 going to insist that you do problems on the exams and get the right answer without knowing what the right answer is. And therefore, if you study by always starting the problem and then saying, golly, I wonder what the answer is, flick, flick, flick, you won't be able to do the exam because that's exactly the point at which you have to know what you're doing. So that point there is where you want to focus on and pretend to be an actor, pretend it is an exam and then see how you do. Then afterwards, grade your exam after you're all done. Say, how'd I do? Gee, I missed that and so forth. And you get much better results. All right. Excuse me. This seems to be a little bit slow. All right. So this lecture has a title. That's because we're filming it. And then you know what the title of the lecture is. Oh, first, you don't have to come to lecture except on exam days. Midterms are always on Tuesday. And we will not start the midterm right at the beginning of class. We'll start about 20 minutes in to make sure we can hand out the exams, have everybody seated, et cetera, et cetera, et cetera. So you have full time to do it. All exams are cumulative. Anything we say at the beginning, we can ask later. They aren't just on little snippets of the material. We have a website. Please don't send email. If it's important, ask me in person. I'll be there every Monday. I'll be here every day after class for a little bit. If it's important, just ask. If you get 500 emails, you just can't deal with any of them, especially if there are minor things. Begins today. The assignment's not yet posted, however, but will be soon. And then, as I say, there'll be a few problems due tomorrow. And you can see the syllabus. We're going to have to cover chapter 5 because the instructor's last term didn't cover chapter 5, which is gases. And we want to make sure that everybody has covered chapter 5 since gases turn up quite a bit. And so we're going to have to step on it a little bit. Therefore, try to keep up. We're going to move pretty quick compared to maybe some other years. The book. You can use the book, or you can just do the problems out of the back of the book to discuss them in the discussion section. You can read the book avidly, or you can not bother reading the book at all and read something else. Each book has its strengths and weaknesses, and they aren't all good in every chapter. So use your own judgment. Grading. This percentage, the 20% for the online homework, 20% for the first midterm, 20% for the second midterm, and 40% for the final. No curve. If you get more than 90% of the points, you have an A. And then we work down on 25 point increments from there. Philosophically, I'm against a curve, period. A curve is treating you like a random variable. I give you more credence than that. And a curve makes somebody else's poor performance equal your good performance, which is not the same thing. And a curve, furthermore, encourages people to not collaborate and help each other. And that's a very poor model for how you have to act in a real job to get anything done. So for those reasons, I don't like a curve. The other thing is a curve is like saying, I'm the fastest miler in town. The first thing people say is, what's your time? What's the objective measure of your performance? They don't know how big the town is. I'm the best student in that class. So what? Who knows who was in the class? So we like to have an objective standard. And then we can say clearly you're above that standard. Much easier. OK. Here's a practice problem. Shout it out. Neuron. So somebody has some. How did you do that? You don't know, right? Not exactly. It's kind of hard to explain. You play around. And then you get something that seems to make sense. And then you lock onto it. And then you check against your vocabulary. And you say, aha, I can make neuron. A couple of things can go wrong. You can give up when you're trying to rearrange it. Or you may not know the word neuron. So you have to have a sufficiently good vocabulary in science. Or you can't do any of the problems. And then you have to be willing to play around. You can't just assume that every problem is plug and chug. Many important problems require sort of a creative inspiration. You have to try things. You can't be afraid of the problem. You have to play around with it and think about it. And then you may discover some way to do it. Now surprisingly, chemistry is pretty much just like this word jumble. Because in chemistry, we're rearranging letters and things and making new things. And then we call them new compounds. And we take lines and we suddenly rearrange them. And then we say, hey, that's a new product and so forth. And it could be the same thing in order to see how to rearrange it. You have to play around with it. You can't just try to memorize all the answers to all the possible jumbled words I give you. That's a very poor strategy. OK, gases. Well, gases are extremely simple. They're as disordered as disordered can be. And for that reason, no living thing that I know of anyway is a gas. And the reason why no living thing is a gas is because a gas can't store any information. And living things need to store information. So condensed phases can store information, in particular, the molecule DNA, which can make copies of itself, sort of a chemical curiosity, took over the entire planet. There were atoms elsewhere, and then DNA came along, and then DNA could make copies of itself, and other molecules can't. And for that reason, we got more and more and more and more DNA over time, inevitably, just from that one fact. No big living thing is a solid, because if you're a big living thing and you're a solid, you can't move. And then something comes along and