 Welcome to module 7 of Chemical Kinetics and Transition State Theory. Today we are going to start with one of the important models that is one of the main objectives of this course. So, we will start looking at kinetic theory of collisions today. A little bit of its history and origins. So, it was introduced by Trottes and Lewis in two different papers in 1918. So, this was essentially a build up over what is the transition state theory, the concept of transition state put forward by Arrhenius. Arrhenius being the genius, he had the vision in late 1800s to postulate that the transition state exists. What he did not provide is how to calculate the rate constant. He gave a formula, but he did not give how to actually get the parameters that are there in that formula. So, that is what preceded after Arrhenius. People were trying to figure out, well you give me a reaction and I will try to find out what that rate constant is going to be. And well remember in this time also the idea of atoms and molecules were also emerging. Dalton happened only a 100 years before that. So, people had figured out Trottes and Lewis particularly one way to find this rate constant from an atomistic picture. So, here I have actually provided you the original paper, you can go at it and look at it. This is a very readable paper by the way and I mean this is not going to be a part of the syllabus, it is too advanced for that at least. But if you are interested in this field of chemical kinetics as general, do read this paper. It is a beautiful paper and it has a lot of insights. It really shows you how the scientists thought and developed theories. Even if this theory is not the most advanced theory as of now, but nonetheless it is 100 years before and it is beautiful how the scientists thought. The resources we are following is basically the chapter 4 of the book by Laidler. I have provided you another reference which is this website. So, you can look at this and this will also has a very very good description. So, let us come to this kind of theory and what is the basic idea? What is our model? How are we going to calculate this rate constant? First thing, this model is only for bimolecular reactions that is there are 2 reactants a plus b. This is on the elementary step. So, I have a and b they are reacting together to give me some products. Products can be as many. The main idea how this theory thinks is that it thinks of a and b as essentially spheres moving around in a box in some box of some volume b. So, I have some box some volume b some temperature t and I have these reactants a and b which are basically spheres in gas phase running around. So, what we assume a very simplistic model we are trying to get the first estimate. So, we are going to make a lot of approximations we want to keep the theory as simple as possible. We want to we are going to assume that these a and b move with constant speed some u 1 and u 2 until they collide. When they collide the speeds can change and when they collide is the point when the reaction happens. So, we are not going to get into the nitty-gritty of bonding there. This theory by the way is before Schrodinger's equation. Schrodinger's equation is in 1920s 1925 and 26 this is 1918. So, the idea of bonding is still not very clear in 1918 by the way. People had some idea, but not very clear. So, this theory says let us not get into the nitty-gritty of bonding at all. Let us say this collide and whenever this collision happens a reaction is going to happen. And so, the rate of reaction is simply equal to the rate of collisions between a and b. So, we have a very simple model with us. We have simplified the problem a lot. We have provided a way for you to think about this problem. So, let us think how we are going to calculate this rate of collisions. To start thinking about it, this problem is of course, in three dimensions or why it is three dimensional as you see it. To simplify let us just think in 2D just for understanding. So, let us start with a simple question. I want the rate of collisions, but let us ask a simpler question. Let us say I have a circle of radius r a. It is moving with some speed u and I have some other circle of radius r b which is at rest. What is the probability that these circles will collide with each other in some small time delta t? And let us assume a uniform density of a and b given by n a and n b where n a is the number density which is the number of particles of a divided by total volume. Well, how do I calculate this probability now? So, this probability is calculated. Well, this probability is equal to the probability that b is present in the region that a covers while it moves. So, I have small time delta t. In this small delta t, a will move a little bit forward. So, I have this a, it will move a little bit forward. And so, it is covering some region and if b happens to be in that region, well your collision will happen between these two circles. These two circles will coincide. So, I simplify my problem a little bit further, one step more. I say well, then I find the area that this a is going to cover multiplying by the density of b that will give me the overall probability. So, I have to calculate this area of this region that a is covering. I call this thing as in general a collision region. So, let us try to think about this. So, this might not look like a perfect circle to you, but let us imagine it is a circle of some radius r a. So, if I let this circle move forward with speed u in time delta t, how much distance will this circle move? Well, of course, u into delta t, that is how you calculate distances. Well, so if the center of b is inside this, of course, these two circles will coincide, they will collide with each other. I want you to note something more. Even if the center is a little bit outside, let us say my center is somewhere here. You note that my god, this will look like the center, but you can still imagine what I am saying. Even if the center is a little bit outside, nonetheless, these two circles are still going to collide with each other. In fact, I will tell you that the circle can be as far as this when they tangentially touch each other. Any further and these two circles will not touch with each other at all. I can do the same thing on the bottom side as well. So, the point is that my b can be anywhere the center of b in this region. So, let me just redraw this figure. I have a here, drawn a length of r b here, this is 2 r a, that is the diameter of a and this length is u delta t. So, if b is present here, the center of b again, if the center of b is present in this region, which a has moved forward or if b is present here, then these circles will collide. So, I want the area of the shaded region. So, a more clear picture is drawn here and I want the area of this blue region that I have drawn. Take a minute or so and solve this problem. It is a very beautiful problem. It is a puzzle. So, take a minute or two, think about it. How will you calculate the area of this blue region? So, pause the video and solve this area. So, hopefully you