 So, now we are going to get somewhere. So, we have derived this beautiful equation Jn is minus beta i SnCn by FnT. Now they are very interesting thing to look into that. Now if I have against n, I am using the corner because I need to talk of this equation quite a bit now. Now this is my free hand landscape and this is my critical thing. Now here at small n near n equal to 1 at the limit and in this scale when n is very large n could be of something like 200 or 300. Typically then in this region particles are trying to go and come back and there is enough of them. So, this is the region it is like equilibrium. So when I go to the limit n going to 0, I am entitled to going to 0 because I am made n to continue on actually it does not go below n equal to 1. But mathematically that does make a difference whether I 2 1 or 0 does make a difference as you will see. So n going to 0 or n equal to 1, i Fn becomes Sn. However when n goes to infinity that means n goes to infinity means here all the way there is nothing there. Before that I have the minimum maximum after maximum it just goes downhill. So n going to infinity this quantity of n t is 0. This is third is the most important thing. In a steady state when there is a flow of train that is what I do flow of train. I again appeal to you to remember that the train is going towards in front of you and one after another the compartments are coming one after another in regular interval. Train has a constant velocity speed or this has a constant speed. What does it mean? That means our case what is happening our compartmental n minus 1 the next compartment is n, the next compartment is n plus 1 there is steady state flow. What does it mean? That means j is independent of n. Now I have this wonderful expression now j is independent of n. So what do I do now and I know the limits. So I immediately now write down now I can do an integration because I know the limits. So I do the integration 0 to infinity. So I bring d here I bring d and here and then I do this integration 0 to infinity and infinity it is 0 and at n equal to 0 infinity f n t at that limit. So there is a d here so it just goes this become evaluated at n equal to 0 and n equal to infinity. Now n equal to infinity is 0 and n equal to 0 f n equal to c n. So I just and there is a minus in front that because it is lower limit I just get 1. So then I get the beautiful relation j equal to I am almost there now. So this is minus 1 in front bracket minus 1 so that means my beta i and beta i and s n because we are in denominator of denominator it comes out. So now I have the following expression so I have to evaluate this thing now. Now let us do little bit so we have done the main thing we have a little faster it I will go now. So this is the thing I said since equilibrium prevails for smallest clusters this f n by c n goes to 1 but when n goes to infinity this goes to 0 because of actual postcritical size is rare actually not rare it is not there then so now I have to calculate the rate. Now what do I do so this is my c n this is beta i s n now I need remember I need the at the critical cluster so s n become s n star or s n c and then I have to evaluate this integral now and so important thing to assume that this is minus note that this particular integral that I have to do this integral that I have to do this is 0 to infinity but I will come to minus into plus infinity then I write c n the minus 1 is there c n is c n equilibrium c then e to the power I say okay I want population here population here so because I have an integration to do I have an integration to do so I now say this I write c n but the because this is and this is a method called method of steepest descent or many other goes by Laplace's formula many methods so basic idea is that around here c n is very very small in this region in this region c n is very very small in this region if c n is very very small then that is the denominator so in the integral over d n these quantity here it dominates and so then I say okay c is I can now consider only a small region around here so the way I do that then I will say okay c n is the c n star c c n star and then I x this is a harmonic I assume it to be harmonic and I write the fluctuation here the n minus n star square and then I need to do the double derivative because this I expand the delta g here delta g n as delta g n star plus delta g's double derivative n n minus n star square by a factor of 2 okay so that is then by 2 and by k B T is here then it becomes these now this so let us explain this again the little tricky I have to do the integral so I have to do I have to do this integral this integral I am going to do and this integral I use the following property that the c n is very very small here and the c n is very small and this is the smallest c n in this landscape which is c n is e to the power minus beta delta g n by Boltzmann distribution and this is maximum the delta g n is maximum is delta g n is maximum means c n is minimum so in this integration c n is minimum here so 1 minus c n is maximum here okay this is I am saying this is used again and again so c n is very rare in a barrier top so 1 over c n it picks and so it is enough in this integration to take care of the region near the maximum that because of this advantage that we are taking that the advantage that c n is this quantity is minimum so 1 over c n is maximum we expand it in a harmonic thing and this is second derivative and then this Gaussian so I can now do this integration completely so everything before I have done that I have the beta is n so beta is n and 1 minus 1 of c n star that will come up so that is so this is beta is n my earlier thing c n star coming because I have making approximation c n is c n star then the Gaussian so that inverse inverse that comes on so this is fully documented by me I know Z is the leftover of the integration and that integration I can do because this Gaussian function now okay so minus infinity to plus infinity I can make it also 0 to infinity I do not care because there is a this integration is so sharply picked the function it is n star it is so sharply picked that I can always make it to minus infinity without any any loss minus infinity as it is given here then this integration is just 2 pi root over 2 pi by a