 This is our seventh screencast of the Chemical Kinetic Series. Maybe I should get some intro music for this. We've moved on to a new topic now, so previously we discussed reaction rates, so now we're going to do the Kinetic Series of Gases. So make sure you do familiarise yourself with the aims and objectives for this section. There is an intro video covering where we're going to go and what we're covering in this section. So do familiarise yourself with that before moving on. Because this is going to be a little bit of a departure from just measuring rates. What we're going to do is cover kind of a little bit of a foundation before we move on to the Kinetic Theory of Gases by looking at something called the Arrhenius equation. So this is something that relates the rate constant K to temperature. So previously I said the rate constant is constant. Now we're going to say that the rate constant varies with temperature. So what is constant between all the directions of the rate constant? It's in fact something called activation energy. It is the barrier that a molecule must hop in order to proceed and react. So that will be a little bit of revision for you. I'm going to give a very quick set of examples and ranges of activation energies as well, just as a check, just so you can recognise them. And then we're going to be able to calculate them graphically. So straightforward. If you can manage those kinetic equations and graphs, you can definitely manage that bit. So let's move on. St. Arrhenius, a real big hitter as far as chemistry is concerned from the late 19th century and early 21st century. You can actually see him appear in the 1922 Solvay conference picture. That famous one with all the key scientists from the day out there. And so he's got quite a track record of doing some really important stuff. He's founded quite a little bit of acid base theory. So a lot of we know about acids come from this. And then ionic dissociation. He kind of proposed that salts will break up into ions, sodium chloride going to things like sodium plus and Cl minus. That's him. And then the greenhouse effects. So he did some of the first measurements that demonstrated the heat capacity and the ability to warm the atmosphere of things like CO2 and water vapor. So you can thank him for a few things, but mostly we're interested in the equation that bears his name, this one. Not so much go too much into the theoretical justification for why this works. That's kind of the subject of this entire section. But we'll go through all the labels in here. So here is obviously in this case the rate constant. We've done rate constants to death in the last set of lectures. Now we're going to apply it in a new way. And then there's the pre-exminational factor, this capsule here that we stick here. This at the moment, as far as you're concerned, is just a factor that converts this neat little bit of theory here to the rate constant. It's just kind of a fudge factor, which is what I say in physical chemistry a lot. I would say that we're relating two values and then we just multiply one value by something to get it right. It does have physical significance, but its physical significance is really the subject of this section of the course. So we will fill that in as we go. Then there's the activation energy. So the activation energy will be a little bit of a revision for you, but we will cover it shortly. It is an energy barrier that molecules have to hop over. Then there is the gas constant. We're kind of familiar with what that is. It pops up in thermodynamics a lot. It certainly pops up in kinetics a lot, and then temperature. So what we're looking at here is a relationship between the rate constant and temperature, and it very specifically follows this equation. Now that equation looks a little bit complicated for you. I think it's worth figuring out how to simplify things. So equations in physical chemistry and physics in particular really do get quite convoluted and really long. So you need to be able to simplify these down. So, for instance, you'll probably be interested in y equals function of x graphs. You probably can do that no problem. You can certainly do y equals 2x plus 3 and so on. This is just saying that there is another function of x here. All I've done is I've replaced temperature with x. I've replaced k with y. So we've got a graph here. And how do we deal with this? How do we convert it to the general shape of the graph? We have to go through the equation and just kind of eliminate what's constant. What isn't a variable? So a here, that's the pre-exponential factor. That's kind of a constant. It's not going to change. The activation energy, again, characteristic of an individual reaction is not going to change. And the gas constant, as the name suggests, is constant. So we can actually ignore the interesting parts of the fact that temperature is, in fact, 1 over. There's a minus here. And it's an exponential function. So all those three things can be brought down and we can plot this. If you have a graphical calculator, you don't want to be plotting that. That's going to be a pain in the ass for you to calculate. You just want to plot this. So if you run to, say, a graphical calculator or to more formalpha, that's a really good site for plotting graphs, you will get a graph that looks a bit like this. It's a fairly straightforward kind of exponential curve. But the important thing is to note that it doesn't just run away with itself up to infinity. So let's substitute that x and y for the real things again. This tells you that the rate constant goes up as temperature increases, but not forever. There is something of a limiting factor and you get less returns for