Black Body Entropy: Solving the Puzzle of q and N

[LCSphysicist]

Homework Statement

I am having trouble to understand how the entropy of a black body was derived .

Relevant Equations

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I thought that would be something like, using similar counts from Einstein solid, $S = kln(\frac{(q+N-1)!}{q!(N-1)!})$
Where q is $hv$
v is frequency

But the results are not similar, so i am little stuck

 

[TSny]

I thought that would be something like, using similar counts from Einstein solid, S=kln((q+N−1)!q!(N−1)!)

Yes. With a proper interpretation of $q$, this expression for $S$ will apply to both the Einstein solid and Planck's blackbody-radiation resonators (oscillators).

Where q is hv
v is frequency

From the above expression for $S$, $q$ must be dimensionless since $N$ and 1 are dimensionless. So, $q$ can't denote the energy $h \nu$. What does $q$ actually represent for the Einstein solid? What does $q$ represent for the blackbody radiation system?

 

[LCSphysicist]

Yes. With a proper interpretation of $q$, this expression for $S$ will apply to both the Einstein solid and Planck's blackbody-radiation resonators (oscillators).

From the above expression for $S$, $q$ must be dimensionless since $N$ and 1 are dimensionless. So, $q$ can't denote the energy $h \nu$. What does $q$ actually represent for the Einstein solid? What does $q$ represent for the blackbody radiation system?

In Einstein solid, it is in fact the energy unit, as far as i know. I never noticed that you pointed. Thinking better, maybe q can be equal to the total energy E divided by the energy unit hv, that is, $q = E_{tot}/hv$?

 

[TSny]

In Einstein solid, it is in fact the energy unit, as far as i know. I never noticed that you pointed. Thinking better, maybe q can be equal to the total energy E divided by the energy unit hv, that is, $q = E_{tot}/hv$?

Yes, $q$ is the total number of energy quanta in the system of N oscillators.