# Wien on Temperature and Entropy

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.

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).

• Could you say the page of the original paper you are looking at? I can’t seem to find the entropy equation you mention. Dec 30, 2017 at 21:32
• 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. Dec 31, 2017 at 5:17
• Okay, when you say 'this important derivation', the derivation of which law? Dec 31, 2017 at 16:45
• 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? Dec 31, 2017 at 20:59
• Hi @ganzewoort. An answer should be in an answer; not in the question. Sep 26, 2021 at 12:36