 Welcome to module 43 of chemical kinetics and transmission state theory. We are going to discuss some neat application. We have been looking at unimole, we have been looking at micro canonical rate constants that is rate constant at a given energy. Now you may ask why have we been looking at it at all. So we will look at the problem that was historically studied why this theories were originally developed which is to understand unimolecular decay. So this is something we discussed in our very first few modules and we will now today look at it in much more details that we understand rate constants a lot more now. So a quick recap this is from module 2 or 3. I have a unimolecular decay meaning I have only one A that is single handedly going to some products B, one reactant only. So this was very deeply studied in 1910s and 1920s. So we are trying this seem to be the easiest problem to start investigating in chemical kinetics and people spend a lot of attention on it alright. So one mechanism that was proposed and that is what we discussed in earlier modules is that rather than A directly going to B that actually does not matches experiment at all. What was proposed is that 2 molecules of A collide with each other to give you A star plus A and this I was not able to put the equilibrium constant very well. So this is my creative way of writing. This is really equilibrium this is k1 and this is k-1 and A star finally decomposes to give you B alright. That is our mechanism. I can build kinetics on top of it. So I can find what is dA over dt, dA star over dt and dB over dt. Then we make the steady state hypothesis which assumes dA star over dt is 0 and dA star over dt you can easily show is given by this expression you get 3 terms because A star appears 3 times one is here, one is in the forward direction here and one in the backward direction. So you get corresponding A star in these rate law equations and the steady state hypothesis is that the intermediate population does not changes with time. So we set that to 0 and finally the rate which is nothing but dB over dt is equal to k2 into A star B appears only here and then we use this equation to calculate A star and eliminate A star out and finally get rate as this because this is something we studied a long time ago. Now finally I can express rate as some k into concentration of A. So this let me define it this way only. It is this is not elementary but I can always define k1 as this. So I can calculate this k1 as then k1 k2A over k2 plus k minus 1 into A I have divided by a capital A on both sides. So clearly this k1 is not independent of concentration it is not a number at a given temperature depends on concentration of A clearly showing that A going to B is not elementary in this model. So today let me just point out that this model is not complete there are few issues with it. So in 1910s and 20s itself some experimentalist had plotted 1 over k1 versus 1 over pressure of A. So let me massage my equation to have 1 over k1. So 1 over k1 is nothing but k2 plus k minus 1 of A. Let me just simplify this and write this as k minus 1 over k1 k2 plus 1 over k1A and basically these experiments were done in gas phase. So pressure is proportional to concentration this ideal gas law. So this is then equal to some constant let me call this C plus some m into 1 over pressure C is nothing but k minus 1 over k1 k2 and m is nothing but proportional to 1 over k1 plus pressure is proportional to concentration of A. So you see what we have got is 1 over k1 is equal to some mx plus C where x is 1 over pressure. So my prediction is at a given temperature if I plot 1 over k1 versus 1 over pressure I should get a straight line correct that is this is an equation of a straight line. This is the experimental result different of you. So I have actually taken the data and plotted it on some graph this is not a straight line it is clearly deviating correct. So we have a problem we get something that is reasonable it is not completely off the charts I mean in some degree I mean this portion is starting to look like a straight line almost but you see something happens here. So we have to try to understand it now why is it what has went wrong in our model. So what did we miss really our model seems pretty good the model is still pretty good there is something very fundamental and very nuanced point that we missed. So let me try to impress that upon you let us just now we have started thinking in the language of energy surfaces we have A here we have B here where is A star that is a pushing you should start thinking pause the video and tell me where is A star okay. So think about it I will give the answer in 5 seconds 5, 4, 3, 2, 1 A star is not this point A star is simply an excited energy level here okay remember transition state is a very specific point on energy coordinate that is one given potential energy A star has can have more energy than the transition state A star can be many different possible structures A star is not the transition state but A star has some high energy E here so the model that I have here is that this is k1 this is k-1 and once I get here this is k2. So k1 pumps me up to the excited state k2 will drive me to the product side and the k-1 depletes my population in state energy E the subtle point is the following we have been taking these k k1 k-1 and k2 independently we are thinking of them as 3 independent numbers as a function of temperature they are not independent that is the point or correlated what does it mean it means the following what is exactly k1 k1 effectively is that 2 molecules of A collide with each other giving me an energized product at a given energy E not at a given temperature T one collision will lead me to one energized state and this energized state can either go towards become a product or have another collision and become de-excited okay. So k1 k-1 and k2 are better thought of as independent numbers but functions of energy temperature is an ensemble average but at a given energy all these 3 things are happening you go up to this energy E and that that energy A you either go forward or you come down okay this is a very nuanced point try thinking about this it is really a matter of time scales at the end of the day if everything was k2 was very very fast then it would not have mattered but these 3 numbers are really interacting with each other so we cannot just make this k1 k-1 k2 as a function of temperature and get away with it okay. So what we actually do this is back to our expression of k1 we so far were thinking of these as independent numbers as a function of temperature at a given temperature this is a bad approximation the correct way to think about it is that we have this graph with us this is let us say EA E star is somewhere here A star this is let us say energy E this is k1 this is k-1 and this is k2 we are going to calculate these things at a given energy and then integrate them over all possible energies and energies of course will range from EA to infinite you cannot have energy less than EA because then no reaction can happen k2 will be 0 below energy EA okay. So that is our new novel idea this is what R, R and K did so we discussed this R, RK theory earlier Rice, Ramsberger and Kassel and that was their major breakthrough they said you know you have been calculating these numbers as a function of temperature but that is not correct you should be calculating them as a function of energy and integrating over all possible energies okay. So we have this I will massage this a little bit to get into a form that is better to understand I will divide both numerator and denominator by k1A sorry k-1A. So I am dividing by let me write k-1A so I will get k1 over k-1 into k2 divided by 1 plus k2 over k-1 and all these things are function of energy I am not writing them as explicitly I think but it should be understood now okay. Just a quick point this quantity that we had now k1 over k-1 this represents the equilibrium constant or between A and A star so that is the reason that I massage my equation to have the form which has k1 over k-1 because that is an easier quantity to calculate. So now the question becomes how do we calculate these quantities? How do we calculate this k1 and k-1 and k2? So there are 2 different treatments that we will look at one is what this R, R and K have done this was in early this was in late 1920s 27 and 28 and then we will follow it up with even better treatment that came with markers or R, K, M and see what even improved treatment can be done. So this is where the 2 theories that we discussed for micro canonical rate constant will be used. Thank you very much.