And this problem. We're trying to figure out how much mass a hot air balloon can lift. Um, so the total mass that the balloon left is the difference between the mass of air displaced by the balloon and the mass of the gas and five, the balloon. Um, we're being told to consider a hot air balloon that has a ah spherical shape, um, in a diameter of five meters, where the air is heated to 65 degrees Celsius in the surrounding air temperatures, 21 degrees Celsius and the pressure both within and outside of the balloon is 745. Tour. Um, first thing we need to do is figure out the volume off our hot air balloon, and the volume of the spear is 4/3 pi are cubed on, and since the diameter was five meters than the, um, radius is going to be 2.5 meters. And so when we substitute in the 2.5, we get that our volume is 65 0.45 meters cubed, which equals 65,000 450 leaders of air within the hot air balloon. We're also gonna want to go ahead and convert our temperatures into Calvin. So of its 65 Calvin started 65 Celsius 65 plus 2 73 which gives us a temperature of 338 Kelvin and then for the 21 degrees would do the same thing. 21 plus 2 73 is 294 Kelvin. And then finally, we're gonna want to convert our pressure from toured the atmosphere. So we divide the 7 54 by 7 60 and we get that the pressure is going to be 0.98 a t m. Okay, Um, so the first thing we're trying to figure out what mass? The Blinken lift, assuming that the average Mueller mass of air is 29 grams per bowl. So first we're gonna figure out how many moles of air we have inside of our balloon. So to do that, we're gonna use PV equals at R T and solving it for M so n equals 0.98 which is the pressure times 65,450 which is the volume divided by our gas constant. 0.8 206 times are temperature and the temperature inside the balloon is 338 Kelvin. So we get that We have 2000 300 and 12.5 moles of air inside of the balloon. Now, since we have, we were told that the mass of the air, the mill a massive airs 29.0 grams per mole. We can now figure out how many grams the balloon is able, or how many grams of air is it inside of the balloon. So Well, get on Muller Mass is going to be the 20 2312.5 moles times 29 which gives us 67,000 63.39 grams. Okay, so now that we know the mass of the air inside of the balloon, we now need to get the mass outside of the balloon. So we're gonna do the same thing we're gonna get from the number of moles, which is gonna be your 0.98 because they both have the same pressure. We're gonna use the same volume because the the air that's getting pushed out is the volume of the balloon times are gas constant eight 206 times are the temperature outside and the temperature outside of the balloon is our 2 94 So, times to 94 and we get that we have a total of 2658.6 two moles and then to get our mass were again gonna multiply that number 658.62 times 29 and we get a total of 77 those in 100 grams. So to find the mass that the balloon can lift, it's the difference between those two numbers. So we're gonna take our roughly seven point 71 times 10 to the fourth grams minus are 6.7 zero times 10 to the fourth grams, and we'll get 1.101 times 10 to the fourth grams can be lifted by the balloon. Now, for part B, it's Afghanistan to figure how much the balloon could lift if it was actually filled with helium at 21 degrees Celsius in 700 and 45 tour. Um, assuming the same volume, etcetera. So let's go ahead and just clear up all this stuff, okay? and start fresh, so their volume is gonna be the same. Um, the mass of the air is going to be the same, but now we need the molar mass of helium. So molar mass of helium is going to be from the period table 4.3 grams per mole. So we're gonna do the same thing we we had from our previous part one apart. A. We know that the the mass that of air on the outside of the balloon is seven point 71 times 10 to the fourth, uh, grams. So now we can go ahead. And if we find the number of moles for helium in the situation, Megan, then just subtract that from the mass of the air. So the number of moles of helium is gonna be our 0.98 because it says the same pressure. We have the same volume, the 65 4 50 and we're gonna divide that by our gas constant and our temperature, which is still is gonna be the same as the temperature of the air. So to 94. And we get that that is, um, acquits to 2000 658 0.62 Moles of helium. And we're gonna take that number 6 58.6 to multiply it by our 4.3 which is the Moler Massive helium. And we get 1.6 times 10 to the fourth grams. So if we subtract that from our 7.71 times 10 to the fourth minus 1.6 times 10 to the fourth, we get an answer of 6.65 times 10 to the fourth grams can be lifted by the balloon if it was filled with helium. And for the final part of this problem, we're being asked to consider the same situation of group one, Um, except her from part A. Except this time we're taking into consideration it being in Colorado, um, with a pressure of 630 tour. So we're gonna have to go through that same process we started with, but this time changing our pressure from 7 50 Our new pressure is 630 tour. So divide that by 7 60 and you get that the new pressure is 0.83 a. T. M. so going through our problem again looking for the number of moles within the balloon we have 0.83 times our volume. 4 65,050 divided by our gas. Constant times are temperature. We're gonna use the same temperatures as we did in part a. So the number of moles inside the balloon equals 1.96 times 10 to the third. Moles take that times 29 and we get if our a mass of 5.68 times 10 to the fourth grams time and then doing this again for the air because again, the number of most for the air is also going to change. With the change in pressure 0.83 times 65,000, 450 divided by 0.8 to 06 times are temperature of to 94. We have that. We get 2.25 times 10 to the third malls. That's why that 2.25 times 10 to the third times 29 and we get 6.52 times 10 to the fourth, um, Grams and then finally tracked those two numbers from each other. So 6.52 times 10 to the fourth, minus five point 68 times 10 to the fourth. And we get a massive 8.3. Whoops. Sorry. 8.4 times 10 to the third grams.