just eats you for lunch, like a pack of crows. So if you were a big living thing as a solid, you wouldn't last long. Viruses, though, are very tiny, crystalline living things, and they can float around in liquid so they can transport. Living things are fluid. We're fluid. We're held together a lot by hydrogen bonds, which aren't so strong, which people who ride around on motorcycles don't seem to understand. You are fragile. And if you have a mishap at high speed, that's it. OK, pressure. Force per unit area. You can get high pressure by having a big force or by having a small area. You see the magician with a big rack of nails. They put it on their chest. They got so many nails, it looks really dangerous. And then some big guy gets on it or they put some heavy on it and everybody holds their breath. But you will never see the magician with one nail right over their breastbone. Because then you put something heavy on it and it just punches right through to the floor. And that's the end of the magician. So it looks impressive to have a lot of nails, but in fact it's making the area big so it's not impressive at all. What's impressive is one scalpel. That's impressive. So PSI, that's how you pump up your bicycle or your car. And usually that's PSIG, which is PSI above atmospheric because atmospheric pressure is 14.7 pounds per square inch. That's just the weight of the column of air above your head. And you want your tire to have positive pressure, so you say above. And G is just what a gauge would read. A gauge is the difference between the atmospheric pressure and the inside. For water, you only need 33 feet and that's why if you go scuba diving, you have to stop and then clear your ears because if you have any air bubbles changing, you'll feel the pressure. And for mercury, because mercury is extremely dense, you use mercury in a barometer because it makes a very compact device. It's only 76 centimeters or 760 millimeters. And a millimeter of mercury in old units is called a torr after torr celling. If you can get a unit in science named after you, you've made it. And second best is to have a hybrid T rose named after you. OK, the SI unit is the Pascal, which is a very wimpy unit. It's hardly any pressure at all. One bar is 100,000 Pascal's and one atmosphere is almost one bar. So for our purposes, the atmosphere and the bar are about the same. They had to redo all the tables, all the thermodynamic tables, to standardize on one bar rather than one atmosphere. And of course, one atmosphere is a nominal value because, as we know, the atmospheric pressure varies. We say we have a high pressure region. The air is pushing down. It's going to keep all the other stuff away and so it's going to sit there while it sits there. If we have a low pressure region, then air is going to rush in. Usually that means there's a storm coming. So if we see the pressure changing, we may want to take some action. And the pressure in the center of the earth is 350 giga Pascal's or about 350,000 bar. That's so high that it's very hard for us to get pressure like that in the lab. And that means it's very hard for us to understand all the reactions that might be going on in the center of the earth because we have trouble simulating them in the laboratory. We can after a certain extent, but there could be big things going on in the center of the earth that we may just not be able to understand completely because we don't know all the chemistry occurring. And the center of the sun is about a billion bar. And that is a very alien environment, 10 million degrees and a billion atmospheres pressure. OK, Sir Robert Boyle, knighted, studied the behavior of gases at constant temperature in a thermostat. A thermostat in those days was some big thing with thermal mass. Usually it was a bunch of ice and water, so it would be at 0 Celsius. And the reason why you like ice and water is because as long as there's some ice there, you know it's at 0 Celsius. And as long as there's some water there, as long as not all the ice is frozen solid, you know it's at 0 Celsius. So it's a nice thing. Nowadays, a thermostat people think of as an electronic device, which uses a feedback loop to control the temperature actively in a room. But this was a passive device in those days. So Boyle studied this. And he found that if you doubled the pressure, which he could do with a device like a piston and just put two weights on it, and he could tell the weights were the same by balancing them, he put on two, he saw that the volume went to half. And if you put on half as much weight, the volume doubled, it came up. And so he concluded that at constant temperature and with a constant amount of material, you can't have a leak. But to be very careful with gases, if you have a leak, your whole experiment is shot. If you have no leak and you keep the temperature constant, then the product of the pressure times the volume is some constant. And furthermore, it doesn't depend too much on what kind of gas it is. Makes it interesting. OK. Which of these plots do you think represents Boyle's law? Shout it out whenever you think. Who thinks it's A? Who thinks it's B? Who thinks it's C? See, a few hands. Is this how you raise your hand? Stick it way up there. Be totally wrong. You'll remember it the rest of your life. And that's the important thing. Not to be right the first time, but to remember what's right. How about D? Some people think D. Well, OK. How are we going to do it? The way I would do it is I'm going to figure out how I can measure the pressure times the volume. And I'm just going to draw these squares here. And I see that when this is 1, this is 2. And when this is 2, this is 3. But this is 1 times 2. This is 2 times 3. That's not constant. 