 So today, we're going to consider a useful relationship that governs the relationship between different state variables in the atmosphere. This relationship is called the ideal gas law. And in this case, the word ideal doesn't really modify the word law. It's not the best law. It's not the ideal law. It basically modifies the word gas. It means that we make some idealizations. We make some simplifying assumptions about the gas that's inside there. And we make laws that apply to those ideal gases. Now as you further your study, you'll find out that this law only holds for very certain conditions. But we're going to stay with the basic assumptions right now as we discuss the law. So back in the 17th century, I believe the 17th century, the late 1600s, there were a number of philosophers and scientists who were studying gases and their relationships. And we have discussed some of these laws in other contexts, if you've seen some other videos or talked about it and been in this course. However, let's talk about how those laws sort of combine. The first of those laws is something called Boyle's Law, named after Sir Robert Boyle. All right, and Boyle's Law looks something like this. Boyle's Law says, if I take a system and I keep the temperature constant, so I don't really allow it to heat up or cool down, I can find a relationship between the pressure, P, and the volume, V. So I've kept the temperature constant. And the other thing about this system is I assume it's a closed system. Closed meaning I'm not adding or subtracting mass gas into the system that I'm dealing with. So Boyle was able to take something where he would take some amount of air, he would put it inside a cylinder of some sort, and then he would compress it. And then he would measure how much he had to push to compress it and how much it actually compressed. So that was a relationship between the temperature, assuming the temperature was constant, that my pressure and my volume, he determined that when we worked them together, the pressure times the volume would stay constant. And I'm going to label that constant with a k and with a little subscript 1 because we're going to be talking about different constants. And he determined that when one of them went up, when you put in more pressure, you would squish the thing more. The volume would go down. And vice versa. So there was a relationship between pressure and volume, and it's what we call an inverse relationship. When one goes up, the other goes down. All right, so Boyle had his law. At the same time, or around the same time in a 100 plus year period, there were a few other people studying these things. Another person created a law. Jacques Charles created Charles law. And Charles law says, if I hold the pressure constant and again, oops, constant, and again, I keep a closed system, there is a relationship between the other two parts that the volume and the temperature have a relationship. But in this case, we take the ratio of those two things and we realize that if you divide one by the other, you get another relationship where those two things are constant. The ratio of those two things stays constant. This is a proportional relationship. We could rewrite this v equals kT if we wanted. And what that says is that as the temperature goes up, if you heat something up, then the volume goes up. It expands as long as you keep the pressure the same. You keep it in a context where the pressure can stay the same. So based on these two laws, you can probably already guess a relationship or at least make some estimate about which things we're going to talk about next. In one law, we took two things, held a third constant, and there was a relationship. So pressure and volume in this law, volume and temperature in this law, so which two are related? Well, now we need to relate pressure and temperature. That third law, named after somebody who bears the name Guilussac. I hope I'm saying my French correctly. All right, the Guilussac law makes the same sort of assumption. This time, the assumption is that the volume is constant, again, in a closed system. And as long as the volume is constant in this closed system, the relationship is that the pressure and the temperature, the ratio between them, is, again, some constant value. Again, a different constant than these other things. All right, but that's another proportional relationship, whereas the temperature goes up if you keep it in the same thing, the pressure goes up. Anybody who has ever seen somebody cook with a pressure cooker, all right, that's the idea. It's the same volume, but as you heat it up, the pressure builds inside whatever system is closed, your closed system. Now, it seems that all these three laws trying to memorize the names that go along with them and the different relationships, they all seem to kind of say the same thing in slightly different ways, maybe not the same thing, but those relationships seem that we might be able to simplify this into one larger relationship. Almost, we're gonna add one more piece here, okay? There's another law, Avogadro's law. Now, you might recognize Avogadro's name, Avogadro's number, is the number of molecules that's in what we consider to be a mole, all