 I still have to discuss evaluation of fugacity coefficients the formula is very simple if the equation of state we have already seen the equation of state for example Vander Waals we have seen did show the corresponding states principle first of all p plus a by v squared into b minus b in reduced form we have p reduced plus remember here there is a constant p reduced squared into v reduced minus 3 is equal to some constant I forget the constant there into t reduced it is 8 by 3 or something like that some constant remember we wrote the cubic equation and equated and showed that as you can show that b is pc by 3 and a is pc vc squared again is proportional to pc vc squared so I think there is a constant here also not necessarily one so there are two constants but the point about the Vander Waals principle was that pr is equal to some function of pr and vr in general you should know that vc is very very difficult to measure critical volume so very often vr is defined is v by vc but vr can also be taken as v divided by RPC by PC and say or either of the it is in some correlations you non-dimensionalize the volume using PC and PC rather than using an idealized critical volume if you like the ideal gas really does not have a critical point because there is no condensation but one of these is used but basically you get this equation this is the equation fundamental equation for corresponding states what it says is that at the same tr and vr the reduced pressure is the same for all substances so universal it is an equation the corresponding state means for every t and v of a reference substance there is a t and v of the actual substance for which the t reduced and v reduced are the same under those conditions P by PC is the same for both substances so it is efficient in principle to measure the data for say argon easy to measure and you make all the measurements for argon put the equation of state the empirical measurements in this form and then you can use it for any other substance provided you measure Tc, PC and vc so really it is a matter of reducing one is universalization second is reducing experimental data but more universally this implies some function of tr and vr implies that zc is equal to constant in the case of van der Waals zc comes out to be 3 by 8 so example van der Waals again zc is equal to 3 by 8 which is much greater than the experimentally observed value experimental value is actually vary more like between 0.2 and 0.35 wide variations at c values and typically for many substances I think for the inert gases about 0.27 so argon krypton neon xenon all these can be correlated very well methane etc and then you start having departures so what people suggested was as this is the first corresponding states this is called the simple corresponding states principle all other corresponding states principle simply add more parameters to this so you write pr is equal to function of tr pr and say alpha where alpha is some characteristic parameter essentially this characteristic parameter should have ideally it should be distinct for distinct substances if substances behave differently and alpha should be correspondingly different it should be easily measurable so only two requirements should be sufficiently different for different substances should be measurable easily measurable historically the first one that was tried was alpha equal to zc itself actually it is taken as zc-zc0 because for simple inert substances zc0 is the critical compressibility factor say for argon for inert gases and methane and so on if you take this difference as the parameter then it becomes very convenient you can then do if you do this you can then write pr is equal to f of you can expand this in Taylor expansion f0 of tr pr, alpha 0 plus the first derivative of this times the value of alpha actually alpha 0 is 0 so plus alpha let us call this f1 so you make a Taylor series expansion you can write pr is equal to this actually normally it is rather than pr you write z itself the compressibility factor which is dimensionless so or z, zc is the critical compressibility factor z is the compressibility factor z is pb by rt zc is the value at the critical point it is just a third parameter it should be different because you would like to use a reason for using a Taylor series expansion is the following if I now measure for argon when alpha is equal to 0 this term will disappear I get enough information from just argon measurements then let us say I do a measurement for carbon dioxide for which I know alpha then carbon dioxide will give me f0 plus this whole thing of which this is known so from the left hand side measurements for carbon dioxide I can calculate f1 this is actually partial of f with respect to alpha at alpha equal to 0 but I do not have to look at it that way just a function of pr and pr and I can get it from experimental data once I get it for argon and carbon dioxide I simply use f0, f1 universally and find alpha for every new substance I have to measure now tc,pc,vc and alpha if I measure four quantities I have got the complete equation of state for any substance because I have already made measurements for argon this comes from for example you do argon data give you f0 then carbon dioxide data give you f1 as being equal to pr for carbon dioxide minus if you