 get started I am going to discuss immiscibility basic idea is the following that the Gibbs free energy is equal to the equilibrium criteria is minimum at equilibrium this is a general thermodynamic result because that constant P and P G should be a minimum for a closed system this is because V G is D of U – T S plus P V this is equal to for P V T systems D U is T D S – P D V therefore this is simply – S D T or they should read this is less than or equal to this is using the two laws where D U is equal to D Q – ? W ? W is P D V – P D V or plus P D V and D Q is less than or equal to T D S right since I will say since so G can only decrease at constant T and P therefore G should be a minimum at equilibrium G is a minimum at equilibrium let me write down G for a system of two phases there is a liquid phase alpha phase and a beta phase these are both liquid phases in the example that we have we had ethyl acetate and water and you are talking of Azadir actin distributed between these two phases so I have two liquid phases G if I write it per mole this is G before mixing this is G for the mixture is G before mixing I am writing per mole of the total contents plus ? G mixing or this is the same as G is the same as G after mixing I am looking first at a single phase liquid mixture I want to know if first of all let me write this out G before mixing is X 1 µ 1 pure plus X 2 µ 2 pure in someone do everything at constant temperature I will divide by RT plus ? G ideal mix plus ? G non ideal or G excess sorry I have G by RT is equal to ideal mixing is X 1 log X 1 plus X 2 log X 2 plus the excess free energy I will take the example in this case I will take example the simplest model would be porters model you have to write a model for excess free energy for the system this case it could be porters model which will then read a X 1 X 2 say µ 1 has some value µ 1 has some value this is µ 1 pure by RT this is µ 2 pure by RT this is X 1 so before mixing this value is the value before mixing it is simply a straight line because you are talking of two pure components it is X 1 µ 1 pure by RT this is the discussion I have done some calculations and plotted numbers if you look at the ideal case this one is always less than 0 right X 1 X 2 are less than 1 so log X 1 log X 2 are negative so that number is less than 0 so the free energy of the mixture at any point is lower than the free energy of this is G before mixing this is G ideal mixture ideal mixture would have a G that is always lower than the G before mixing so ideal mixing means all components are always miscible in all proportions because at any given T and P if you fix the composition the free energy of the mixture is lower than that of the separate components so if you keep them together and allow them to mix they will mix because this has a lower G than this and according to thermodynamics the system comes to equilibrium at the lowest possible G so this will be if you take any line here fix the composition this is the G before mixing this is G after mixing if the mixing was ideal that means as far as ideal mixtures if the ideal mixing theory is right all components are miscible in all proportions and the equilibrium state is always that of a mixture the idea in statistical thermodynamics is that the system in this state still has the choice of being like this I mean I mix the components they do not have to mix they can stay apart but that is not a equilibrium state because that state has higher free energy than this but if you take real mixtures there is interaction between parameters this has to do only with entropic effects effectively essentially it can rearrange itself it has more configurations available than it had before if you mix then you can have a component one in the middle two around it and so on various configurations whereas here it was one and one only one around it two and only two around it and no interactions whereas the that configuration is still available to it it has additional configurations in which it can rearrange any number of twos around the one therefore this is larger entropy and if there is no energy of interaction this will always be the equilibrium state equilibrium state is one in which there will be more in states available microscopically for the system then there were before mixing but if you add this energy contribution a x1 x2 for example if a is if a is less than 0 a x1 x2 is negative then again you get a completely miscible solution if a is less than 0 you will get one more addition here could shift things so this is a less than 0 if a is greater than 0 and sufficiently small you will get somewhere in between this is a greater than 0 but small I will define what a small in a minute but is a is large a greater than 0 and sufficiently large in what you could get initially of course it will be below the line but this is the ideal mixing line right so this case is a greater than 0 a greater than 0 you will still get to begin with it will go like this I suppose I have to draw this we will get a curve like this because a x1 x2 is a maximum at x1 equal to x2 equal to half so at x1 equal to x2 equal to half it can become positive or it can become positive at slightly so there is a range over which the net result can be positive so this will be the result when a is greater than 0 and sufficiently large so look at this mixture for example fix this x1 as the you put in a certain amount of 1 and 2 this is a is the point here a b c this is d this I should have chosen another point here let me let me redraw this curve to illustrate it and then it does not this can stay so I have a b c d e of which b c d e are all less than the free energy of the separate components so up to this point at this composition this is x1 let us call it a small a at that composition they are still missable in all proportions but if you go to this composition here for example draw a line here x1 b so we will say first at x1 equal to