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Kinetic Theory of Gas problem

  1. Feb 11, 2015 #1
    1. The problem statement, all variables and given/known data

    Helium gas with T1 = 500K and P1 = 0.02MPa in a rigid container with volume V = 1 cm^3.
    Then Helium goes through a process where atoms with kinetic energies greater than kB*T1, where kB is Boltzmann constant, are instantaneously removed from the container.
    Atoms remaining in the container attain a M-B velocity distribution with a final Temperature T2.
    Calculate T2 and the final pressure P2.

    2. Relevant equations
    (1) PV = N * kB * T
    M-B speed distribution:
    (2) f(v) = 4 * pi * v^2 * (m / (2*pi*kB * T))^(3/2) * exp( - m * v^2 / (2 * kB * T))


    3. The attempt at a solution
    First, I find the number of molecules using (1) and the total number of atoms in state 1 is 2.987e18.
    Then find the limiting velocity by setting E = kB* T = 1/2 * m * v^2 to solve for v,
    integrate from 0 to v ( f(v) dv ) to find the fraction of the atoms up to this speed v = 0.463.
    The remaining atoms will be this fraction multiply by the number of atoms = 1.34e18 atoms remaining.
    Then I am lost on solving for T2 with this remaining atoms that attain M-B velocity distribution.

    Any help will be appreciated. Thanks!
     
  2. jcsd
  3. Feb 11, 2015 #2

    DrClaude

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    Staff: Mentor

    I'll give you a hint: when you remove the atoms, you are also taking away their energy.
     
  4. Feb 11, 2015 #3
    That means if I can find the total energy in state 1, I can also find what is left in state 2?

    Thanks for your fast reply DrClaude.
     
  5. Feb 11, 2015 #4

    DrClaude

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    Staff: Mentor

    Since you have figured out the cutoff velocity, it should be easy to get the kinetic energy of the atoms that are left.
     
  6. Feb 11, 2015 #5
    Hmm sorry DrClaude, I am still a bit confused.
    Does it mean sense that if I integrate from 0 to cutoff velocity ( f(v) dv) and set it = 1, I can find a temperature that satisfy this?
     
  7. Feb 11, 2015 #6

    DrClaude

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    Staff: Mentor

    What you need to do is use the distribution function to find the energy of the atoms in the velocity range 0 to cutoff at T1. Then you should find a T2 that gives you the same energy.

    Note that calculating the energy is not simply integrating f(v) dv. (This should be in your textbook.)
     
  8. Feb 11, 2015 #7
    My bad. From the text I found, the energy is:

    E = integral ( 0 to cutoff velocity) ( 1/2 * m *v^2 * f(v) dv) , then uses the value to find T2, where T2 = E/kB.

    And I have T2 and number of remaining atoms, getting P2 should be no problem.

    Thanks for the help!
     
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