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What fraction of molecules in an ideal gas have velocites between φ1 & φ2 and θ1 & θ2

  1. Mar 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Approximately what fraction of molecules of a gas (assumed ideal) have velocities for
    which the angle φ lies between 29.5° and 30.5°, while θ lies between 44.5° and 45.5°?

    2. Relevant equations



    3. The attempt at a solution

    What does the question even mean geomtrically?
     
  2. jcsd
  3. Mar 25, 2012 #2
    Re: what fraction of molecules in an ideal gas have velocites between φ1 & φ2 and θ1

    Since the Maxwell Boltzmann distribution for a free diluite gas depend only on the modulus of the velocity, the fraction of the particles that have velocity between the angles defined by [itex]\phi_1[/itex] and [itex]\phi_2[/itex], [itex]\theta_1[/itex] and [itex]\theta_2[/itex] is simply the fraction of the area of part of the spherical surface determined by those parameters and the surface of the sphere; namely

    fraction= [itex]\frac{1}{4\pi}(\phi_2-\phi_1)([/itex]cos[itex](\theta_1)-[/itex]cos[itex](\theta_2))[/itex]

    all the angles are expressed in radiants.
     
  4. Mar 26, 2012 #3
    Re: what fraction of molecules in an ideal gas have velocites between φ1 & φ2 and θ1

    Thanks mate, is there somewhere I could get a detail analysis and derivation of this, because it wasn't in any lectures and isn't in my text at all.
     
  5. Mar 27, 2012 #4
    Re: what fraction of molecules in an ideal gas have velocites between φ1 & φ2 and θ1

    You are welcome; the best place where one can study all this stuff at the introductory level is Huang's textbook "Statistical Mechanics". Best,
    f.
     
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