your temperature increases as it goes on. So that is, if you're all right with reading big, long equations like this, this is going to seem really stupid, but I know a lot of people do look at maths notation and really struggle. The trick is just to look at it and try to break it down to its component parts just like this. Cut out the stuff that's constant and just convert it to something that looks a bit friendlier for you. So let's recap the Uranius equation. What does it do? It relates the rate constant k as a function of temperature. It's very straightforward and it includes the gas constant as a constant for physics and physical chemistry. It includes the activation energy, which we're going to cover soon. And it has a pre-expansion factor, which is effectively the subject of this entire section of the lecture course. So let's move on to activation energy. You should be used to seeing diagrams like this. You start with a reactant on one side and you move to the products. And up on the y-axis you have energy. So increasing energy goes up. So to react from iodine and hydrogen to form hydrogen iodide, it has to go up over this hill here. This is kind of an energy diagram and it has to get up that high in energy. Now things like the reaction coordinate, you might be used to as being said to be kind of abstract, but you can think of them as having real properties of that reaction coordinate, that axis. For instance, imagine two atoms are stuck together and they start to stretch apart. That distance is the x-axis. So you can imagine the energy increases, increases, increases, and then they've broken when the energy decreases again. So as you plot the energy along the bond length, you see an activation energy appear. So these activation energies are measured relative to kind of their starting materials. So they can work in two different directions. This delta e here or ea here, ea refer to two different directions. So this is if we are going backwards in the reaction and this is if we're going forwards in the reaction. Now, this says that our rate constants are going to be different in each case. So the Arrhenius equation relates the rate constant based on the activation energy. So for a set temperature, we know this forward reaction is going to go faster than this backwards reaction. But of course, the actual rate is proportional to concentration, remember. So as the reaction proceeds forward, we reduce the concentration of reactants, so the rate slows down, but we increase the proportion of products. So the backwards reaction increases and eventually they meet each other and hence why we have equilibria. So this will discuss a little bit more when we do the relationship to thermodynamics. But for now you can kind of tell that because the activation energy areas are different in each direction, we get different kind of rate constants in each direction. But because of their proportion, the rates proportionality to concentration, they will eventually kind of meet up and equilibrate. Now, activation energy, we can have a look at it in another way about how do molecules cross it. So this is sort of something of a distribution function. If we have a number of molecules in this energy and a number of molecules with that energy, their relationship is here. This is not quite the same as what we can think of it as the same as quantization of energy from quantum mechanics, or we can just treat it as just an arbitrary set of energies. Imagine that's molecules that are moving at 200 meters per second, these are molecules that are running at 400 meters per second, all that kind of speed. What's the energy difference between those two molecules doing that? What's their difference? And then we've put it into this distribution and we can find that out. So if we do a little bit with that equation and give it more of a kind of a plot as a distribution function instead, and we'll get onto this later, we can get different values or different plots, depending on different values of temperature. So here at low T, you can see there's a lot more molecules that kind of low energy, because energy is on the x-axis here. Higher temperature, it starts to move along. Higher temperature again will probably get something like this. So what you can see is that there are more molecules above this activation energy. So this is just marking the value. It's got no real bearing on the equation at all. This is just marking the value where the activation energy should lie. So anything to the right of that line will react. Anything to the left of that line doesn't have the energy to, so it can't react. So you don't necessarily need to memorise this kind of distribution equation. It is useful to be familiar with it and understand where it comes from and what it's doing. So it's just some examples of activation energy. So if I tell you to calculate an activation energy barrier and you come up with half a million kilojoules per mole, you should be aware that that's wrong, that's way too high. The typical value is between 10 and about a couple of hundred kilojoules per mole. This is the chemical range we are dealing with. Anything higher than that and you're talking things like nuclear chemistry. So it's clearly not. And if you've got huge and negative numbers like minus 10,000, clearly something's wrong. So this sort of number range is what we're after. So these are examples of activation energy. So obviously these are some of them are higher than others. So they're going to go slower than the ones with lower activation energies. And there are some kind of weird effects involved. So this is not a typo that is actual negative activation energy. And I just bring this up because it's evidence of