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Total Internal Energy

Patrick Ford
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Hey, everybody, welcome back. So in our problem here, we have a tank of gas that has the most volume of some kind of an ideal mono atomic gas, we're told the amount of moles and what the pressure is and ultimately we want to figure out what's the total internal energy of the gas in this tank. So in terms of our target variable, it's pretty straightforward. We're looking for the total internal energy which remember is just E internal. So we're looking for E internal. The best thing we can do is probably start off with our E internal equation. Right, It's probably a pretty safe bet. So let's go ahead and get started here. E internal is equal to three halves. N. R. T. Now, before I get started, I want to mention that there's actually two different ways to solve this problem. What I'm gonna do is I'm gonna show you how to solve them using both ways, because I feel like they're both really important to know. So, I'm gonna show you the first way. So here, if we're looking for the E internal, right, So we need to know to out of the three variables in this equation, remember are just a constant. So we already got that. So we need to know the number of moles and the temperature were actually given straight up what the number of moles is, it's just tense. That's my end. But what about the temperature? Well, let's see here, I've got the tank, it's got there's a 0.3 here, that's the volume is at the point a here, remember that's the pressure. So we're not given the temperature directly but we are given the pressure and the volume. So whenever you get stuck with one of these variables that have to do with you know, gasses like pressure, volume, molds or temperature. The best bet is to use the ideal gas law PV equals NRT. So to figure out the temperature here when you get stuck, you're just gonna go over here and solve it and try to solve it by using the ideal gas law. Alright, so we've got here is to solve for the temperature, I'm gonna move this stuff to the other side. So we've got PV divided by N. R. And that's gonna equal t. So I'm just gonna go ahead and plug a bunch of stuff in. So I've got the pressure first but the pressure is 0.8 atmosphere. So before I plug it in I actually have to convert it really quickly, which I'm going to do over here. So I've got 0.8 atmospheres. And then to get in terms of Pascal's I can just use this a conversion factor over here. So I'm gonna multiply it by 1.1 times 10 to the fifth pascal's per one atmosphere, cancel that out when you'll get is 8.8 times 10 to the fourth Pat scales. That's a little quick conversion over there. No problem. So this is gonna be 8.08 times 10 to the fourth. Now, what we've got here is the volume and the volume of just given straight up in meters cube. So I don't have to do any conversions. This is going to be 0.3 and then I'm gonna divide this by n times are so my end here is gonna be 10, my R is going to be 8.314. When you work this out, what you're gonna get is a temperature of 291.6. That's in kelvin's. Now, remember that's not All right. And that's not our final answer. We actually have to plug this back into our E internal equation and then we'll have our answer. So this E internal here is gonna be this is gonna be three halves and now we're gonna have N again. So this is gonna be 10 times are which is 8. times the temperature that I just found, which is 2 91.6. When you work this out here, where you're gonna get is an e internal of 3.64 times 10 to the fourth jewels. All right. So that's the answer. If you want to go ahead and skip to the next video, you totally can. But I remember, as I mentioned, there's two different ways to solve that. I want to show you really quickly how you can also get this a different way. Alright, so I'm gonna I'm gonna put here or you know, this is another method of doing this. You can start off with your internal equation. So E internal is equal to three halves N. R. T. All right. So, we've seen this these three variables N. R. T. And another equation. We actually just used it earlier in the video. Remember N. R. T. Also pops up in the PV equals NRT equation. So, here's what I'm gonna do, right? If this equation says that P times V is equal to N. R. T. Then what I can do here is that can come to my internal equation and I can say, well if E internal is three halves times N. R. T. This is really just three halves times P. V. Right? These two things mean the same thing according to this equation. So instead of NRT I could just replace it with P TIMES V. Now the really sort of cool thing about this is that if I do this, I no longer actually need to go and figure out what the temperature is By going to the ideal gas law, I can actually just plug into the pressure and volume straight into this problem. And I should hopefully hopefully get this number again. So, I'm gonna do three halves times the pressure 8.8 times 10 to the fourth and then times the volume, which is 0.3 and wouldn't you know it what you're gonna get here is you're gonna get um 3.64 times 10 to the fourth and that's in jewels. All right. So I mentioned those are the two different ways to get the same exact answer. Hopefully this makes sense. Um And we'll see you the next one.
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