Boltzmann's version of 2nd Law -log relationship

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SUMMARY

Boltzmann defined entropy (S) as S = k log W, where W represents the number of microstates in a system and k is a constant. The logarithmic relationship arises because when combining two systems, the total number of microstates is the product of the individual states (W_a * W_b), necessitating the use of logarithms to maintain the additive property of entropy. This ensures that S(container A & container B) = S(container A) + S(container B). The discussion also touches on the transition from Boltzmann's entropy equation to Planck's energy equation E = Hf.

PREREQUISITES
  • Understanding of thermodynamic concepts, particularly entropy
  • Familiarity with statistical mechanics and microstates
  • Knowledge of Boltzmann's constant (k)
  • Basic grasp of quantum mechanics and energy quantization
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celal777
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Hello List,

I understand Boltzmann defined entropy as log W where W is the number of microstates in the system time a constant "k" ; hence S= k Log W.

Can someone explain to me where the logarithmic relationship comes from please ?

Many thanks in advance,

Celal Berker
London, England
 
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You want the Entropy to be a quantum that adds up when you're looking at more than one system: E. g. if you have two containers filled with gas, you want the relationship S(container A & container B) = S(container A) + S(container B).

But if the system A has W_a states and the system B has W_b states, the combined system has W_a W_b states: a product, not a sum. So youo have to take the logarithm of this number to get the relation above.
 
Many Thanks Bruno (is that your name?)

So how did Planck, by what thought or reasoning processes did he go from the relation S= k log W to E=Hf ?

--Celal
 

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