 Welcome back to Screencast 9. The last one, that semi-option one, did a lot of derivation about why we're interested in root mean squared speeds and pressure relating those two things together. This is going to be, thankfully, a little bit more qualitative, less derivation, but still quite a bit of analysing equations. So we're going to introduce something called the Maxwell Boltzmann distribution. You'll probably come across a little bit before I've mentioned it, but we're going to relate to things like a distribution of molecular speeds. We're going to specifically look at that version of this, and then we'll see how mass and temperature affect it, and then finally go into a few measures of speeds. So just a bit of plot history. The Maxwell Boltzmann distribution, this comes from really 19th century physics, as you can probably tell from the the age of these pictures and the fact that they are dead bearded white guys, which science tends to name everything after. The one left is James Clark Maxwell, really, really big hitter in physics, derived something called Maxwell's equations of electromagnetism, and found that the speed of light is constant as well. So as most people probably are familiar with Einstein's famous equation, that uses the fact that C, the speed of light, is constant, and that drives from Maxwell's equations. So he determined through just studying how magnets and electricity interact, that there must be a constant speed, and that speed happens to be light, which obviously, as far as relativity is concerned, gets a bit strange. Again, that's another kind of physics, but that's what Maxwell did, and he also did a lot of kinetic theory gases. The fiddle triangle gases and got a bit of what we're about to discuss together, and then that was all refined by Ludwig Boltzmann. So we name a lot of things after Boltzmann. There's the Boltzmann distribution, there's the Boltzmann constant, for instance. If you get a constant named after you, you know you've been quite successful with your science. So the Boltzmann constant, that's KB, comes from that guy. So this is a bit of the Maxwell-Boltzmann statistics, which is not quite the same as the distribution. They are related, but we're not going to go too much into the stats. And it's the idea of this sort of function of how many molecules are in one state versus another as a function of their energy difference. So you can see at a really low temperature, they're all huddled around at low energy state. As you increase the temperature, they start getting enough energy to jump into higher energy states. So you can consider these as quantized boxes, like in quantum mechanics, or you can consider it as a continuous energy distribution in terms of classical mechanics. So it holds true in both cases. Push the temperature up even higher, if they go even higher and higher energy levels and so on. So you can calculate that out from this. But we're actually interested in something more distribution. So this is not in fact the equation that we plot for this graph. The equation has to be in that form in order to make this shape. The statistical distribution here doesn't get us in. So we're going to look at the equation that actually does plot this. And there are a few bits related to it. There is a little bit of a relationship, but we're not going to go into the statistics too much. We're just going to be interested in the Maxwell-Boltzmann distribution equation, which is this utterly huge monstrosity of an equation here. But hopefully you shouldn't be too overwhelmed by this. There are certain things that all we can familiarize ourselves with. And there are certain things we can do to simplify this down. Now this is an equation, and an equation is always just a y as a function of x. So we have yx. Well, there'll be a little squiggle line to represent it. That should be straightforward. Now, if you're not familiar with it, this whole notation of f with something in a bracket means a function of we can also even do g as usually the next one up. But this with a followed by bracket just means a function. It just means, well, it just means it's an equation basically. And this is representing the distribution. So this is our function of velocity, velocity being a variable. That's how I look at it and feel the things in this. There is a four pi involved in it. Where does pi come from in this? These pop up a lot in these physics equations. And it mostly has things to do with when you're generalizing from any direction to a single direction. So you know that the area of the surface area of a sphere should be four pi r squared. So these four pies tend to come from that. If we're going from something that's quantitized in just one direction to generally around an entire sphere or in any direction, we multiply by four pi and divide by four pi from the other case, if we're going from a general any direction on case to a single direction. So there is there is a reason for them to appear there. We also have a function of the mass. So the mass of the molecules going to be involved with this. And then we also have kb, that's the Boltzmann constant, and then temperature as well, temperature mass. We are doing it to the power of two over three. That's a bit weird. But think that three kind of comes from the fact we're dealing with three dimensions. So when we were generalizing from one directional velocity to three, remember, we had three involved in it. So that's where that comes from. Then we've got the velocity squared. This is the first interesting part of this equation, because this is our variable. And the variables are what we're going to be extracting the moment, an exponential. So we're raising oil is constant to a particular power, in which case it is minus, again, mass Boltzmann constant temperature. So you can kind of see very similar things happening here. There's a two kvt here, two kv2, a mass times a velocity and mass times a velocity there. In fact, both cases are velocity squared. So there's quite a bit going on here, but we can interpret what's going on. And one of the best ways of interpreting this is to break it down into multiple components or and then kind of simplify down what we're not interested in. So if I wanted this in one of the standard ys of function of x equations, I want this y equals something x. And I want to get rid of all the stuff I'm not interested in. So four pi, well, that's constant to constant Boltzmann constant constant pi, we can cross all of these off from not that interested. And again, over here as well. Then for any particular molecule, any particular temperature, these are also going to be constant, the temperatures and masses. So what we're really interested in is just this. It's an x squared. And then we're interested in the E because that's to find the fact we've got an exponential, we can't just cross that off. And we're interested into the fact that there's x squared here, or converted v into x. And then there is a minus. So if you type this equation into a graphical calculator, you should get that characteristic shape out. And it will go on for nothing forever. Now in this case, I've kept the T's in just to see that we can plot different versions for different temperatures. So now let's have a look at what happens for these two halves of these equations. Why is that shape appear? So what I've done is I've broken it down to y equals an x squared type function and a y lead to the minus x squared function, multiply those two together to get Boltzmann distribution. So this first one is x squared. That is a parabola. So you've probably plot this kind of thing, y equals x squared. And it goes kind of symmetrically around zero there. Now y equals e to the minus x squared, you've