 Maths and derivations, well at some point we've all got to do something we don't like so this is going to be one of those mostly mathematical things Now on the back on the bright side, I don't expect you to memorise all of this This is just going to be deriving what we mean by root mean squared and why is it important? Again, I know I keep banging on about this. If you remember that whole Johnstone's triangle thing. Oh Oh This really does hit all of these sectors so what we're going to start looking at is the microscopic world of molecules and so on and then we're going to try and derive a macroscopic property from it, i.e. pressure, but we're going to go through a little bit of mathematics to justify it So we really are going through this entire cycle Because once we can define say a root mean squared speed or a speed of molecules we can get pressure out and between it there's mathematics, so if you're mathematically literate you could probably skip through and just read these slides and Not listen to me garble about it If you really really don't want to and you just want to pass an exam You can skip this and just learn the equation at the end You will be asked to apply it however, so I don't think this is entirely optional But Let's go ahead and see what we're going to actually cover today One pressure and concentration this will be a minor amount of revision if you've already done a bit of a level kinetics before I'm just trying to Get a relationship between these two down before we move on and then velocity momentum of molecules so this is kind of How does this relate to pressure and then finally the whole concept of this root mean squared speed idea working out a particular Sort of measure of molecules velocity In a gas so quite a bit of maths bit of heavy on the theory side But hopefully on can take it slowly bit by bit and I hope you can follow it So first pressure and concentration previously in the first half of this the first topic We mostly looked at kinetics from the idea of concentration. So Right constant times a concentration of some description. So we wrote concentration down in these square brackets For gases we don't quite define concentration like this We actually define it in terms of pressure and now if you look quickly at these two diagrams here You can see sort of an analogy between concentration and pressure It really is just the number of units per particular volume. So if we have a Set volume of liquid and we have this many Molecules in clearly. This is a high concentration This is clearly a low concentration. Similarly, this is a low pressure and this is a high pressure. There are more molecules in there One thing that you'd be really more interested in with gas phase reactions though is something called partial pressure So before I explain that I'm gonna look at an analogy involving concentration So if you were to do a solution based reaction that involves breaking things into ions, for instance something had to split into a plus and b minus at some point and you want to change the concentration of this You have to keep your ionic strength similar in order to make the reactions Considerably the same so you would add something like a sodium chloride that would Be relatively neutral towards your reaction, but it would keep the ionic concentration exactly the same So the overall concentration of anything with a charge remains the same because you've Made it up. So in this case, for instance, you can see there are very few of these smaller molecules on This lower concentration side. There are a lot more on this side here Yet the total number of molecules is the same so their overall concentration of everything Remains the same but the constant the relative concentration of what you're interested in changes The same is very true with partial pressure. You want to keep your overall pressure the same That's pressure is defined with P and You can count them up there all the same number of molecules in both of these boxes one two three four parts However many I've added in but you can see the difference. So there's only three of these green ones I've added in only a handful of Them here and a lot more green for instance So the overall pressure is the same but the partial pressures of each of these two gases differs it sort of a molar concentration of each So just to review that section quite quickly Our concentration is defined as a number of discrete entities for volume Remember, we are not interested in grams as much in kinetics as all the people might be And pressures also defined is the exact same thing. So there is a an analogy between the two So we can do kinetics pretty much as we've done previously in the first topic with rate constants times partial pressures That's going to come up at some point if they are pretty much directly analogous So let's move to Speed and momentum of molecules So I'm going to cover just a few assumptions we make in this kinetic theory of gases section One is that a gas consists of molecules in ceaseless motion. That means they don't stop They just wander around they don't just suddenly stop out of their own accord and then stop with it again After all, why would they you might think oh, they're so down because of air resistance, but Kind of on a molecular scale the molecules themselves are the air and air resistance is a kinetic kind of collision. So Without a collision, they don't stop. They just keep traveling on without stopping. So you will later cover on What's the average distance they can travel before they collide it can be sometimes a lot longer than you might think And we're also going to assume that molecules have a negligible size their diameter is much less than just the distance between them Now this isn't to say that they are point particles without any Any kind of extent whatsoever. So if you're used to a little bit of mechanics and physics You'll be used to the concept of a point particle that is a point and