In this lecture, we're going to continue our study of the Kinetic Molecular Theory building on the conclusions from the last lecture. in which we laid out the postulates of the theory and then walked through some of the implications of those. So you'll recall that our postulates based upon observations from experiments are as follows. That in a gas, the gas mart-, particles, whether they are molecules or atoms are in constant and random motion. And the pressure that we observe for that gas is due to the collisions of those particles with the walls of whatever the container is that we're looking at. We have, on the basis of the experiments, concluded that gas particles are very far apart from one another. We'll get a good estimate of just how far apart in this lecture. And, that because they are so far apart from each other, that despite the differences between the types of molecules that might be in a gas. they don't actually exert any forces on each other and again that's because of the great distances between them. From the previous lecture we drew two important conclusions here. Conclusion one, really was essentially sort of a replication, at least of Boyle's Law, if not of the Ideal Gas Law in general. Specifically, that the pressure times the volume is equal to a constant provided that the temperature is a constant, and the number of particles is a constant. And in fact the pressure times the volume, is proportional to the number of the particles, consistent with the Ideal Gas Law. But, unlike the Ideal Gas Law where the temperature would be sitting, we found this term here that looks like the kinetic energy of the individual particles. And from that we were able to conclude that the total kinetic energy of the particles is proportional to the temperature of the particles as well as the number of moles of the particles. Let's think through a bit now what the implications of these conclusions might be. Remember we discussed in the previous lecture, that from this we can now answer the question in the Ideal Gas Law. Why is it that the density of the particles in terms of numbers and moles per volume, is proportional to the pressure? And the answer is now, because increase in the density of the particles, the number of particles per volume, increases that frequency with which those particles hit the walls. Because the pressure is proportional to the force generated by those collisions, then increasing the frequency of those collisions proportionally increases that force, and therefore also the pressure. Why does increasing the temperature increase the pressure? Well, as we've just seen, increasing the temperature increases the speed of the molecules. And increasing the speed increases the pressure of the particles. But what's interesting is, of course, that what we've observed here, is that increasing the speed increases the pressure as v squared, not as v. And in the temperature internal, so, is proportional with v squared, not v. Why v squared? The answer is that by changing the speed of the particles, we change two factors. Also, we increase the frequency with which the particles encounter the walls. so they hit the walls more often when they are moving more rapidly. And in addition, because they're traveling with greater momentum, they impact the walls with greater force as they're accelerated in the collisions with the walls and that also increases the pressure. So when we increase the speed of the particles, that has two factors, which is why the pressures proportional to v squared and why the temperature is proportional to v squared. Let's think about another implication here having to do with when the Ideal Gas Law is not valid. The Ideal Gas Law actually is useful under some circumstances but not under all circumstances. When the Ideal Gas Law is valid then we've shown, actually, that we believe that the particles are not interacting with each other. A consequence of that is, if the Ideal Gas Law is not valid, then apparently it must be true that the particles are interacting. Because when we went through the derivation assuming that the particles were not interacting, we wound up with the Ideal Gas Law. Let's take a look then, and ask the question, when is the Ideal Gas Law not valid? Well, one way to look at that would actually be to plot the pressure as a function of the number of moles per volume. Now of course, the pressure is equal to nRT over V. We can partition out the n over V and write this as RT times n over V. What that means is we expect that the pressure should be proportional to the particle density in moles per volume. And the proportionality constant should simply be RT, as we change the temperature. We would change the slope of this graph, but nevertheless it would be a straight line for each temperature the pressure proportional to n over v. That's of course the Ideal Gas Law, we could then actually take for any given gas, add a specific temperature. When we measure the number of moles per volume and also measure the pressure. We could plot these two against each other and that's what's shown in this particular graph. Shown here, you'll notice, is the Ideal Gas Law graph in blue, as well as data for three real graph real gasses. If we look in this area of the graph down here, we notice, in fact, we get exactly the proportionality that we were expecting back over here as predicted by the Ideal Gas Law. Really can't, looking there, tell the difference between the blue line, representing the Ideal Gas Law and the red or the green or the purple lines representing real gases. By the way, if you're wondering what these densities correspond to, for example, 0.05 moles per liter. You can get that by simply noting that one atmosphere pressure seems to correspond about 2.05 moles per liter. So, there's our measure. As a consequence, we