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(a) How many atoms of helium gas fill a balloon of diameter $30.0 mathrm{~cm}$ at $20.0^{circ} mathrm{C}$ and $1.00 mathrm{~atm}$ ?(b) What is the average kinetic e...

Question

(a) How many atoms of helium gas fill a balloon of diameter $30.0 mathrm{~cm}$ at $20.0^{circ} mathrm{C}$ and $1.00 mathrm{~atm}$ ?(b) What is the average kinetic energy of the helium atoms? (c) What is the root-mean-square speed of the helium atoms?

(a) How many atoms of helium gas fill a balloon of diameter $30.0 mathrm{~cm}$ at $20.0^{circ} mathrm{C}$ and $1.00 mathrm{~atm}$ ? (b) What is the average kinetic energy of the helium atoms? (c) What is the root-mean-square speed of the helium atoms?



Answers

You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1.25 atm and the temperature is 22.0$^\circ$C. Assume that all the gas is N$_2$, of molar mass 28.0 g/mol. (a) Find the mass of a single N$_2$ molecule. (b) How much translational kinetic energy does an average N$_2$ molecule have? (c) How many N$_2$ molecules are in this balloon? (d) What is the $total$ translational kinetic energy of all the molecules in the balloon?

Okay, In this problem, we're going to be finding the total tannic energy in a mole of ah, of gas at 27 degrees Celsius. And then we were asked to find the equivalent speed of a person about 65 kilometers and mass running that would have the same kinetic energy is that mole of gas? So the main equation that we're gonna use is the kinetic energy, the average kinetic energy of a particle in a gas at a certain temperature, which is three abs times. Katie, it's all going to do is multiply this by the number of particles, get the total kinetic energy of the one ball of gas. So we could say get total kinetic energy. We could multiply the average by the number of particles in the gas, and we could actually combine. Begin with little K to get you gas. Constant are so we would be left with three halves. Artie, let's move over. Doing it knew Tad. So if we have total kinetic energy of three halves, Artie, where ours be gas constant and we know the temperature. When you get into just Kelvin. However, these are playing things in. We have 3/2 times gas constant is 8.314 jewels per Mel's Calvin. Okay, however, we Valerie, we've eliminated this. They're no more moles, and we multiply this by 273 degrees Robert 273 Kelvin. Those are 27 degrees the gas. This is conversion into Elvin from Celsius. And if we do this, computation will be left with 3740 Jules, which is the result for the amount of energy for Barbie were asked to find the velocity of a 65 kilometer kilogram mass that would be equivalent to this amount of kinetic energy for this gas. So we know the formula for kinetic energy of a mass moving is 1/2 V squared. It's over. Isolating V. We would have V equals the square root of two que over m. We can play everything in two times 3740 Jules over 65 kilograms for this average human and we would end up with about 11 meters per second and that's it

We know that we're dealing with helium atoms and we can say that for party, the average too broadly wavelength we can say Lambda average would be equal to the planks constant divided by the average momentum. This is gonna be equal to plants constant divided by the square root of two times the last times a kinetic energy average. And this would be equal to we can say h divided by the square root of two times the mass times three times Boatman's constant times temperature over to. And we can rewrite this and say that the average wavelength would be equal to planks, constant times the speed of light divided by two times I'm c squared, divided by Katie. And at this point weekend, say that here my apology. At this point, we can say that the average wavelength would be equal to 1240 electron volts nano meter, and then this would be multiplied by the square root of three times four times 938 mega electron volts. And then this would be multiplied. Bye. Divided my apologies by 8.62 times 10 to the negative Fifth Electron volts per calvin and then divided by whether multiplied by 300 Calvin extend the square root and four party, we find that the average the broadly wavelength would be equal to 7.3 times 10 to the negative 11th meters. Or we can say 73 pico meters. We can then say for part B, the average separation would be equal to one over the third route of n. And this would be equal to one over the third route of the Mo Mentum divided by the bullets, man. Constant times the temperature t. So this would be 1.38 times 10 to the negative. 23rd. This would be Jules per Calvin and then multiplied by 300 Calvin. And then this would all be divided by 1.1 times 10 to the fifth. Pascal's my apologies. This isn't, um I said that this was momentum. This is not momentum. This is actually pressure. My apologies. And we can then take take this, take the third route, or we can say this entire term to the 1/3 power. So taking the third route or sending it to the 1/3 power of course, this is going to be the same answer, and the average separation is found to be 3.4 nano meters. This would be our answer for part B. And then finally, for part C, the question is asking us, Can the Adams be treated as particles under these conditions? And yes, ah Adams can be treated as particles because the average dobro glee wavelength is much less than the average separation. And that would be the condition for them to be treated as particles. That is the end of the solution. Thank you for