like pr for argon remember pr for argon is exactly this f0 is pr for argon if I take this difference and divide by alpha CO2 I get f1 I do have to measure pr as a function of f,r and I keep saying pr here this should be vr sorry this is vr I have got pr on the left hand side it is a more convenient way of doing it is to measure z as a function of pr and pr any two variables on the right hand side the left hand side should be a dimensionless quantity so in principle it is sufficient for me to measure carbon carbon dioxide data and argon data and then measure four quantities for each substance tc,pc,vc and alpha now this has been done for example in great detail by Haugen in Wisconsin is a book by Haugen Watson and regards it is more of historical interest but it is a good book the three volume book on chemical engineering principles I think volume one you use in stoichiometry in your course right so there are two other books which second book is on thermodynamics which contains extensive calculations using this see once I have an equation of state I can measure I can calculate p all I have to do is a lot of algebra I have to find v at every p and t differentiate with respect to n i in a mixture and do this I have to do the same thing for a mixture from mixture alpha will be will have a suitable combining rule every parameter that appear in the case in the equation of state will have a combining rule but if you do this first the first thing that you do is do this for pure substances and then for pure substances of course RTL ln p this is for pure substances or I write pure 0 to p v- sorry v by RT-1 by p dp v reduced by t reduced I have to multiply by p that is I do p by pc and then I have a pc inside here then I have p v by RT-1 d ln p p v by RT is the same as this it is equal to p equal to 0 to p this is pc vc by RTc pc into p reduced v reduced by t reduced here-1 d ln p reduced which are a p reduced therefore the limits will become pr equal to 0 to pr equal to if it is a two parameter equation of state like the van der Waals this is known is only three by eight so I can do this calculation once and for all and I can give the plots or the tables in the book and you can just use the tables directly all you have to do is find what p by pc is you can then plot p versus pr for various values of pr it will always go to one in the limit as p goes to 0 I think the plots go like this this is for different TRs this is the simple corresponding states simple corresponding states principle SCSP if you like for two parameter equations of state if there are only two parameters then ZC is a constant the two parameters can be determined in terms of PC VC NTC and you can have a universal plot I think this is given to you in Smith and Van Ness if you have a second parameter here a then you will get a fee 0 and a fee one so you get two plots you can otherwise have let us call this fee 0 to indicate that it is for substances that are described completely by a two parameter equation of state typically this would be obtained from argon data or methane argon and so on then there will be a second parameter you will plot fee one which corresponds to F1 there against PR for various values of TR I do not remember what the curve looks like you get a series of graphs and you will get fee 0 fee is equal to fee 0 plus alpha fee 1 so the easiest way is to do this to simply take the fee 0 graph take your T and P of the substance divided by TC PC produce a reduced coordinates at a given PR you go and with the correct TR and read the value of fee fee 0 similarly read the value of fee 1 and do this calculation so all I need is one table that gives me PC PC VC and alpha now there are many many parameters alpha that have been used but the most successful one is called pictures a centric factor picture is a chemist it is introduced in a centric factor on the face of it it is amazing how one would come come up with something like this that is the definition of an eccentric factor it is as argument actually came from this you know that if you plot log of the vapor pressure P saturation versus 1 by T it is well known that you will get a straight line as the temperature increases you move this way and the pressure increases so you get a straight line like this this is experimental observation it is also a consequence of using the Clausius Clapeyron equation and the ideal vapor phase you have DP by DT is equal to ? H by T ? V for vapor liquid equilibrium this is approximately equal to ? H by T V vapor and this is equal to for an ideal gas this is ? H by RT squared by P so you get D log P by DT is equal to ? H by RT squared and if ? H is constant this is along the saturation curve when vapor and liquid are in equilibrium so log P plotted against 1 by T should give you a straight line for a given ? H if ? H is constant actual practice ? H will vary with temperature the variation is fairly small this has been known experimentally and if you do this by in reduced coordinates that is if you plot log PR saturation versus 1 by TR still get a straight line and TR equal to 0.7 is approximately the normal boiling point and you get a straight line for all for simple word simple begs the question simple gases typically inert gases and again methane spherical molecules you get nice straight line it turns out