Hey, everybody. So let's work on this problem together here. I've got a spherical balloon that I'm told some information about. I'm going to just draw this out real quickly. I've got this spherical balloon and we're told that the volume is 43 meters cubed, and the pressure is 1.2 atmospheres. Now remember, whenever we are given units of atmospheres, we can always convert this to pascals by using this conversion factor over here. Now we're told here also that the average kinetic energy of the particles inside the balloon is this number over here, and ultimately we want to figure out is the number of moles of gas inside the balloon, basically how much gas we have. So the variable for that, remember, is going to be little n. So which equation do we start off with? Well, remember we have one equation involving n. It's going to be PV= nRT. Alright? So let's go ahead and start off there. So we've got PV=nRT. So if you want to figure out n, then we just have to move everything over to the other side, and we've seen this before. So basically, what happens is you end up with PV/RT=n. Alright? So now what I'm going to do is just start plugging in some numbers here. So the pressure, before I plug it in, I can't plug it in as 1.2. I'm going to have to convert this really quickly here. So this 1.2 atmospheres, I can use this conversion factor to get it in terms of pascals. Basically what I'm going to do is I want to divide by units of the atmospheres on the bottom so that it cancels. Well, the conversion factor is 1 atmosphere and this is 1.015 Pascals. So this unit will cancel. What you'll end up with here is 1.215 Pascals. Alright? So that's just the number that I'm going to plug in here, 1.215 Pascals. Now I've got the volume, which is 4-3. Now I'm going to divide by the R, which is the gas constant, 8.314. And now finally, I'm going to look at the temperature. So what is the temperature? Well, actually, I'm not really told in this problem what the temperature is. So I can't just go ahead right away and plug it in. So I'm going to have to figure this out. The only other information that I know about this problem is that the average kinetic energy is just this number over here. So basically what they're giving me in this problem is they're actually giving me k average. And remember, this equation here for k average is related to the temperature. So that's how we figure out what T is equal to. Alright? So basically, I'm going to go over here. So I've got that k average is equal to 32kbT. Alright? So basically, this just says that the average kinetic energy is related to the temperature. So that's the sort of relationship, but that's the link of how we get the temperature. Okay? So here's what we're going to do. We're going to do this k average, and then I'm going to just divide by this other stuff over here to get the temperature. So this is going to be 7.2×10-21 divided by, this is going to be 32, and this is going to be 1.38×10-23. That's just the Boltzmann constant, which is listed right over there. That's equal to the temperature, and if you work this out, what you're going to get here is you're going to get 347.8 Kelvin. Alright? So this is the number here that we plug in for this over here. So now we're just going to multiply by 347.8 or sorry divide, and that's going to give you your final answer for moles. And if you work this out, what you're going to get is 167 moles. Alright. So that's it for this one, guys. Let me know if you have any questions.
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21. Kinetic Theory of Ideal Gases
Average Kinetic Energy of Gases
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