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Thermodynamic entropy of system of any size. 
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#1
Aug1811, 03:24 AM

P: 159

After a bit of calculation, I came up with the following quantity for the bitentropy of a thermodynamic system.
We have the following assumptions: 1. System at thermal equilibrium. 2. Ideal gas. 3. Monatomic gas (i.e. no internal degrees of freedom for particles). 4. All particles have equal mass. 5. Units are such that k_B (boltzmann constant) normalized to 1. Using just informationtheoretic arguments (no assumptions from thermodynamics!) I calculated the raw entropy of such a system to be: S = (Q/T)[(log(T/T0)  1) + (log(V/V0)  1) + log(Q/T)] (T=temperature, Q=thermal energy, V=volume of system, T0,V0=unknown normalizing constants). This can be simplified to: S = (Q/T)[log(Q/T0) + log(V/V0)  2] Further, I suspect it works for any system size, even systems that wouldn't be called 'ensembles' in the thermodynamic sense (like just a single particle, in which case Q=0. In general, we take Q = total energy  kinetic energy of center of mass of system). In addition, we find that dS is proportional to dQ/T (i.e. Clausius law of entropy), in the limit where Q >> T (which is always true in thermodynamic ensembles) and volume is held constant. Yet another interesting thing about this is that the entropy is not zero at the limit of T=0 (because then Q=0 too). Thus it appears the third law of thermodynamics need not apply from a purely informationtheoretic standpoint. Is my formula correct? 


#2
Aug1811, 04:17 AM

Sci Advisor
P: 3,564



#3
Aug1811, 04:51 AM

P: 159

Thanks for that link, I didn't know about that. I was just trying to satisfy my own curiosity.
The SackurTetrode equation appears only to work under the assumption of the uncertainty principle, whereas I derived my formula without any QM assumptions. Therefore I don't think they can be directly compared. However, this part: 


#4
Aug1811, 06:41 AM

Sci Advisor
P: 3,564

Thermodynamic entropy of system of any size.
If I am not wrong, Sakur and Tetrode derived their formula in a classical mechanics context (in 1912 QM was not yet discovered), i.e. without involving the uncertainty principle.



#5
Aug1811, 07:05 PM

P: 159

No, the SackurTetrode equation requires Planck's constant, which was of course discovered in 1899.
From http://en.wikipedia.org/wiki/Ideal_gas : 


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