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# Re: [Phys-l] polytropic Sackur-Tetrode

This formula is more commonly known as the 'spectroscopic entropy'. It is basically the derivative of the free energy in terms of T with V held constant. Λ is h/(2 pi m k T)^1/2. The formula differs from yours only by the inclusion of a factor representing the weight of the lowest electronic state, w.

S/N V/N γ
----- = ln w----- + -----
k Λ^3 γ-1

Wilson: Thermodynamics and Statistical Mechanics, Cambridge, 1960

Bob at PC

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Wednesday, January 20, 2010 12:44 AM
To: Forum for Physics Educators
Subject: [Phys-l] polytropic Sackur-Tetrode

Alas on 01/17/2010 04:17 PM, I mentioned the Sackur-Tetrode
formula in connection with the entropy of air:

Λ >
which is not correct. Equation [1] only applies to
monatomic gases. My apologies.

This led me to derive a generalization of this formula
for use with polytropic gases. To wit:

S/N V/N γ
----- = ln ----- + ----- [2]
k Λ^3 γ-1

V/N Df
= ln ----- + 1 + ----- [3]
Λ^3 2

where Df is the effective number of degrees of freedom,
to the extent that any such thing exists.

I cannot imagine that equation [2] or [3] is original
with me. Something this simple, this useful, and this
easy to derive must have been derived 100 times before
now. For all I know, Sackur and/or Tetrode derived it
circa 1912.

On the other hand, I must say that I don't recall ever
seeing it before. I spent a few minutes googling for
it without success. So this is a mystery; I would have
expected every thermo reference on earth to include it.
If anybody knows of a relevant reference, please let
me know.

In any case, I thought some folks would find it interesting.
Even if you don't remember the details, it's nice to
know such a thing exists, and can easily be rederived
if/when needed. For additional details, including a
sketch of the derivation, see
http://www.av8n.com/physics/thermo-laws.htm#eq-s-monatomic
http://www.av8n.com/physics/thermo-laws.htm#eq-sackur-polytropic
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