1 is 6. The other is 2. This one is 2, but then it's 1 times 4. That's not constant. This one, however, when it's 1, it's 2. When it's 2, it's 1. That one works. And this one, when it's 1, it's 3. And when it's 3, it's 1. So C and D appear to be the ones that satisfy Boyle's law. Is Boyle's law correct? No. It's only approximately correct. It's going to be correct under the conditions with Boyle's laboratory. It's not going to be correct in the center of the sun or some other condition he didn't study, because he never would have come up with Boyle's law if he had studied all those things. So part of science is to try to simplify things so that you can get something done. If you just say, gee, gases are so complicated, they have all these different values, the problem is it's hard to make any progress. So you make an approximation. You say, under most circumstances, if we're reasonable and it's a gas, it's going to behave pretty much like this. And I have to be prepared for a little bit of give and take. I can't interpret these things literally. For example, at very high pressures, a gas will liquefy. And once you make a liquid, liquids are incompressible. That means you squeeze on them more. They don't change volume, at least not much. So in that case, the pressure's going up, but the volume's staying the same. Well, it's not going to be a constant. One of them's the same number. The other one keeps increasing. You said pressure times volume's a constant. So once it liquefies, the constant won't be a constant. It'll change. So PV will deviate whenever the pressure's sufficiently high. So that constant K, whatever it is, will increase. Why are condensed phases incompressible? They're incompressible because once you have a condensed phase, a solid or liquid. Liquids are more useful, however. The atoms begin to touch, and their electron clouds start to overlap. And then there's a lot of repulsion. And then as you try to squeeze those in, there are huge electrical forces fighting you. And they are very powerful forces, so you can't win. It's very, very hard to squeeze down on something. And that's, of course, the principle of a hydraulic jack. If you take your car into the shop, they put it on a lift, they punch a button, and the car magically goes up like a magic carpet. How do they do that? Well, easy. They just basically equalize the pressure. So I have a long fin thing here, which in this diagram I've just simplified. And I have a weight of 100 pounds on a square inch. So I have 100 psi with this weight. And then I have this thin tube here. And then over here I have a big fat guy, because as you know, when you see the car go up, there's this big steel thing that comes up. It's not tiny. It's a big, wide thing. And so if I'm willing to go down here 100 inches, and this area is 100 square inches, then I can raise 10,000 pounds, 1 inch, because the 10,000 pounds distributed over 100 square inches is 100 psi. And if I drop this down 100 times as far, I can bring this up 1 inch. And so what I do is I get a pump over here, and I just pump on it, boom, boom, boom, boom, boom, injecting more and more liquid, not too high pressure, and then slowly the car lifts off. And I have to have all these pipes and things able to hold 100 psi, but nothing more than that. And it's very nicely controlled. It's a way you can use fluid to do work. It's a great invention used everywhere. Elevators, you name it. OK. While the Englishman Boyle was busy studying pressure and volume, Charles in France was busy studying volume and temperature. And what he was doing was making sure that the pressure was constant. So he could have a cylinder with a constant amount of pressure, and it could change volume. And he changed the temperature and found that the volume was proportional to the temperature. And so if I plot volume versus temperature, I get a straight line. Of course, I can't make the temperature too low, because then I get a liquid. And in 1848, Lord Kelvin, after whom we named the absolute temperature scale, he realized that the lowest temperature attainable would be when the gas appeared to have zero volume. Because how could the gas have negative volume? What would that mean? So you take a bunch of gases, and at temperatures you can access, you measure their change in volume, and then you plot a straight line. And then you plot that straight line, and you extrapolate it to see when the volume would be zero. And what you find, if you do that, is that they all converge on one point. You have to be careful how you do the experiments. If you're sloppy, they'll be all over the place. But if you're a good experimentalist, measure the temperature accurately, you ensure that the temperature everywhere in the gas is the same, because it can be different momentarily in different places, especially if you're changing it in the experiment. You have to be patient and wait for it to come to the full temperature. Then you plot these things, and there you go. There appears to be some point there at which the gas, and that would be a special point. We know that no real gas will have zero volume, but still that seems to be a special point. That's as cold as we'd predict a gas could possibly get. And it turns out that's as cold as anything could get. So we call that absolute zero on the Kelvin scale. So this is Charles' law. And if we know the volume at any temperature, we can calculate k, this v divided by t. And then we can calculate any other vt pair. If we know the temperature, we can predict the volume. Same amount of material and same pressure. Then we move further south, Avogadro. So they were all working with gases, because that's what they could work with. That's what they could afford to have in their labs, and