 So this is our tensor screencast. It's quite a nice round number, isn't it? Now we're gonna get into something a bit more about collision rates building on a little bit of things We know previously about speed So we know that the speed of the log obviously imparts a bit of energy to something to get to the activation energy Go by the activation barrier and so on But it also has an effect on the number of collisions that could possibly happen and for this We really need to focus down on what a molecule sees and then where's it moving? So that means going to collision cross-section Messing around with our speeds a little bit to have relative speeds and then figure out something called the collision rate So that collision right? It's helpful so Just once again we are building on things so once we define collision cross-section and Relative speeds for instance the collision rate comes from putting these two things together because Obviously the size of a molecule will depend will influence the number of collisions the speed that the molecules moving will influence the number of collisions So we obviously get that on the previous two. So keep that in mind whilst we're going through some of this First of all, let's talk about collision cross-section So collision cross-section is effectively defined as an area of a sphere or circle and you might be thinking That's a bit weird molecules aren't spherical are they? You'd be right. They absolutely are but they do tumble around a lot So a molecule like this could actually tumble in all sorts of different areas in different orientations And as it tumbles through and moves to a solvent or move through a gas Its orientation will change So kind of on average A collision could be considered spherical because that orientation the molecule is utterly random So on average this doesn't let's don't say that our cross-section is exactly literally like I've drawn here that says It must be a sphere that encompasses the entire molecule. It just means that it can kind of average down to a sphere But to kind of get the link across I'm gonna just draw the molecule in there So if this thing tumbling around It's sort of average look as it smeared out would be roughly spherical So here's a sphere and it obviously has things like a radius. It obviously has a diameter It has a circumference and so on So if we're dealing with this kind of thing you might expect pie to start appearing and so on So let's have a look at what collision cross-section does So here's our molecule It is surrounded by this sphere that we are modeling it as so it doesn't matter what the molecule is We're just gonna pretend it's a sphere and when it goes forward Through a load of other molecules that could collide with it hits them There you go. It hits three flies off. It has missed Three of them and it has hit three of them So as you can see the ones that are really halfway into this kind of This tube here. It's sort of means that the molecules sweeping out Sort of it tubes it or cylindrical shaped area As it passes through these gases. So anything inside that is hit or things aren't So this comes across to our definition called collision diameter So this is obviously the collision diameter and we give it D As a label and that is equal to the two colliding objects as radius is added together or R1 plus R2 That's probably an overly formal way of dealing with it It gets us this diameter and then we want to get the collision cross-section which is defined by sigma so once again we We do run out of Greek letters very very quickly almost as fast as we run out of Latin letters So collision cross-section in this case is also sigma in addition to everything else the sigma signifies So this pi D squared is an area and it is this area. It is slightly outside The radius of this so it's not defined as the radius of one molecule It's defined as the radius of two of them added together. So if we had another molecule here You know its center is outside of that dotted red line It is That distance between them is greater than D the collision diameter So this is how we define it. It's got to be this R1 plus R2 kind of thing going on So let's just Really quickly review that was a very quick one wasn't it? Collision diameter defined by adding the average radius of molecules together The cross-section is from squaring the collision diameter and multiplying by pi So that's just getting a circle. So it is a circle It is an area And then the number of collisions is going to depend on this cross-section and the speed of the molecule And this is what we're going to get on to next So let's have a look at relative speeds Relative speed is what we're interested in this case We've defined kind of root mean squared speed and some other speeds here. We're just interested in relative So we got C. That's our average root mean squared speed I'm kind of the average speed independent of direction here and We want a relative speed of collision because actually if two molecules are kind of coming very closely to each other You know one's going slowly the other ones just coming to gently that relative speed is kind of small If they're coming head-to-head with each other, it's much much faster So here in this case, we clearly have to see this is coming forward at speed C This is coming forward at speed C. They're gonna hit this if You know They're almost not going to touch each other. At least if they do it be very glancing and we can also work it out By sort of a collision thing here. So this is just straight forward pythagoras theorem. If that's See there That's a square root of the other side sums together so It's all right to kind of leave it there for now We're just interested in kind of a sort of a relative speed and we calculate it using more like this equation So I would eight Boltzmann constant temperature pi mu Okay, so where does that mu come from? You may have come across it before But one is just spend some time on that and this mu is the reduced mass So this pops up all the