One cylinder in the diesel engine of a truck has an initial volume of 600 cm^3. Air is admitted to the cylinder at 30°C and a pressure of 1.0 atm. The piston rod then does 400 J of work to rapidly compress the air. What are its final temperature and volume?

Video transcript

Hey, everyone in this problem, we're told that turbo boost allows heat engines to take in air at pressures above atmospheric pressure. So a 0.3 L cylinder takes in air at 1.6 atmospheres and 25 °C and 650 joules of work is done compressing the air adiabatic. We're asked to find the final temperature and volume of air in the cylinder. We have four answer choices A through D and option A and B have that. The final temperature is 1590 Kelvin. Option C and D have that. It is 1891. Kelvin. Then if we look at the volumes, option A and C have that the volume is 2.96 centimeters cubed and option B and D have that, that volume is 4.55 centimeters cubed. And so we have different mixes of those answer choices. Let's start by writing out some of the information we're given. So we're given this initial volume vnat of 0.3 L and we can write this in terms of meters cubed. So this is gonna be 0.3 multiplied by 10 to the exponent negative 3 m cubed. Here we have our initial pressure p not 1.6 atmospheres again, converting to our standard unit. This is 1.6 multiplied by 1.01325 multiplied by 10 to the exponent five pascals. We have our initial temperature T knot of 25 °C, which is equal to 298 Carlton. OK. Adding 273 and we have our work. OK. And the work done is to compress because we're compressing in an adiabatic way, the work is going to be negative. OK. So we have negative 650 jewels. And that's really, really key to this question is realizing that because we're compressing, that work is negative. OK? We're looking for TF we're looking for VF now we have an adiabatic process. So let's think about some of the relationships we have with work pressure, temperature volume in an adiabatic relationship. OK. So we have that the work W is equal to negative N multiplied by CV, multiplied by delta T or we can write this as N multiplied by CV, multiplied by T, not minus TF. OK. So we've taken that negative into delta T and switched the order of delta T. And we also have that the work can be written as one divided by gamma minus one multiplied by peanut V, not minus PFVF. So if we look at these equations. Which one can we use? Which one's gonna help us out? Well, taking a look at the first one. OK. We'll need the number of moles, we'll need CV. But we know tina. So if we can get N CV, then we'll be able to calculate the final temperature. OK. So that's gonna be what we're gonna do with that first equation. The second equation, we don't have the final volume, we don't have the final pressure. And so using this equation is gonna be really difficult. OK. So let's start by finding that temperature first and then we're gonna think about how we can relate that to that volume we're interested in. OK. So again, before we do that, we need to find the number of moles and, and we can use the ideal gas law to do that. Can I recall that P V is equal to N RT? And we're dealing with air, we can assume that we have an ideal gas. We're using subscript not to indicate that we're talking about this initial point. And if we wanna isolate for the number of moles, and this is just going to be peanut vno divided by RP not, and we have all of those values. So let's go ahead and substitute them in N is going to be equal to 1.6 multiplied by 1.01325 multiplied by 10 to the exponent five pas multiplied by 0.3 multiplied by 10 to the exponent negative 3 m cute all divided by 8.3145 jewels. Her Mo Calvin multiplied by 298 Calvin. When we work this out, we get that the number of moles and that we have is going to be equal to 0.01963 moles. And this is gonna be the same throughout this entire compression. OK? So we have our end value for this problem. Now, the other thing we needed was CV. So we can go ahead and calculate CV as well. And CV, recall we have a diatomic gas, OK? We can, we're dealing with air oxygen is a diatomic gas. And so our CV value is just going to be equal to five halves multiplied by bar. OK? And again, because we have a diatomic gas, OK. So now we have everything we need to use that first equation to find our final temperature. Let's go ahead and do that. So we're gonna start by finding T. Wow. So again, using the equation work is equal to negative and DV, delta T, what do we have? Negative 650 joules? He is going to be equal to 0.001963 more multiplied by five halves multiplied by 8.3145 jewels per mole. Kelvin multiplied by our initial temperature 298. Kelvin minus our final temperature. Now, we're gonna divide both sides by this great big mass. 0.001963 multiplied by five halves, multiplied by 8.3145. Then we're gonna be left with 298 Calvin minus TF on the right hand side and then we can rearrange, we can move TF to the left hand side and all of that other stuff to the right hand side. When we do that, we're gonna have that our final temperature TF gonna be equal to 298 Kelvin plus 650 jewels. All divided by 0.001963 moles multiplied by five halves, multiplied by 8.3145 joules per mole Calvin. All right, we have this great big long expression. You can plug it into our calculators and we're gonna get a final temperature value of about 1891 Calvin. Mhm. All right. So we have our final temperature value. We're done half of this question. Let's take a look at our answer choices. See if we can narrow it down at all. So A and B had a temperature of 1590 Kelvin. This is not what we found. So we can eliminate those B or C and D. So we have that correct temperature 1891 Calvin that we found. So those ones could be the correct answer. Now, we're gonna move over to finding this volume. Now, remember once again, we're dealing with an antibiotic process. We said we couldn't use this equation dealing with work because it has two unknown values. But there is another equation that we can use. I recall in an adio process that TNA multiplied by V knot to the exponent gamma minus one is equal to TFVF to the exponent gamma minus one. We now know the final temperature. So we know T knot and VNA or sorry TNO and TF and we know V not, that means the only unknown is VF as long as we can find this gamma value, which we know we can. So this is how we're gonna find the final volume. We just need to find gamma. Now, gamma is gonna be related to that CV value. We already found gamma recall is equivalent to CV, sorry CP divided by CV. For the heat cap, the ratio of the heat capacities. And we know that CP is equivalent to CV plus R L CV. We found to be five hats are five halves, R plus R that gives us seven halves R and so gamma is going to be equal to seven halves or divided by five halves are, the RS will divide out, divided by twos will divide out. We're left with seven, divided by five or 1.4. OK. So we have that gamma value. Now we can get back to finding VF BF is gonna be the only unknown in our equation. Now, let's go ahead and substitute it knot 298. Kelvin multiplied by V knot 0.3 multiplied by 10 to the exponent negative 3 m cubed all to the exponent 1.4 minus one. This is gonna be equal to the final temperature 1891. Kelvin multiplied by VF to the exponent 1.4 minus one. We're gonna divide by 1891 Calvin on both sides and then simplify, we're gonna be left with VF to the exponent 0.4 is equal to 0.006 1429 meters C to the exponent 0.4. OK. So the units are a little bit messy here but they're gonna get fixed in just a minute. Now we have VF to this exponent of 0.4. We want to undo that and to undo that we raise it to the exponent of one divided by 0.4 and 0.4 divided by 0.4 gives us one. And so we're left with is an exponent of one, which is what we want. So what we have is that VF is going to be equal to 0.0061429 meters C to the 0.4 all to the exponent of one divided by 0.4. And when we work this out on our calculator, we get a final volume VF of 2.957 56, multiplied by 10 to the exponent negative 6 m. Cute. Now, if we take a look at our answer choices, these are in centimeters cute. So we need to convert what we found two centimeters cubed. We're going from meters cubed to centimeters cubed. So we're gonna multiply by 10 to the exponent six. And so our final volume VF is just going to be equal to 2.96 approximately centimeters cubed. OK. So now we have our final temperature and our final volume. And if we compare what we found to our answer choices, we can see the correct answer is option. CCF is 1891. Kelvin and VF is 2.96 centimeters cubed. Thanks everyone for watching. I hope this video helped see you in the next one.