have paused the video and now you are back. Hopefully you have calculated the area. If not, please do give it a shot. It is a very beautiful puzzle. How do I solve this? Well, the problem is this semi-spherical region. It is hard to calculate this. You can do it a bit more manually, but a simple idea is as follows. You ready for it? It is a beautiful idea. What we do? We take this little region that is here. Take it out. We remove it and we paste it here. You know that this will fit perfectly here. So, I will get a figure that will look like this. So, I have moved this blue space out and put it in the white space here. Now, you see the problem is much simpler. You simply get a rectangle length u delta t height 2 into r a plus 2 into r b which is trivial to find the area. So, it is just 2 equal to 2 r a plus r b into u delta t. So, I have found the area of collision region which is 2 into r a plus r b into u delta t. So, now I can get the probability equal to 2 r a plus r b u n b delta t. So, that was 2d. That was just so that we have an understanding of how to calculate these areas and build a basic picture. Reality is in 3d. So, the question remains the same. Now, I have a sphere of radius r a moving with some speed u and let us assume again b is at rest. We will come to it. Do not worry. I know you have questions. We will come to that. But let us start with assuming b to be at rest. Bear with me. What is the probability that these spheres will collide with each other in some time interval delta t? Again you have the number densities and the total volume is v. Idea is the same. Probability will be the volume of collision region into n b. So, again I have to do the same trick again and find this collision region. Yeah. Now, let us see how good I am drawing its spheres. That was my abilities of drawing circles. So, this is my depiction of a sphere. Much more beautiful picture can be found on the right. But the idea is I have a sphere which is moving forward just like circles. Same analog. Well, a collision will happen if b is found inside this region or if b is a little bit outside, b can be here as well. Worst come worst, b should be here. So, b is also a sphere. So, what I get is essentially a cylindrical looking shape like this. And b must fall inside this cylinder. So, let me clean this up a little bit. This is my a. There is basically this cylinder here. This a had length radius r a. While this cylinder has radius r a plus r b. b can be as far away as r b outside this region, the center. And this is moving forward and I get this cylinder of length u delta t. And here is again my a. So, once more a little bit of a is poking out of the cylinder here and a little bit of gap is left here. So, we do the same trick there. We take out this patch from here and put it back here. Fill it in from the side. So, the area effectively will be the area of this cylinder ok. So, the area will be what is the area of a cylinder? Area of the cylinder is the area of the circle my bad I should be using the word volume now we are in 3D. So, volume is area of circle into length of cylinder. So, this will be equal to area of circle you all know very well is pi r square and here r is r a plus r b and the length is u delta p. I have used dt here it should be delta t. So, a much cleaner picture you can find here it is taken from this website. It is the same resource I provided earlier. This is my a it is shown as a circle and I get this cylinder of size r a plus r b and this length is u delta t here. So, if the center of p is inside this cylinder you have a collision if it is outside it is a miss. And so again getting back I have the probability is the volume of collision region into n b and volume of collision region we found to be pi r a plus r b square u delta t. So, I get probability of a collision is equal to pi r a plus r b whole square u n b delta t good. So, what though we wanted the rate what am I doing why am I calculating these probabilities it will come helpful yeah I promise you. So, this probability the rate the rate of collisions by definition is the number of collisions per unit time per unit volume. Remember this this is how rate is defined of any quantity what is the number of that events happening per unit time per unit volume. Every single time for every single thing anytime anybody asks you rate this is what they mean by the word rate this is what I have to find. But the number of collisions happening per unit time is the total number of molecules of a that I have multiplied by the probability that a will collide in unit time delta t in this small time d t this is where probability finally enters you see it now. So, you are convinced that the number of collisions is equal to this I find a total number of a's and I find how much is this 1 a is going to collide what is the probability that this a 1 a is going to collide with b. Well probability of collisions we have already found and I am just showing you here pi r a plus r b square n b u delta t this is what we derived. And the total number of molecules of a is nothing but n a into v remember n a is the particle density if I multiply it by v I get the total number of particles. So, the number of collisions per unit time I multiply these two quantities and I divide by delta t this is the total number of collisions but I wanted to find it per unit time. So, I divided by delta t good and so the rate of collisions or note that here delta t cancels and so the rate of collisions was the number of collisions per unit time which is here per unit volume. So, I take this pi what I get from here and I divide by volume. So, again volume cancels and I end up with pi r a plus r b whole square u into n a into n b. So, that is the rate of collisions that is what you basically get in kinetic theory this is our model. I know you have still some questions one is of course what is u as an experimentalist one would come to me and tell me a reaction he will not tell me what is the speed. There is also you may notice there is no temperature here anywhere. I was performing a reaction at a given temperature I mixed two reagents at some volume and temperature. Temperature is also not present something is fishy and last thing you also notice that I had assumed b to be stationary and a to be moving of course that is also not true why should you assume that. So, these questions we will address in the next module. In summary for today what we have developed is a very very simplified idea of how to calculate rate constants which is called the kinetic theory of collisions. And within this kinetic theory of collisions the rate is equal to the rate of collisions of two hard spheres of radius r a and r b. There is no real notion of a bonding here and what we have shown that if a is moving and b is stationary and a speed is u this rate of collision is given by the following formula pi r a plus r b square u n a n b. In the next module we will get into more details we will see how temperature enters. We will look into how u is calculated and we will look at how do we include motion of b. Thank you very much.