it's a particle But it doesn't have a dimension. It is literally infinitely small no matter how big you zoom in It doesn't have any dimension No, what we're saying is that its size compared to the distance between them is small. So we're looking Situations like this. Yes, you can see that they do have an actual Size and an extent, but this distance between them is huge But we're not going to focus on cases where So there's size and they're a lot closer together The distance between them there is almost on par with their size So that would be something more like you would see in a liquid This is more you would see in a gas the distances here are huge and then molecules interact by elastic collisions This basically means that the collisions are perfect and they transfer energy around perfectly I'll not spend too much time on that definition of elastic Please just go look it up if you're confused by it. It just means that energy is transferred efficiently and instantaneously so We are stepping a little away from reality. We're treating the moments like hard perfect snooker balls here They don't slow down or interact in any kind of squidgy way But do look that definition up in your own time if you want to know a bit more about it It does have some very interesting specifics So what can we get from the speed momentum of molecules? We're going to define pressure And so we're gonna get pressure from the number of molecules We're going to relate this microscopic property of just molecules moving around to pressure, which is something we can measure in on that So I'm going to start with just almost a thought experiment here So we're going to find a box with three dimensions here I don't know you can call it x y and z dimensions And one of these walls is going to have an arbitrary sized area on it and we're going to label that area a And we're also going to get a distance away from here And we're going to define this as the velocity in the extraction times by a particular time again It will become clear why I'm not giving this a time That the delta t it's just an arbitrary period time. It could be a millisecond. It could be a second But to explain that a little further if our velocity is in meters per second Our time is in seconds those cancel out that is therefore a distance in meters, so what we've got here is another sub box based on That's kind of defined as you know related to the speed of a molecule So any particular molecule heading in the extraction at a particular speed Can hit the wall here Now just a kind of further confusion. I'm using Vx here to mean velocity in this x-direction We could also have v y which means in this direction or v z which means in whichever direction However, you want to define these coordinates. Actually, I have I should probably Set these labels right there we go x y and z directions there arbitrary, but it's sort of breaking out Breaking down a velocity in a particular direction into smaller units. So if our Molecule was heading in this direction up in there then Well, that's vx This is v y and why we're breaking that up will become a bit clearer in a moment We're only interested in its direction. Let's velocity in the extraction What we also interested in is momentum and now strictly speaking that should also be px because it's momentum only in one direction and Momentum comes from mass times velocity. So if you're not familiar with this concept in momentum is The value that is conserved when two things collide So a large object that's moving slowly will hit a small object and the small object will move off quicker Because momentum is conserved Something with a smaller mass must leave with the higher velocity And now if that oxygen molecule hits and bounces off the wall It will change its momentum and it changes it by 2m times the Attitude of the velocity that looks a little bit more scary But let's break that down It's because the meant momentum is going positive in one direction and then it comes back Perfectly opposite in the other direction. Remember, these are perfect elastic collisions. We're receiving no energy is lost It just gets the wall comes back with the exact same speed So it comes from positive Of this value going in that direction and then it comes back with the exact same So it has changed by 2 in total as far as the total magnitude change is concerned And now how many potential collisions it comes back to this whole vx times t thing We have n which is the total number of molecules We have per volume area and then we want to calculate the volume of the number of molecules that would Be in the area that could actually Make this collision here. So what how many molecules can hit that? Well, we know their relative concentration We just take that area and then we times it by this distance So that is the number of molecules in this little sub box here And why not point five? Well 50 percent of the molecules are going to be heading towards the wall They're going to be heading in this kind of direction. Maybe they're not Heading straight to it, but maybe they're going to glance at it Hence why we've broken the sum to vx and the other half we're going to be flying off in this other direction They're not going to be going anywhere near the wall. So we half it Statistically speaking only half are going to be heading in the right direction to make that collision. So we have that So when we multiply them together Can you hear that? That's Glenn Lee's computer does that all the time. I don't know why So we multiply those together and we want to get the change in total momentum that occurs So we take how many molecules we get and we take the individual molecule and we multiply those together So we can calms down a few things if we rearrange these a little bit. We notice the two Let's go back We notice these two and two cancel out for instance and we get the total change in momentum It's equal to the number of molecules as in a concentration of them apply