 and welcome to module 20 of chemical kinetics and transition state theory. Now we have developed some of the language of statistical mechanics specifically partition functions. Now we want to direct our attention towards calculating rates for using these partition functions. So before doing that let me tell you our general prescription on how we are going to calculate the rate. So this is going to be a small module but it is an important module. So we are actually going to not think about molecules today. Let us think about something better. Let us think about cars and I have a puzzle for you today and we are going to solve this puzzle. Imagine I have a road and on this road you have a lot of vehicles that are moving. Now you have a distribution of speeds. Yes, so you are really thinking about some highways, many cars are moving, you have different lanes and just for simplicity let us assume you have cars moving at 30 meter per second, 20 meter per second and 15 meter per second, only 3 speeds. Just as an example and let us say 50% of the cars are moving with 30, 30% are moving with 20 and 20% are moving with 15 meter per second. Now imagine I have some signpost somewhere, just a spot on the road and I ask you the question what is the rate at which cars will be crossing this line, this blue line here. And we have to assume some density of the cars and let us say that density is some 2 cars per meter. That is actually a lot of cars but fine every kilometer you have a 2000 cars. So a very, very busy highway I guess but let us assume it whatever it is a number. So this is a puzzle for you. What I want you to do is this is one of the most important things actually. If you can solve this puzzle you understand the essence of all rate theories then. Everything that will forward here from this point on you then understand. So I really want you to take your time, pause this video and get this answer. Most important I want you to understand is the logic behind how to solve this problem. So pause the video and solve this problem and do not rest until you have solved this problem. Please pause the video now. Okay, hopefully you are back and hopefully you have tried to solve this. If you have not please pause the video and do attempt to solve this one. This one is critically important. Okay, we are going to solve it together now. So we are going to divide this problem. The total rate will be the rate at which cars moving with speed equal to 30 meter per second plus rate of cars traveling with 20 meters per second plus rate of cars traveling with 15 meters per second. So what I will do is then to find these three individual rates. So let us focus on rate at 30 meters per second. How do I solve this problem? What is rate? So we have to get back to our fundamentals. Rate is the number of cars crossing the signpost per second. So if I wait for one second, I have to observe how many cars passed which were traveling at 30 meters per second. Okay, so imagine you are sitting there at that toll booth at the signpost and every time a car traveling at 30 meter per second crosses by you have a counter and increase the counter by 1. And you find the number of cars that passed in 1 second. That is the rate. Okay, by definition. Okay, okay. So how do I calculate this number? Well, the idea is in one second cars traveling with 30 meter per second will travel well of course 30 meters. Okay, the speed is 30 meter per second in one second you have 30 meters of distance. What it means then is if I take a length of 30 meters here, cars that were in this 30 meter would have crossed this blue line. And any car that is outside this 30 meter line will not have crossed this blue line in one second. So in one second time only the cars that fall within this 30 meter range will be able to cross. So this number, so this rate of 30 meter per second will be number of cars in 30 meter length. Okay, so this will be equal to density of cars with speed equal to 30 meter per second into 30 meter. So I find the density in per unit length which have this for following speed and I multiply it by 30 meters per second. That will give me the rate. So this density is the density of cars into fraction of cars with speed equal to 30 meters per second into 30 meters per second. The density of cars with a particular speed equal to total density of cars per unit length multiplied by the fraction of cars that were traveling with 30 meters per second. Yeah, so that is how I will find the density which are with a particular speed. Because think about it very carefully what have we are doing? The total rate at 30 meters per second will be the number of cars in the 30 meter length because those are the cars that will be able to cross this signpost in one second time. Fine, but the number of cars in this distance is nothing but the density of the cars into this distance. The density of the cars will be the total density of all kind of cars multiplied by the fraction of this particular kind of car. So this rate at 30 meters per second is then density, the total density into fraction of 30 meters per second into 30 meters per second, the speed. The density is 2 cars per meter, that is the total density. A fraction for 30 meters per second is given to be half, 50 percent. 50 percent is nothing but half into 30 meters per second. So if I multiply this, I am going to get 30. So I can do the same for 20 meters per second too. This will be the same as a density of total cars into the fraction of cars travelling with 20 meters per second into 20 meters per second. The same logic in one second, the total number of cars travelling with speed 20 meters per second, well those cars better be in this length of 20 meters. Same logic. So this becomes 2 cars per meter into the fraction here is 0.30 into 20 and I can calculate this, this is equal to 12 and the corresponding rate for 15 meter per second will again be the total density into the fraction of 15 meter per second which is 0.20 into 15 meters per second. So this is equal to 6 cars. So the total rate is then equal to the sum of these 3 which is nothing but 48 cars per second. So if you sit at that sign post, you will observe 48 total cars passing. So we want to generalize this a little bit. The total rate, if I have in general some number of cars with some number of speeds will then be equal to sum over all speeds at which the cars are moving. The total density of cars per unit length into fraction id u. So if you are travelling with some speed u in each second the cars have to be within the length u meters. So I have to find the number of cars in this u meter length but the number of cars in this u meter length is nothing but the density of cars, total density of cars into the fraction of density which are moving with speed u multiplied by u meters. So that will give me the rate in per second. I can generalize this a little bit. If I have a distribution of speeds rather than summation, this summation will become an integral. So imagine you have a cars at all possible speeds travelling not only at 30 but you also have at 30.001, 30.002. At every speed there is a distribution that is how molecules behave right. So I am trying to get to the molecules now from cars. So the fraction basically gets replaced by some density into density into u where d is total density per unit length, rho of u du is fraction of cars speed u. So this is the expression I get. Rate is equal to integral and I had forgotten to do one very important thing which is the limits. So we are only looking at positive speeds. For integral goes from 0 to infinity, all cars that are moving with positive speeds. So these limits are very important. So in writing this expression, we have made two important assumptions. One is only forward speeds. That is that is that is cars are later on not turning around and coming back. Then your rate decreases, the forward rate decreases. So we are assuming all the cars once they go through this goalpost, keep on going forward. And the second is importantly we are treating cars classically obviously. This becomes a bit more tricky when you think of atoms and molecules. Cars are of course described by Newton's laws very well. Atoms molecules most likely they are but you can have exceptions. So today I just want to give you this important expression for the rate. Any time you calculate rate actually this fundamental expression is used. Rate is an integral over speeds, the density multiplied by the probability that I am at given speed u and multiplied by u. So this u is called the flux. So I am at a given goalpost. What is the flux multiplied by its density? That is how you calculate rate and in doing that we have written two assumptions here. We are really thinking of a speed and when you even use the word speed it is somewhat classical. In quantum mechanics you have to be very careful when using the word speed. So we will just end here and in the next module we will actually derive transition state theory. Thank you very much.