x1 a mixture always has lower free energy than the individual components before mixing this is the individual components before mixing were separate their free energy was at capital A it reduced to b because of the ideal mixing it reduced to see in the case of a mixture where a is less than 0 it reduced to a point d where a was greater than 0 but small it reduced till reduced to a point e when a was large was sufficiently large to cause destabilization but not enough at this composition but if you move to a composition like x1 b we will draw the same points a b c d e a is here b is still here below c is still here and then d is here but e is above in this case at x1 equal to x1 b point e has a g greater than g before mixing thermodynamics tells you that that cannot be the equilibrium situation at the same time if you look at this curve I can draw a tangent here you why I need to draw a tangent there are two minima in this curve and then a maxima if you look at this two minima these two constitute solutions of this composition x1 low and x1 high we will just call it x1 low x1 high there is a mixture of composition a homogeneous mixture of composition x1 low is a homogeneous mixture of composition x1 high and if you kept these two as the pure components and mix them that is suppose you always had solutions of mixture x1 composition x1 and solutions of mixture of composition x1 high and they formed a physical joint entity rather than a single solution then the free energy would lie on this line right the free energy would be number of moles of this times the free energy here plus the number of moles of the second phase times the free energy here because the total moles is one I always have a fraction so along this line the free energy is lower than here so this point here f f has x1 equal to x1 b a heterogeneous mixture of two phases of composition x1 low and x1 high has a g less than g at f I am sorry g at e or g before mixing you can take the case of g before mixing which would be this point a g after the real mixing because of the value of a the energy would have been at e had it continued to remain as a homogeneous phase this is higher than the separate components so it will stay at a but then there is another state where the free energy is even lower that is the point f here which corresponds to a mixture of two phases x1 low and x1 high therefore thermodynamics tells you that it is settled for that that is you will get two phases this one will have x1 low and the other one would have x1 high I am talking only two components here suppose I had ethyl acetate and water and I was not talking about azadiractin for example and have ethyl acetate rich layer and a water rich layer where say one is ethyl acetate two is water when I discuss the problem azadiractin I was talking about a liquid-liquid mix two liquid-liquid phases plus a solute distributed between them our interest was in azadiractin I realized that I hadn't discussed immiscibility itself so basically what we find is sometimes a system splits into two parts and this will explain why a system should separate into two phases what I have given you there is an example this A x1 x2 is only simply to do the algebra let me examine the algebra of that system a little more look at if you want to know whether there is a minimum and a maximum and all that you are going to do the differentiation g' by RT at constant temperature I take the derivative of that with respect to x1 I get µ1 pure – µ2 pure by RT that is the first term the next term will give me x1 if I differentiate log x1 I get 1 by x1 so I get 1 then I get x2 if I differentiate log x2 I get 1 by x2 into – 1 so that is a – 1 that will cancel so I have to only do log x1 – log x2 in the last one will give me I will write it below this a x2 – x1 if you look at g' by RT this term if I differentiate this further with respect to x1 this will vanish this will give me 1 by x1 plus 1 by x2 and this will give me a x2 will give me – 1 this will give me – 1 so – 2a so if you look at this case that we are looking at if I am looking at a sufficiently large I get a curve like this I get one minimum one maximum and another minimum so mathematically I should pick up 3 points there right g' will be equal to 0 at 3 points but if g' vanishes here the slope keeps on decreasing and then the slope increases and then it decreases again and so on so I want to know if there is an inflection point I will ask when is g' equal to 0 this will happen if a is equal to ½ of 1 by x1 plus 1 by x2 but a is not composition dependence dependent this will happen if the maximum value on the right hand side you can verify that the maximum value occurs at x1 equal to ½ because if you take 1 by x1 plus 1 by x2 then f' is – 1 by x1 square plus 1 by x2 square this is equal to 0 if x1 equal to x2 and since is equal to ½ so at x1 equal to x2 equal to ½ this is a minimum I will say actually this is minimum because when x1 equal to 0 it obviously goes to infinity so what you are discovering is not a maximum you are discovering a minimum here for this function so the minimum value of a is 1 by 2 into 1 into 4 in other words as long as a is greater than 2 that is if a is greater than 2 greater than equal to 2 g' is equal to 0 in the range for x1 equal to 0 to 1 you g' will be 0 at least at 1 point at a is equal to 2 it will be 0 at exactly x1 equal to x2 equal to ½ that means if the energy of the interaction is such that this constant a is greater than or equal to 2 in this model you will get you will predict immiscibility again for immiscible mixtures for example the porters model may not actually describe the system at all very well you have to try out different models in order to see which one is satisfied because there is no unique model but there is some interesting features of the mathematical models that you have one of the interesting models is the Wilson model which describes alcohol water systems hydrocarbon