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(II) A gas consisting of $15,200$ molecules, each of mass $2.00 \times 10^{-26} \mathrm{kg}$ , has the following distribution of speeds, which crudely mimics the Maxwell distribution: (a) Determine $v_{\mathrm{rms}}$ for this distribution of speeds. (b) Given your value for $v_{\mathrm{rms}}$ , what (effective) temperature would you assign to this gas? (c) Determine the mean speed $\overline{v}$ of this distribution and use this value to assign an (effective) temperature to the gas. Is the temperature you find here consistent with the one you determined in part $(b) ?$

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(a) 710 $\mathrm{m} / \mathrm{s}$(b) $241.2 \mathrm{K}$(c) $654.2 \mathrm{m} / \mathrm{s}$ $243.6 \mathrm{K}$

Physics 101 Mechanics

Chapter 18

Kinetic Theory of Gases

Temperature and Heat

Cornell University

University of Michigan - Ann Arbor

University of Sheffield

University of Winnipeg

Lectures

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(II) A gas consisting of $…

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(II) A gas consisting of 1…

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(a) How do average speeds …

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( a) How do average speeds…

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15. The distribution of sp…

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Use the Maxwell distributi…

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(a) Compute the rms speed …

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You have two identical con…

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(a) Using the Maxwell spee…

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Figure $19-28$ shows a hy-…

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The molecules of a given m…

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Figure $19-28$ shows a hyp…

Okay, so we're doing Chapter 18 Problem 21. So it says the gas containing an inn of 15,200 molecules and they each have a massive two times 10 to the negative. 26 kilograms has the following distribution of speeds and number of molecules here. So this is what we have on this table over here. The distribution of speeds. Um and it says this crudely mimics the Maxwell distribution, so it's gonna be important. So for part A here, we want to determine the r. M s for this distribution. So when we already have the speeds here in the proportional molecules here, we can find the B r m s as one over total end, all square rooted. And then the sum of the I velocity. Then how many molecules air their times, the speed of the velocity squared. So now we can solve for this, and this is gonna be a lot of writing here, So it's got to one over, actually. All right with this is 15 200. Okay, so this is 1600 times to 20 square. Those 41 100 times for 40 square lost 4700 times 6 60 square plus 3100 times. Aye, aye, D squared cost 1300 times, 1100 squared and us 400 times 13. 20 square. Cool. Plug all of that in and take the square root of it. And we should see that this comes out being 706.6 meters per second. We can approximate this to 7 10 and for the future. Okay, Part B, Given your value of the r. M s, what effective temperature would you assign to the guests? Okay, so we know that the are a mess, Kevin by three k b t over him so we can rearrange this for teeth. And this is just in the r m s square over three k b cool. So let's go ahead and use our mess and see what are effective temperature. So this is the more mass is three times are two times 10 to the negative 26 kilograms and rvr mess was six or 706 26 square it all over three times 1.38 times 10 to the negative. 23 jewels for Kelvin. Cool. Plug over it in should come out to be 241 point to Kelvin. Awesome. Okay, someone out? A new page here for part C. So part see, determined means speed of this distribution and use this value to assign in effect, to measure. Okay, so the means speed here can be calculated similarly, but some over each speed times the number of molecules in that speed ben times its people cool. And now this becomes one over in sometimes. So we have 1600 to 20 plus 4100 times for Rory Plus for 706. 60 plus thirty, one hundred 8 80 plus 1300 times 1100. Lastly, we've got 400 times 13. 20. Cool. And that's all divided by any coming. 15,200. So our average are mean velocity here ends up being 6 50 meters per second. So Oops. Sorry. That's the approximate version. So pre actually plugged this in. We should get 6 54.2 years. Cool. So this is, uh, a little bit slower than the arm s. But now, let's see if the temperature the effective temperature using this speed corresponds to the same effective temperature we found in part B so for average speed different than our MST, we can relate this as eight over pi times K b t over the Moeller Memphis. So it's equations a little bit different, but again, we could just be arranged for temperature here. So temperature then becomes high times m times the average squared over eight K beatings. So let's plug. Listen, so we have pi times two times 10 to the negative, 26 kilograms times 6 50 Bruce, this is 6 54.2 squared all over. Eight times 1.38 times 10 to the negative, 23 joules per kill. And this was meeting for a second. The velocity cool. So if we plug all this in your calculator, let's see what we get Mrs to 43.6 Calvin, which is very close. If we go back to the first page here, we've had to 41 3 Kelvin. So this is just roughly around to help at all. So these corresponding are very consistent. Sensor just average rough estimates. Here, it's me

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