that TR equal to 0.7 this point here this value I am plotting now log to the base 10 for convenience this was just numbers log to the base 10 PR saturation at TR equal to 0.7 PR saturation is approximately 0.1 this is an experimental observation for all simple substances I am sorry I take this back not normal boiling this is close to the normal boiling point okay I take this back TR equal to 0.7 is close to normal boiling point it is close to but it is not exactly at TR equal to 0.7 PR saturation is equal to 0.1 for inert gases so log to the base 10 PR will be minus 1 this is plus 1 minus 1 it will be 0 so ? is approximately 0 for inert gases for all other substances ? is different from one ? is very many characteristics first data is readily available this is PR saturation so you take saturation pressure vapor pressure of the substance at TR equal to 0.7 then you get ? is readily available is very measurable it varies between 0 and 0.4 approximately which is very nice for a Taylor series expansion because you are doing a Taylor series expansion with ? is a parameter if ? is less than 1 then you have guarantees of convergence as a Taylor series expansion and so it should be small that is another criteria that people came up with if a is very large then you get large deviations and you cannot do a Taylor series expansion conveniently and thirdly it is a measure ? is a measure of a centric nature of the intermolecular force that is the intermolecular force effectively is like you take two molecules and model them like spheres then you have to have a centre of mass for each and a centre of force for each because you have to know what the line of interaction is and if the centres of force and centres of mass do not coincide the molecule is said to be a centric the centre of force not being centrally located with the centre of mass a centric if you like so what is actually discovered is that ? for example for carbon dioxide is almost 0.38 I forget the exact value but it is quite large so what has happened is ? this has become a very useful correlation this is the most popular correlation so you write plus ? ? ? and in effect this is the only three parameter correlation that you have to know it is widely used for gases and it gives you excellent correlations this combined with Lewis and Randall rule that is if you are dealing with mixtures Lewis and Randall rule tells you that Vi bar is equal to Vi therefore Vi is simply Vi pure and Vi pure is given by this Vi pure is Vi 0 plus ? Vi 1 almost find 90% of treatment of gas phase non ideality uses this rule and therefore you have to calculate only Vi pure Vi pure is almost invariably calculated as Vi 0 plus Vi there is a variation of this in the literature is what is called a leak Kessler equation of state because it has been very successful you have to know about leak Kessler equation of state leak Kessler equation of state uses ? exactly like this what it does is makes the equation of state interpolated because ? is 0 for inert gases and some almost 0.4 for carbon dioxide all other gases behave in an intermediate fashion so you use an interpolative equation of state which essentially gets all the data for any substance between that of argon and carbon dioxide so the leak Kessler equation of state essentially has a detailed equation of state but it also calculates exactly like this ? 0 and ? 1 are calculated for the leak Kessler equation this has been done for several equations of state in particular for the leak Kessler equation of state that I will give you that it is given I think in Wallace the detailed treatment is given in Wallace and there are also much more complicated equations of state all I want to say about equations of state and calculation of fugacity coefficients we will just look at vapor liquid equilibria of various forms in this case when I am talking about liquid I am talking about solvent solvent mixtures because if you have solvent solute mixtures you are either talking about solubility of a gas or solubility of a solid I will discuss those topics separately but if you look at solvent solvent mixtures the final diagrams are plotted like this let us look at binaries if you plot for example isobaric data these are plotted often like this all you do is take experimental data you take a mixture liquid mixture this can be X or Y I take a liquid mixture of this composition let us say point A is a liquid mixture of composition X A at some sufficiently low temperature I increase the temperature to some point and at this point you begin to have vapor formation in the vapor that comes out has this composition I should have plotted it the other way round which is normal let me explain why normally if you have a binary you choose this X and Y so that the more the X composition refers to the more volatile component so that the vapor is usually in the liquid forms this is the liquid this is the vapor so this liquid is in equilibrium with this vapor here vapor line is on top so the composition Y is larger than the composition X so I boil this liquid at this point the first bubble of vapor forms in this point is called the bubble point if it was a if it was the pure substance there will be only one point at which the vapor and liquid will be in equilibrium but I am talking about a mixture this point is