and as a Gaussian integral and that gives now the delta g square 2 is there at the 2 pi KBT so I now get this beautiful expression the rate of nucleation so everything is determined I have now I have rate of nucleation is given by Zeldovich factor Z rate of impringement beta i that comes from kinetic theory surface area and the critical concentration that has to be there we know that and it is just reproduced beautifully and c n star is given by Boltzmann concentration of monomer because I start here and go there this is the monomer and this is the n star so that is given by n star my probability of being here is given by this expression and my impringement surface given that and these way Zeldovich factor is called the non-equilibrium effects that take into account of going the both ways this is exactly we do in chemical kinetics we sometimes call that Smokowski equation approach and they all these things happen very similar time and it is exactly what is Cramer's theory also of chemical kinetics very well known theory of chemical kinetics that we will do little bit hopefully someday so now what is Zeta what is Z Z is this beautiful expression let me write down that that delta g second derivative by 2 pi KBT and rate is Zeldovich factor impringement surface ratio C n star this is the rate of the rate of nucleation the reason I did it in detail in such great detail and more detail is given in the book and of course I think what I did here little bit more straightforward than what is given in the book big book is little bit I think some more detail steps and little bit more talking has been done which is not necessary so basically the summarize the whole thing is that we did a calculation of the rate of barrier crossing you know it is a very standard thing in chemical kinetics and many other thing but Zeldovich did it without knowing anything but around the same time Smokowski did this barrier less cohesion Lindemann did Michael S. Benton did all these things all in the same time the same thing and this very general approach the formula of these 2 pi KBT root over this thing is essentially goes as Laplace's formula and so it is a few more comments we have done so the this is what I was talking about the temperature dependence a very strongly temperature dependent phenomena the supersaturated this sometimes it is not just a temperature but pressure and super saturation that means you increase the pressure you see nucleation that is a way to see the nucleation in a cloud chamber and these are the different super saturation the super saturation S3 is more than S2 than S1 and then when the super saturation increases the size of the nucleus comes down size of the nucleus comes down and barrier comes down so activation free energy decreases the reason is that that if you transfer into free energy landscape then what is happening is that as you increase the super saturation then this Delta GV which is Delta GV this quantity is increasing as this quantity is increasing then r star r star is decreasing and Delta G star 16 pi by 3 gamma cube by Delta GV square this is also changing but this is changing much faster so as we increase super saturation or in lower temperature we will call it increase super cooling like when the water instead of 0 degree centigrade I am going to minus 10 degree minus 20 this you can say minus 5 minus 10 minus 20 below below the freezing and then the barrier comes down as I said and the size comes down then what is really interesting you find that nothing is doing nothing is doing nothing is doing you go on suddenly it comes explosive like it is shown here rate of nucleation suddenly increases because the barrier has to come down to a level where thermal fluctuations can access it and that exactly happens so this is the picture very important picture of the nucleation theory next is that as both free energy difference between bulk phase and surface known for several systems notably for water and many other systems a quantity of equation is possible and it is there are certain limitations they are very detailed things like the surface tension we are using is that for coexistence but we are using it in a out of equilibrium so there is certain limitation there then we assume it as a sphere that is a limitation there and many times they are little elliptical simulations have showed and then core of the nucleus is really liquid like but surface you know nucleation we are assuming this is sharp boundary and core is that of the new phase and this is the old phase that is not quite like that because this is some extent it is cold like but then there is a diffuse region and that is also an important thing that we have not discussed and that is done by theory called density functional theory of expression that we can do only when do the density functional theory which I will explain little bit density functional theory later when you do the density functional theory and there is this beautiful problem heterogeneous nucleation and that heterogeneous nucleation is described like that you are on a surface then the basic idea is that surface tension of the you reduce the exposure of the liquid to the old new phase to the old phase and the surface tension of these could be much less and as a result you get a free energy barrier that is the heterogeneous nucleus which is the barrier gets substantially reduced and the amount it gets is used is a quantity psi theta w which is that is given by this I will not going to derive it is another long derivation but theta w is determined by this theta w is determined the surface tension of these surface tension and these surface tension and that is the expression that is given so that this heterogeneous nucleation barrier comes down this is homogeneous nucleation but to describe heterogeneous nucleation you need homogeneous nucleation pure homogeneous nucleation and so that expression is not given here but it is kind of said that if the angle is 90 then barrier becomes half if it is 3 by 8 then barrier becomes much less 0.375 so this is a an important important thing and there are some many other things which we would be describing later. So we will start next little bit of Ostwald's step rule then we go over to some other aspects of statistical mechanics and that will be in the next class.