A gas cylinder holds 0.10 mol of O₂ at 150°C and a pressure of 3.0 atm. The gas expands adiabatically until the pressure is halved. What are the final (b) temperature?

Video transcript

Hey, everyone in this problem, an adventurer brings an insulated airtight cylinder containing 0.15 mole of gas to assist in inflating a portable shelter. The initial temperature and pressure of the gas are 120 °C and 2.5 atmospheres respectively. As the climber inflates the shelter, the gas is going to expand adiabatic until the pressure drops by 40%. And we're asked to determine the Argan gases, final temperature. We're given four answer choices. Option A 47.4 option B 84.3 option C 54.5 and option D 48.1. All of those answer choices are in degrees Celsius. So let's start by writing out what we know about the situation so far. So we were told the number of moles and is 0.15. We have the initial temperature which we'll call T I, this is 120 °C and we can convert this to our standard unit of Calvin by adding 273. So this is gonna be 393. Kelvin. We have the initial pressure which we'll call P I is 2.5 atmospheres. Again, we can convert to our standard unit. And in this case, that's past gals by multiplying by 1.01325 multiplied by 10 to the exponent five. And this gives us an initial pressure of about 2.533125 multiplied by 10 to the exponent five pass go. Now we want to find the final temperature when the pressure drops by 40%. OK. So that means that our final pressure PF is going to be 60% of the initial pressure. So PF is equal to 0.6 P I. Hey, the pressure has dropped by 40%. So there's 60% of that initial pressure remit. And we want to find again that final temperature TF Now we're told that this happens in an adiabatic way the gas expands idiotically. So we have an antibiotic process and because we're dealing with Argan, we can treat this as an ideal gas. And so those two things together give us a couple of equations to use. So we're call that in an antibiotic process or an ideal gas, we can write that the initial temperature T I multiplied by the initial volume V I to the exponent gamma minus one is gonna be equal to the final temperature TF multiplied by the final volume to the exponent gamma minus one. And we have a very similar relationship between pressure and volume. So we can write piv I to the exponent gamma. Did he go to PF VF to the exponent gamma? So looking at these equations, OK. This first equation has that final temperature TF that we want to find. OK. So we're gonna want to use this equation. The problem is we don't have the volumes yet and we can actually calculate the initial volume using the ideal gas law because we know NT and P at that point. So we can calculate V I but we still don't know VF so we're gonna have two unknowns there. We won't be able to solve for the final temperature just yet. So what we can do is use the second equation. We know the relationship between the pressures. We can find the initial volume that will allow us to calculate the final volume, which we can then use in the first equation to find the final temperature. OK. So a good idea in these problems is to write out all of the equations that you might know and try to piece together how you can get the information that you need. So let's start first by finding the initial volume V I. And again, we're gonna use the ideal gas law. So we're called that the ideal gas law tells us that the pressure multiplied by the volume is gonna be equal to the number of moles multiplied by the gas constant R multiplied by the temperature. OK. So in this case, piv I is equal to N RT I. And we wanna solve for V I so we can isolate and find that our initial volume is going to be N multiplied by R multiplied by the initial temperature T I divided by the initial pressure P I substituting in our values, we get that this initial volume is going to be equal to 0.15 M multiplied by 8.31 45 jewel. For more cover multiplied by 393 Calvin. OK. The unit of Calvin will divide out, the unit of mole will divide out. We're gonna be left with the unit of jewel and the numerator. And then we're dividing by our pressure of two 0.533125 multiplied by 10 to the exponent five pascals. And when we work all this out, we get an initial volume V of about 0.0019 3 4 9 m. Cute. OK. So we have our initial volume. Let's go back to our equations. Hey, remember that we're gonna use the second equation first. OK. We know the relationship between P I and PF we know the initial volume V I. So we can use this equation to find VF the only other thing we need first is this gamma value. OK. So for gamma, this is the ratio of heat capacities. And so for gamma, we know it's gonna be the ratio of heat capacity CP. OK. So the heat capacity at constant pressure divided by CV, heat capacity at constant volume. This is a mono atomic gas that we're dealing with. We have Argan, this is mono atomic. And so we know that CV is going to be equal to three halves multiplied by the gas constant R. We also know because this is an ideal gas that CP and CV are related. And so CP is going to be equal to CV plus R. This gives us three halves of plus R which is just equal to five halves R. OK. So we know CV, we know CP, we can find gamma by dividing the two. So we're gonna have five habs are divided by three halves are maybe ours will divide out. The division by two will divide out and we're just left with gamma is equal to five thirds. All right. So we have V I, we now have gamma, we can use equation two to find the final volume. So let's do that. And again, we're just working through this step by step, getting those pieces of information that we need in order to get to that final temperature that we want. OK. So we have the equation piv I to the exponent gamma, it's equal to PFVF to the exponent came up. Now we could substitute in the value of P I. We know that exact value the numerical value we're actually gonna leave it alone for now because we have a relationship between P I and PF and we're gonna see is that, that P I value is actually gonna cancel out. So we don't even need to write it out. So we're gonna have P I multiplied by the initial volume. We just found 0.0019349 m cubed to the exponent of five thirds that gamma value, this is going to be equal to our final pressure. And, and we know that this is 0.6 multiplied by the initial pressure P I. And they multiplying by the final volume VF to the exponent five thirds. And now you can see, OK, we can divide both sides by P I. We divide both sides by P I. It's gonna divide out and we're gonna be left with VF to the exponent five thirds is equal to 0.0019349 m cubed all to the exponent five thirds divided by 0.6. Now to undo this exponent of five thirds, OK? We want to turn this into an exponent of one. OK. Remember your exponent rules. If we raise the entire term on the left hand side to another exponent, we can multiply the exponents together. So we have five thirds, we wanna get an exponent of one. So we're gonna raise this to the exponent of 3/5. OK? Five thirds multiplied by 3/5 is just one. So on the left hand side, we just get the F and on the right hand side, we get 0.00 1 9 3 4 9 m C to the exponent five thirds divided by 0.6 all to the exponent 3/5. And we can work this out on our calculator and get that the final volume. VF is going to be 0.00 26 two 8 8 6 2 m. Cute. right? We have our final volume. Let's go back to our equations. Remember we're looking for the final temperature. We can now use the first equation. We have the initial temperature, the initial volume and the final volume we know gamma. So the only unknown in our equation now is TF we can finally get what we were looking for. So let's do our very last step here and that is to find the final temperature. So again, we're using that first equation tiv I to the exponent gamma minus one is equal to TFVF to the exponent gamma minus one, substituting in our values 393. Kelvin multiplied by 0.0019349 m cubed to the exponent of five thirds minus one. That's gonna be equal to this final temperature. TF multiplied by 0.002628862 m cubed to the exponent five thirds minus one again, which gives us two thirds. So our final temperature TF if we rearrange is gonna be equal to 393. Kelvin multiplied by 0.0019349 meters cubed to the exponent two thirds divided by 0.002628 8 6 2 m cubed all to the exponent two thirds again. And when we work this out, we get a final temperature TF of 320.371. And our unit here is Cal ok. So we have our final temperature in Calvin. The answer choices give us the temperature in degrees Celsius. So we want to convert this back into degrees Celsius by subtracting 273. So we get our final temperature in degrees Celsius. It's 47.37 degrees approximately and that's it. That's the final answer we were looking for. We're gonna go back to the answer choices and round, ok, to three significant digits. And what we find is that the Argan gasses final temperature is going to be option a 47.4 °C. Thanks everyone for watching. I hope this video helped see you in the next one.