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Can anyone identify the approximation used in this solution?

  1. Feb 27, 2014 #1
    I am trying to follow the reasoning of the last problem in the set linked below. I can't figure out what approximation they used in step 24. Thanks.

    http://www.physics.fsu.edu/courses/spring08/phy5524/sol1.pdf

    It looks like it might be an approximation based on a geometric series, but there is no summation.
     
    Last edited: Feb 27, 2014
  2. jcsd
  3. Feb 27, 2014 #2

    D H

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    They're using the approximation ##\frac 1 {1-x} \approx 1+x##, or even more appropriately, ##1+x \approx \frac 1 {1-x}## where ##x=\frac {N-1}2 \frac {v_0} V##.
     
  4. Feb 27, 2014 #3

    Simon Bridge

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    There is a summation at step 20.
    The approximation comes in at 21-22

    But you want to know how you go from $$\frac{P}{k_BT}\simeq \frac{1}{V-(N-1)v_0/2}$$ to $$\frac{P}{k_BT}\simeq \frac{1}{V-Nv_0/2}$$ ... well ##(N-1)x \simeq Nx## if ##x<<Nx## right? Presumably N is a very big number?

    [edit] DH has it formally :)
     
  5. Feb 27, 2014 #4
    Thanks for your help! DH, I assume that approximation is only valid for small x correct? It seems as if N is very large, then the approximation wouldn't be a good one.
     
  6. Feb 27, 2014 #5

    Simon Bridge

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    Hmmm... I didn't check it properly.
    We seem to be saying different things after all - I am pointing out that for large ##N##, ##N - 1 \approx N##.

    You can get some very hokey-looking approximations in physics.
     
  7. Feb 28, 2014 #6

    D H

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    Yes, N is large, which makes step 25 valid, where N-1 is simplified to N. The assumption that makes step 24 valid is that ##Nv_0 \ll V##. In other words, this assumption says the volume excluded by all of the gas molecules is much less than the total volume of the system.
     
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