possibly come across exponential functions like this before e to the minus x things do pop quite a bit in physics actually. And that function kind of comes down this way. Make sure that it comes down here. Now the important thing about the e to the minus x squared is that they never reach zero. They go off to infinity before they reach zero. So these are the two functions. What happens when we multiply them together? Well, just get rid of some bits right at the beginning. Well, if x is zero, this term is zero. So it must begin here. If you've done physics at some point, they will tell you to look at the extreme behavior. What is it when x is zero and what is it when x is infinity and ask you to interpolate in between. So when x is zero, it must be zero because of that. And then it will go up like a parabola again. But here's the important thing you need to know about the exponential functions. When they're x to the e to the minus x, as x increases, they begin to approach zero way faster than this begins to approach diverging away from zero and approach infinity. So obviously when x is infinite, why is infinite? But when this is infinite here, this becomes zero. So that dominates towards the latter half of the equation. The exponentials approach zero far faster than the x squared can diverge from it. So this dominates at the beginning. And then if I control this relatively smoothly, it sort of interpolates in between. So there's the x squared keeps going up. Here is our e to the minus x squared coming down. And they sort of swap over in which dominant. So you can take a really complicated equation like the Maxwell Boltzmann distribution and break it down and see what it does. And that's really the important skill when it comes to physical chemistry and mathematics in general, as far as we're concerned at science, we're interested in graphs, what's the shape of the graph? How do we relate one quantity to another quantity? And even though we have this huge quantity of information here, how can we break it down to just the simple bits? So that is the skill you need to know. So the remaining part, let's see how it's affected by these masses and temperature. So first, I'm going to look at temperature. As you can see, we're dividing through by temperature. So things are kind of an inverse of temperature. So as t increases, this number gets smaller and smaller and smaller. And so what we can see is that the v squared or x squared term kind of gets a bit smaller and smaller. So this becomes less capable of dominating at the beginning end of the graph. And so it shifts in that direction. At low temperature, the opposite happens. So high temperature, we increase the speed and distribution. Now for the mass, this is our other variable for any particular molecule. As mass increases, this part of the term obviously increases and becomes larger. So that x squared bit dominates a bit more. So it kind of shifts the graph in this direction. In the other case, for mass is really hard, high this dominates. And so the graph against us moving in that direction. So high mass shifts us one way, low mass shifts us the other way. High temperature goes that way, low temperature goes that way. So those two effects kind of balance these two other out. Now this makes some kind of intuitive sense as well. Heavier molecules require a bit more energy to accelerate them around. They're going to be sluggishly moving around and you need a lot of energy to accelerate them. So you need higher temperatures to get all things being equal to the exact same graph for a heavier molecule. And when there are higher temperatures, they are whizzing around a bit more. They are capable of being much higher in energy and much higher in speed. So that is why these graphs are shaped like this. And they are not entirely symmetrical as well. So the things like the median, median, median, mean and mood of their speeds are going to be different. So finally, we're going to just cover some measures of speed. So how are we going to measure speed a bit? And we spent a lot of time driving this one. It took 20 odd minutes. And this is our, this is our root mean squared speed. This is kind of an expectation value. What should the root mean squared speed of it being? And the reason we do this convoluted thing is because the average, there's random motion. It means that this average speed here is not going to be the same value if you just took an average velocity. For instance, half molecule is going that way, half molecule is going that way, they're on Virginia at zero. So we actually have two different ways of calculating it. So this is root mean squared speed at the top and then average speed at the bottom, which is calculated in a very different way. And it gets us two different ways of doing it. And then we also have most probable speed, which is this is where we look at the peak of that distribution diagram. So we want to work out what is the very peak of that? What is the most probable speed that we'll be finding? Sort of a measure, another measure of average. So these are three ways of kind of measuring average speed. And there are three sets of different equations. We're not going to spend too much time deriving them or just define them, just kind of accept them for now. And there's also a way of doing it between molecules. So here, molecules. As you can see, the Boltzmann constant appears. That tends to be when we're dealing with individual molecules. We use the Boltzmann constant and then the massive and individual molecule. In this case, this is for moles. So this is probably going to be more useful for us from a chemist's perspective. Here we just replace the Boltzmann constant with the gas constant. That's R. They're related terms. And then M4 molar mass. So we can see that there are two ways of looking at this. We can have a look in terms of moles and think of them as a bulk entity or can really focus down on one individual molecule. Get to see exact same result in any case. Let's just review this one. So our Boltzmann distribution relates temperature and rate constant. I've clearly copied that over from another one. This is not quite right. I should proofread these things. So the Boltzmann distribution is all about getting energies out based on this energy and the number of molecules and the Maxwell Boltzmann distribution of speeds actually gives us our speed and then kind of a percentage of molecules going upwards. So it's this kind of shape that we want. 20x squared. It also implies there is no upper limit. I mean statistically speaking there could be a molecule with infinite velocity. Obviously that's a nonsense. But the distribution as far as the theory is concerned implies that. You would have to have an infinitesimal number of molecules that have exactly infinite speed. So it's not entirely absurd but that's what the model says. And it also says that zero molecules are stationary. It's got a bit more of a physical significance because think about absolute zero for a moment. That is the temperature in which kiss in which everything stops moving. So absolute zero is when nothing moves. We can't get to absolute zero so there are no molecules that are perfectly stationary. That's a neat little consequence of that equation as well. And then mass and temperature. So a low mass gives us a wider and higher distribution. It sends the graph in this direction. And then higher temperature is a wider and higher distribution as well. Again sends the graph in that direction. So that's the Maxwell- Boltzmann distribution and it's quite important for kinetics especially when we're starting to look at the microscopic theories behind it. So we'll see you next time in the lecture where we'll do some problem solving.