A cylinder contains 0.100 mol of an ideal monatomic gas. Initially the gas is at 1.00 * 10^5 Pa and occupies a volume of 2.50 * 10^-3 m^3. (b) If the gas is allowed to expand to twice the initial volume, find the final temperature (in kelvins) and pressure of the gas if the expansion is (i) isothermal; (ii) isobaric; (iii) adiabatic.

Video transcript

Hey everyone in this problem. We have a container with a movable piston. Okay, it has 0.22 moles of helium. The initial pressure of the helium is 1.1 times 10 to the five pascal's with a volume at that pressure of 0.575 m cubed. Okay. The gas undergoes a process where the final volume is gonna be 1.5 times the initial volume and we're asked to find the final temperature in Calvin. Okay. And the final pressure. If the process is so thermal. Alright, now let's think recall that ice a thermal. It means that we have constant temperature. Well, if we have constant temperature then T to the final temperature we're looking for is just gonna equal the initial temperature. T. one. Alright, well we don't have the initial temperature but can we find it? Okay, we have information about pressure and volume. We want to find information about temperature. How can we relate all three of these? Well, the ideal gas equation to the ideal gas law. Okay. Talking about helium so we can treat it as an ideal gas. Okay. And so the ideal gas law is going to give us PV is equal to N. R. T. Okay, now we're trying to find T. Two but we don't know P two yet. We know that T. Two and T one are equal. So let's go ahead and find T one since we know P one and we know V one. Alright, so P one. The initial pressure we're told is 1.01 times 10 to the five. My skills. Okay. V. One. The initial or volume. Sorry not velocity. 0.575 m cubed. Okay. N is the number of moles which we're told is 0.22. This is 0.22 more times R. Which is the gas constant 8.314 joules per mole kelvin eight times T one. And again T1 is that temperature we're looking for because we know that it's equal to our final temperature because we're in an icy thermal process. Alright. So simplifying we get 580.75 on the left hand side K. We have pascal meters cubed. So that's gonna be unit jewel on the right hand side. The unit of mole and Permal will cancel. We're left with 1. jewel per Calvin times T. one. And when we divide the unit of jewel will cancel when we left with T one is equal to 317.51 Calvin. All right now we're told T one is equal to T. Two. Okay, so this is going to be equal to T. Two. And if we round to the nearest Calvin we get 318 Calvin. Okay, so that is the final temperature that we were looking for. Okay now let's go ahead and find the final pressure. So we can do it in a very similar way again. We have an ideal gas now that we have a final temperature, we have the final volume. We can find the final pressure using the same equation. The ideal gas law. So now we have P two. V. Two is equal to N. R. T. Two. Just gonna move to give ourselves some more space. Alright, so p. two. Well, that's what we want to find that final pressure. Okay, what about the final volume? Let's go back up. We're told that the final volume is going to be 1.5 times the initial volume. Okay, so the final volume Is going to be 1.5 times the initial volume which was 0.00575 m cubed Is equal to end the number of moles. 0.22. They are the gas constant same as before. 8.314. And the unit is jules per mole Calvin K times T two which we know is 317.51 Calvin. Alright, so on the left hand side we're gonna have P2 times 0.0086 - five K units meter cute. On the right hand side. We're going to get 580.7511908. Okay. And our unit here will the mold will cancel with the Permal. The Calvin will cancel with the per Calvin and we're just left with jewel. So P two. When we divide we get jewel divided by meters cube, which is gonna give us pascal. Okay? We're going at 6.73 times 10 to the four pascal. Okay. And so that is the final pressure we were looking for. So if we look at our answer choices, we have a final temperature T two of 318 Calvin. A final pressure P two of 6.73 times 10 to the four pascal's. And this is going to correspond with answer. A thanks everyone for watching. I hope this video helped see you in the next one.