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As far as I understand, Wien defines entropy, which he uses in his derivation, as

$$ S = \text{v} \int\limits_0^{\infty} \varphi(\rho, \nu) d \nu, $$

where $\text{v}$ is the volume occupied by the radiation, $\nu$ is the frequency of that radiation, $\rho(\nu)$ is the radiation density and $\varphi$ is a function of the variables $\rho$ and $\nu$.

Unfortunately, I can't find an English translation of his seminal paper

Wien, W., "Temperatur und Entropie der Strahlung", Ann. Phys. Chem., 52, 132-165 (1894),

where he has developed his ideas on the matter. I would appreciate it if someone who has dealt with this problem could shed some light on this important derivation and/or possibly give a link to an English translation or perhaps a translation in Russian of the paper.

ANSWER:

One can at once see how this integral can be entropy by considering $\varphi = \frac{\rho(\nu)}{T}$, where $\rho(\nu)$ is radiation density. So, then, we will have

$$ \text{v} \int\limits_0^{\infty} \varphi(\rho, \nu) d\nu = \frac{\text{v}}{T} \int\limits_0^{\infty} \rho(\nu) d\nu = \frac{U}{T} = \frac{Q}{T} = S$$

because the process is isochoric.

Unfortunately the above formula for $S$ cannot be used to recover the II law for an isochoric process

$$\frac{dS}{dU} = \frac{1}{T},$$

so $S_\nu$ must be used, expressing the entropy for a given frequency and not over all frequencies as in the integral shown in the OP, which I don't believe Wien used in any of his works (unless, someone can show otherwise).

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    $\begingroup$ Could you say the page of the original paper you are looking at? I can’t seem to find the entropy equation you mention. $\endgroup$
    – auden
    Dec 30, 2017 at 21:32
  • $\begingroup$ That's the problem. I just guessed that the definition in question of (S) might be in that paper but I can't find it in English (or Russian) to see. It very well may be that said definition is in some other of Wien's papers. I wasn't able to find Wien's Collected Works either, if such collection exists at all put together. $\endgroup$
    – ganzewoort
    Dec 31, 2017 at 5:17
  • $\begingroup$ Okay, when you say 'this important derivation', the derivation of which law? $\endgroup$
    – auden
    Dec 31, 2017 at 16:45
  • $\begingroup$ Wien's blackbody radiation law in the form $\frac{\partial \varphi}{\partial \rho} = \frac{1}{T}$. Where in Wien's works and how is this expression shown to be connected with his law? $\endgroup$
    – ganzewoort
    Dec 31, 2017 at 20:59
  • $\begingroup$ Hi @ganzewoort. An answer should be in an answer; not in the question. $\endgroup$
    – Qmechanic
    Sep 26, 2021 at 12:36

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If you started to think of radiation in a closed volume like gas molecules and apply the ideal gas law to the radiation photons you might get a start on how to derive the origin of the expression you are querying. I remember some time ago reading that was how Wien derived the formula.

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  • $\begingroup$ I don't believe Wien used that definition of entropy anywhere in his works. $\endgroup$
    – ganzewoort
    Jan 3, 2018 at 22:57
  • $\begingroup$ To ganzewoort: how did you make out pv\T where p(v) =radiation density is an entropy expression? Should it not be V(PV?)/T $\endgroup$
    – user177923
    Jan 15, 2018 at 16:34

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