right? So, Avogadro's law basically says that if you have some amount of gas, let's see here, if you have some amount of gas, then the volume divided by the amount of gas is a constant, okay? Where n is an amount of gas. Now, what does that do? Well, that basically means that the system is no longer closed. We have a system where we can add or subtract the gas, and when we do, you've all experienced this. If you take helium and put it in a balloon, you add more helium, the balloon gets bigger, okay? And that relationship assumes some relationships of the other two things, temperature and pressure, constant. So now we have all these relationships that come together, and if we look carefully and we realize that these different constants, if we take the temperature and realize that it's constant, maybe we can put it in where the constant is over here, or the pressure's constant, maybe we can put it in over here, or the volume's constant, maybe we can put it in over there. We can combine all these, recognizing their constant nature and put them in with the constants that exist over there. And in that case, we create this thing called the ideal gas law. And here's what the ideal gas law looks like. Ideal gas law. The very typical way of writing the ideal gas law looks something like this. P v equals n r t. Now hopefully you recognize the different parts, p being pressure, v being volume, n being r, n is the amount or the number of molecules. T is our temperature, and then the question is, what's this r? Well, the r is just like all these other parts, it is a constant value. It's a constant value that depends on whatever the gas is that you are applying the ideal gas law to. Whatever the ideal gas is, that ideal gas, if we can come up with a constant that represents that, then that's what that constant value is. And there is our standard ideal gas law. Okay, let's take a look at that for a second. Just compare that to the other laws here. Look at the first one, look at Boyle's law. P v equals a constant. Well, that's if we have a closed system, n is constant, and the temperature is constant. Well, a constant times a constant times a constant is gonna be something constant, and so you just have P v equals a constant. So hey, Boyle's law is in the ideal gas law. Similarly, you can make the same arguments. If I have my v over t, if I take this t and move it over there, take the p and move it over there, I can recognize that everything on the right side becomes constant, and v over t is a constant. So all of these other laws are encased in the ideal gas law. All we have to do to reduce the gas law to these laws is to fulfill the conditions that each of them came with in the first place. And that's what makes this law so lovely and powerful. However, there are a few rules that we need to observe if we are going to use the ideal gas law. And I'll take a minute to erase this and we'll talk about those. All right, so there are a set of rules that you need to remember if you're going to use the ideal gas law and apply the ideal gas law. The first of those rules looks something like this. We recognize that something like the volume, if I take the volume, if I think about what a volume is and I go from a really big volume to a really small volume and I keep going smaller and smaller and smaller and smaller and smaller, I get to a point where the volume is equal to zero. Volume is what we call an absolute scale. Zero has a meaning, there isn't even really a negative meaning to negative volume. I mean, there might be some mathematical arguments about what that means. But in our physical experience, volume going to zero means that everything here, when everything goes to zero, like when something goes to zero, everything should become zero, okay? Well, that's not entirely true of how we typically learned how to use temperature, for example. Zero degrees can be very cold, but zero degrees is cold or different levels of cold depending on which system or scale you're using. Are you using Celsius or are you using Fahrenheit? Also zero degrees has a very different meaning because you can get colder than zero degrees. And so what would happen in a scale like this if you started having negative values or if you were at zero, what would actually happen? We know that things don't shrink and disappear when we get to zero degrees, if it's the zero degrees we're commonly familiar with in Fahrenheit or Celsius. So one of the rules is we have to use a different scale, a different system. So our first rule for using the ideal gas law is that temperature must be absolute. And what that means is that it must be on a scale that goes to zero when it's very, very cold to a theoretical value known as absolute zero, where it's so cold that effectively the volume goes down to zero. Now, again, we're gonna be dealing with this usually in a range that we're not anywhere close to that value. Otherwise, the ideal gas law rules, other rules don't really hold. But the main idea is if you're going to apply the ideal gas law, the temperature must be in an absolute scale. In other words, you're not allowed to use Fahrenheit or Celsius. Instead, what you're asked to use is degrees Kelvin, which is a scale that has an absolute