 You write the first law for closed systems there is an assumption that you have elementary concepts of q and w straightened out but you are always in conflict about them. There is an assumption of continuum that it is not made up of molecules it is actually made up of molecules but you do not recognize that simply say that any time you are dealing with the system you are dealing with such a large number of molecules that any change in density is can be treated as continuous actually the molecules are either that many a plus 1 or minus 1 you cannot change it continuously but the change of plus 1 or minus 1 in 10 to the power 23 molecules is so small that you can treat it as differential. So you make that assumption and treat as far as the classical thermodynamics is concerned treats matter as if it was continuous all the variables are continuous you assume that there are measurable quantities like p, v, t and you have the empirical evidence that reproduce the state of a system you need to measure in a pure substance only two of the three quantities if you measure p and t all these are not written down here all these are implicit in your entire treatment. So then you can say given all that in addition to all that I discovered that there is a quantity u such that du is a perfect differential although it is the difference of two quantities which are not perfect differential that is an achievement to say that took nearly 2000 years because people had sort of vaguely groped and said various things then june came along with this calorie meter and asserted that that was true. So it became a hypothesis and so far it is not been contradicted at all the second law was even more subtle to discover it took but after the discover although the second law came before the first law entire discovery of the concept of entropy energy although it was not well defined it was understood reasonably well people were using energy as if it was a state variable they did not define it in calculus but they understood it quite well entropy was much more difficult concept it is really amazing that somebody should after a lot of work cumulation of knowledge discover that there is a differential quantity called ds which can be produced by dividing delta q by absolute temperature. So the whole idea of introducing an absolute temperature and proving that it is the only factor that will produce a perfect differential out of a non perfect differential delta q that is sort of asymmetric there is nothing common about it no if not delta q then cannot you do the same thing with delta w you cannot nobody has done it. So if you discover another property where you take a path dependent quantity divide it by an absolute quantity and produce a perfect differential you can apply for a noble price right away. So far nobody else has done it in the whole history of mind kind that it is the only case where you have found a perfect differential therefore you have found another property s with which you can deal exactly as you dealt with p v t etc. Then there are elementary properties like p v t ideal gas that came from gailus axe measurements you know you take the volume and you produce an absolute temperature and that absolute temperature for an ideal gas at which the volume is 0 is – 273 degree C so that is an empirical information number of volts again you think you can identify the species you can measure them you can do it reproducibly. So those are elementary properties I must also say q and w are taken as elementary concepts I should have drawn a box there in the first one you can add a box elementary concepts like q and w the difficult thing there again in the development was that q and w are equivalent according to joule but are different and there are exactly two ways in which energy can be exchanged with the surroundings no more than two ways otherwise you could not have written down this law if there was a third way you have to produce another symbol and add to it and again you are ready for a noble price if you find it then you have you applied it to Carnot's ideal engine you applied the concept of entropy you came up with reversible engines in the maximum possible efficiency and then it turns out that T was equal to T ideal gas that that already been of course the whole of thermodynamics will go through if T was equal to alpha times T ideal gas or alpha is any constant it is just luck that alpha was chosen to be one then work done in PV systems is always defined by PDV it is a concept of mechanical work and we put that concept in and then derive various properties we also introduce auxiliary properties we have u and s plus p v t etc by combining those in various forms you produce other properties these are not independent properties they are combinations of existing properties now these properties you use you write for close systems therefore you are able to write differential equations first order differential equations describing the change in internal energy the change in enthalpy change in Helmholtz free energy and so on and by putting down the inequality sign which is what the second law told you the second law is basically in inequality you are able to come up with fundamental equations for close systems having done that in order to extend it to open systems you allowed change in the number of moles so you simply wrote dg is equal to for example minus s d t plus v d p plus mu i d n i right simultaneously you went back to calculus and wrote the other equation for any extensive property that is d m is partial of m with respect to t d t plus partial of m with respect to p d p plus mi bar d n i one example of that is this now you have a criterion of equilibrium you ask under t and p when is the system at equilibrium and you showed that for example if t p are constant g should be minimum or dg is 0 incidentally in all of this there is another thing that you have taken on faith which is the law of conservation of mass which again has not been contradicted so far so you have to put that in you have to put the criterion of equilibrium and derive expressions for phase equilibrium simply put two phases together and ask that the combined two phases be at equilibrium similarly you asked about chemical equilibrium if formulation is always mu i mu i is equal to 0 or if you have multiple equations mu i j mu i is equal to 0 and you found RT l n k is equal to minus delta g 0 delta g 0 contained everything that you do not know but you isolated quantities that are functions only of temperature on the right hand side so essentially what you said was k at one temperature is a constant regardless of the pressure and that is a very useful concept to have then you have d of delta g 0 by t is equal to delta h 0 and then similarly d of delta h 0 by dt is delta c p so you have to measure delta c p c p 0 is the ideal gas specific it you have to know it is a function temperature you have to know delta h 0 at one temperature c p 0 you have to know at one temperature for all as a function of temperature you have to integrate that to get delta h 0 c p 0 you have to know as a function of temperature at all temperatures as far as phase equilibrium is concerned you got rid of mu 0 and therefore you had all measurable quantities you simply play games with these equations because the equations I have written down here are for solvent systems for solvent solute systems you have to give different models in do the equation now on the right hand side again the number