a more complicated reaction scheme going on. And we will cover that towards the end of the lecture course. All you need to know from now is if you do see an activation energy with a negative value, it just means something more complicated is going on. It doesn't mean that that activation energy graph has reversed and that we're doing this. With that is now the activation energy barrier and that's the reaction coordinate. It doesn't mean that otherwise if they would just go down to the lowest energy and be sinking to a transition state, that makes no sense. It's just evidence that something more complicated involving the forwards and backwards reactions and this equilibria are going on. And it's kind of an edge case where I'm kind of intuitive understanding what's going on kind of falls down a little bit. So let's just recap activation energy. It is the energy required for a reaction to progress. And it is defined as the energy difference between the reactions and its transition states. So it is that energy there or because it can go in both directions forward and backwards, it's that energy. So we can have two different values and they contribute to where an equilibrium would lie. Now, how do we determine it? I want in social intelligence by doing this too slowly. If you were comfortable determining the rates graphically from the previous section, you should be more than capable of doing this. So the Arrhenius equation is an exponential form. Physical chemists don't like that. We hate things that are experiments. I'm not sure you can do the least squares regression thing with an exponential decay factor and solve it using solver, but let's do it the simpler way and we'll just linearize it. So you take a logarithm of both sides of that and you get this equation on the right. Now, I have mashed that up a little bit. That's not usually how you would see it written down in a textbook. The reason I've rearranged it a bit like this is to emphasize the following. That is y equals mx plus c. It is a straight line. In fact, there's a straight line, but it does shot ahead a bit too quickly. So what we're interested in is we have different temperatures that we've recorded a reaction at. We have a rate constant that has been determined at each of those temperatures. I think when I'm scribbling left and right, its power point is taking that as me wanting to move or something, which is a bit weird. But we can't plot, anyway, we can't plot temperature versus just rate constant here. It doesn't give us anything useful, but we can get one over temperature, temperature to the minus one, and we can get log k. So if we plot those, it really is jumping today, isn't it? If we plot those, we get our different values out that we can then actually go ahead and plot on a straight line. So this is our straight line graph. As you can see, we've plotted that y equals mx plus c version was reiterated here. Always useful to write this kind of thing down so you can see it. And here we also have that y equals mx plus c. So we can figure out the value of m, the gradient. So that m is equal to minus activation energy via r. Then to get the activation energy out, we just simply take r to the other side, take the minus to the other side, and we get that is equal to minus the gradient times r, which we can calculate here. We take that gradient multiplied by the gas constant and work it out into kilojoules per mole, really straightforward. We can also, if we get rid of the log and do it, eat the power of something, we get the pre-exponential factor out as well. So the reasonable range for pre-exponential factors is quite literally in the millions. Reasonable ranges for activation energies are in a couple of kilojoules per mole to a couple of hundred, depending on what the reaction is. And as you progress as a chemist, you probably get an instinct to understand what things should have a high activation energy and what shouldn't. Really simple reactions. We'll have a little activation energy that will go ahead quite easily. So let's just review this entire section now. I've spent quite a bit of time on it compared to what I was planning. We did the Arrhenius equation, so that is relating k, the rate constant, to a pre-exponential factor times a to the power of minus activation energy over RT. So it is a function of temperature to get your rate constant. The activation energy, EA, that is the energy required for molecular rearrangement. And that goes in both directions. So a reactant can form a product, and the product can actually come back to a reactant. Remember, the universe doesn't care that we call them reactants and products. They are just two chemical entities, and they will jump back and forth between each other. Kind of it will, but at least obeying these equations. And only a certain proportion of molecules will have the energy to hop over that barrier. So determining the activation energy, well, we linearise the Arrhenius equation. We take logs of both sides of it. Then we find that our variables become log k and 1 over t. And if we plot them together, we get a straight line. So if we've taken four or five experiments, we can plot them, do a best fit line, and get the gradient of it, and that gets us our activation energy. So that kind of covers the basics of what we want to do. Later on in the lecture, we'll probably be solving these as problems and going into a little bit more detail about maybe some errors, maybe. But until then, hope this was useful. Hope it wasn't too much revision. I do try and remember all the equations as well, especially the Arrhenius equation. You don't want to forget that at any point or think that it's the wrong thing. So see you soon at the lecture.