called the bubble point so this is the composition Y in equilibrium with so this will be Y A star the star indicates that in equilibrium with X A if you plot this Y A star versus X A you can do this at every liquid composition if you plot Y A versus X A usually plot it just Y A this is the 45 degree line you could get for a curve like this you could get something like this this is your equilibrium curve incidentally conversely if you had started with complete vapor of this composition let us say this point is B if you had started with vapor of this composition and you had reduced the temperature you had cooled it at constant pressure all this is done at constant pressure you would have come to this point at which the first drop of liquid will form this is called the dew point after the fact that dew forms from air early in the morning when the temperature drops or very early I am talking about not only maybe one o'clock two o'clock so two points characterize mixture the dew point and the bubble point if I have a vapor you talk of the dew point if you have this is one of concerns in chemical engineering and pumping for example if you are pumping a liquid and you do not want vapor to form you are using a rotary device and if vapor forms the vapor bubble breaks on the blade and causes erosion so you do not you want it to be completely liquid you do not want it to be vapor so you will have to make sure the temperature does not go above this point as long as you keep the temperature below this point you would not have any vapor formation and similarly if you are using a device for the vapor and you do not want liquid droplets then you will make sure the temperature is above this point so that you do not have so determining a liquid dew point and bubble point will determine the operational temperature or for example the pressure that is this is the temperature diagram if you did an isothermal calc this is isobaric and this is of importance industrially because most industrial columns operate at constant pressure every large fraction is atmospheric distillation the other case is isothermal operation in the isothermal operation if you plot P versus X, Y and you get this curve this way you get the other way around so at constant temperature if you go from this point and increase the pressure at some point you will have condensation now this is the liquid this is the vapor so this is the liquid in equilibrium with this way or if you start with a liquid for example and come down reduce the pressure finally there will be a vapor formation of this composition life is not quite as simple because this diagram can take a thousand shapes for example there are mixtures where the Y does not always lie above I can plot either of these in this form at constant pressure or constant remember you are talking about binary system I have to establish that let me establish degrees of freedom the phase equilibrium problem simply tells you T a is equal to T beta T a is equal to P beta also tells you mu a is equal to mu beta or mu i a is equal to mu i beta say the general phase equilibrium problem so what are the variables variables in the problem are T P and mu i a the whole set which is actually to or T a P a in mu i a so 2 plus r components a is the number of phases into pi phases this is number of phases this is number of components so the number of variables is simply equal to 2 plus r into pi what is the number of equations this is I fix alpha beta can run from 1 to pi minus 1 so this is pi minus 1 equations this is also pi minus 1 equations this is again pi minus 1 for each component into r equations so the number of equations is simply pi minus 1 into r plus 2 so the number of degrees of freedom is simply the number of variables minus the number of equations so r plus 2 is something wrong okay this is alright is equal to I have got some r plus 2 what is wrong and some more constraints is this all the number of equations I claim this one more set of equations that I have been talking about nothing but those equations no no that is for specific a phases right I am talking about chemical potentials what governs the composition dependence of the chemical potential there is a Gibbs-Duhem equation right so this one equation for every phase so I have plus pi which is one Gibbs-Duhem equation for every phase so you get minus pi that is if you treat the chemical potentials as independent variables you are talking of r intensive variables to describe each phase but each of those r components is governed by a Gibbs-Duhem equation so you have take this into account so you get your famous rule is equal to r plus 2 minus pi you can either do this or simply write this set of equations as mu i alpha is equal to mu i beta you put a model for chemical potential that is you solve the Gibbs-Duhem equation which is how we do it we assume a model for GXS derive expressions for mu i in terms of x1 x2 xr minus 1 and write this equation as if it is an equation governing r minus 1 composition variables not r chemical potentials that is the other alternative is t alpha is equal to t beta p alpha is equal to p beta and mu i alpha which is a function of tp and x1 alpha to xr minus 1 alpha is equal to mu i beta which is a function of p beta p beta x1 beta to xr minus 1 beta now the number of variables is