 Hopefully, if you've been paying attention, you're now getting quite comfortable with the idea of kinetics and some of the terminology we use and hopefully with a bit of mathematics as well, hopefully you can start seeing how equations work. So if you weren't necessarily strong with that, I hope it's getting a little better. If not, we will work on it. Just do send me an email or ask me and we can break things down a little easier. So we're going to round off the second topic now, which has all been about the kinetic theory of gases and we kind of started with the Arrhenius equation. I went through a bit about what it means for a gas to move at a particular speed and collide and sweep out areas through its collision diameter and so on. So now we're going to round that off and connect it all together. And for this, we're going to look at simple collision theory, some equations that those theories predict and then we're going to relate this to the Arrhenius equation. So we've had the Arrhenius equation where the rate constants is equal to our pre-exponential factor and then the exponential of an activation energy over RT. That is a empirically derived formula. It's a nice little bit of theory about what activation energy should do, where there's kind of a constant that multiplies it up to get to the rate constant that we can derive experimentally. So first topic was all about this. Our second topic has all been about trying to interpret that and now we're going to finally put those two together. So collision theory, this goes back to that very first lecture. Collision theory just states that molecules have to collide and then a reaction occurs. So specifically what did we say that must occur? Well, one, a collision must occur, a collision of some kind and the last topic or so has all been about getting the speeds of molecules. So we can figure out how often do they collide and we get this sort of formula here. So remember, this is our collision cross-section. It is sort of an area. Then we have a speed. We can get a speed out based on a few factors such as the mass and the temperature and then density of the two things that are colliding. So this is for a reaction of A plus B. They're going to smack into each other and react. So we have, say, if they are gases, two concentrations. Now we don't necessarily want that in terms of just what was how many collisions are there. We want to kind of generalize it into something that's right. So instead of having this ZAB factor term here, we have Z0. So I've put this down here as Z0. This just means that this is the number of collisions that could happen for any particular given concentration. So that means we can now relate it to k, the rate constant. Because remember, rate is equal to k concentration. So if we scratch out these concentrations, we can scratch out that and we're now interested in just k. So that's the simple idea. So we are interested in just these two factors, cross-section and speed. That's an area and that is the mean speed of the molecules. The second part from collision theory, there must be sufficient energy. So what we need to do is how many molecules are on this side? So if they collide, do they have the right energy or are they going to hit each other head on with enough energy to get over the activation barrier or not? So we have this Boltzmann factor, which is the number of molecules that are higher than a particular energy. I'm not going to go into the equation of how we derive that. It's a little bit of integration, obviously. We're not going to cover it, but we need to figure out how many molecules does that part of the graph represent. I will just give it to you. It's a little simpler than it seems at first sight. And then, of course, it must be the correct orientation. So this is quite important. Generally speaking, if this substitution reaction wants us to collide here, it's going to go ahead. Every cloud here is going to bounce off. There's nothing it can do. There might be a transient reaction, but it's not going to happen. Nothing's going to occur here. So these are the three factors from simple collision theory. We must have the right orientation, there must be the right energy, and there must be a collision in the first place. So what we get at the end is this equation. So I'm going to break this down a bit. We have this P. This is the steric factor. So when I said all about the orientation, this factor effectively says what proportion of the collisions are actually in the right orientation. Now, you might naively think, well, it's got to hit the atom on this side. So the steric factor must be 0.5% or 50% or something like this. That's not necessarily true. We have to derive these empirically. All we can do is get the rates out and try and figure it out from data. Later on, when you do transition state theory, you can start putting a better number on that. But until then, this is just empirical. What can I say? I didn't sign snot spelling. We must have this in here, the number of collisions. So this is fairly obvious. The more collisions that happen, the higher the rate. So finally, the molecules with the right energy. So that is our factor. We have that Boltzmann factor here, and we divide it through by the Boltzmann constant and temperature. And it e raised to the whole thing. So we denote that, if you've noticed, not with just k, but ksct or k simple collision theory. So that means we've derived this semi-empirically sort of with a bit of theory involved. So this is the main equation we're interested in. So this is what we're all here for. We figured out that our three components from just thinking about what happens when molecules combine gives us this equation at the top. And then the Arrhenius equation, something that was mostly empirically derived, that can get us k and activation energy and so on, is this. And hopefully you should be able to see some similarities here. Obviously we've got k on both sides. That much should be clear. Then we've got an exponential term. Again, something that should be quite clear here. And then notice they are to the minus, something good. We're looking very similar here. And then look at these terms here, ea over rt at the bottom, and then our Boltzmann factor over kbt there. So the Boltzmann