Hey, guys. So in earlier videos, we've talked about different types of energies for ideal gases. 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 gonna be if you have a collection of particles. Let's say that's just n particles. So really, really sort of simply here, the basic difference is that when you calculate this, this is the average kinetic energy of 1 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 wanna 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, 3 halves kBT, and we have our constants over here just for reference. So this average kinetic energy is just gonna be 3 halves times 1.38 times 10 to the minus 23, and then we're gonna multiply this by 300. When you work this out, what you're gonna get is 6.21×10-21J. So that's the average kinetic energy per particle. Now, if you wanna calculate the total internal energy and if you have 10 particles, all you have to do really is you just have to do, this E internal here. It's just gonna be n×kaverage. It's just gonna be 10 times the average kinetic energy, 6.21×10-21, and then you end up with 62.1×10-21. Notice how all we've done here is we just shifted the decimal place to the right by one space. It's just 10 times greater. Alright? So that's the fundamental difference between them. So I wanna 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 u, but here at Clutch, we don't wanna 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 3 halves, big N, then kBT. Notice how all we've done here is we just added an n inside here, and that's just basically another way to rewrite this. Now some textbooks may also rewrite this equation again using the relationship that we've seen before. We've seen that nkBT, so nkB is equal to nR when we talked about the ideal gas law. So we can use is this relationship here, and you could rewrite this equation again as 3 halves nRT. Anywhere these equations will work. You'll just use this one when you have the number of particles like we did in our first example, and you'll use this one when you 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 monoatomic gas. So this only works for you when you have single atom type gases, and most of the problems will tell you whether it's monoatomic or not. So let's take a look at our second problem now. So now we have total internal energy of a gas, and we're just gonna assume it's monoatomic, 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 2 times 10 to the 4th. And now we wanna calculate the number of moles. That's actually just little n. So which one of the forms of this equation do we have to use? Well, it's just gonna be the one that has the moles inside of it. This is gonna be 3 halves nRT. So what we're told here is that this E internal is just 3 halves nRT, and this is equal to 2 times 10 to the 4th. So now all we have to do is to just go ahead and solve for this moles of gas here. Remember this R is just a constant that we have over here, and we have the temperature already, and we obviously have the energy. So the n is just gonna be 2 times 10 to the 4th, and this is just gonna be divided by 3 halves times 8.314 times, and this is gonna be 401 Kelvin.