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The hot air V aifiled the with balloon < after it is heated? of an ideal Ji 1 cool orning ea5 21 ftori balloon: js heated t0 12] Whal 15...

Question

The hot air V aifiled the with balloon < after it is heated? of an ideal Ji 1 cool orning ea5 21 ftori balloon: js heated t0 12] Whal 15

the hot air V aifiled the with balloon < after it is heated? of an ideal Ji 1 cool orning ea5 21 ftori balloon: js heated t0 12] Whal 15



Answers

The mass of a hot-air balloon and its occupants is $320 \mathrm{kg}$ (excluding the hot air inside the balloon). The air outside the balloon has a pressure of $1.01 \times 10^{5}$ Pa and a density of $1.29 \mathrm{kg} / \mathrm{m}^{3}$. To lift off, the air inside the balloon is heated. The volume of the heated balloon is $650 \mathrm{m}^{3} .$ The pressure of the heated air remains the same as the pressure of the outside air. To what temperature (in kelvins) must the air be heated so that the balloon just lifts off? The molecular mass of air is 29 u.

In this example, we want to figure out what temperature we have to heat the air inside a hot air balloon in order for it to just barely lift off the ground. So the balloon has a volume of 650 meters cute, and the pressure inside the balloon is the same as the atmospheric pressure outside. So it has to lift a total of 320 kilograms of material. So that's the basket, the people inside the fabric of the balloon. And it's doing this in air with chest, a density of 1.29 kilograms per meter right here, and were also given the molar mass of their okay. So what does it mean if we're just barely going to float off the ground or be suspended in air? Well, in terms of Newton's laws, that would mean that the force of buoyancy of our balloon in the air is just barely beginning to overcome the force of gravity holding you to the ground. So the force of buoyancy is just barely bigger than the force of gravity. So the point at which this is going to start happening is when they're exactly equal buoyancy force equals F G. And that's just from Newton's second law, right? The sum of all forces would be equal to zero at that moment. And we just rearrange that equation to say that the buoyant force is equal to the gravitational force. So that equation, uh, FB equals F G. Let's break that down a little bit. We know the buoyant force is going to be equal to the density of the displaced air times the volume displaced. So that would be the volume of the balloon times gravity. And this is going to be equal to to make ourselves a little bit more room here. Actually, here, we'll just move down. This is going to be equal to MGI. Where here we go. There's going to be equal to MGM, where m is the mass of all the stuff in the balloon and the people and all that, but it's also the mass of the air inside the balloon. So we'll put mass Hey, for the master here, Times G. Okay, so we're gonna have jeez cancel out here and we're left with this equation. So we'll hold onto this for a second. Um, right off the bat. We can see that we know the density of air and we know the volume of our balloon. And we knew the mass of the material in the balloon. So this equation will allow us to solve for the mass of the air insider balloon. So we're gonna see why this is gonna be necessary in just a moment. So remember, original question was to find the temperature necessary for liftoff to occur. So if we go back to the ideal gas law equals and r t, it relates all these properties of an ideal gas to each other, right, the amount of temperature, pressure and volume. And we're specifically interested in finding the temperature. So we'll solve for that by dividing both sides by NR. And what will be left with is that the temperature is equal to P. B over and are Okay, So what do we know here? Well, we knew the pressure of the air inside the balloon, right? I think we're calling it d be be a or P B. They're gonna be the same. We know the volume. The only thing we don't know is the number of moles here. However, if we knew the mass of the air, we could certainly get the molds of their right. So the equation that relates moles and mass is the mass is equal to the number of moles of a substance times being molar mass of that substance. So if we had em, we could definitely get in Right, which divide both sides of this equation by the Miller Mass. Began on the left and is equal to Little Emma over bigger. And we could just substitute that in our equation here and be good to go. All right, So if that's the case, though, we're still gonna need to figure out with this massive areas right here. So we'll go back to this equation we first got from Newton's second law. And so for the mast year. So let's just write it cleaned up here. Density of Aaron times. The volume of balloon is equal to the massive the balloon, plus the mass of the air inside the balloon. So all we have to do is subtract and be from both sides in order to isolate the mass dia. And we'll have that the mass of the air is equal to rows of a visa be minus the massive A balloon. And if we plug in those values into that equation, we will have that the mass of the air is equal to 518 0.5 kilograms. So now what we can dio is go ahead and solve for the number of moles, Then we'll have and then just plug everything into our equation for the temperature and call it a day. So the number of moles Well, there is going to be the massive year divided biting ruler Massive here. Remember the molar mass of the air that we were given Waas $29 or 29 grams per mole. Since we're gonna be working in kilograms weaken, Just convert this toe kilograms by moving the decimal place three times back. Right, since there are 1000 grams per kilogram so that would give a 0.29 kilograms per mole. So now we should be ready. Yeah, 519.5 over a 0.29 So 518.5 kilograms over a 0.0 29 kilograms per mole. You can see how the units will get moles there. And if we do that, we will get it. Are you, uh, 17,000? 879 moles. So quite a bit. Okay, so now all we have to do is plug in this little in here to our master equation. Our pressure, the balloon, which is just one atmosphere. The volume of the balloon, which waas 650 meters cube and then are the ideal gas constant. If we do that, we will end up with a temperature of 441 0.6 Kelvin. So quite a bit harder than room temperature. So there you have it. We just figured out the temperature to which we have to heat up air inside a hot air balloon in order to lift up all the material of the balloon in the basket such that the balloon just barely starts lifting off the ground.