by an area by the mass by a velocity squared So that term is going to become Useful soon and then by this delta t which is arbitrary. So the next section I'm going to say why that delta t is arbitrary There goes good bouncing again So our total pressure is equal to this as we've just worked out Now we want to work it out in terms of force now force is defined as a rate of change of momentum And now you might just think force is equal to mass times acceleration But kind of break it down here. That's force is equal to mass Times your velocity per time again. So you've got a Mass and a velocity here so that the momentum delta t so force is a rate of change of momentum as well as it is Just acceleration So we can cancel out that delta t. We don't need it And it gets us here now we want to get in terms of pressure now pressure is defined as force divided by an area So we can cross out that a as well So we've crossed out the delta t we've crossed out the a and we get pressures equal to m N number of molecules in per area times the mass times this velocity x squared So now we've got actually all of these arbitrary terms like the delta t and the a and so on Start to cancel out. We don't need them hence why I never defined them in the first place so if we Break this down We now have pressure is equal to the number of molecules Times their mass times their velocity squared now just quick note on this notation This is sort of a bra ket notation and this means Sort of average value expectation value you see the long quantum mechanics But you also see a lot in statistics as well. It's just when you see this as far as you're concerned right now That just means average value It appears in other things, but you're just interested in the terms average value So their average value of their speed is equal to pressure because obviously some molecules are going to be quite sluggish Although they're not going to go a lot faster But they're going to average out and when we measure all the molecules on the macroscopic level in the lab We're just going to get an average. We can't Really get a difference between we can't detect the difference between really fast molecules and slow molecules very easily We're just measuring an average So let's just review this for a minute. Why pressure? The pressure is to find us the force that comes from molecules that hit the sides of a container Not necessarily a container by the way. That's just sort of an abstract thought experiment It can still be a force outwards into another gas for instance Into the force that could cause work to happen For instance, if you think back to your thermodynamics and we can calculate it from certain molecular properties, namely their mass and their speed So To run through that derivation very quickly again, we want to calculate a force and pressure So we find the momentum change of a cloudy molecule Remember that momentum change is double that so it's that way than that way Then we want to find the number of molecules that can strike an area So that's their velocity times a just an arbitrary length of time And we multiply them up by the number of molecules the area their mass and so on to get this whole macroscopic force. It's happening Then because force is the rate of change of momentum We can cancel out the delta t because pressure is per area we can cancel out the a so we get this final equation here That relates pressure to the number of molecules and their mass and their velocity squared So that velocity squared becomes a bit important later So Now speed and momentum of molecules That's not the end of the story because as I said, we have vx was our interesting property But these molecules will be flying in any direction and they will be broken down to a particular vyz and vx Again, not too convinced by that. I imagine it was flying off in this direction We could break that down into two components of the vy and the v x If it was entirely flying by coincidence or design in just one direction Then obviously it's overall velocity in any direction is exactly the same as vx because it's constrained entirely into one direction So do you spend some time wrapping your head around this kind of vector thing? It's really useful in physics and physical chemistry to be able to break up Directions into the three component parts of x y and z We would think of a velocity in the x y z direction as vx plus vy plus vz, for instance, it just breaks up or We might even notate it like the vx v p y vz we might notate it is that By the way, you just need to know that it breaks up into three x y z So let's bring back to our equation we had last time Pressure is equal to the number of molecules times mass times that velocity squared or it's proportional to a one-dimensional velocity squared But we are interested in their average speed in all directions. So we're now going to label this c I don't know why we're labeling this to be the lecture notes I've picked up from last year or c you might see it differently in different textbooks All you need to know is that this is speed it is independent of direction Now on that first Maths in where I talk about differentiation. I said velocity has a direction. It could be positive or negative speed doesn't so this speed idea here Is completely independent of direction. We're just interested in speed But it can't be negative. It doesn't have direction associated with it And we're going to square it y squared Well, have a look at here. I've drawn a right angle triangle So hopefully you're familiar with Pythagoras's theorem that whole a squared plus b squared equals c squared thing in three dimensions Well, it just turns out that b squared plus b squared plus c squared equals d squared so You can satisfy yourself of line your own time, I think but For now, you just need to realize this is the base of your Pythagoras's theorem So we take our x y and z velocities square and we get the speed squared So now let's start seeing