Hey, everybody. Welcome back. So in our problem here, we have a tank of gas that contains some kind of an ideal monoatomic 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 32nRT. Now before I get started, I want to mention that there are actually two different ways to solve this problem. What I'm going to do is I'm going to show you how to solve them using both ways because I feel like they're both really important to know. So I'm going to show you the first way. So here if we're looking for the E_internal, right, so we need to know two out of the three variables in this equation. Remember R is just a constant, so we already got that. So we need to know the number of moles and the temperature. We're actually given straight up what the number of moles is. It's just 10, so that's my n. But what about the temperature? Well, let's see here. I've got the tank. It's got there's a point three here. That's the volume. Is it the point eight here? Well, 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 has to do with gases like pressure, volume, moles, or temperature, the best bet is to use the ideal gas law, PV = nRT. So to figure out the temperature here when you get stuck, you're just going to go over here and solve it and try to solve it by using the ideal gas law. Alright? So what we've got here is to solve for the temperature, I'm going to move this stuff to the other side. So we've got PV divided by nR, and that's going to equal T. So now I'm just going to go ahead and plug a bunch of stuff in. So I've got the pressure first, but the pressure is 0.8 atmospheres. 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 Pascals, I can just use this conversion factor over here. So I'm going to multiply it by 1.01×105 Pascals per 1 atmosphere, cancel that out. What you'll get is 8.08 x 10⁴ Pascals. So that's a little quick conversion over there. No problem. So this is going to be 8.08 x 10⁴. Now what we've got here is the volume. With the volume, I'm just given straight up in meters cubed, so I don't have to do any conversions. This is going to be 0.3. Then I'm going to divide this by n times R. So my n here is going to be 10. My R is going to be 8.314. When you work this out, what you're going to get is a temperature of 291.6. That's in Kelvins. Now remember, that's not our 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 going to be this is going to be 32, and now we're going to have n again. So this is going to be 10 times R, which is 8.314 times the temperature that I just found, which is 291.6. When you work this out here, what you're going to get is an E_internal of 3.64 x 10⁴ joules. Alright? So that's the answer. If you want to go ahead and skip to the next video, you totally can. But remember, as I mentioned, there are two different ways to solve that, and I'm going to show you really quickly how you can also get this, a different way. Alright. So I'm going to I'm going to put here or, you know, this is another method of doing this. You can start off with your E_internal equation. So E_internal is equal to, 32nRT. Alright? So we've seen these three variables nRT in another equation. We actually just used it earlier in the video. Remember nRT also pops up in the PV = nRT equation. So here's what I'm going to do, right? If this equation says that P times V is equal to nRT, then what I can do here is I can come to my internal equation and I can say, well, if E_internal is 32 times nRT, this is really just 32 times PV. Right? These two things mean the same thing according to this equation. So instead of nRT, I can 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 going to do 32 times the pressure, 8.08 x 10⁴, and then times the volume, which is 0.3. And wouldn't you know it? What you're going to get here is you're going to get 3.64 x 10⁴, and that's in joules. Alright? So I mentioned these are the two different ways to get the same exact answer. Hopefully, this makes sense, and we'll see you in the next one.