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Calculating velocity using maxwell distribution

  1. Oct 8, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider helium as ideal gas with a Maxwell distribution of speeds.

    (a) Investigate the maximal value Fmax at the peak of the Maxwell distribution F(v):

    Calculate this value for He at T = 300 K and at 600 K, and for N2 at 300 K

    b) For He at 300 K, obtain speeds v1 and v2 for which F(v) = Fmax/2, and then calculate the fraction of particles with speeds v between v1 and v2



    2. Relevant equations
    F=4*(M/(2*Pi*R*T))^(3/2)*Pi*v^2*exp(-M*v^2/(2*R*T))

    3. The attempt at a solution

    Okay so I got A) already, however for B I am stuck. For B I found the Fmax (using the equation above and plugging in most probable velocity as V, i am on the right track?) and then I divided by 2. So now I have the value (fmax/2) and I tried rearranging this equation to find V1. Am I on the right track because when I tried isolating for V^2 I ended up with v^2 ln v^2 and I don't can seem to get it into more simpler form.

    I apologize if I posted this in wrong section, this is my first time posting here and I have no idea where this belongs. This question was from the subject:physical chemistry
     
    Last edited: Oct 8, 2016
  2. jcsd
  3. Oct 8, 2016 #2

    Charles Link

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    You find the maximum of ## F(v) ## which I think occurs at ## v_{max}=(2 kT/m)^{1/2} ## (set ## dF(v)/dv=0 ## ). Next, find ## v ## where ## F(v)=(1/2) F(v_{max}) ##. There will be two solutions for ## v ## where this occurs. Also, note your ## R ##, which is the universal gas constant is ## R=Nk ##. ## \ ## I should write for you ## v_{max}=(2RT/M)^{1/2} ## where ## M ## is the mass of a mole of helium atoms. ##(M=4 \, g=.004 \, kg. \ R=1.987 cal \cdot (4.184 \, joule/cal)/(mole \, deg \, K)) ## . My ## m ## above is the mass of one helium atom. ( ## N ## is Avagadro's number)
     
    Last edited: Oct 8, 2016
  4. Oct 8, 2016 #3

    BvU

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    Hello KK, :welcome:

    Well, what did you find for the most probable v (let's call it vmp) ?
    Can you post your work for part b?
     
  5. Oct 8, 2016 #4
    oh I hope its like that then I can do it easily. But isn't asking to count the Y axis (or have I interpreted this thing wrong?). Does Fmax refer to X axis? If it did wouldn't it just say calculate the most probable velocity ?
     
  6. Oct 8, 2016 #5
    Well not much, I tried isolating for V from that equation I provided above but I couldn't get it by it self. And, the V(most probable) I got = 1116.74m/s
     
  7. Oct 8, 2016 #6

    BvU

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    No. You were doing the right thing: solving for v at the v where the y (F) is half the Fmax
     
  8. Oct 8, 2016 #7

    Charles Link

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    The most probable velocity ## v_{mp} ## is a better name for it, and it occurs at the maximum of the function ## F(v) ##. You can find the ## v_{mp} ## and ## F(v_{mp}) ## by setting ## dF(v)/dv=0 ## and solving for ## v ##. This gives you ## v_{mp} ##. ## F_{max}=F(v_{mp}) ##.
     
  9. Oct 8, 2016 #8

    BvU

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    Numerical value isn't interesting yet.
    Can you post your work for part (b) ?
     
  10. Oct 8, 2016 #9

    BvU

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    I gather he/she has done that - worked it out numerically.
     
  11. Oct 8, 2016 #10

    Charles Link

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    I'm responding to the OP's question in post #4. Note to the OP: ## F(v) ## is the distribution (density ) function for the speeds of the atoms.
     
  12. Oct 8, 2016 #11

    BvU

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    Yes and you gave part a away in post #2. I'd like to read what KK found there -- and I also would like to have the full text of the exercise. Why do they ask for Fmax ? Or don't they ? What's KKs answer (including the dimension) ?
     
  13. Oct 8, 2016 #12
    Yea I did that for part A) but how am i suppose to count velocity at Fmax/2
     
  14. Oct 8, 2016 #13

    Charles Link

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    It should be easy enough to compute. Set ## F(v)=(1/2) F(v_{mp}) ## and solve for v. I think I'm giving you about as much or more than the Physics Forum rules permit. You need to put a little effort into it and also show your calculations please.
     
  15. Oct 8, 2016 #14
    alright so this is what I have so far

    Original equation and doing some rearranging, I get:
    upload_2016-10-8_22-25-33.png
    upload_2016-10-8_22-28-29.png
    upload_2016-10-8_22-29-58.png
    upload_2016-10-8_22-30-51.png
    upload_2016-10-8_22-37-14.png
    upload_2016-10-8_22-38-43.png

    got this far?? Is this correct so far? I cant seem to isolate V
     

    Attached Files:

  16. Oct 8, 2016 #15

    Attached Files:

  17. Oct 8, 2016 #16

    Charles Link

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    The constant in front of the expression will divide out (cancel) on both sides. Try using the expression ## F(v)=(1/2)F(v_{mp}) ## and solving for ## v ##. You can at least get a numerical answer. You should get two roots ## v_1 ## and ## v_2 ## for the temperature they gave you. I don't know that you will be able to get a simple algebraic expression for ## v ##. A suggestion would be to sketch the graph of ## F(v) ## and estimate what ## v_1 ## and ## v_2 ## are. If you tabulate the function ## F(v) ## using a spreadsheet and about 1000 points, that may be your simplest way of solving this.
     
  18. Oct 8, 2016 #17
    yea I did use that equation and when I rearrange it would leave me with this:

    upload_2016-10-8_23-6-4.png

    so now I just gotta solve for V tho? I am on the right track? correct?
     
  19. Oct 8, 2016 #18

    Charles Link

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    Please read my post # 18. I edited it slightly, so be sure to read the last couple of sentences.
     
  20. Oct 8, 2016 #19
    yes that makes sense but I am not sure we are allowed to do that. But still ,I will use your method if I don't get it resolved by tomorrow. Anyways, do you think there is a way to to isolate for V (from my work above)? you don't have to tell me the answer, yes/no will suffice as I don't really know what is allowed to be answered here and whats not.
     
  21. Oct 8, 2016 #20

    Charles Link

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    My first impression is no, there is no simple algebraic form. I only looked at it quickly, but the form ## v^2 exp^{-av^2}=C ## doesn't (as far as I can tell) have a simple algebraic solution for ## v ##. A sketch will allow you to estimate the ## v_1 ## and ## v_2 ## points very quickly, and a good tabulation could get you very accurate numbers for ## v_1 ## and ## v_2 ##. Once you estimate ## v_1 ## and ## v_2 ##, you could tabulate with higher resolution around those points, by first getting ## v_{mp} ## and ## F(v_{mp}) ## to high precision.
     
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