Hey, everybody. So let's check out this problem here. We have the internal energy of a monoatomic ideal gas is given by this equation here, 32nRT. Some of you may have already seen this equation. That's totally fine. If you haven't, that's also fine. Just know that this is an equation for the internal energy, where n here is the number of moles in a gas and T is the temperature.
Here's what's going on in this problem: We're going to add 1300 joules of heat to some amount of gas, and its temperature is going to increase from 270 to 320. We want to calculate how much work was done by the gas. Ultimately, we try to figure out what W is, the work done by the gas here. How do we start things off? Well, we're going to have some heat, some work, some internal energy, so we're going to use the first law of thermodynamics. We're going to write out this equation here, the change in internal energy of the gas equals the heat added to the gas minus the work done by the gas.
To calculate what the work done by the gas is, I'm going to go ahead and rearrange some stuff. What I'm going to do is move this to the left side and move the internal energy to the right side. What you end up with here is that the work done by the gas is equal to the heat added to the gas minus the change in the internal energy of the gas.
Now, what about the heat that's added to the gas here? Well, if you look at the problem here, we're going to add 1300 joules of heat to the gas. The change in the internal energy of the gas is really what's going on in this problem. We're going to have to figure this out. Remember that the change in anything really is always equal to final minus initial. So it's efinalmomeinitial.
Now, what happens is we have a new equation to solve for this: it's just 32nRT. The n is going to stay the same because it's just the amount of gas that I have, so that doesn't change. But what happens here is that we have some kind of increase in temperature. We have to use this equation to figure out what the change in the internal energy is, and then once we figure this out, we can just plug that back into this equation and solve for the work done by the gas.
We are going to start plugging some stuff in. This is going to be 1.5, that's the number of moles that I have. This R constant, remember, is just 8.314. Then T final. So this is just going to be 320 minus 3/2 1.5 * 8.314 * 270. Now you could have rearranged some stuff, condensed this equation, but I just plugged everything in. What you're basically going to get here is that the change in the internal energy is equal to 935 joules.
Remember, this number here is what we plug into our original equation. So now we just have to go and plug this last step in here, which is the work done by the gas is equal to q2, remember this is just the 1300, that's the heat added, minus the change in the internal energy here, which was just 935. So what you end up with here is that the work done by the gas is equal to 365 joules, and that is the final answer here.
Alright? So that's it for this one, guys. Let me know if you have any questions, and I'll see you in the next one.