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

[Kilian Stenning]

Homework Statement

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 .

Homework Equations

f(z) = Cexp(-mgh/kt)
<z> =∫ z fz dz/∫ fz dz

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'hospital's rule w.r.t T but I can't seem to get the right answer

 

[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 !

 

[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).

 

[Kilian Stenning]

thanks got it now!