Homework Help: Average height of molecules in a box as a function of temperature

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1. Feb 23, 2016

Kilian Stenning

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. Feb 23, 2016

BvU

Hello Kilian,

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 !

3. Feb 23, 2016

TSny

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).

4. Feb 23, 2016

Kilian Stenning

thanks got it now!