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.
