21. Kinetic Theory of Ideal Gases

The Ideal Gas Law

# Solving Ideal Gas Problems With Changing States

Patrick Ford

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Hey guys, so now that we've been introduced to the ideal gas law and some of the problems you're gonna run across, you're gonna have to compare an initial and final state of a gas in the example, we're gonna work out down here, we have some amount of moles of gas in a container and were given with 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 we're gonna try to figure out is what's the final pressure. So basically what happens is we're gonna try to we're gonna take a gas in its initial state and we're gonna change it. We're gonna figure out which one of the variables changes as a result. Now, in order to do that, we're gonna use an approach kind of similar to how we use energy conservation. We know the equation for ideal gasses is PV equals N. R. T. Now what happens is this is sort of like sometimes called an equation of state, it's sort of like a snapshot of these four variables at a specific moment in time. If you change the gas however, from an initial to final, this PV equals NRT has to remain sort of conserved if you will. So we're gonna do is we're gonna stick subscript on each one of these letters here, so we have P initial, the initial equals and initial R. T. Initial. Now remember this R. Is the universal gas constant. Right? So it doesn't actually need a subscript and then we have p final the final equals and final R. T. Final. So what happens is these two things are equal to each other? We're gonna 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 we're gonna do is we're gonna divide by N. S. And T. S. On each side. And what you end up with is this equation over here, P initial the initial over an initial T. Initial equals P final the final over and final T. Final. So this is the one equation that you need to solve any kind of problem or an ideal gasses change from one state to another. Now let me show you how to use this. Using a step by step approach, that's gonna get you the right answer every single time here. So the first step, if we're just gonna use this equation is to actually write the equation. Now we're just gonna write this equation every time. It's always gonna work. So the first step here is P initial the initial over an initial T initial equals P. Final the final over end. Final T final 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 gonna have to cancel out the constant variables. So what ends up happening here is that in most of these problems, two out of the four variables are going to remain constant, which is actually just gonna let you cancel it 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 two moles of an ideal gas. So we know that N equals two, 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 zero. That's the same amount of gas. So that means that N can just be canceled out. So, for instance, if we put two for two moles inside of the left and right and sides of the equation, it's just going to cancel out, right, Those things are gonna 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 T. I. Is equal to T. F. Whatever that number is, it doesn't really matter that we know it or not because it's just gonna be canceled out. So 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 0.05-0.01. 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 PF 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 p initial, the initial over the final equals p final. So we just move this down to the other side and now we just start plugging in. So the initial pressure is one times 10 to the fifth. The initial volume is 0.5 and the final volume is 0.1. When you work this out, what you're gonna get is five times 10 to the fifth pascal's and that is the answer. All right. So that's how to go through these steps here. Now, if you think 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 five. And as a result, our pressure went from one times 10 to the fifth to 10 5 times 10 to the fifth. So it increased by a factor of five. So what we say here is that P and V are sort of indirectly or in uh yeah, sorry, in inversely proportional to each other. And that brings me to this point here, which is that when scientists were studying these gasses 100 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 boils, Charles and gala sacks 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, 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 we end up with, is this relationship which we just saw, right, if one goes down by a factor of five, the other one goes up by a factor of five. 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 gets sort of canceled out like this, then what happens is you just end up with V over T on both sides. Now these are directly proportional even though it's a ratio and think about what's happening is that if this increases by a factor of five, Then this also has increased by a factor of five in order to keep that number, whatever it is constant. The same works by the way for gala sacks law. If V is constant, basically cancel out these two things here, then you just get P over T and you have the exact same relationship so you might need to know that. Um, 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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