The engine of a Ferrari F355 F1 sports car takes in air at 20.0°C and 1.00 atm and compresses it adiabatically to 0.0900 times the original volume. The air may be treated as an ideal gas with g = 1.40. (b) Find the final temperature and pressure.

Video transcript

Hey everyone welcome back in this problem. We are told about piston air compressors. K devices used to compress air to high pressures for storage and tanks. The device takes air in atmospheric pressure of one atmosphere and at 18°C, the compression is very fast. So we can assume that this is idiomatic. Hey, if the final volume is 0.12 of the initial volume, the gas or the air story behaves like an ideal gas. Okay. With gamma equal 1.4, the ratio of heat capacities. We are asked to determine the temperature and pressure at this instant of the compression. Okay, so let's just start by writing out what we know. Okay, this is an idea. Batic process when we hear idiomatic we think no heat transfer. Okay, we know P one. Okay, The initial pressure controlled is one. Oops one atmosphere. We have T one. The initial temperature is 18°C. Which is going to be equal to if we add 273.15, this is going to be 291.15 Calvin. We have V one. We don't know the value of gamma is equal to 1.4. Now for the subscript two values P two. We want to find this T two. We want to find this as well. V two. We aren't told of value but we are told that it's going to be .1 to the initial volume. Okay, so this is going to be 0.12 times. V one. Alright. Okay, so again, we want to find p. two and T. two. Now let's start with temperature. Okay, this is an idiomatic process. What do we know about temperature? Well in an idiomatic process we have the relationship given by the following. Okay recall that we can write T. One. V. One to the gamma minus one Is equal to T two. v. 2 to the game of -1. Okay. Alright, so we're looking for T. two. We know T. one. We don't know V. One and V. Two but we do have a relationship between the two. Okay, so let's start filling in that information and see where that gets us. So on the left hand side we're still gonna have T one V. One to the gamma minus one. On the right hand side. We're gonna have T two V. Two which we know is 0.12 times V one. So we have 22 times 0.12. V one to the exponent gamma minus one. Okay. All right, well when we have things multiplied together to an exponent we can break that up. So on the left hand side we're gonna leave it alone for now on the right hand side we're gonna have T. Two we're gonna have 0.12 to the exponent gamma minus one times V one. Gamma minus one. Ok now we see we have this V one to the gamma minus one term on both sides. We can divide that out Solving for T. two because that's what we want to find. We're gonna have T to Is equal to T one divided by 0.12 to the gamma minus one. Ok now we see we have a relationship between the temperatures and the only thing this depends on is Gamma. Okay so we've gotten rid of the dependence on volume by knowing the relationship between those two volumes. Okay so now we can substitute the information we know. T one is 291. Calvin 0.12. Virtual gamma is 1.4. So we have 1.4 -1 and this is going to give us a temperature of 679.9 Calvin Okay or 406.75 degrees Celsius K. So we add back or sorry we subtract back the 273.15. So our T. Two here 406.75°C. Alright so we found the temperature. Now let's go ahead and do the same for the pressure. Okay so let's give ourselves more room and just like for temperature in an idiomatic process we have a relationship for pressure. Okay so we can write he won V one to the Gamma is equal to P two. V two to the Gamma. Okay so we call for an innovative process. We have this relationship. It's similar to the one for temperature and again we don't know the direct values of V. One and V. Two but we know the relationship between the two. So let's use that information and see if we can simplify it in the same way we did for the temperature. So we get P one V. One to the gamma on the left hand side. P two times V two which is 0.12 times V one to the gamma on the right. Okay. We're gonna leave the left hand side alone for now. We can split up that multiplication. We get 0.12 to the exponent gamma time is v. One to the exponent gamma. Okay. V one to the gamma is on both sides. We can divide that term out and solve for P. Two and we get P two is equal to P one divided by 0.12 to the exponent gamma. Okay, so just like for temperature, even though we didn't know the direct or the exact value of V one and V two, we were able to use the relationship between the two to simplify our expression in terms of just P one and gamma. Okay, substituting in what we know. Okay and you'll notice if we go back up to the top here, the possible answers are in atmosphere. So we're gonna leave our pressure and atmospheres when we do this calculation because it's just the pressure divided by a number. Okay, so we can choose whichever unit we want. So we're just gonna leave it in atmospheres and again R. P. One is one atmosphere. Okay so scrolling back down to do our calculation, we have one atmosphere Divided by 0.12 to the exponent γ K. The ratio of heat capacities which were told as 1.4. And this gives us a pressure of 19. atmospheres. And so that is that pressure that we were looking for. So going back up to the top we were asked to find the temperature and pressure when we're doing the compression and the answer is going to be a. K. The temperature we found is 407 degrees Celsius with a pressure of 19. atmospheres. Thanks everyone for watching. I hope this video helped see you in the next one.