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[15 polnts| Durlng physlcs experiment, helium Bas Is cooled to temperature of 10.0 K at a pressure 0f 0 1C0 atm. What are (a) the mean free path In the gas, (b) th...

Question

[15 polnts| Durlng physlcs experiment, helium Bas Is cooled to temperature of 10.0 K at a pressure 0f 0 1C0 atm. What are (a) the mean free path In the gas, (b) the rms speed of the atoms, (c) the colllslon Irequency of atoms using speed found In (b).

[15 polnts| Durlng physlcs experiment, helium Bas Is cooled to temperature of 10.0 K at a pressure 0f 0 1C0 atm. What are (a) the mean free path In the gas, (b) the rms speed of the atoms, (c) the colllslon Irequency of atoms using speed found In (b).



Answers

During a physics experiment, helium gas is cooled to a temperature of $10 \mathrm{K}$ at a pressure of 0.10 atm. What are (a) the mean free path in the gas, (b) the rms speed of the atoms, and (c) the average energy per atom?

Do identify the gas. We have to figure out what is the atomic number or mass divided by massive proton. Now Mass is given by one of our MP. Now the new waiter, Moss. Often Adam is given by density of density, of the material times volume off one. Adam who it is really on n now v over n is the Oliver's off the the number density. So that comes out to be Rover MB times Katie Overbey. Now we can put the numbers in. We get, we have the density is 0.8 or do massive of broad on 1.66 times 10 to the native 27 k is 1.38 I can stand to the native 23. We're working at 300 Calvin, and the pressure is given by 50 kilo Abascal. So that is 50,000. So we get best before, which means that this is his idiom. Now we're trying to find the RMS Speed, which is Road three gaby D over M equals three times 1.38 times 10 to the negative, 23 times 300 divided by masses for the mass number times the Proton Mass. One foreign 66 times, 10 to the native 27 and that comes out to be 13 74 meter for a second. In order to calculate the mean free path, we have to use the formula, Lambda equals inverse off four route Dubai and over V. Times are square in Worse of that so that he's four route do by and over ve is be over, Katie. Times are square. In the words of that now, putting the numbers in, we get the mean free paths to be for Dubai. The pressure is 50,000 basket divided by que 1.38 times 10 to the native 23 and the temperature is 300. Kathleen times are square. Ah, good estimate for the Mona Atomic Molecules lease 0.5 times 10 to the negative. He doesn't they tend meter square of that in the worst. So doing the calculus that this comes out to be 1.86 time. Stand to the native six meter

Hello and welcome the chapter 16. Probably number 44 on principles of physics. So in this question, we're gonna be something for a couple different things. We're going to use these equations here. The left. We have our ideal gas equation. Um, we have our average kinetic energy. We also have the RMS velocity equation here. We'll also need to know the volume of a sphere. Continue for this v here. But let's just jump right into it is a relatively long problem. Uh, so to start things off, we need to calculate the volume of this sphere that we're interested in. Um, so we're gonna be using the Krieg normal equation. 4/3 pi radius cubed. But in this case, were given the diameter. Not only that, but the diameter and centimeters. So I have that kind of shortcut it here we have centimeters converts two meters by dividing by hundreds of 30 about 100.3. And we also need most find out by 1/2 to get the radius and said of the diameter on, then we cube. So doing that equation will give us a volume of 0.14 cubic meters. However, this equation, I'm gonna stick with using leaders so relatively straightforward again, we just multiply that by most by that by 1000 to get us 14.14 about leaders. So once we have our volume, we can solve, We have to solve our Let's go back up here. You solve our ideal gas equation for this end here. So you just bring Aarti over. So doing that we have n is equal to pressure times volume times are times t Sorry about that. Um are divided by r times team. So pressures and atmospheres volume isn't leaders are ours because our standard constant here and the temperature is given to us in Celsius. So if convert that Kelvin So it's just a matter of adding to 73 to 22 93. That's when I blogged all those numbers in and do our equation. Will have 0.588 molds of gas here. So we're on our way. We're almost done with this part I'm going to do is convert this molds into number of Adam's now. And that's a priest report equation, right? We multiply that by Abu God Rose number. So, by doing that we're going to get 3.541 times, 10 to the 23rd Adams. And then that's the answer for our first part here part, I am. So for the next part here, we have to find the average kinetic energy. So that equation is three habits, Katie, with the cave being at normal, constant and temperature has to be in Calif in here. So really, just plug our numbers in because we have everything we need. Three halves times 1.38 central lines May 3rd Time's up to 93. Remember, that's in Kelvin this time. So you multiply those numbers together, we're going to get an average kinetic energy of ah, 6.651 exactly times 10 to the minus 21 jewels. Um, so that's not very very. But that's our average kinetic energy for each of this, These particles and so, uh, all that's left here is to find our RMS velocity, which is gonna be route three r t divided by the molar mass of helium. But in this case, we're gonna be more massive of helium and kilograms per mole. So that's four times 10 to the minus three. So, um, again, at this point, we pretty much, however, numbers here we just in terms of plugging them in. So three times are, uh this are against its wherein Kelvin and Kilograms is our is 8.314 times that to 93 divided by four times 10 to the minus three on. Then we take, you know, one hat. Taking to the power of 1/2 is the same thing Is taking a square root. I would really like this rotation eso Once you do this equation here with all the numbers in, we're gonna in arm s velocity of 1351.67 meters per second.