of properties required to specify the state of a system is a completely empirical quantity told you there is a Gibbs phase rule but the phase rule assumes that you know the number of components in the number of phases and you may not be able to in macroscopic thermodynamics you may not be always able to take the number of phases for example in solid solid as long as a distinct phase solid liquid vapor you are very clever just look at it and say there are three phases but if there are three solid phases you are in trouble you will know simply the structure may be different you only if you look at it in the microscope you will know so the number of properties required to specify the state of a system is always an empirical quantity we have known from experience at a pure component single phase system there are two variables that required to specify the state of a system similarly in each case you know the number you can use the Gibbs phase rule if provided you know the number of components in the number of phases then of course the Gibbs phase rule is very rigorous then you then again big make the postulation that some quantities are intensive some quantities are extensive intensive variables do not vary with the size of the system T and P are alright Mi bar the partial molal property is also an intensive variable that is an assertion in fact in nano systems where the number of molecules is very small this is no longer true in the whole of classical thermodynamics will break down in the way in which you use it if you go to systems with small number of molecules because then the changes are not continuous you no longer have the applicability of the continuum but on the does not mean you have to throw away all the concepts you have to rework it and a lot of it withhold but with some reservations for example the fact that Mi bar the partial molal property is independent of the size of the system will no longer be true so you cannot do the simple integration that you already did so you have to divide properties in intensive and extensive and what you do is then you are able to integrate the fundamental equation we wrote the fundamental equation DG is equal to for example minus SDT plus VDP plus mu IDNI you integrate this equation and produce the Gibbs-Duhem equation I have got the next step is you have you prove that M is equal to Ni Mi bar for all macroscopic systems and then we have asserted in particular Mi bar for M equal to G is the chemical potential it is so important that we give it a name it comes again and again we have we have already asserted that Mi bar is a function only of TP X1 to XR minus 1 only of the mole fractions not of mole numbers given that you can derive a Gibbs-Duhem equation which is the equivalent of composition variables of the equation that gives you for composition variables sum over XI equal to 1 instead of that you get a more complicated equation for chemical potentials but remember while the Gibbs-Duhem equations as I have written them down are actually R minus 1 equations because for every J you will get one equation the solution to it lies there is only one independent function there so if I know G XS I can calculate all the others so the nature of the equation because the first order ordinary differential equations they lead to a single solution provided you know one function as a function of a composition which is delta which is G XS or G so if you know one mixture property as a function of composition then you can calculate all the others you think the Gibbs-Duhem equation therefore mixing becomes very important is central in thermodynamics so you ask what are the models for mixing I have that here in the bottom row you have models for G XS so that essentially has to do with mixing this is for the condensed phase for the gas phase you avoid the whole thing because solving the Gibbs-Duhem equation for the chemical potential is equivalent to solving the Gibbs-Duhem equations for the partial model volume and the partial volume is measurable so you do not bother to solve the Gibbs-Duhem equations for the gas phase the partial model volume can be solved for either from empirical data you can measure it or from equations of state you can go to Van der Waals equation of state the Ridley-Kohm equation of state and derive expressions for VI bar and from VI bar you can calculate VI so you got the chemical potential model which satisfies the Gibbs-Duhem equation so if a vapor you write mu i is equal to mu i 0 plus RT ln Fi Fi is simply P yi times Vi and RT ln Vi is integral Vi bar minus RT by P dp so if you know VI bar you know mu i bar you know Vi and therefore you know mu i for liquid you write mu i star plus RT ln x1 I am going to do the next one from the next class I will discuss this case of electrolytes or solutes where it is more convenient to use molality as a measure unit of measurement rather than the mole fraction but in all these cases you have liquid 1 gamma i equal to 1 when xi goes to 1 that is the solvent components the solute components gamma i goes to 1 when xi goes to 0 and finally the third type of solid components and molality xi goes to 0 it means very dilute solutions a molality of 1 is a very dilute solution you are talking of 1 mole of solute for 1000 grams of solvent therefore you have the molality units and a very dilute solution you are saying gamma i goes to 1 so use that and then all the time you can do manipulations because you have introduced a redundancy in thermodynamics by introducing three additional functions H A and G at originally you had only U and S along with PVT you introduced H A and G so instead of two variables you have introduced three others so there is a Phi C 2 redundancy in thermodynamics H A G are additionally introduced they are only combinations of U S and other properties so to that extent every equation in thermodynamics has Phi C 2 equivalent in fact if you allow permutations that is if you put one in the denominator doing asking for G with respect to P you can ask P with respect to G so if you allow permutations you are talking of Phi C Phi P 3 is even larger number so that is the number you have in terms of combinations and permutations actually in principle what you do is take PVT H A G U and S means 8 out of 8 any three variables are the independent variables so there are 8 C 3 ways of writing every equation in thermodynamics in principle that is why when people list Maxwell's relations it is meaningless it is absurd to go on listing them you simply recognize that if you take any three variables as the base variables and write the differential equations you can produce Maxwell relations of any kind you want so Maxwell relations you produce whenever you need them you write down what the independent variables are and in the purpose of all this manipulation is first thing is to understand the overall picture as I told you thermodynamics is so general that even the all the models in if you ever go to astrophysics we will discover that their model for the universe is that of an ideal gas and therefore when they calculate your background radiation they do it from PV power gamma is equal to constant because the planets are very small compared to the intervening space so you can treat them like point particles it is all a matter of scale and in that scale you have a large number of non-interacting point particles practically ideal gas that you can describe systems the way they are in fact I do not know I mentioned it.