that's what they could understand. You don't work with things that you can't have the technology to deal with in your lab. That's still true today. What Avogadro found was that if you double the mass of the gas, then the volume doubles if you keep the pressure and temperature the same. That sounds pretty obvious. If I've got 1 kilogram of gas and it has a certain volume, it makes perfect sense that 2 kilograms of gas would have twice the volume. It almost seems that how could it be anything other than that. But sometimes, if you have interactions, it doesn't work. For example, if you double the number of employees at a company, you don't necessarily double the output. Because if the employees argue with each other or get confused or spend time talking instead of working, you get less done. On the other hand, if they work together, and one person's extremely good at this and the other person's extremely good at that, and they both do the jobs they're very good at, then your productivity can be far higher than double. It can be much, much, much higher. The mass of the gas depends on the number of moles of gas. And recall, a mole is just a number. That's all a mole is. It's a number. It's the number of carbon atoms that weigh 12 grams. So if we write this down, we can say the volume's proportional to the number of moles. And here, we understand we haven't changed the temperature and pressure. We've just added more moles of gas. If we put all these relationships together, the English, French, Italian, now we can come up with a master equation called an equation of state that will work for gases only and only under the conditions that these scientists studied the gases and found that their equations worked, not under some other conditions, and certainly not for a liquid or solid. When we start with Boyle's law, we write p times v is a constant. We don't know what the constant is, but it's a constant. That's for a fixed amount of material and a fixed temperature. So therefore, we can put some function of number of moles and a function of temperature over there, because they're fixed. And then the constant, I call it k prime now. So this function of n times the function of t times k prime is k. Long as I don't change n or t, I don't know what the function is. But the other guys did change n and t, so we can get n, the function of n, f1, and the function of t, f2, by using Charles's law and Avogadro's law. We can get those. So if we write it this way, and we stare at it a while, we say, hey, Avogadro told us that if we fix p and t, then these things don't change. And this is supposed to be a constant. Avogadro told us that volume's proportional to number of moles. But that means the function n is just n. So we put in n. And if we fix the number of moles and the pressure, then Charles told us that the volume's proportional to temperature. So this is fixed, this is fixed. That's a constant. The volume's proportional to temperature, it must be that the function of temperature is just temperature, t. So we put those things in. And historically, the missing constant is not called k prime. It's called r, the gas constant. And so we write v is equal to nrt upon p. Or make it scan better, pv equals nrt. And this is called an equation of state, an ideal gas law. So when we speak of an ideal gas, we're speaking of something that will follow this equation. Now pressure is forced per area, but that's also energy per volume. Because if I have newtons per meter squared, I can turn that into newtons times meters per meter cubed. So pressure is an energy density. Just like I can have a mass density, so many grams per cubic centimeter, gold is more dense than water. I can have an energy density. High pressure things are energy dense. That's what a bomb is. It's a high pressure thing. That's trying to get out. And it causes a lot of damage when it does so. And if you have a chemical plant that has high pressure parts, you have to be extremely careful that none of them suddenly give way unexpectedly. Or you have a real problem. So p times v then, energy density, energy per unit volume times volume, just has units of energy. So p times v is a measure of energy. If somebody says, I want to store wind power by compressing air into a giant cave underground that we dug out. And so when the wind's blowing, I'm going to run this compressor that runs off the electricity the wind gets. I'm going to pump air in there. Hope it doesn't leak out. And then later, I'm going to let the air come out and spin some generator at night when the sun isn't shining or when we're in a lull. You can figure out how much energy they can store by the pressure they can put in that cavern times the volume of the cavern. And if that turns out to be bigger than the volume of the earth or something like that, then that scheme is not going to work. Or if the pressure is unduly high, pumps work OK until you get up to extremely high pressure. And then the pump can't pump in any more gas. Just like when you pump your bike tire, you pump, pump, pump, and then it gets pretty hard. Then you quit. Well, the other side has units of energy too then because they're equal and things that are equal have the same units. Exactly the same, not different units, both energy. Exactly the same units. Otherwise, they aren't equal. So R, whatever it is, has units of energy per mole per kelvin because R times moles times kelvins is equal to energy. The SI unit is 8.3144, et cetera, joules per mole per kelvin. But more common for doing gas problems involving pressure and volume is to use R in liter atmospheres, in which case it's 0.028057, et cetera, liter atmosphere per mole per degree kelvin. You have to be very careful when you do problems that you get the units right. You will not get the units right if you do not write them down. Let me repeat that. You will not get the units right if you do not write them down. And you won't catch a mistake you've made either. So always, always, always write the units. It seems like a real pain until you get zinged on an exam with a bagel, which is what I'm going to give you, when the units are wrong. Or they aren't listed. The answer is 3. 