can also look at this graph and say, that there isn't really much deviation, except for ammonia. All the way through to the point where we are dealing with a pressure that maybe is around 0.35. Which would correspond to something like seven atmospheres of pressure, pretty high. As a consequence, the Ideal Gas Law is really quite valid. Works really, really well for very high pressures. Al-, although on this graph, you can start to see the deviation of the pressure of ammonia from the Ideal Gas Law. Whereas the pressure of helium and nitrogen, away from the blue line representing the Ideal Gas Law, is really pretty hard to see. Often, we like to draw this graph in a way that somewhat exaggerates the differences between the real pressure and the ideal gas pressure. And we're going to do that as follows. Remember, PV is supposed to be equal to nRT if the Ideal Gas Law is valid. That means if we write PV over nRT, that should have constant value equal to one, if the Ideal Gas Law is working okay. If it works. So, one thing we could do, since we can see deviations from the Ideal Gas Law in this graph as a function of the number of particles per volume. Let's plot this quantity here versus n over V. And that might actually make it easier for us to see the deriva-, the deviations. Here's the associated graph. Notice on the y axis we've plotted PV over nRT and we expect that that should be equal to one if the Ideal Gas Laws is working perfectly. We're going to plot that as a function of n over V and remember that 0.05 here is something like about an atmosphere. What we see on this graph is that the helium graph deviates very, very slightly from one. The nitrogen graph actually even all the way up to like seven atmospheres of pressure is deviating only slightly. And let me call attention to the fact that the axis over here has been exaggerated the low into the axis is not zero but 0.9. So for nitrogen, we're seeing deviations of maybe 3%, maybe 2%, probably more like 1%, away from the Ideal Gas Law for very high pressures. Ammonia on the other hand, is deviating significantly. Notice that on this particular graph, that ammonia is actually varying down. So that the pressure of ammonia is less than the pressure predicted by the Ideal Gas Law. But let me stress, that doesn't mean the pressure of ammonia is decreasing. Let's go back look at the previous graph, and it's clear on this graph that the pressure of ammonia is rising very rapidly. Just not quite as rapidly as we thought that it would as we increased the den-, the the density of the gas. What we will refer to that, is as a negative deviation from the Ideal Gas Law. Meaning the pressure is less than we thought it was going to be. For some gases that's the case, for many gases it's actually the case. The question is, why would it be less than the pressure? The answer is, we're about to show, is that it must be that the molecules are attracting one another. Why would we say that that's true? Well, if the gas molecules don't interact with each other we know that the Ideal Gas Law is going to be valid. What if they do interact with each other? Let's draw a picture here, of a wall with a gas molecule working it's way towards that wall. It's going to generate a force against that wall which will contribute to the pressure of this gas. But apparently, the force with which is going to hit the wall as well as the frequency with which these particles will hit the wall. Will be lower because the pressure is going to be lower when there's a negative deviation. What would cause it to hit the wall less frequently or with less force? Something must be decelerating this particle as it approaches the wall. Perhaps it's another particle nearby attracting that particle. There may be attractions there. And if this particle is attracted to this particle, it will actually slow down as it attempts to move away from that particle, slowing it on its approach to the wall. That will reduce the force of its impact with the wall. And it will cause particles which are near the wall to impact that wall with less frequency. Consequently, when we see a negative deviation, as we have seen on this particular graph. We can reason from that, that that deviation is actually due to attractions. That this region in here, we're seeing that reduction of the pressure. Or, for that matter, here we're seeing this reduction of the pressure as a consequence of the attractions of the particles towards each other. It's also pal-, possible though for positive deviations to occur. That is, it's possible for PV over nRT to be greater than one rather than less than one. That means that the pressure of the gas is actually higher than it would've been if the gas molecules were not interacting with each other. What would cause that? Look back at this picture again here and say, here comes a gas molecule approaching the wall. Apparently, since the pressure is higher, then this particle must hit the wall with greater force than we were expecting, if this molecule and this molecule didn't act with interact with each other. That suggests, in some regards, this particle's being forced up against the wall by its interaction with another particle. That must mean that these two particles are in fact repelling each other. So if we look at now, positive deviations from the Ideal Gas Law, they are a consequence of intermolecular repulsions. Whereas on the previous slide, it's clear that negative deviations are the consequence of intermolecular attractions. The origins of these attractions or repulsions is something that we will be studying in lectures later on in this semester. But it's important now to realize that when par-, particles interact with each other, they result in deviations from the Ideal Gas Law. And in general attractions will work to lower the pressure, relative