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Pipes, resonating frequencies and yeah some gases

  1. Apr 23, 2015 #1
    1. The problem statement, all variables and given/known data
    A closed organ pipe resonates in its fundamental mode at a frequency of ##200Hz## in ##O_2## at a certain temperature. If the pipe contains 2 moles of ##O_2## and 3 moles of ##O_3## are now added to it, then what will be the fundamental frequency of same pipe at same temperature?
    [Given answer is ##172. 7Hz##]

    2. Relevant equations
    The relevant equation according to me is:
    ##v =√(\gamma P/\rho) ## (in a gas speed of a wave)

    3. The attempt at a solution
    What I did is:
    Velocity of a wave in a gas=
    $$v=√(\gamma P/\rho)$$
    So using ##v=\nu\lambda##
    We can say that
    ##\nu_1/\nu_2=√(\rho_2/\rho_1)##
    And we know that
    ##\rho_1=4×16/V ## where V is the volume of pipe
    And ##\rho_2=3×3×16+2×2×16/V=13×16/V##
    Therefore my frequency is coming out to be
    ##110 Hz##
    Which is wrong.. I know the whole of this seems to be wrong from the start... so how to do this?
     
  2. jcsd
  3. Apr 23, 2015 #2

    BvU

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    Are you assuming P and ##\gamma## remain the same ?
     
  4. Apr 23, 2015 #3
    First what is the gamma for diatomic gas?
     
  5. Apr 23, 2015 #4
    Okay, ##\gamma## Will also change for ##O_2## and ##O_3##.
    For a diatomic gas it is 7/5 and for a triatomic gas it's 4/3.
     
    Last edited: Apr 23, 2015
  6. Apr 23, 2015 #5
    Thanks I got it now:smile:
     
  7. Apr 23, 2015 #6

    BvU

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    Care to enlighten us with your workings ?
     
  8. Apr 23, 2015 #7
    Yup... why not...
    I just put the values of ##\gamma ##
    too, instead of cancelling them while comparing the resonant frequencies in the two cases.
     
  9. Apr 23, 2015 #8
    But γ would not be 4/3 when you mix both of them.
    What would be equivalent γ then?
     
  10. Apr 23, 2015 #9
    Hmmm.. ##\gamma## can be calculated by comparing the internal energies of the gases in the two cases. :
    ## n_1C_v1ΔT+n_2C_v2ΔT=(n_1+n_2)C_{net}Δ T##
    Cancelling ##ΔT##
    ##2×(7/5)R+3×(4/3)R=5×R/(\gamma-1)##
    Cancelling ##R##
    ##\gamma=59/34##
    But it's giving the answer 123. 51 Hz
     
  11. Apr 23, 2015 #10
    What about P?
     
  12. Apr 23, 2015 #11

    BvU

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    I would expect ##\gamma## to come out between 1.4 and 1.3, not at 1.74 :rolleyes:
     
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