3 what? If I have to say what, then my answer is here's your score, 0. I can't be asking what the units are. Small mistakes cost big money. And they cost these people their jobs. 1999, NASA lost $125 million Mars Orbiter because one engineering team used metric units, while another used English units. And guess what? Neither team wrote the units. Which way are we going? Three to the left. Got it. What a crash. Aren't you guys working in metric? No, we've never worked in metric. We've always worked in metric, says the other team. Write the units down. The head of JPL doesn't care that this is a small mistake. What they care is it's $125 million loss that makes them look terrible. Who didn't write down the units? Was it you? Yes, sir. You're gone. That's how that works. I hope you're going to be doing something important, like the Mars Orbiter. But if you are, you have to be careful that you don't make dumb mistakes. Everybody will make them anyway, but try to minimize them. If you write the units, the other guy says, oh, this guy's working in miles or something. Thank goodness they told me that. They're in the stone ages, but I'll convert it to meters. And then we won't crash the ship. Now, just like Boyle's law, the ideal gas law is only approximately true. Nobody would believe it literally for all cases. However, as long as the conditions aren't too extreme, it works pretty well. So it's a good first rough guess. But at high pressure, PV is too large. The molecules or atoms begin to repel. And at low temperature, PV might be too small because the molecules or atoms might stick together. At low temperature, we get dew on leaves. We get water vapor condensing out of the air onto the leaves of plants in the morning as the temperature drops below the dew point. In fact, at high and low temperatures, if they're really high and low, the predictions are just always way off. It's always pretty bad. At low temperatures, it may condense. And at high temperatures, the gas may react, decompose, or ionize. In the center of the sun, there are no atoms. Too bad. All the electrons boiled off. There's just the nuclei and then electrons everywhere, like one big mess. That's what a plasma more or less is. So we wouldn't expect that to work in the center of the sun or anything like that. But much like we cannot actually physically draw a perfect circle, we can easily imagine one. And that's the beauty of imagination, is that you can imagine things that are impossible to realize in real life. But that's also the danger. Everybody who entered the lottery imagined that they were going to win. They didn't imagine that they were going to get hit by lightning 50 times this year, even if that's the same likelihood. Because people have a tendency to imagine good things being more likely than bad things. That's human nature. If it's something good, it could happen. If it's something bad, no way. Will I crash my car? No. Boom. You know how that works. So we can imagine gases that would exactly follow that. And we'd just say, well, there's such a thing as an ideal gas. There's no real gases is really going to behave that way, but we can solve problems, and we can use an ideal gas, this imaginary gas, and we can ask what it would do in real things, like if we had it running an air conditioner or something, we can ask questions if we had a heat exchanger, for example. And the properties of an ideal gas tell us the following. Because when you contract the temperature to zero, the volume is zero, that means the particles have zero volume. That's hard to imagine how something has mass and zero volume. But anyway, that's what we're saying. And likewise, because the gas always fills the whole container no matter how low the temperature is, it never turns into liquid on the bottom. That means that the molecules have no attractive forces. They don't tend to stick together. All real molecules are a little sticky. They stick together. If they're slow enough, they stick together. And they crash to the bottom. When you get a whole bunch of them, the whole drop crashes to the bottom as a liquid, condenses. So there are no size, there's no repulsive forces, and there's no attractive forces. That's basically what the ideal gas equation says. There's a better equation, the Van der Waals equation, much better at modeling gas. And then there are even better equations that chemical engineers use. And then if you really have to know what the volume of a gas is supposed to be for something critical, like a power plant, there are steam tables. You look up what the answer is, because somebody, an engineer, measured it. And they know what pressure it is at that temperature, and what volume it will occupy. So we can take into account that the gas cannot shrink to zero by subtracting something off from the volume of the liquid or solid, let's guess. And the thing we subtract off, we multiply by the number of moles, because the volume is going to depend on the number of moles. We know that from Avogadro's law. And we want the constant in the table that we make up to not depend on the amount of material. We don't want to have to say how much gas we have. We just want to say, look, for this helium, this value is this. Now it makes sense that small atoms and molecules should have small values of B. So now this is getting