where the whole squared part of this root mean squared term is coming from The whole squared thing is because well, we have this vx squared and then we can do Pythagoras to get a speed Uh, now these molecules are going to be moving randomly. Remember, they're not going to be Something like a jet that's they're predominantly flying in one direction So all of this random ceaseless motion means they go in any direction whatsoever. So An average speed in the x-direction is going to be equal to the average speed in the y and the z directions Vice versa. So it doesn't really matter which one we collapse in We can replace this with v x squared plus v x squared plus v x squared All three of those combined together So it becomes c squared is equal to three times just a one dimensional velocity Now we can just rearrange this or we bring in some of our previous pressure equations And realize that some threes and some dividing things come in So let's have a look at This we're now dividing at pi three Because we're substituting our vx for c here And what we find what we can now start bringing into the ideal gas equation Now once again, this pen has decided to there we are now it's back Uh, I can draw again So p v equals n r t This is one of those equations that in physical chemistry should be practically tattooed onto your eyeballs You should know this off by art You know you can replace n for the number of molecules and the avogadro's number and you should be able to Know you can replace r with the Boltzmann constant to be related So we can actually replace p v equals n r t with this and rearrange slightly And so we end up substituting Um, we end up getting a formula for our pressure in terms of number of moles avogadro's number Sorry Boltzmann constant temperature and volume Here's our original, uh, so Here when we want to We originally defined previously that n is the number of molecules per unit volume So that means that whole n times avogadro's number Divide by volume equals the big n that we had previously so we can substitute that into there Have a look at this one and we realize we've now can get Pressure in two complete in terms of two completely different equations here one in terms of Number of molecules Boltzmann constant temperature and another in terms of total number of molecules there mass and their speed squared So this is the one that comes from What we've just done involving speeds and the microscopic properties and this comes from mostly the ideal gas law And then we can equate them together Do a little bit of rearranging and then finally routing stuff together. So we've got We bring the three of it to that side Then we bring the mass over to that side. So we get things in terms of the speed squared Speed squared is great, but we probably want to route it. So here is our final equation after all of that 20 minutes worth of derivations and explanation here We have finally this one Formula so this is the formula you need to be aware of this is the one that you will have to apply And this is basically saying that speed of a molecule is related to only two factors one It's mass to the temperature And this is one of the neat things about gases the Boltzmann constant and the gas constant say they're all gases effectively behave The same it doesn't matter what they are and it's one of the great things about kinetic theory of gases Granted those are in ideal conditions such as relatively low pressures where they All those those three assumptions are laid out earlier called true But that's really great means you can do an experiment for one gas substitute for a completely different gas and you get similar results So let's review the mean squared again. So where does that three come from? Well, the last calculation we only did was in terms of v x one dimension. So that whole thought experiment with the box We were only interested in that velocity in one direction the random movement implies that All those velocities broken into three directions should be the same so we can just substitute for Three times it Ends a typo shouldn't be there The root part so the last calculation showed pressure was related to the velocity squared So when we want to get the speed we label it c and then we don't see putting a square root in to get it back And the k, b and t while we want to substitute in from the ideal gas law so that n and r can be Replaced by the Boltzmann constant and the number of molecules as our big n that we had originally And then when we combine the two equations and we can start cancelling out all the things We don't need we don't need to know the number of molecules We don't need to know the size of the volume or the areas or anything All we need to know is the Boltzmann constant temperature mass And there we go. That is our equation That we need to be aware of So finally our last review of the entire section the last 25 minutes pressure is analogous to concentration. So in kinetics, we want to be All these um Velocity and pressure what pressure is a force exerted on a container or on molecules that are surrounding something And so we can find out the force of an individual molecule and then multiplied up to get the macroscopic value and multiple The means to be spared the means for a speed C can be calculated from the number of molecules and just the temperature via this equation So that is the one you need to know And we will try to apply it in the lecture. I will have you ship some values into it To some other problem solving and we'll introduce the other Related ones and later screencasts and lectures as well. So Hope you did manage to follow that Hope it was useful in helping derive where this equation comes from. You don't need to memorize this derivation Just be aware of the thought process behind it Certainly, you need to know that this just doesn't come out of nowhere We do genuinely have a reason for using it and for why it looks like that. So until next time. Goodbye