21. Kinetic Theory of Ideal Gases

The Ideal Gas Law

# Doubling Pressure & Temperature

Patrick Ford

384

2

1

Was this helpful?

Bookmarked

Hey everybody. So let's get started with our problem here. So we have an ideal gas in a sealed container, we're told the the initial volume pressure and temperature are, and we're told that the pressure and the volume are both going to double. So let me just go ahead and draw out sort of a sketch of what's going on here. So imagine that I have this kind of like sealed container like this, but I have a sort of plunger that I can move up and down, so this thing can sort of go up and down like this. So what happens here is that this plunger? Imagine it's kind of like halfway through this container, which means that the gas is sort of contained inside of just this area over here. So this is the gas, we have an initial volume pressure and temperature. Now, what happens here is I'm gonna move the plunger up so that it goes like this, so that's basically at the top of the container. So then I have all the gas has now effectively sort of doubled in size. Right? The container is now sort of bigger than it was before. So that's what's going on in this problem here, we have a final volume, final pressure and final temperature. What we're trying to find this first part here is what is the temperature final Alright, so we've got an ideal gas, we're going to change some of the characteristics or variables like pressure and volume. We want to figure out what's the final temperature. So in order to do that, we're just going to stick to the steps here. The first step we're gonna do is write out our changing ideal gas equation. So I'm gonna do P initial v initial divided by n initial T initial equals P final the final over and final T final. So we're looking for is T final here. Now what I want to do is we want to cancel out any constant variables because they'll just cancel out from both sides of the equation. So we'll look through our variables, we'll pressure and volume clearly don't remain constant because we're both told that they both double. However, this is a sealed container. So if it's a sealed container, what happens is that nothing is allowed to go in or come out. So what that means here is that the number of moles, the amount of gas that's inside the container does actually remain constant. So that's what happens any time you have a sealed container just means nothing goes in or out and remains constant. Alright, so now that that steps down, we're gonna go ahead and solve for our target variable. So there's a couple of ways to do this, but you can basically just you can flip the equation or what you can do is you can move this over to the other side and what you'll get is p initial the initial times T final equals p final v final uh sorry, I also have a T initial over here. So then what you have to do is you have to move everything else over to the other side. So the P initial the initial comes down, T initial comes up like this. And what you end up with his here is T final is equal to P final. The final over P initial V initial times. This is going to be T initial. Alright, so that's what the equation works out to now. Notice how we haven't plugged in any numbers yet and if you look the numbers are kind of strange. We've got you know, volumes that are in different units like leaders instead of meters cubed. We've got pressures in terms of atmospheres and things like that. So there are a couple of instances a couple of problems where you may be given in units of leaders and units of pressure and atmosphere. So there's a couple of conversions but we're actually gonna go ahead and shortcut this for just a second here because we can actually sort of solve this problem in an easier way because we know that the pressure and volume of both double. So what that means here is I can sort of rewrite the initial pressure of the final pressure is as multiples of the initial. What do I mean? It means that p final if it's just double the initial pressure, I can write this as two P initial right. I can do the same exact thing for the volume. The volume also doubles. So the tv final is just two times the initial. All right. Now. The reason that this is really helpful is because then I can just cancel out the P initial V initials. Right. I can just cancel out both of those things in the numerator and denominator and then I just end up with basically are just the multiples on the outside. So really what happens here is a T final just becomes four times T initial and you can kind of like sort of reason through this when you look at PV equals Nrt remember this is an equation. Whatever you do to one side you have to do to the other. So what's going on here is you're gonna multiply both of these things by two P initial and V initial. So then what happens is in order for this equation to stay balanced. The temperature has to increase by four times. That's what's going on here. All right. So that basically all I have to do is just take the initial temperature which is 40 degrees Celsius and then multiply it. Right, so what happens here is my T initial here is Celsius. So I'm gonna add to 73 to it and I'm gonna get 313. So really mighty. Final just becomes four times This is going to be 3:13 and I'm gonna get a final temperature. That is 1252 Kelvin. Alright, so you could have actually written all the numbers outdone all the conversions and you'll actually have just gotten the same exact answer over here. This is just a shorter way to do this. All right. So, let's get started now with part B. So, that's basically what the temperature is. It jumps by four. So, let's look at part B. We're supposed to figure out how many moles of gas there are. So remember the moles of gas is just gonna be little N There's actually a couple of different ways. I can do this very if I'm looking for little end that was the only variable that I canceled that in my equation. I can actually just go ahead and use just PV equals Nrt to solve for this. So, you can use PV equals N R T. Now, here's a question. Do you use the initials or the finals? It actually doesn't matter. You can use P initial, the initial equals an initial R. T. Initial or you could use P final v final equals and final. Our final t final. Right. So basically these things are gonna be the same. Remember this end remains constant. So it never changes. So, you can use either set of variables. The choice is up to you, but you'll still get the exact same answer. So, I'm just gonna go ahead and use the initial because for whatever reason. Right. So, I'm just gonna use the initial. So, what happens here is your P initial over here? The initial divided by r Times T initial is going to equal N all right. And then if you chose final you just use all the final versions. So the initial pressure here is going to be 0.15 atmospheres. So I'm gonna, what I'm gonna do is I'm gonna come over here and I'm going to convert this. My initial temperature are pressure is 0.15. And what I'm gonna do here is multiplied by a conversion factor. And I've actually listed a couple of common ones that you might need to know. So one atmosphere is 1.01 times 10 to the fifth. So I'm gonna take 0.15 and I'm gonna multiply it by 1.01 times 10 to the fifth. And what you'll get is let's see, let's scoot us over here. So have a little bit more room Which you end up with here is 1.515 times 10 to the 5th. In pascal's. Alright, so it's a little small but what about the initial volume? But the initial volume is in terms of leaders. But remember we need this in terms of uh in terms of meters cubed because of our our constant over here. Remember so a common one that you'll need to know is that one leader is equal to 0.1 m cubed. So if we're given 2.8 liters and we're just gonna have to multiply by Um let's see the conversion is one leader divided by zero Over 0.001. Actually, I'm sorry. This is gonna be is 0.1. This is gonna be meters cubed divided by one leader. So then the leaders cancel what you'll end up with here is 0.0 um 0 to 8. Alright, So 0.0 to eight. Alright, So if you work this out here, what you're gonna get here is 1.515 times 10 to the fifth, Then you'll have 0.00-8 divided by the R. Which is 8.314 times the initial temperature which we already know is 313. So, if you go ahead and work this out, what you'll get is uh 0.016. And that's mold. So that's your final answer. All right. So these are your true answer choices over here and let me know if you guys have any questions

Related Videos

Related Practice

Showing 1 of 10 videos