interesting, because if we can determine B, which is just measuring gases that anybody can do, we can say something about how big the particles are, what their volume is, on an atomic level. So here, this equation, we have to have a different value of B for every kind of gas. That's no big deal. So I've compiled some values. Now, word to the wise, I will never, ever, ever expect you to know specific numerical values of data. If you work with things a lot, you'll know the molar mass of silver, or the oxidation states that chlorine can have, et cetera. I was forced to do that as a kid. I had to memorize where every element on the periodic table was. Honestly, that's a complete waste of time. If you're going to be a chemist, you'll know where they are, the ones you use, and the ones you never use. You won't care where they are. That's how that works. If you aren't going to be a chemist, you're cluttering up your head with all this nonsense. You may as well just memorize everybody's phone number in this class, if you aren't going to call them. It doesn't make any sense. So I will list data, but don't spend a lot of time writing down the data, because if I want you to do it on an exam, I'll give you the data. I'm not going to ask for specific values. And likewise, I'll give you the values of the constants, R, and how to convert Kelvin, et cetera. Not important. So helium, we get 2.37 times 10 to the minus 2. That's something like the molar volume of helium atoms, although we have to be careful. Argonne's bigger. Xenon's quite a bit bigger. Oxygen, hey, that's interesting. Oxygen has a smaller value of B than xenon. Xenon's an atom, and oxygen is O2, so that B xenon is kind of puffy, maybe like a big cotton ball. Chlorine then is bigger than xenon. And carbon tetrachloride, very far from an ideal gas, this used to be used as a dry cleaning solvent and is no good for your liver. They switched away from it to something else, which, unfortunately, is also toxic TCE. And then they switched to something else, and then supercritical CO2, who knows. If you dry clean blue jeans, they never fade, which is interesting. They just look brand spanking new all the time. I know that because there was one kid in Utah who always had new blue jeans. We couldn't figure it out. It's having them dry cleaned every time. They never changed color. Kind of interesting. Now look at neon. Neon is smaller than helium. This doesn't make too much sense. It might be true, but I don't think it's true. I think helium's smaller than neon. It's directly above on the periodic table. So what we have to understand is that the way you get the value of B and A is you get a bunch of data, you get the equation, and you pick the values of A and B that give the best match to the data over the range you've got. So you don't want the data to be over too big a range, or A and B are bad. You don't want it to be too small, because then you don't push the parameters far enough. So there's an art to it. So B values are just a best fit. I mentioned A, but I probably got ahead of myself. So gases also can attract. How can they attract? All molecules attract. Well, let's take two atoms. Here they are. They're neutral. They have no electrical forces, but they have some number N of electrons, and then a plus N positive nucleus that's much heavier. The electrons are like little gnats, and there are N of them, and they don't have any brains. They just all do whatever they want to. And so you can ask yourself, what's the probability that the electrons are always, always 50-50 at all moments? If I have 10 electrons, there's always five on this side and five on that side. And then if I cut it this way, there's five here and five here. If I cut it this way, there's five there and five there. And you can start to see that it's probably impossible at all times if the electrons are doing their own thing for that to be true. And you would be right. It is impossible. And so here's what happens instead. We get a charge fluctuation, and we get a dipole. And what happened here was that we had a little bit the electrons scooted over to the outside on the atom, just for a picosecond, just a small amount of time. And so the total atom is neutral, but this side is a little negatively charged, and this side is a little positively charged. It's not going to stay that way, but it has to happen. And it keeps happening in all directions. Positive, negative, positive, negative, on and on. Now supposing I'm nearby, and I'm an electron in this guy, and I'm really light, and I'm negatively charged, and there's a positive charge near me. Aha, I'll go that way. So then they get sucked over there. But in that case, we have to get an attractive force. Because this negative charge and this positive charge pull each other in, it's going to cause the two to start moving toward each other. And then the spell may be broken, fluctuate again. But still, on average, any atom nearby the other one will tend to move toward it. And then of course, if it gets bashed at top speed by some other freight train, it just says, forget it, it's away. Still a gas. But as we cool it down, and they spend more time moving slowly, and they spend more time close to each other, it's like a night out. First you have dinner, then condense into a liquid phase on the bottom of the container. Same idea. So we take into account the attraction by reducing the pressure. And the way we do that is we add an attractive term. That makes the pressure plus the attractive term equal to the old pressure. And that means that the new pressure is lower. And we expect the pressure to be lower for the following reason. If I have a gas, the pressure is basically the gas pushing out against the sides of the container. If the