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Equilibrium temperature distribution of ideal gas in homogeneous gravitational field

  1. Jan 5, 2006 #1

    mma

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    Could somebody tell me, how temperature of an ideal gas varies on height in homogeneous gravitational field in equilibrium?
    I mean a gas column perfectly isolated from its environment.
     
    Last edited: Jan 6, 2006
  2. jcsd
  3. Jan 6, 2006 #2

    mma

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    The interesting in this question is, that a thermodynamic system in equilibrium has only one temperature: T=dU/dS, so our gas column has only one temperature.
    But, if we divide our gas column into horizontal layers, then considering these layers as thermodynamic systems, they will have different temperatures, because the average speed of the molecules is greater on the lower layers as in the upper ones. It is a necessity, because every molecule moving upward loses from its speed.
    On the other hand, neighboring layers are connected with each other thermally, and therefore they must have equal temperature in equilibrium, so we come again to the other consequence, that our system has only one temperature.
    This is a contradiction. What is the solution?
     
  4. Jan 6, 2006 #3

    pervect

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    Staff Emeritus
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    In non-extreme situations (i.e. planetary atmospheres), where there is also very little heat conduction, you can use the approximation of the adiabatic atmosphere.

    This uses the "adiabatic gas law" for the rate of cooling as the gas expands, plus the usual hydrodynamic equilibrium equations.

    See for instance

    http://farside.ph.utexas.edu/teaching/sm1/lectures/node56.html
    http://daphne.palomar.edu/jthorngren/adiabatic_processes.htm

    If you are interested in exotic situations, relativistic effects may become important.
     
  5. Jan 8, 2006 #4

    mma

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    No, thank you, the non-relativistic approach is enough for me. But this adiabatic approximation seems too heuristic for my taste.
    I've found an article for the single-particle distribution for an ideal gas in a gravitational field:
    http://gita.grainger.uiuc.edu/IOPText/0143-0807/16/2/008/ej950208.pdf
    But how can I calculate the temperature distribution from this?
     
  6. Jan 10, 2006 #5

    mma

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    OK. Then what about this? :
    http://www.iop.org/EJ/abstract/0143-0807/17/1/008
    What is surprising, that concrete mathematical analysis also shows that (1) is the correct answer in the thermodynamical limit.
    Isn't it interesting?
     
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