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21. Kinetic Theory of Ideal Gases

Internal Energy of Gases


Internal Energy of Ideal Monoatomic Gases

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Hey guys. So in earlier videos we've talked about different types of energies for ideal gasses. We've already talked about the average kinetic energy that was per particle, that was this equation over here. But in some top problems you're gonna have to calculate something called the total internal energy for an ideal gas. And that's what I want to show you how to do in this video. So I want to show you the basic differences between this average and total type of energy. And then we're gonna go a little bit more into the conceptual understanding of what this total internal energy actually represents. So let's get started here, basically the difference between the average kinetic energy and the total energy has to do with how many particles you're looking at. This average kinetic energy was per particle. The total internal energy is going to be if you have a collection of particles, Let's say that's just end particles. So really, really sort of simply here. The basic difference is that when you calculate this, this is the average kinetic energy of one particle. But if you have multiple, you just multiply by however many particles you have. Let's just do a quick example here, we have 10 particles of a gas that's at 300 Kelvin in a container. So in the first part we want to calculate the average kinetic energy. Remember all you need to calculate the average kinetic energy is the temperature. So remember we have this relationship that three halves K B. T. And we have our constants over here, just for reference. So this average kinetic energy is just gonna be three halves times 1. times 10 to the minus 23 then we're gonna multiply this by 300. When you work this out you're gonna get a 6.21 times 10 to the -21 jewels. So that's the average kinetic energy per particle. Now, if you want to calculate the total internal energy and if you have 10 particles, all you have to do really is you just have to do this internal here. It's just gonna be N times K average. It's just gonna be 10 times the average kinetic energy 6.21 times 10 to the minus 21. And then you end up with 62.1 times 10 to the minus 21. Notice how all we've done here is we've just shifted the decimal place to the right by one. Space is just 10 times greater. Alright so that's the fundamental difference between them. So I want to point out just real quickly here that the symbol we use for in total internal energy is gonna be E internal. So some textbooks will also write this as you but here a clutch, we don't want to confuse you with the potential energy and so we just write this as E internal, it's always gonna be written that way. Now there are other variations of this E internal equation. We saw there was just N times K average. So one way you could just rewrite this is you just stick an N in front of this equation over here. So this is three halves big N than K B T. Notice how all we've done here is we've just added an end inside here. And that's just basically another way to rewrite this. Now. Some textbooks may also rewrite this equation again using a relationship that we've seen before, we've seen that N K B t, N K B is equal to N R. When we talked about the ideal gas law. So we can use is this relationship here and you can rewrite this equation again as three halves N R T. Any one of these equations will work. We'll just use this one when you have the number of particles like we did in our first example and you use this one, we have the moles of a gas. And so the last thing I want you to know is that this equation only works for a single atom type of gas, which is also known as a mono atomic gas. So this only works for you when you have single adam tight gasses and most of the problems will tell you whether it's mon atomic or not. So let's take a look at our second problem now, so now we have a total internal energy of a gas and we're just gonna assume it's mono atomic is 401 Kelvin. And the energy is this and we want to calculate the number of moles in this gas. So we have that T. Is equal to 401 Kelvin. We have that the E internal is equal to two times 10 to the fourth. And now we want to calculate the number of moles. That's actually just a little end. So which one of the forms of this equation we have to use? Well, it's just gonna be the one that has the moles inside of it. This is gonna be three halves and R. T. So we're told here is that this E internal is just three halves N. R. T. And this is equal to two times 10 to the 4th. So now all we have to do is just go ahead and solve this moles of gas here, remember this are is just a constant that we have over here and we have the temperature already and we obviously have the energy. So the end is just gonna be two times 10 to the fourth. And this is just gonna be divided by three halves times 8.314 times and this is gonna be 401 kelvin. When you work this out, you're gonna is exactly for moles. So that's how many moles of gas that you have in this container. Alright, so that's how you basically use this equation here. Notice one thing here, is that the the amount of energy that was the total internal energy of this moles of gas Was way bigger than the energy that was per particle, right? We had only 10 particles. So the difference really has to do with the scale and thermodynamics a lot of times we use the total internal energy of a gas because it's easier to measure. And it's similar to but kind of different than the mechanical energy that we studied way back in earlier physics chapter. So, I kinda want to go over this really quickly here. The basic idea, the basic difference is that the mechanical energy, the one we've already talked about a lot is called a macroscopic energy. Macroscopic. Remember that means big, big scale, large scale. The idea here is that that the mechanical energy was the sum of kinetic and potential energies but of the entire object. So if I have a box that has a bunch of gas inside of it, but it's also attached to a spring and it's moving. Then the mechanical energy only is really concerned with the the kinetic energy of the object as a whole. And not the velocity of the gas particles and the potential energy of the whole object, let's say it was attached to some spring or something like this. Now, the internal energy is a microscopic energy that means very small scale. And basically what this means here is that it's still kinetic and potential energy, but it's of the particles that are inside of objects, not the object as a whole. So in this case it's the opposite. We don't care about the spring, we don't care about the box as a whole as it's moving. The only thing that this internal energy is concerned with is the velocity of the gas particles and then any potential energy of the gas interactions of the molecules. Alright, so that's it for this one, guys. Hopefully that makes sense. Let me let me know if you have any questions.

A container filled with 2 mol of an ideal, monoatomic gas is has a total internal energy equal to the kinetic energy of a 0.008kg bullet travelling at 700 m/s. What is the temperature of the gas in Kelvin?


Total Internal Energy

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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.