gas is attracted to itself, so it's making a smaller, then the pressure is less. It's not pushing out as much. If the atoms were repelling each other somehow, which they cannot do, then it would push out, unless they're charged. If they're all charged, they can't. So we have to add an attractive term, and we can see what it should be, something, and then something that depends on how close the particles are, which is the number of moles per volume for the first guy and the number of moles per volume for the second guy, because we need two guys to get the attraction. Two, not one. For the b, we needed one guy, volume of one guy. For the attraction, we need two guys, so it's squared. Density squared. And then that gives us our equation, which again, if I want you to deal with this equation on an exam, I will give it to you, but it's pretty easy to remember. I remember a for attraction, and then b is for repulsion. But I don't remember b. I remember a is for attraction, end of story. Always try to remember the minimum. Don't put another thing in there, because if you put another thing in there, you may get mixed up. The values of a for the attraction are tabulated, and they make sense when we understand it to collect their forces. So let's look at the values of a for various materials. I would expect the following. I would expect a small atom that has tight control of its electron. It's like a little golf ball. I would expect that it would have a small value of a, because there wouldn't be any electrons moving around, flopping all over the place. I would expect a big atom, like xenon, with a lot of electrons. I would say that's very unlikely that xenon can be 50-50 all the time. It's going to have a big value of a. And so if I were given materials on an exam and the values of a, and I were asked to write something intelligent, I could. Because I could say, aha, look at this high value of a. That means this atom is very polarizable. Look at this high value of b. This atom occupies a slightly bigger volume than that one, et cetera. So I could make statements about atomic properties just from these things. So again, these are just for interest. We don't care what the actual numbers are. Helium, 3.47 times 10 minus 2 in liter squared per bar. Argon, wow. Much more attractive. Well, liquid helium is 4.2 Kelvin. It's extremely cold before it liquefies. Argon is more like the temperature of liquid nitrogen. And that's reflected in this much larger attractive force. It's much easier to liquefy argon than it is helium. Xenon, much bigger, 4.28. Let's do the other guys and see how they, 1.38. So oxygen, again, is smaller than xenon in terms of attraction and smaller in terms of apparent volume. And then chlorine is much higher. And then carbon tet is, wow, that's almost 20. So that's very, very high. And of course, carbon tet's a liquid at room temperature. The others are not. And now for neon, we see that neon, which has more electrons than helium, 2 versus 10. So neon has a much higher attractive, about 10 times higher anyway than helium. And so our conclusion is that the neon nucleus controls those 10 electrons much less tightly than the helium nucleus controls its 2 electrons. And so it has a much larger value of A, the Van der Waals constant for attraction. OK. Now solving for P or T, if you're given the other variables, is easy. No problem. But solving for N or V, if I said a gas occupies 1 liter at 100 Celsius and 50 bar, how many moles of gas are in there if it's a Van der Waals gas? If you haven't tried that, that'll be like neuron. You can stare at it a long time, and nothing will click. So if you haven't played around with it at all, played around with the equation, made it your friend, it'll be your enemy. It's just a disaster when you try to deal with it. Likewise, V is tricky. Now, a mathematician would tell you to expand it out and then write down the roots of a cubic equation, because mathematicians like to do that. Believe it or not, they love to do that. The more difficult, the better. Makes them feel good. Coffee tastes great, so on. Look at this equation. It's unbelievable. But we solved it. Quite a thrill sometimes. But we usually don't want to solve it for every possible volume. We have an exam to do. We're given the temperature and pressure, and we're at, and one mole, let's say, and we want to know the volume. And we want to do it efficiently. So we want one number, not the totality of all solutions in the entire thing. So in chemistry, usually we only want one numerical value. We don't want a full analytical solution. And so it's always crazy to take a problem and make it harder, and then solve it, and then get the answer. But if you write down the roots for a giant equation, which is for all possible values of P and T, it's much more complicated than just sticking in the numbers and seeing what you get. So never make a problem in chemistry or any other science more complex than the starting problem, and then try to solve it. Unless it's already known how to solve it. Some complex problems are known, in that case, fine. Usually not, though. Usually you've made it worse. Try to simplify it. Though the roots of a cubic are tedious and long, you can write down there's an exact solution. There's an exact solution for a quartic. By the time you get to five, something weird happens. There's no solution. You can't write it down. At least not in all cases. OK, here's practice problem three. So we've got our dry cleaning solvent. What volume would you predict at 100 degrees Celsius and one atmosphere for one mole of carbon tetrachloride if it is treated A as an ideal gas? Read follows exactly the equation PV equals nRT or B as a van der Waals gas. Read follows exactly the equation