(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show how the Boltzmann entropy is derived from the Gibbs entropy for systems in equilibrium.

2. Relevant equations

Gibbs entropy S= - [tex]\int[/tex] [tex]\rho[/tex](p,q) (ln [tex]\rho[/tex](p,q)) dpdq

where [tex]\rho[/tex](p,q) is the probability distribution

Boltzmann entropy S= ln[tex]\Omega[/tex]

where [tex]\Omega[/tex] is the number of microstates in a given macrostate.

3. The attempt at a solution

1. Well, when the system is in equilibrium (ie when the Boltzmann entropy can be used) all microstates have equal probability. So this means that each microstate has a probability of 1/[tex]\Omega[/tex] and the probability distribution [tex]\rho[/tex] will have a constant value regardless of what p and q are.

2. I tried putting [tex]\rho[/tex]=1/[tex]\Omega[/tex] and subbing it into the Gibb's equation

S= - [tex]\int[/tex] 1/[tex]\Omega[/tex] (ln [tex]\1/[tex]\Omega[/tex]) d[tex]\Omega[/tex]

using d[tex]\Omega[/tex] since we want to add up over all the microstates and there are

[tex]\Omega[/tex] of them. But I can see that this won't give me the Boltzmann entropy.

Any ideas?

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# Show how the Boltzmann entropy is derived from the Gibbs entropy for equilibrium

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