The interior of a Boeing 737-800 can be modeled as a 32-m-long, 3...

Question

The interior of a Boeing 737-800 can be modeled as a 32-m-long, 3.7-m-diameter cylinder. The air inside, at cruising altitude, is 20°C at a pressure of 82 kPa. What volume of outside air, at −40°C and a pressure of 23 kPa, must be drawn in, heated, and compressed to fill the plane?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A gas turbine engine has a cylindrical shaped combustion chamber with a radius of 0.25 m and a length of 1.0 m. While operating the air inside the chamber is at a temperature of 1000 an 800 degrees Celsius and a pressure of 1.6 mega pascals calculate the volume of air at a temperature of 25 degrees Celsius and a pressure of 101.3 kg pascals required to fill the combustion chamber. So our end goal is to calculate the volume of air. So we're given some multiple choice answers. They're all in the same units of cubic meters. Let's read them off to see what our final answer might be. A is 0.45, B is 0.65 C is 0.91 and D is 1.8. OK. So first off, let's make the following assumptions. Let's assume that air is an ideal gas. The air particles have a meal volume and that there is no molecular forces between the particles and that the cylindrical chamber is rigid and does not change in volume as air is added or removed. We also need to recall and use the ideal gas line equation which states that pressure multiplied by the volume is equal to the number of moles multiplied by the universal gas constant multiplied by the temperature. Ok. So to begin, we need to calculate the amount of air in moles needed to fill the combustion chamber based on the interior conditions. Then we need to determine how much space the amount of air will occupy under the exterior conditions. Ok. So essentially, we need to look at the outside of the chamber and the inside of the chamber. But we're gonna start with inside and then outside. So considering this, we can rewrite the ideal gas line equation to consider the interior and exterior conditions. So let's do that. So the pressure interior multiplied by the volume of the interior divided by the temperature of the interior is equal to the number of moles multiplied by the universal gas constant, which is equal to the pressure of the exterior multiplied by the volume of the exterior divided by the temperature of the exterior. So we need to rearrange this to sulfur the volume of the exterior. So let's do that. So rearranging that equation to sol for the volume of the exterior, we get that the volume of the exterior is equal to the pressure of the interior multiplied by the volume of the interior, divided by the temperature of the interior multiplied by the temperature of the exterior, divided by the pressure of the exterior. OK. So at this stage, we can plug in all of our known variables. But before we do that, let's take a second to make the following notes, we need to convert the degrees Celsius to degrees Kelvin. So let's do that for both the interior temperature and the exterior temperature. So we're given the interior temperature as 1800 degrees Celsius. So in order to convert that to Kelvin, all we have to do is just add 273.15. So when we add those two numbers together, we get 0.15 Kelvin. And then for the temperature exterior using the same steps as above, it was the exterior temperature was 25 degrees Celsius. And then we add 273.15. So when we add those two together, we get 298.15, Kelvin, we also need to note how to solve for the volume of the interior. And the. So for the volume interior, we need to consider and remember the volume of a cylinder which states that pi multiplied by the radius squared multiplied by the height Awesome. So now that we have all of our helpful notes, let's officially plug in all of our known variables and solve for the volume of the exterior. OK. Let's do this. OK. So our pressure interior of the interior was 1.6 mega pascals, but we need to convert mega pascals to pascals. So let's use dimensional analysis to do that. So let's note that there's one million past skills in one mega Pasco multiplied by pi multiplied by the radius squared which the radius was 0.25 m squared multiplied by the height which was, they gave us a length of 1.0 m. OK. All divided by the interior temperature in Calvin which is 2073.15 Kelvin. And let's scroll down a little bit multiplied and we ran out of room. So let's write this down below for a second multiplied by the temperature of the exterior which was 298.15. Kelvin divided by the pressure of the exterior which the pressure of the exterior was given to us as 1.13 multiplied by 10 to the fifth power past scales. And this is after you convert. So using dimensional analysis, we can convert 101.3 kg pascals to pascals. So we can use dimensional analysis to do that, but I skipped ahead and did it for you. So when you plug this onto a calculator. The volume for the exterior should be when you round 0.45 because when you meters cubed. So when you plug it into your calculator, you'll see 0.446 but will round to 0.45. Awesome. Mm. Ok. Hooray. We found our final answer. So that means that our final answer has to be a 0.45 m cubed. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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The interior of a Boeing 737-800 can be modeled as a 32-m-long, 3.7-m-diameter cylinder. The air inside, at cruising altitude, is 20°C at a pressure of 82 kPa. What volume of outside air, at −40°C and a pressure of 23 kPa, must be drawn in, heated, and compressed to fill the plane?

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