In part a of this problem. We want to complete arrest in a tick in our job. Helium particle The expression far and risking your technology up Particle is kinetic energy today about to go equal to three upon Do indoor boards minutes constant into temperature In governments ago, my wife relating different values here. Three upon do well you off Boardman's constant unease. One point she had times attended a power minus 23. You'll for I mean in the room in Britain, off helium gases doing descended that is equal to during people. Us. So 73.15 in held in skill. So this gives us every asking and ignite your particle equals toe. Five Piety times 10 to the power minus doing doing you're number. The second part we want to call glued every the recipe job helium particles by using the relation kinetic energy equals do half times mass Indo. So get up subete disabuse us. It's a bit equal toe, so get a load off toe into kinetic energy. A farm must no way for letting different values, so this gives us original city equal toe toe and do and risking your dignity of particulate is why Appointee times 10 to the power monitor. Do one you upon, mess up This particle off him is four part 00 three time standard obor minus 27 kilogram. So the US US A. Recipe off each particle equal toe one fire 63 times into the polar, three meter or skin. Knowing this Parsi off this problem, we want to pull pull it thermal energy off helium gets inside the blue. So the mission far end particles inside the balloon for caramel energies adding to rest in additional job Each particle that is see upon to into care do you so we can write this and is number almost into regardless, member into three Upon to K to we can write total number of particles equal toe number almost inside the balloon into irregardless number that is equal to and in due 3.2 Gorris number in drew care times to yes and regardless number into war demands constant is equal to general universal gas constant there is our so we can write in tow trip on two are in duty So this gives us thermal energy equal toe three upon toe and are. Do you by using Arial? Yes, good. And we can write three upon to anarchy that is equal to be no y polluting different values here she upon to assume the pressure inside the balloon, is equal toe the atmospheric pressure that is one day in one 2 to 5. Paschal Indu while you mawf Miriam Inside a balloon is zero point 01 zero giving me down. So it is gives us tournament energy up. Civilian gears inside the balloon. Equal toe 15 one line. Why's air 75? You're There's another problem. Thanks for watching.

So in this example, we have this sample here are gone and helium gas in the same container and they're at a temperature of 150 degrees Celsius. And we want to determine what the average kinetic energy is of each of those gas molecules and also what the RMS speed is for each of those molecules. So, first we should recall that the average kinetic energy for an ideal dass is equal to three halves. Holzmann's constand times of temperature. So the average kinetic energy is actually independent of the type of molecule for our ideal gas equations here. So this should be the same for both are calm and helium. So if we convert the temperature appear to Kelvin before re performer calculation, it'll be 150 plus to 73.15 Hey, is 423.15 Children, and we just plugged that in. Now we will have our average kinetic energy for those helium and argon. So this comes out to be 8.77 times 10 to the minus 20 first jewels. Awesome. So now we're going to see that when we calculate the armas velocities that they actually do not have the same magnitude. And that's because the masses of each of these molecules are different. So again for an ideal gas, we have that the RMS is equal to this square. Ah, three k b t over and not that's the mass of the individual molecule. Or alternatively, we could use three rt over eat molar mass. So either one of these will get you there as long as you stay consistent with all your units. So if we calculate TV armas for Neil Young, we will end up with 1624 meters per second. And similarly, if we plug in the values for are going, we'll end up with 514 meters percent ing. And just in case we, uh, I forgot in our here is a point 314 I believe joules per mole, Kelvin and the boatman constant is 1.381 times 10 to the minus 23 and the units are always a mess on this Yeah, kilogram meters squared her second square to kill them other. So let's see if these results makes sense. We got that the rooming squared velocity for helium is much higher than that for are gone. And this makes sense because if the temperature determines the kinetic energy, think of my can extract Kinetic energy is 1/2 MV square. The temperature determines the kinetic energy, and these both have the same kinetic energy on average per molecule. Then, if the mass is much, much greater, then we will require a smaller velocity to have the same kinetic energy. So these results are consistent. So there you have it. We determined the average kinetic energy and the root mean square philosophy of a sample are gone and helium gas.


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