 Welcome to module 41 of chemical kinetics and transition state theory. In the last module we derived the transition state theory estimate of rate constant at constant energy. For the rest of the course we were deriving transition state theory at constant temperature. So, in this module I want to connect these 2 ok. So, these are the 2 expressions we have derived at constant temperature this is the expression that we have played around with over several modules and this is the expression we derived in the last module of the rate constant at constant energy. So, today let us look if we can derive one from the other particularly I will go from micro canonical to canonical. If I have to calculate k of temperature one way of writing this is actually I can integrate over all energies find the density of energies and k of E. So, I find the rate constant at given energy find what is the density at that energy and just integrate the product over all energies that will give me the current constant at the given temperature. So, k of E we will use the transition state expression E minus E A what is rho that is the question. So, we have been writing rho of x comma P as e to the power of minus beta H of x comma P divided by the normalization constant. If I want to write it as a function of one number energy well you notice that this H is nothing but energy. However, there can be degeneracy of energy there can be a density of energy and the density is given by G. So, think about it at a given energy what is my rho what is the density well the density at x comma P is this this is my true Boltzmann distribution and H is nothing but energy. But we note that rho of E is nothing but e to the power of minus beta E, but I have several possible x comma P where the energy is equal. So, I must multiply e to the power of minus beta E by the density at that given energy by the number of states I have at that energy and G essentially represents the number of states at that energy. So, at some energies I might have many many more states. So, my rho is higher my overall density is higher than right. So, it is a product, but I must also integrate over also I must normalize it properly I get this. Let us just look at this denominator I just wrote very quickly. So, in the denominator I wrote now we will replace the expression of G of E that we had written last lecture I always forget H to the power of 3 n. So, let me write that very first 0 to E G of E is integral over all x integral over all P delta let me write d E here of H of q comma I am sorry I have changed my language. So, let me just write x here P here minus E e to the power of minus beta E I will take the integral of E inside e to the power of minus beta E I look at this expression and I realize this is equal to what I should be integrating over E prime I am sorry otherwise things become very confusing I was using both E to be the same I should not be doing that well E prime lies between 0 and E. Therefore, this H I can simply replace here. So, I am using this relation of Dirac delta function where E prime is the same as x my function is e to the power of minus beta x and this is my Dirac delta function. Now what is this integral equal to anybody remembers till what is it partition function. So, integral from 0 to E G E prime E to the power of minus beta E prime d E prime is equal to q. Now here when you are doing this integral this rho was supposed to be an integral only over reactant space. So, basically what we then do is we consider this to be only reactant partition function because again it is the same idea as always we are only thinking that my entire population is limited to reactants there is nothing in the population in the product side. So, I am integrating only over reactants everything at this R sub label ok. So, k of t I have to find this formula and I have shown that this thing is equal to nothing but q R that is good ok. So, I get 1 over q R g R of E k of E now k of E I will use this formula what we had derived in the last lecture this cancels here that makes me happy integral 0 to infinity d E I will do an integration by parts here I will call this as my first function this as my second function. So, I will write this as W dagger let me before I do that let me do one more thing I realize that this integration that I am doing should be only from E A to infinity because if E is less than E A k T S T of E is actually equal to 0. So, the reaction happens only if energy is greater than E A the entire micro canonical transition state theory rate that we are deducing is for energies above E A this W dagger has no sense if E is less than E A ok. So, our integral of energy is only from E A to infinite why because energy less than E A my k of E itself is 0. So, I can replace this 0 to E A directly all right. So, I am integrating by parts now I am calling this one as first this one as second. So, I integrate second one I get this from E A to infinity minus integral I again write 0 I do not know why d E in minus integral of differentiation of first function integral of second function like this look at the first term this is actually equal to 0. So, I will let me put infinity here I realize E to the power of minus beta infinity is 0 I put E A here I realize W dagger of E A minus E A equal to W dagger of 0 which is equal to 0 W again is a total number of states for energies less than that energy. But if my energy is 0 then there are no states. So, my W dagger is 0. So, this term happily vanishes all right. So, this is equal to then I see I have a negative sign here and a negative sign here I make that a positive I take beta as 1 over k t you start seeing hints of transition state theory do not you k t over h 1 over k r similar similar. Now, what is d W dagger over d E g ok. So, you can look that in the last module as well d W over d E is the density all right. So, I get this expression all right what do I do now I do replacement of variables I define E prime as E minus E A over h q r ok d E prime is d E let me find the limits when E is equal to E A E prime equal to 0 when E equal to infinite E prime also is equal to infinite. So, my limits here are 0 to infinite now d E is the same as d E prime E to the power of minus beta E is nothing but E prime plus E A g of E prime let me just few more steps almost there over I will write this as q r here and I will take E to the power of minus beta E A outside the integral. Now, you realize this integral just a few slides ago we showed is equal to partition function and this partition function will be the transition state partition function because this g is the transition state I had forgotten to write this I remember I am following this g here ok. So, q dagger over q r. So, we have derived the transition state theory at canonical ensemble that we had derived much earlier before, but from a new perspective this time we had first derived a transition state theory at constant energy in the last module and if we just integrate over all energies with the correct thermal density multiplied by this k of E we see that we get the same expression back. So, this is good this is more or less of a really a consistency check what we are doing today. This in deriving both these expressions we have made the same set of approximations the same model except that the ensemble is different here I have canonical here I have macro canonical, but the physics is the same. So, I should be able to go from one to another and to going from one to another is this formula if I do that I should be able to get from one expression to another and that is what we have done today. Thank you very much.