Internal Energy of Gases: Study with Video Lessons, Practice Problems & Examples

In thermodynamics, the total internal energy of an ideal gas is calculated using the equation E_internal = 3/2nRT for moles or E_internal_avg = nkT for particles. This energy reflects the microscopic kinetic and potential energies of gas particles, contrasting with macroscopic mechanical energy, which considers the whole object's motion. Understanding these concepts is crucial for analyzing gas behavior and energy transformations.

Internal Energy of Ideal Monoatomic Gases

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.

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?

• A. 123.2 K
• B. 78.6 K
• C. 235.7 K
• D. 11.2 K

Total Internal Energy

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.

Here’s what students ask on this topic:

What is the total internal energy of an ideal gas?

The total internal energy of an ideal gas is the sum of the microscopic kinetic and potential energies of all the gas particles. For a monoatomic ideal gas, it can be calculated using the equation:

$E_{\mathrm{internal}}=\frac{3}{2}nRT$

where $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the temperature in Kelvin. This equation is valid for monoatomic gases and reflects the energy due to the motion of the gas particles.

How do you calculate the average kinetic energy of a gas particle?

The average kinetic energy of a gas particle can be calculated using the equation:

$E_{\mathrm{avg}}=\frac{3}{2}{k}_{B}T$

where $k_B$ is the Boltzmann constant and $T$ is the temperature in Kelvin. This equation shows that the average kinetic energy of a gas particle is directly proportional to the temperature of the gas.

What is the difference between average kinetic energy and total internal energy of a gas?

The average kinetic energy of a gas refers to the energy per individual particle and is given by:

$E_{\mathrm{avg}}=\frac{3}{2}{k}_{B}T$

In contrast, the total internal energy of a gas is the sum of the kinetic energies of all particles in the gas. For a monoatomic ideal gas, it is given by:

$E_{\mathrm{internal}}=\frac{3}{2}nRT$

where $n$ is the number of moles. The total internal energy considers the entire collection of particles, making it a macroscopic property.

How does temperature affect the internal energy of an ideal gas?

The internal energy of an ideal gas is directly proportional to its temperature. For a monoatomic ideal gas, the internal energy is given by:

$E_{\mathrm{internal}}=\frac{3}{2}nRT$

where $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the temperature in Kelvin. As the temperature increases, the internal energy increases proportionally, reflecting the increased kinetic energy of the gas particles.

What is the relationship between internal energy and the number of moles in an ideal gas?

The internal energy of an ideal gas is directly proportional to the number of moles. For a monoatomic ideal gas, the internal energy is given by:

$E_{\mathrm{internal}}=\frac{3}{2}nRT$

where $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the temperature in Kelvin. This means that if the number of moles increases, the internal energy also increases proportionally, assuming constant temperature.