In this case we have to find the number of moles in the balance at the U. N. Temperature. But this number of moles is a small land. In order to calculate this number of moles in the balance, we need to use the ideal gets question that is written as a peewee is equal to small and R. T dispute the pressure with the volume. Are is the gas constant in the streets that temperatures. So from here we can widely screen for N as N equals two PV divided by R. T. Call it equation number one. Let's put the values into this point. So it will be small and is equals to the pressure that is equals two one atmospheric pressure. So we have to convert it into possible by multiplying it with 101000s of 302 5 Pascual where atmosphere pressure into the volume that is equal to 12 m Cube divide by Which is equal to 8.314 y'all per kelvin per mole into The temperature that is equals to 40 plus 273.15 into Calvin. So from here we will get the value for this small and as small and as equals two 466 point 905 most. This is a required answer. Thank you.

The volume of this balloon at the ground at sea level in order for the balloon to take off his 1500 cubic meters at sea level, the pressure would be one times 10 to the fifth Newton per meter square. This is at one a team, so that's one of the assumptions that were making. The other is at the sea level. The temperature is 293 killed it. So to calculate the amount of mass, let's first find a number of moles. So we're going to use the fact that we're assuming ideal gas. So another assumption here, if you're keeping track of assumptions that are being made because that's one of the answers to the questions, is the ideal gas law applies, so PV equals NRT. So in the number of moles, death portion off here, the values that were given So we're going to say PV equals NRT pressure. Volume is equal to the number of moles times the ideal gas called a constant multiplied by the temperature. So we're using Kelvin here, obviously, so in is equal to PV over party. But in these values and our is 8.3 jewels per mole. Kelvin, this is equal to 6.2 times Temple of Formals. Uh, this is to the power for that kind of looks like it's times 104. So let's fix that. That's better Malls. Okay, so we can calculate the mass, then, since we're assuming helium, not hydrogen, Right, Because if we used hydrogen, that's very flammable, and that's what them I'm sure you can look up some historical events about that. So this isn't in times the, uh, mass for helium, which is four times 10 to the minus three kilograms formal. So Okay, so plugging that value in and carrying out this operation, this comes out equal 2.5 times 10 squared kilograms. So again, some of the assumptions that we made our the volume the volume that was given is an assumption. But the pressure and the temperature at 1 18 AM Aziz well, is the fact that the ideal gas constant applies here