time in physics especially in spectroscopy and when you're interacting two bodies together reduced mass is really really useful But it means Effectively, you and you need to use one number for the mass of something you have two objects interacting with each other You can have a reduced mass. You can just combine them together And you can also kind of give them a center of mass as well So even though this diagram here is like two things rotating around each other You can treat it as one thing and then you can solve it perfectly. So reduced mass is really useful And it's this equation ma by mb ma plus mb Now if you often forget which way around that goes one is multiplied one as I Added you might get confused and think that maybe it's this at some point Again followed Mb you might think it's that way around but think of the units here reduce mass as a mass So it can't be this because we have a mass squared At the top and this well they add so that must be mass So they are now canceling out mass. It doesn't matter what these units are We just need to know that the dimensions cancelled down So do not get bogged down with it thinking always the equation this way up the other way Work out the units. I'm honestly Like so much of physical chemistry if you just look at the units If you all the consequences of the whole reduce mass concept if the masses are very similar Reduce masses about half of what each of the individual values is you I mean you can work this out imagine 20 times 20 divided by 40 What is that 400 by 40 equal to 10 so and then imagine something like a Thousand times one on the top divide by a thousand plus one You can kind of see that that's like a thousand over thousand It's equal to one roughly equal to So the reduced mass is When there's a big discrepancy in masses is Roughly approximate to the smaller unit That approximation will come really useful when you do a bit more physics because think of the atom for a moment a single electron is about 2000 times smaller than the mass of The nucleus or the nucleus is about 2000 times heavier So you can use the reduced mass of a proton and electron or we can use the roof just the mass of the electron They approximate to very similar things So we've used mass gets used a lot in physics. It's really really useful. So do get used to this equation It's what we obviously want for the relative speeds and it pops up so often That is really worth gauge grips with if you haven't already So let's just review that bit Like in the menu of reason we are interested in the relative speeds of molecules and because they can combine together or pencil Pointing together. It's obviously a tire than the relatively just the normal speed if they're kind of glancing Their collision speed is slightly less. We're interested in more the relative speeds of each of the molecules And we calculate that using a reduced mass of the two things together So we have Object a object be colliding with each other We can from their reduced mass and a few other bits of calculation get their Relative speeds. So that's one of the other measures of speed that we are interested in in this case Now collision rate per second. We'll go back to this diagram We have one two three. Well, that's three collisions in a couple of seconds. So What you probably tell us if it moved faster than the rate per second would increase and if it wasn't So here I'm gonna speed it up. Here's the relative collision speed going up There you go much faster you've hopefully the sound recorded the clicks as well there But if we increase the density go back down to the same original relative speed You can hear it go There we go much higher collision speed so all collision frequency, sorry, so we have got two factors that come into this Relative speeds and the density in addition to the cross section so We've looked a little bit of this so far. We figure out what the cross section is We also had a look at what relative speed was and we can work out density based on Partial pressures and a little bit of the ideal gas equation. So what we get is a much bigger Much bigger equation here, but it is still just the three things multiplied together Don't be bogged down by the fact that this looks really huge This is just working out our effort speed. This is just working out the collision diameter And this is working out partial pressures. So this is starting to become something closer to a Microscopic justification for why things go to a certain rate If there's more of them physically more of them that density increases the rate goes up If they're moving faster that increases the rate goes up and if their cross sectional collision area goes up The number of collisions also increases So we do have a few more things to bolt on to this sterics in particular Now let's just go back over this and review All those three points again. So we did collision cross section. That is if we have a molecule here, then it's actual cross sectional area is a little bit outside it and it's based on the radius of whatever else it's I'll redraw that one and make it actually look like it's the right size for that. There we go That is B its center has to cross this line So the area is a little bit outside the mean diameters of each and the relative speeds So that's C rel is what we're interested in because Head-on collisions are going to be slightly faster than glancing ones We really want to kind of get an average of that speed Not one of the root mean squared speeds of all the molecules involved and our collision rate therefore depends on that cross section The relative speed and obviously the density which comes from partial pressure. So multiplying all those factors together gets us our Collision rate constant our collision rate at least So that's it for this one do a little bit more on this in the next screen cast There's a little bit more detail we need to do including mean free path and a few others But we're nearly at the end of this whole microscopic kinetic theory of gases section