Hey, guys. So now that we've been introduced to the ideal gas law, in some of the problems you're going to run across, you're going to have to compare an initial and final state of a gas. In the example we're going to work out down here, we have some amount of moles of gas in a container, and we're given what the initial pressure and volume are. Then what happens is we’re going to compress the gas using a piston or something like that, and what we’re trying to figure out is what's the final pressure. So, basically, what happens is we’re going to try to take a gas in its initial state and we’re going to change it. We’re going to figure out which one of the variables changes as a result. Now, in order to do that, we're going to use an approach similar to how we used energy conservation. We know the equation for ideal gases is PV=nRT. Now what happens is this is sort of like sometimes called an equation of state. It's sort of like a snapshot of these 4 variables at a specific moment in time. If you change the gas, however, from initial to final, this PV=nRT has to remain sort of conserved, if you will. So what we're going to do is we’re going to stick subscripts on each one of these letters here. So we have PinitialVinitial=ninitialRTinitial. Now remember, this R is the universal gas constant. Right? It doesn’t actually need a subscript. And then we have PfinalVfinal=nfinalRTfinal. So what happens is these two things are equal to each other. We’re going to set these two equations equal. And in doing so, what happens is that the R is going to cancel. Right? It's just a constant that cancels out from both sides. And this equation has a lot of equal signs. So what we’re going to do is we're going to divide by n's and T's on each side, and what you end up with is this equation over here. PinitialVinitialninitialTinitial=PfinalVfinalnfinalTfinal. So this is the one equation that you need to solve any kind of problem where an ideal gas is changed from one state to another. Now let me show you how to use this using a step-by-step approach that's going to get you the right answer every single time here. So the first step, if we’re just going to use this equation, is to actually write the equation out. We're just going to write this equation every time. It’s always going to work. So the first step here is PinitialVinitial/ninitialTinitial=PfinalVfinal/nfinalTfinal. Notice how, again, the R is missing because it actually gets canceled out when you set these things equal to each other. Alright. So the next thing we have to do is we’re going to have to cancel out the constant variables. So what ends up happening here is that in most of these problems, 2 out of the 4 variables are going to remain constant, which is actually just going to let you cancel even more things inside of this equation. So let's go through the our variables and figure out which ones we can cancel out. Now the first part of the problem tells us that we have 2 moles of an ideal gas. So we know that n equals 2, but we’re also told that no gas can leak out or in. So what that means here is that the change in number of moles is just 0. It's the same amount of gas. So that means that n can just be canceled out. So for instance, if we put 2 for 2 moles inside of the left and right sides of the equation, it’s just going to cancel out. Right? Those things are going to cancel each other out. Alright. So the other thing we can do is we can say that the container is then at constant temperature. So that's the other variable that has remained constant. So we know that Ti is equal to Tf. Whatever that number is, it doesn't really matter that we know it or not because it's just going to get canceled out. So what we're left with is really just these four variables over here. Now we know that the initial pressure is going to change to some final pressure, and we’re told that the initial volume goes from 0.05 to 0.01. So that means here that we have the initial pressure with the initial volume. We have the final volume and we want to calculate what is. So all these variables are going to change. We can't cancel them out. And now the last thing we do is we just go ahead and solve. So what happens here, is we just end up with PinitialVinitial/Vfinal=Pfinal. So we just move this down to the other side. And now we just start plugging in. So the initial pressure is 1 times 10 to the fifth. The initial volume is 0.05, and the final volume is 0.01. When you work this out, what you're going to get is 5 times 10 to the fifth Pascals, and that is the answer. Alright? So that's how to go through these steps here. Now if you think about what's happened here, what we've had is that the initial volume went from 0.05 to 0.01. So what ended up happening was we had the volume that decreased by a factor of 5. And as a result, our pressure went from 1 times 10 to the fifth to 5 times 10 to the fifth. So it increased by a factor of 5. So what we say here is that P and V are sort of indirectly or inversely proportional to each other. And that brings me to this point here, which is that when scientists were studying these gases a hundred years ago, hundreds of years ago, they actually sort of came up with these three relationships that are historically called the gas laws. And they're called Charles or Boyle's, Charles, and Gay Lussac's law. Now the thing is these are actually special cases of this ideal gas law which we now know how to use. So I want to go over them quickly just in case you need to know them. Basically, what they did is they held the number of moles and one of the other variables as fixed, and they found that these relationships, between the gas variables. Now what we actually just saw was Boyle's law. Boyle's law says that if the temperature is constant, basically if there is no change in the temperature of the gas, then what happens is that P is inversely proportional to V. That's what we just saw here. And basically how you get to these equations is you just cancel out the constant variables and what we end up with is this relationship which we just saw. Right? If one goes down by a factor of 5, the other one goes up by a factor of 5, so they’re inversely proportional. Now the other one's called Charles' law and basically if you held the pressure constant, V is directly proportional to T. So if you have n and P that get sort of canceled out like this, then what happens is you just end up with V/T on both sides. Now these are directly proportional even though it’s a ratio. And if you think about it, what's happening is that if this increases by a factor of 5, then this also has to increase by a factor of 5 in order to keep that number, whatever it is, constant. The same works by the way for Gay Lussac's law. If V is constant, basically cancel out these two things here, then you just get P/T and you have the exact same relationship. So you might need to know that, but basically those are just the gas laws. So now that we know how to solve the ideal gas law problems, let me know if you have any questions. That's it for this one.
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21. Kinetic Theory of Ideal Gases
The Ideal Gas Law
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