Question gives us this kind of equation that describes the pressure as advances in a balloon with a diameter that increases the diamond has given as d. We've got this constant. See that we don't know on a constant the one that we are given, as well as the atmospheric pressure P, Nor we told the D equals four meters. The pressure is 400 killer Pascal's with a temperature that remains constant at 20 degrees Celsius. So the first part of question asks us for the maximum pressure inside the balloon anytime during this inflation process. So to start with where we need to find out what the constancy is. So in order to do that, we take this information on the bottom line Here in this introduction section that at D equals four meters are pressure is 400 killer Pascal's. So we can just in put that into the equation. So R P is equal to 400 which is equal to P north, which is our 100 killer Paschal's plus our constant one minus and then D one over these just 1/4 multiplied by 1/4 again because we have the D one over the factor on the outside So we can rearrange this so that we can say that while we have 400 equals 100 and then multiplying this allow we have plus C and then that will be times by 3/16. If you expand everything out and then you can say from that that r C is gonna be equal. Teoh rearranging everything again. We have 1600 this is Dimension Lis. So therefore, our pressure is gonna be equal. Teoh Ah, 100 plus our constant, which is 1600 And then we have one minus the one over D times by the one over day. So to make things easier, we're gonna have to differentiate this. So what we're gonna do is we're gonna rewrite the one over D to be equal to X to the minus one, as this will make differentiating it just a little bit easier, so different cheating it. Now we have dp over the d because it's when we're looking for the maximum diameters. We need to differentiate with respect the and that's gonna be equal to our constant C multiplied by Well, we have a D one over the here in the bracket. So that's going to be minus X to the minus tube. And then we're gonna have X to the minus two. So that's going to be equal. Teoh. The two comes here and then we have X to the minus three. And then that's gonna be a low over our the war because that the one is a factor that comes out. So now we know that at a maximum that's going to be equal to zero. So therefore minus X to the minus two plus two X to the minus three is equal to zero. We can be arranged that by having well two X to the minus three is equal to you X to the minus two. And then we can divide and say that our X is equal to two. So that's gonna be the ratio of D to the one is going to be related by this two factor on the outside. So therefore, um, at our maximum value of pressure. So Max P de diameter is equal to two times the one from that ratio, which is equal to while two times one meter is two meters. So therefore volume it can be given by pie, Uh, for three times by the diameter over to Cube. That's just the equation. But the volume of a sphere. Andi, once we input all of those values in we have four pi over three times by our diameter, which is to over two cubed. Of course, that just comes out is four pi over three, which is equal to 4.18 meters cubed. So finally, we can say, what are P. Max is gonna be equal. Teoh, which is gonna be equal Teoh 100 from the equation that we described at the start, plus our constancy, which is 1600 and then we have one minus are ratio off B one to D Well, the's equal to two D one, which means that the one over the is gonna be 1/2 so half, half on that is equal to 500 killer pass. Yeah, 500 Killer Pascal's. That sounds for the first part of our question. So now we need the pressure inside the tank on. This is why we did this volume calculation before that, we need the volume off that balloon. So to start with, we're gonna want the mass in the balloon. So the mass in the balloon you're gonna cool MB is gonna be well using the ideal Gasol. We have PV over rt. Where are is our That's constant, Andi. Well, the precious 500 killer Paschal's multiply by the volume that we just worked out, which is 4.18 meters cube over are expressed in killer Paschal's per etcetera, etcetera so we can cancel out with the kill Paschal's in the pressure. If you look that up on literature tables in textbook for helium you confined that is 2.771 multiplied by our temperature which we're told is 20 degrees Celsius. So that is equal to 20. Plus the conversion to Kelvin, which is adding 273.15 on that comes out as 3.44 kilograms. So that's the mass of the blue. We're gonna need massive the tank So the mass of the tank when it's full is going to be equal to P the over rt again, where the is the volume off our tank which we are told is 12 meters cube. So the pressure is equal to two mega Pascal. So in Killer Pascal's, that is 2000 killer Paschal's Times by 12 which is our volume of the tank over the gas constant, which is the same as before 2.771 multiplied by our temperature, which is the same as before, which is 20 plus 2 73.15 which is 293.15 on. That gives us a mass off 39.416 kilograms. However, this is the mass off the tank when it's full. We want the mass of the tank when the balloon is inflated to its maximum. So we figured out that the master we require in that case is 3.44 kilograms. So simply the massive the tank at the stage that we want is going to be the maximum massive helium in the tank, minus the mass in the balloon that we went out at the start of the question that maximum pressure. So that's going to be equal to, um, put the intel calculated. We get 35.976 kilograms and then finally to work out the pressure inside the tank at this time We simply need to put this value of mass back into our ideal gas equation. So the pressure in the tank is gonna be equal to what? We have em in the tank multiplied by what we have our tea over the Andi putting in some values. There we have. Well, we've got 35.976 times by our our, which is to your 0.771 times by our temperature, which is 293.15 over the volume, which is, um 12 Andi That will come out at an answer off 1825.5. Killer Paschal's. And this is our answer.


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