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Temperature inside a box as particles escape a small hole

  1. Nov 2, 2016 #1
    1. The problem statement, all variables and given/known data
    Edit: Thanks to gneill for showing me the LaTeX ropes. Equations should work now.

    A perfect gas enclosed within a container escapes into vacuum through a small circular hole. The particle flux through the hole is ##\frac{nc}{4}## and the energy per particle escaping can be found to be 2kT. Assuming that the volume of the container is fixed, find an expression for the temperature inside the box starting with an initial temperature of ##T_o##.

    2. Relevant equations
    $$J_n = \frac{nc}{4}$$
    $$E_{avg} = \frac{3}{2}kT$$
    $$E_{esc} = 2kT$$
    $$R_{eff} = \frac{PA}{\sqrt{2\pi MRT}}$$

    3. The attempt at a solution
    Temperature is simply an average of the kinetic energy of all the particles. So before any particles escape, the average kinetic energy in the box is

    $$E_{0} = \frac{3}{2}kT_o$$

    The average kinetic energy will drop by the rate of effusion times the average kinetic energy of the escaping particles, or

    $$KE_{box} = \frac{3}{2}kT_o - \frac{PA}{\sqrt{2\pi MRT_o}}*2kT_o$$

    I don't think this is quite right though. It needs an integration somewhere from ##T_o## to 0? Any nudges in the right direction are appreciated.
     
    Last edited: Nov 2, 2016
  2. jcsd
  3. Nov 2, 2016 #2

    gneill

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    Staff: Mentor

    To have your LaTeX syntax recognized you need to surround it with tags. The tags for inline equations are double hash marks (#). If you use double dollar signs ($) instead, the equation will be placed centered on its own line and in a larger font.

    Here's an example:

    The following equation is inline in this line of text ##J_n = \frac{nc}{4}## as you can see. The next one will be placed on its own line $$E_{avg} = \frac{3}{2}kT$$ all by itself and with a larger font.

    If you Reply to this post you'll be able to see the embedded tags in the quote.
     
  4. Nov 3, 2016 #3

    rude man

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    Kind of out of my line, but why would the particle energy leaving the box be greater than the particle energy inside the box?

    And what happened to nc/4 and what are the symbols and units? Isn't THAT the effusion rate rather than your equation for Reff?
     
  5. Nov 3, 2016 #4

    haruspex

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    The temperature is falling, so the rate of energy loss falls over time. Write an expression for the rate of loss of energy as a function of the temperature at some arbitrary moment.
    What are n and c, or is it a single variable nc?
     
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