constant is there, the gas constant is here. So that tells us we're really working on kind of a molecule scale, and the gas constant basically means we're working on a molar scale. So this one is very theory driven. It's based on what happens when two molecules collide. This one's empirically driven. We're asking what happens when we're actually measuring concentrations in terms of moles per decimeter cubed and so on. Of course, this looks a little bit different. We have a here. We call that a pre-exponential factor earlier. But here above we've actually split it into two terms. So the number of collisions. Remember that's the number of collisions irrespective of concentration because obviously the rate is related to the rate constant times the concentration and then the steric factor. So what you can kind of see is when I said a, the pre-exponential factor was sort of a fudge factor to get us to a rate constant from a bit of theory. The steric factor is sort of doing the same thing here. We can work out how many collisions there should be. We can know what the energy should be and so on. We can calculate the steric factor from that. So a couple of caveats obviously. This is sort of an approximation. They are theories. Remember the point of theories in science is to make predictions. If they make very good predictions, they're good theories. If they make terrible predictions, they are bad theories. These have to be very good theories. They match it quite nicely. But there is a little bit of, you know, give and take between them. So a, the pre-exponential factor is slightly, very slightly temperature dependent. It's not going to be, that Orrhenius equation is not 110% perfect. So this won't necessarily bug us for our purposes because we're going to be working over kind of small temperature ranges as far as the Orrhenius equation is concerned. We're not going to be doing something at 10 degrees and a million degrees for instance. Things will definitely start falling apart that much. But when you're doing something at 10 degrees and 20 degrees and 30 degrees, it holds true enough. And steric factors are empirical. So like I said earlier, you can't just naively assume that it must be, the steric factor must be at least 50% because this part of the molecule is blocked off and this is active. We've got to work it out. Sometimes these steric factors can actually imply, require the other molecule to orientate correctly as well, that way instead of this way or something. So there is a way to work those out properly. It's called transition state theory. And transition state theory is something you get on to much later on in a chemistry course because it requires a little bit of quantum mechanics. It requires a little bit of almost computational chemistry and it requires a lot more detail about energetics and so on. So you will get onto this eventually but you don't need to know it now. Let's just put it into a black box and forget about it. So just give you some sample values for the various things that we can get out here. For instance, we can figure out what A is according to our experiment and we can figure out what it should be according to the simple collision theory and activation energies and then get the steric factors. So what you can see is that A, as we expect, there are 10 to the 6, 10 to the 12, that sort of value. These are about 10 to 11 and so on. The activation energies are anywhere between 0 and a few hundreds kilojoules per mole. And P, here we go. Look at the values here, 0.16. So you might think 16% of collisions are in fact the right orientation. But look at some of these ones. These are really quite small. That's not very intuitive. You might think that that reaction there is actually comparatively simple. You should think that, oh, most collisions should go ahead. That's not necessarily true. We have to derive this from data and things like transition state theory and looking at slightly more complex systems. We'll get onto slightly more complex systems. Later, I can explain why that is. And some, in fact, are greater than one. So maybe the theory doesn't match perfectly here. So obviously, very simple reactions, particularly first order ones. P seems quite intuitive. But that's the range of values you would be expecting for these. So don't much memorize these. Just get familiar with them. Take a look at the numbers. What are you expecting to see? You're expecting to see the 10, 7s and 6s and 12s around here and so on and hundreds for this sort of value. And anywhere between one and almost zero, in fact, very close to zero for P. It's quite a wide range of things, but that's where we are looking for. So let's just kind of review this. Simple collision theory says we need a collision of the right orientation and of the right energy. So when we build that into an equation, the collision of the right orientation of the right energy. There we go. The three components of our simple collision theory equation. And the relationship to the Arrhenius equation, well, we can say K is equal to Ae minus activation energy over RT. There we go. You can see some rough equivalence here. Our Boltzmann factor is very similar to the activation energy. P and Z0 there, or they come to the pre-experimental factor here. And again, we just switch the Boltzmann constant and the gas constant, which just means we're dealing with either an individual molecule or the molar scale. So that is pretty much it for this second topic. It was reasonably quick to actually link the Arrhenius equation to this simple collision theory. But the main point of this entire topic has really been to get over the point of collisions, the scale of molecules, their complexity, their speed, and so on. Because that all goes into what was, eventually, just one term in this equation. So do make sure you can revise that. This is effectively just the punchline of the entire thing. So hopefully you'll be able to apply it and understand how these two relate. Next topic is experimental methods. So we're actually going to do some of this and not necessarily go personally into a lab, but we will get some data out. We will look at how we would go about doing this in the lab. What experiments can we do? So hopefully, really practical applied chemistry will be coming up.