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I Ideal gas pressure from Maxwell-Boltzmann distribution

  1. May 19, 2017 #1
    Hi everyone

    I'm having trouble with solving an exercise in statistical physics. I need to argue why the average number of particles with a velocity between ##v## and ##v+dv## that hit a surface area ##A## on the container wall in a time interval ##\Delta t## is $$N_{collision}=v_{x}A\Delta t f(\mathrm{\textbf{v}})dv_{x}dv_{y}dv_{z}$$ where ##f(\mathrm{\textbf{v}})## is the Maxwell-Boltzmann distribution. Consider the gas as an ideal gas.

    I don't quiet know where to start so...

    Thanks for your help!
     
    Last edited: May 19, 2017
  2. jcsd
  3. May 19, 2017 #2

    stevendaryl

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    Are you sure that there isn't an additional factor of density (number of particles per unit volume) involved?

    What I would say is this: You have a wall that is oriented perpendicular to the x-axis. So for particles traveling at velocity [itex]v_x[/itex] in the x-direction, consider all particles that will hit the wall in the next [itex]\delta t[/itex] seconds. Clearly, for a particle to hit the wall in that time interval, it must be closer than [itex]v_x \delta t[/itex] in the x-direction. So there is a certain region of space that contains all the particles that could possibly hit the wall in the next [itex]\delta t[/itex] seconds (traveling at velocity [itex]v_x[/itex] in the x-direction). What is the volume of that region? The number of particles in that region is proportional to that volume.
     
  4. May 19, 2017 #3
    Thank you very much for your answer!
    I guess the number of particle traveling in x-direction would be: $$n_{x}=\int_{0}^{\infty} f(\mathrm{\textbf{v}}) dv_{x}$$ right? But how do I include the infinitesimal time Intervall ##\Delta t##?
     
  5. May 19, 2017 #4

    stevendaryl

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    Can you answer my question: what is the volume of the region of points such that a particle at that point will hit the wall in the next [itex]\Delta t[/itex] seconds?
     
  6. May 19, 2017 #5
    You said that:
    So by finding the number of particle which will hit the wall in the next ##\Delta t## seconds I can find the volume? The number of particle which will hit the wall should be equal to: $$dn_{x}=\Delta tv_{x}f(\mathrm{\textbf{v}})dv_{x}dv_{y}dv_{z}$$ with $$\mathrm{\textbf{v}}=
    \begin{bmatrix}
    v_{x}\\
    0\\
    0
    \end{bmatrix}$$
    Now i need to link this with the volume right?
    I'm not that good in statistical physics so what I just stated could be completely wrong...
    Thanks for your help!
     
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