P plus An squared over V squared, et cetera, equals nRT. This might be something that you might want to figure out. Well, for the ideal gas, we just use V equals nRT over P. And I see that it's at 100 Celsius. And I say right away, what units? Always Kelvin, Lord Kelvin. Always quote temperature in Kelvin. It's a beautiful day. It's 294 Kelvin. Yeah, you're eccentric, but you're never wrong. But if you do chemistry problems in units other than Kelvin, you're wrong. And you get killed if you just substitute Fahrenheit or some other hokey units into these equations without thinking about it. And it's not enough to write degrees or something because degrees what? Now, in actual fact, when you solve a problem, the guy says, what is the something in Fahrenheit? And they don't know any science. And you convert to Kelvin, you solve it. And then you don't speak to them in jargon. You don't say what it is in Kelvin. You convert it back to Fahrenheit or bushels of corn or dollars and cents and tell them the answer they want to know. That's what you do. You don't leave it in your intermediary language of science and try to simplify it. Yes? This is supposed to be, it should have been a capital L, but that's supposed to be liters. But yeah, you're right. I'll make it a capital L so it's clearer. OK. And so what I get is 30.6. And then I put, you see how I put a parenthesis around the 2? That's like tying a special knot on a tie. You don't have to do it, but some people will notice. That means that I'm putting that there, but that digit is uncertain. So if I'm going to calculate further, I'm going to use 30.62. I'm not going to round to 30.6 and then go further. However, I don't really believe the 2 is significant because I've measured the other things well enough to know if that's 2 or 1 or 5. That's my uncertainty. This is always the safest way to quote things. If you look up the constants in NIST in the table, every constant, speed of light 2.99735. OK. So for the Van der Waals gas case, I looked up, and now I do have a capital L, 19.96 liters squared atmosphere per mole squared for A, the attraction, and 0.138 liters per mole for the volume. And N is one mole exactly the way the problem is worded, I guess. So we have this equation. Now, the first thing you do is you just put in all the numbers you know right away. There's no sense in even wasting any time simplifying this equation symbolically by moving letters around and playing around. All that's going to happen there is that if you make an algebraic error, you'll start out with the wrong answer from the get-go. Put in all the numbers you know and then continue. I'm going to put in all the numbers. I know what RT is. That was the 30.62 from before. And so I write the equation as a numerical equation with one unknown v. And my claim is that we solve this by guessing. Guessing. Simple. Or if your calculator will do it, you can turn it over to your calculator. I'll accept that on an exam in all detail in a minute. But if your calculator does it and your calculator does it wrong, then you're on that $125 million space orbiter and you're going down with the exam. So you better make sure that it does it right. So here's what I'm going to do. I'm going to set up, I'm going to call this numerical thing with v in, some function of v. And the right hand side was RT was 30.62. And I'm going to plug in values of v. And I set up a table. You can do this on a spreadsheet easily. And it's duck soup. And it really organizes your thoughts to set up a table. Don't just scribble things at random angles on the paper. Well, I'm going to first guess it's an ideal gas. An ideal gas has a volume of 30.62 liters. I shove in 30.62 squared and 30.62 here. And I turn the handle and out comes 31.13. Now I want 30.62. And then I have a comment. I always have a comment. Because if I come back six months later and look at this work, the comment is going to be important to me sometimes. It'll jog my memory. It'll put everything in context. I'll remember what the whole problem was about and what I was doing. If I just have numbers and things, sometimes I can't remember what I'm doing, then I have to start over. Well, OK, it's too big. So I guess I should lower the volume. How much? Well, it was about half a liter too big. Why don't I lower it to some round number? So I went from 30.62 to 30.0. And now it's still about half a liter too big. Excuse me. It's half a liter bigger than what I put in. But it's 0.1 liters less than what I want. So now I say, well, gee, I guess I'll go up by 0.1 liters. And then bingo. I get the right answer. It's amazing, but if you get good at this, it rarely takes more than three or four guesses, OK? Rarely, because you'll see what values you're getting and you'll home in. Also, if you punch a bad button, so I get 30.62, 30.52, 50, I don't accept the 50. I go back and do the calculation again. I probably hit a wrong button. But if I'm doing some method where I only do it once and I hit a wrong button, I just get this bogus answer and then I just write it down, OK? We're sort of lucky to arrive at the answer quite that quickly. But even 10 guesses is much easier than using the roots of a cubic. And here's my comment then. And then we'll stop. If your calculator will do it, I'm fine with your calculator if it'll graph the function and you can find where it's equal. No problem. Do it that way. You won't need to have a high-powered calculator to do well in this course, but if you have one and you know how to use it, use it. I'm not going to shackle you with your hands behind your back. But you don't have to. However, check your answer, please. Put it back in, see if it solves it on an exam especially to make sure it's right. See you on Thursday.