1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Average height of molecules in a box as a function of temperature

  1. Feb 23, 2016 #1
    1. The problem statement, all variables and given/known data
    A circular cylinder of height H is filled with monatomic gas molecules at temperature T. The cylinder stands on the surface of the earth so that the gas molecules are subject to the gravitational field g.

    (a) Find the average height, z , of the molecules in the cylinder as a function of temperature. (Hint: The probability of finding a molecule at height z is governed by the Boltzmann distribution). Show that for

    T → 0, z = 0 , and for T → ∞ , z = H/2 .

    2. Relevant equations
    f(z) = Cexp(-mgh/kt)
    <z> =∫ z fz dz/∫ fz dz

    3. The attempt at a solution
    let B=mg/kT
    Ive got an answer for <z> to be (1/B)*(1-(e^(-BH))*(BH+1))/(1-e^(-BH))
    To get the limits I've tried to use l'hopitals rule w.r.t T but I cant seem to get the right answer
     
  2. jcsd
  3. Feb 23, 2016 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Hello Kilian, :welcome:

    The brackets tend to make it hard to read. $$
    (1/B)\ * \ (1\ \ -\ \ (\exp(-BH))*(BH+1)\ \ \ )\ \quad /\quad (1-\exp (-BH))
    $$Do I see $$ <Z> \ = {kT\over mg} \ \
    { 1 - \left ({mgh\over kT}+1\right ) e^{-{mgh\over kT}} \over
    1 - e^{-{mgh\over kT} }} \quad ?$$
    In which case I see the ##T\rightarrow 0## limit, but not the other one !
     
  4. Feb 23, 2016 #3

    TSny

    User Avatar
    Homework Helper
    Gold Member

    For the T → ∞ case, you will need to use l'Hopital's rule twice. Or you can expand the exponentials to second order in BH = mgH/(kT).
     
  5. Feb 23, 2016 #4
    thanks got it now!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Average height of molecules in a box as a function of temperature
Loading...