take the weight fraction the polymer will be very high but if you take mole fractions the polymer will be very low because the polymer molecular weight will be 100,000 the other molecular weight will be 10 so the factor of 1,000 or 10,000 so he was looking at special case of polymer solutions and he derived the theory that is a beautiful theory but it is one of the earliest as far as so flurry is another person it is called the flurry against theory against a student but anyway coming back here all theories flurry for example said U excess is effectively 0 it is negligible it is the entropy change that is most important and many other theories assume that S excess is 0 it means the mixing is ideal but the energetics are important so Van Laar for example said Van Laar's assumptions are the following assumed first that ? V is approximately 0 it is a very valid assumption I will say valid almost always and then he assumed S excess is equal to 0 this is not always valid certainly not valid for solutions of mixtures of substances that are very different in size but this is not always valid I will say for example not valid when sizes differ significantly so he finally said therefore ? G is equal to ? U you can relax some of those assumptions it is not trivial but you can but you are now going back in 1898 he went back to this equation he did not do it exactly like this this is also equal to CV so he starts with this equation du is equal to you are talking of constant temperature process so this is 0 in the real interest was in mixing of liquids that is of interest even now in thermodynamics because in mixing of gases not much happens gases behave reasonably well unless you are talking of very very high pressures so what Van Laar was then did a very clever thing he said he wants to mix 1 and 2 let me say x1 moles of pure 1 and x2 moles of 2 separately in 2 boxes from here you have to go to mixture of 1 and 2 of composition x1 this is liquid this is also liquid and you want to know what is ? G the progress we made is ? G is the same as ? U but I do not know ? U yet so I have not made much progress what he said was at this is a pressure P at T P this is also at same temperature and pressure he said we will go here T, P is equal to 0 it is simply this is a thought experiment so you just imagine that you evaporate completely 1 and 2 go to P is equal to 0 you have to be below the critical temperature actually for this you do not I am sorry let me just get back here so you just go to P is equal to 0 then you mix the 2 this is ideal mixing so you produce a gaseous mixture of 1 and 2 and then you condense it so this is evaporation this is condensation so you are replacing mixing this is ideal mixing this is mixing so the process of mixing you replace by evaporation ideal mixing and condensation I was looking at it step by step if you look at this you want only ? U for each of them ? G for this is the same as ? U ? G for these three steps because G is a function of state ? G does not matter if I calculate ? G from here to here or from here to here then to here then to here I add these so we will call this a b c d so first is ? G a to d is equal to ? G a b ? G b c now he said you can replace this with ? U a d actually he did not say this sorry this is ? U – T ? S please correct this but he said G excess is equal to U excess because ? S for the real process he said was the same as ? S for the ideal process the ideal process entropy you still have to take into account anything I want to stop here and repeat what I said because I said a lot I will continue next class actually the calculation steps are fairly straightforward imagine first of all the quantity of interest is the chemical potential for all phase equilibrium calculations for reaction equilibrium because on the one hand for reaction equilibrium I am going to use the fact that G should be a minimum and G again is given by Ni ? I so if I know ? I as a function of composition then I can play games with this if I want G to be a minimum I use the usual calculus but I have to use it in terms of measurable quantities so I have to express G is equal to Ni ? I in terms of composition variables so I need chemical potential as a function of composition now thermodynamics will give you the temperature dependence of the chemical potential through the enthalpy measurements it will give you the pressure dependence of the chemical potential through volume changes measurements but it cannot give you the composition dependence except through the Gibbs Duhem equations the Gibbs Duhem equations as I told you constitute N – 1 equations for N variables so there is 1 degree of freedom the way to deal with the degree of freedom is to look at the mixing process and derive all the results in terms of change in G during mixing so that is all I have done instead of looking at change in G I looked at change in any extensive property because in particular I am not only interested in change in G I am also interested in change in enthalpy and change in volume the change in G is given by I we derived those expressions and you know how to get e1 bar e2 bar once I know the change in G I have to know change in G as a function of composition so that is all I am looking at what Van Laar said was we do not understand liquid mixing it is too complex but we understand ideal gas mixing so he will go from here evaporate this in order to evaporate this all he has to do is integrate this from P is equal to 0 to P is equal to P right or the other way around P to 0 if you integrate this you just write this as doh v by doh p dp and integrate it if I have an equation of state I can calculate all these quantities it is not ideal after all you are going on increasing the pressure it is going to be non-ideal so you put in the Van der Waals equation of state for P do these differentiations and get this difference in particular you will get a by v squared Van der Waals equation says P plus a by v squared into v minus B is constant so when you do P with respect to T you will essentially end up here with a by v squared so if you integrate that you get directly a nice expression in terms of a you will get ? u so the change in internal energy here is measured by that constant a11 for pure 1 and a22 it is got two subscripts because in a mixture a changes then you do the ideal mixing there is only entropic change when you do ideal mixing and calculate that entropic change separately so ? s ideal is known ? u ideal is 0 then I have to do this calculation for condensation again Van der Waals equation had to be used it is a mixture a and b are functions of composition and what functions of composition you have to use an empiricism there I know how mixing occurs for hard spheres for example the size of the mixture the mixture here is treated as if it was one molecule or one substance with certain parameters certain size and certain energy so here again Van der Waals already had mixing rules we use those mixing rules and calculate the condensation energy it will also come out in terms of a mix so the whole process this plus this plus this will give you this expression ? once I have ? g I can get ?1 ?2 by differentiation you already written those equations.