pi again it is pi minus the number of variables is pi variables pi phases each phase has 2 plus r minus 1 now the variables are not mu i's but these now you already solve the Gibbs-Duhem equation because you have got composition dependence this is obtained from the Gibbs-Duhem equation in each phase for example if you are writing the vapor phase you will write this in terms of fugacity coefficients you will calculate fugacity coefficients from an equation of state so it automatically satisfies the Gibbs-Duhem equation because the fugacity of coefficients are obtained from vi bar vi bar is obtained from equation of state so those satisfy the Gibbs-Duhem equation in the liquid phase you may use an excess free energy model so these are given I have given you this expression as a function of composition so you only this many variables now the equations are exactly as we have written down pi minus 1 to r plus 2 so if you take v minus e you will still get the same equations f is simply equal to r plus 2 minus pi because pi into r plus 2 minus pi into pi minus 1 to r plus 2 is exactly r plus 2 then you have a pi left minus pi actually this is the way we solve the problem that is these are obtained from solutions of the Gibbs-Duhem equations how you solve it is a detail for the vapor phase you do not solve the Gibbs-Duhem equation you simply go to the equation of state get vi bar get v for the liquid phase you go to g excess model so in a sense this information is really empirical except that in most cases the number of components and the number of phases fairly obvious so it is a very good rule it is an exact rule but you do not know r and pi exactly all the time and therefore f is not known exactly so coming back here we are discussing binary two phase equilibria so you have here r is equal to 2 and pi is equal to 2 so the number of degrees of freedom is 2 minus 2 plus 2 is equal to 2 degrees of freedom so if you fix temperature or pressure then the only freedom you have is either with liquid composition vapor composition so if I fix temperature and liquid composition everything else is fixed or if I fix for example pressure in this case I fix the pressure and I fix the liquid composition automatically is determined so number of degrees of freedom is you have to look at it carefully that will tell you what measurements you need to make here for example very often you can measure t x and y fairly straightforward that means you have one measurement more than is required fact you measure p also you have two measurements more than is required so you can actually use this data to verify the Gibbs-Duhem equation now then they started actually up to about 40s by 40s they started believing it then they said verifying the internal consistency of experimental data if the data do not satisfy the Gibbs-Duhem equation you tell them no this data is not right send it back to the fellow and ask him to measure it again and then now it is just used people do not measure additional data because such a new sense is assumed the Gibbs-Duhem equation is valid into the analysis so let me get back here so the principle of all separation processes is to take an x if you let us do the isobaric case for example in a distly crude oil distillation column you take a certain composition x and you find the vapor comes out is why then you have a series the actual way the physical column is set up you would not recognize this because normally what you do is take a still in the laboratory at constant pressure and increase heat it till the two come to equilibrium and you measure this composition this composition that is how you do this in the laboratory but in actual practice the distillation column does not look like anything like what you have all you need is good mixing between liquid and vapor and then you get an equilibrium so the actual distillation column as I said last time looks like this you have a column you have n plus 1 tray and the nth tray the liquid from the n plus first tray comes down here this is liquid and the vapor comes all over the place this liquid flows out this way and then flows down this way so this vapor coming up bubbles through you must make sure that there is good contact between liquid and vapor here then the vapor leaving here and this liquid this is Ln this is called Vn this is the vapor leaving the nth tray is the liquid leaving the nth tray these two are in equilibrium the composition of this if it was a binary system this Xn and Vn is what you read here if this is the nth tray liquid this liquid is in equilibrium with vapor which is much richer in component 1 in the first component so as you go up you get a vapor that contains more and more I am sorry this is vapor this is Yn this is Xn as you go up you get a vapor that is richer in richer in the more volatile component and finally at the top if you have sufficient number of trays you will get pure component 1 if it is a binary you will get pure component 1 pure component 2 at the bottom that is all you do in binary distillation if it is a multi component distillation very often it is treated like a pseudo binary there is the most volatile component will be treated as one component all the rest will be lumped together as equivalent second component and you can do the treatment all the way down the column.