systems extremely well I can now show that the Wilson model cannot predict immiscibility that is there is no value of the constants in the Wilson model that will give you immiscibility. So you can look at a model and ask if it is applicable or not if it is not applicable then you do not use it for such a system that is all if you have Wilson's model you hit it on the head any number of times in the computer you will never get parameter values that will predict immiscibility so if the mixture is actually immiscible then you can eliminate some models is being impractical for example the most successful model for immiscible mixtures not involving electrolytes is the NRTL model okay I will take an example I will take the Wilson and the NRTL models and show you Wilson does not predict immiscibility the procedure is the same all you do is take write down the expression for G which will involve this term always this will change this is G excess by RT this is model dependent term so you will always get this straight line you will always get the ideal curve this depending on the values of the parameters in the value of G excess by RT as a function of composition you will get different curves if the free energy net free energy always lies below the ideal mixing line or below the before mixing line you will have complete solubility any time it crosses the line of G before mixing you will have immiscibility so we will take a look at G excess models I will take two examples one Wilson because it was one of the first models that was proposed and was very successful for many mixtures when people discovered that it does not predict immiscibility at all it has this logarithmic form I will just write them down G excess model this is G excess by RT incidentally his notation for capital G and my notation small g are the same it small g is a specific property for him capital G itself is a specific property he uses n times that value for me number of moles this is Wilson and then NRTL it is also got another 2 x1 x2 x2 where 2 1 2 is a constant at constant temperature this is constant all these parameters are constant at constant temperature he uses capital G here it is part of the trick in the derivation I will explain to you what happens but I have used lambda that is all this is a molecular parameter which is of this form which so this is also another constant but because of the alpha here this is different from the other one so effectively this is a two parameter equation 2 1 2 and 2 2 1 3 alpha is a third parameter so alpha 2 1 2 and 2 2 1 there are 3 parameters in this equation there are 2 parameters in this one interest is that if you go to the limit as x1 goes to 0 for both of these I will write the limit as x1 goes to 0 is x1 goes to 0 this term will go to 0 this term this will go to 1 so you get G excess by RT is equal to x2 will go to 1 x1 goes to 0 so you will get log 1 you are getting the limit I am sorry I should look at I do not want to look at this you have to look at gamma I will come back to it sorry I wanted to discuss the limiting activity coefficients gamma 1 and gamma 2 I will do that later because I have to again copy the expressions for gamma 1 gamma 2 but let us just get back to what we are discussing now this is the G excess by RT model so all I need is G excess prime and G excess double prime I will do that I will essentially show you that with G excess double prime here I will never get G after mixing less than G before mixing I mean greater than G before mixing therefore this Wilson always predicts complete solubility so we will take Wilson's something you have to remember essentially you have to examine if the model is good enough for the situation if you know that there is a range of immiscibility then there is no point using Wilson's equations for such systems on this diagram that I have given there actually this is for I forget which system I think it is isopropyl alcohol benzene or something there is immiscibility data available and this is simply a calculation it was actually one of the quiz paper some time ago the curve is peculiar because that is the way excel plots the curve that is all it is actually a smooth curve you can see the three that straight line is x 1 mu 1 plus x 2 mu 2 by RT mu 1 by RT I have taken as 0.3 mu 2 by RT as 0.2 but those constant values make no difference the calculation because by the time you do G double prime those constants will be mu 1 pure mu 2 pure do not appear so G double prime depends only on the ideal mixing which will give you 1 by x 1 plus 1 by x 2 and then you have G excess model so if you look at G double you would look at the equation for G which is the one on top you essentially the excess free energy and its derivatives those are what are plotted there G double prime is plotted on top this n is a mistake I say that is actually x 1 that is the way I calculated x 1 was 0.05 plus n minus 1 into 0.1 it just give it steps you should have removed n and plotted x 1 x 2 etc. But this is illustrative all you need to do is the same calculation for different values of the parameters in this case I have taken a is equal to 2.5 if I had taken less than 2 you would not have got immiscibility because the energy is not sufficient to compensate for the entropic change it occurs when you mix it is if the 1 2 interactions are lower all interactions molecular interactions interaction energies are negative that is the way they are all formulated so if the 1 1 interaction and the 2 2 interaction are larger than the 1 2 interaction negative terms then you will they would prefer to stay in their neighbourhoods rather than in the 1 2 mix that is when you have this problem this a is actually dependent on 1 2 1 1 and 2 2 it is actually the difference between effectively the difference between 1 2 2 times 1 2 and 1 1 and 2 2 actually we will derive a small theory called the flora against theory in which that term will be very very important.