Solving party of this problem. So here I can write develop B. B. Is equal to an arty. Simplifying it further by putting the value 1.13 Multiplication 10 to the power five, multiplication, four by three. Multiplication by multiplication, argue that it's 0.15 Q is equal to and multiplication 8.314 Multiplication 293 So finally I get the value of N is equal to three points 54 Multiplication 10 to the power 23 Adams. Now I'm going to solve part B. So in part B I can write the value of K is equal to three by two. Our baby and three by two are by and a multiplication P which is equal to three by two. Multiplication 8.314 Multiplication 293 Bye. 6.23 Multiplication 10 to the part 23 which is required to 6.7 Multiplication 10 to the power minus 21. Now I'm simplifying part C. The year I will use the funds to be our message equal to root under three R B by M. My duplicates than any, which is required to loot under three multiplication, 8.314 My application to nine today by 0.4 which is equal to 1350 m per second as the answer.

In this problem. We're going to be looking at this sphere of helium gas here, and given the geometry of that sphere, the pressure of the gas inside and its temperature, we're going to determine the number of helium molecules in that spirit. So the first thing will probably want to do here is determined with the volume of this virus. Uh, and we can recall from our geometry knowledge that the volume of the sphere is 4/3 pi are cute. Uh, in our case, are the radius of this fear is going to be half of the diameter, which is 15 centimeters or 0.15 meters. Okay, so now to actually determine what the number of molecules of helium is inside this year here we will refer to one of our equations for an ideal gas that we derived from the molecular model of gas, which was that the pressure is 2/3 times and over v times the average kinetic energy. Um, and of course, the average kinetic energy for or of the molecule in an ideal gas is three halves Primedia. So if we substitute in our value for okay here, well, actually, first let's solve for um And so if we solve for and will have three p the over to okay, sequel to em then we can substitute in Ah, our expression for K appear a lot and equals three p. And we'll leave the out to cause I'm gonna substitute that and as well, so divided by ah one over. Okay is going to be 2/3 times one over clear Meteo. So we gotta cancel these out and then he is going to be 4/3. Hi are cute. So if we simplify that now we will have that end is equal Teoh p over k b t A times Aurthur inspire a huge But sir So now all we have to do is make sure that we have the right units for everything we've already converted are two meters which you convert our pressure to Pascal's. So I believe we had one atmosphere worth of pressure, which is just equal to about 1.1 time instead of the fifth Pascal's on. Our temperature is in degrees Celsius, so we want to convert that to kill them. Oh, so that's gonna be equal to 293 Wayne. One of five degrees killed them. Okay, so everything's converted. All we have to do is plugging these values now. And if we do that, the net result here for big end going to be three. 25 4 times 10 to the 23rd molecules, just about half of all. All right. Now, given that we know this, let's try to determine what the average kinetic energy is of a molecule in this helium sample here. So one thing we could do is we could refer back to this equation and just sold for K now instead of, um P I'm sorry instead of in, or we could just use this three halfs k b t here, and they should come out to the same thing using three have scabies, Probably slightly, slightly more simple calculation. So we'll just do that. It's OK. Equals three has Kenny t. Where the Bozeman Constant is 1.3 a one times 10 to the minus 23rd. And if we calculate that, we will get 6.21 actually, 6.7 Read that all from times 10 to the minus 21st. Jules. All right, no one last calculation. Let's look at what the RMS speed of a molecule in this gas is. So the RMS rooming squared velocity is just equal to the square root. Um, either three K B t over the mass of an individual molecule or three K B t. There are three rt over the molar mass, which I tend to prefer this one here because the molar masses generally easier to remember. So for Culio, the molar masses just four grams per mole. Or of course, we convert that two kilograms we will have a 0.4 kilograms per mole, and then everything else is in its proper units. So if we plug in juggle rules numbers in there, we should get that the root mean square velocity is 1352 meters per second. So you haven't we determined what number of molecules were in this helium sample, given its geometry pressure in temperature on. And then we did some more calculations after that, Specifically, the average kinetic energy of a molecule in that gas and the rooming square velocity of the molecule in that gas


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