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

The equation developed independently by Sackur and Tetrode
can easily leave one queasy, on examination.
It leads to unrealistic values of entropy at low temperature,
and plays into the unhappy construction that
"A difference that makes no difference, is no difference"
paraphrasing Gibbs, when referring to his eponymous paradox
whereby, if you partition a box filled with a homogeneous gas
in equilibrium, the partition increases the entropy as defined...
So, ignore it, until you find a case where the properties of the
partitioned gas DO differ. A sort of observer effect...
(I am writing slightly tongue in cheek, no doubt.)

Perhaps Bob's recognition of the label, spectroscopic entropy can
be associated with the name for Λ - the thermal wavelength?

Brian W

LaMontagne, Bob wrote:
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. /snip/