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

AI Thread Summary
The discussion focuses on calculating the average height of gas molecules in a cylindrical container as a function of temperature, using the Boltzmann distribution. The average height, z, is derived from the probability distribution of the molecules, leading to the equation <z> = (1/B)*(1-(e^(-BH))(BH+1))/(1-e^(-BH)), where B = mg/kT. The limits of z are established, showing that as temperature approaches zero, z approaches zero, and as temperature approaches infinity, z approaches H/2. Participants discuss the application of L'Hôpital's rule to find these limits, particularly for the T → ∞ case. The conversation emphasizes the importance of correctly manipulating the exponential terms to arrive at the desired results.
Kilian Stenning
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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'hopitals rule w.r.t T but I can't seem to get the right answer
 
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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 !
 
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).
 
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Likes BvU
thanks got it now!
 
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