to what it would have been as an ideal gas. And repulsions will raise the pressure, relative to what it would have been to as an ideal gas. One of the things that this causes us to discuss is the idea that molecules will actually collide with each other. That is, if there are attractions and repulsions, we might imagine that over the course of the molecule moving its way through the gas. They will in fact run into each other, even under the ideal gas assumption. So assuming the molecules don't interact with each other, how often do they just accidentally run into each other? There are a variety of ways that we can measure that. One of the ways we measure it is by measuring something that we'll call the mean free path. Let me draw a picture of what this means here. The mean free path is, if I have a group of molecules in the gas very far apart from one another. And a gas molecule is attempting to move amongst them. For the most part it can move quite freely and actually pass long distances before it encounters any other molecules. But eventually it may travel in such a way that it encounters, in fact runs into, another gas molecule. Of course, these molecules are all in motion. But as they randomly move about, they will randomly run into each other. The distance, on average, that a molecule might travel before it runs into another particle is called the mean free path. Because it's the average distance that it can travel freely before it runs into something. We've labelled the mean free path here Lambda. It's a common terminology for it. And notice that it depends upon a variety of factors. It depends upon the temperature, which we wouldn't be too surprised by because faster molecules will cover ground more rapidly. And therefore run into each other more often. Let's see, the pressure is related to the density of the gas. Notice in general that these terms here look a little bit like the Ideal Gas Law. P over RT or RT over P should be something like the same as V over n. And if it's the same as V over n, then the lower the density of the gas the longer the mean free path is going to be. Looking at this value d here, actually has to do with the size of the molecules. Remember these molecules are not actually points. They are little, they, they, they're individual particles that have volume. The larger the volume of each individual particle, the easier it is for them to run into each other. Notice that when d is a larger number, the mean free path is smaller because it's more likely that the molecules will run into each other. Let's just do a quick calculation of what this turns out to look like. Sort of a typical particle diameter for a small particle. A small molecule, for example, might be about 300 picometers. If we take that, insert it into the above equation at room temperature and room pressure. We wind up with a mean free path, which is about 100 nanometers. Remember a picometer is ten to the minus 12 meters and nanometer is ten to the minus nine meters. So where the order here of 300 times the size of an individual diameter of a molecule before these molecules can interact with each other. Think that through. Imagine that you are a person who is say, a meter and a half uh,uh, tall. Kind of a common height perhaps. Then 300 times that is 450 meters, about half a kilometer. So in fact, the distances between the molecules are comparable as if each individual were surrounded by empty space, for a diameter of something like a kilometer, or a half a kilometer. Very low density indeed. And very large distances between molecules, and this mean free path helps illustrate that point. How often do the molecules collide? That's another measure of the importance of collisions and this will actually be important to the rates of chemical reactions. The time between collisions could be calculated by the distance that particles travel in before running into each other. the shorter that distance, the more frequently they will collide. And furthermore, by on average about how fast are the molecule's going. And we can measure that by the root mean squared speed, which is a measure of the average speed. From the previous slide we can insert the mean free path, from the previous lecture we can insert the average speed of molecules. And if we just take, sort of an average amount here, using for example oxygen's diameter and its average speed at room temperature. Then the time between collisions, is in fact, two times ten to the minus ten seconds, less than a nanosecond, about a fifth of a nanosecond. That means they're running into each other very frequently. That's a fascinating result, because the molecules are very far apart from another. They travel long distances relative to their own individual sizes, before they run into each other. And yet they run into each other very rapidly. That's a consequence of how fast they are moving. Notice that the average speed of these molecules is actually quite high, 480 meters per second. The molecules are really traveling rapidly. And that's why, in fact, they run into each other quite frequently. This number here is actually a useful number. Because one thing we will discuss in this class, is that the rate of a chemical reaction can be no greater than the rate at which the reacting molecules run into each other. So in some regards, the time between collisions is a nice measure of the time that it will take for a chemical reaction to occur at its fastest. A collision does not guarantee that a reaction's going to occur. But it is a minimum condition for a reaction to occur, so this is a useful measure. We're going to begin then, to now that we have developed this kinetic molecular theory for gases. Proceed on to talk about kinetic molecular theory involving, for example, liquids and we will take that up in the next lecture.