zero value, or degrees Rankine, R is Rankine. In fact, I'll go ahead and write Rankine and Kelvin. So if you're ever going to use the ideal gas law, you need to make sure you've done conversions for Fahrenheit and Celsius into one of these temperature scales. Our second rule for the application of the ideal gas law, pressure must also be absolute. Well, what do you mean by that? Isn't pressure something that can go down to zero pressure? Isn't that a vacuum? Yes, that's true. We can get to zero pressure, effectively a vacuum where there's nothing pushing. Okay, but the question about that is how you go about measuring. Often when we measure pressure, for example, have you ever put pressure into your bicycle tire or into a soccer ball or a football? When you do that, you pump up and you put a bunch of air into that tire or into that ball and you measure it using a gauge. Well, that gauge doesn't tell you how much pressure is in the ball. It tells you how much pressure is in the ball more than the pressure of the atmosphere. You're adding until you balance the atmosphere and then you add more and it measures that difference in pressure between the atmosphere and where you are, okay? That level of pressure, that's a relative pressure or a gauge pressure, okay? So I'm gonna put not gauge. In this case, you often have to listen for the context or read the context of the problem. Is there a reason to believe that this was measured with a gauge or is there a reason to believe that this system, that this measurement or whatever we're taking, it was compared to a zero value or an absolute value. Very often, if your measurement is in atmospheres, it is usually taken to be, I've already subtracted off to one atmosphere that was out there. But you have to pay attention to the context. So make sure that your pressure is measured such that it is an absolute pressure. If, for example, I was doing something with a bicycle tire, I might look and say, oh, this is 40 PSI. Well, that 40 PSI would be the bicycle tire and then I would add the additional 14.7 PSI that comes from the atmosphere itself to get an overall pressure that was inside. Because as we heat or cool or otherwise affect that system, it's going to affect the total pressure, not just the gauge pressure. And then the third rule or a third application piece is probably a better way of saying this, okay? Is you can, in many contexts, N can often or can also be M. You can sometimes exchange mass. You can measure the amount of stuff that's in the system by measuring the mass that's in the system. So instead of counting the molecules, if you happen to know what's in there or you could take whatever gas it is, assuming it's an ideal gas, you can measure the mass of what's put in the system and you can put M in instead of N. Now notice that will change your constant. Usually the constant that's being used here is often a value that's measured by chemists or measured by physical scientists to figure out how that goes in with certain types of gases. And then you need to know the context. Was that measured based on the mass of the gas or based on the number of molecules of the gas? But often we can exchange when we're doing this mechanically and can also be mass and we can exchange those as long as we're consistent in what we use. One last thing I'd like to say about the ideal gas law, very commonly as engineers, we will not really care about this constant R. Most of the time what we're gonna do is we're gonna recognize that there are two states of the system, maybe a before and an after state or an inside and outside state or some relationship that takes two situations and compares them. To do that, what we're gonna do is we're gonna do a little bit of algebra here and rewrite this in terms of pV over Nt is equal to R where there's that constant. And when we do that, what we'll then recognize is okay, what if I take a system at some time or place or situation one, maybe before I heat it up or maybe before I compress it. We'll recognize that's gonna be equal to a constant value, but once we adjust one of these variables, then the system itself will rebalance so that everything, so this relationship between all these variables remains constant. So I like to call this the engineering version of the ideal gas law. Notice there is no constant in there. We've ignored the constant. What we're going to typically do is we'll say okay, can we identify some subset of these things? Maybe we know the, maybe we hold the pressure constant so we can cancel those out. Maybe we know one volume and we're trying to find the other volume. Maybe we cool it down so we know two temperatures and we assume it's closed so we cancel out the N2s. We'll often do something where we'll think about the system that way and then eliminate the different pieces of the system. And that's how we go about solving problems. So depending on what kind of use you might see, you'll either want to remember this version, PV equals NRT is very commonly taught in high schools, or you might want to practice having in your head the engineering version, P1V1 over N1T1 is equal to P2V2 over N2T2. And don't forget, you'll need to remember your rules for application of the ideal gas law.