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possible to determine the black body radiation from the function φ. Conversely, through integration one can obtain φ from the black body radiation law keeping in mind that φ vanishes for ρ = 0.

Limiting law for the entropy of monochromatic radiation at low radiation density

Admittedly, the observations of "black body radiation" so far indicate that the law that Mr. Wien originally devised for the "black body radiation"

$\rho =\alpha \nu ^{3}e^{-\beta {\tfrac {\nu }{T}}}$

is not exactly valid. However, for large values of ν/T experiment completely confirms the law. We shall base our calculations on this formula, keeping in mind that the results will be valid within certain limitations only.

First, we get from this equation:

${\frac {1}{T}}=-{\frac {1}{\beta \nu }}\lg {\frac {\rho }{\alpha \nu ^{3}}}$

and then, using the relation obtained in the preceding section:

$\phi (\rho ,\nu )=-{\frac {\rho }{\beta \nu }}\left\{\lg {\frac {\rho }{\alpha \nu ^{3}}}-1\right\}.$

Let there be a radiation of energy E, with a frequency between ν and ν + dν. Let the radiation extend over volume v. The entropy of this radiation is:

$S=v\phi (\rho ,\nu )d\nu =-{\frac {E}{\beta \nu }}\left\{\lg {\frac {E}{v\alpha \nu ^{3}d\nu }}-1\right\}.$

We will limit ourselves to investigating the dependency of the radiation's entropy on the volume that is occupied. Let the entropy of the radiation be called S₀ when it occupies the volume v₀, then we get:

$S-S_{0}={\frac {E}{\beta \nu }}\lg \left({\frac {v}{v_{0}}}\right).$

This equation shows that the entropy of monochromatic radiation of sufficiently low density varies with volume according to the same law as the entropy of an ideal gas or that of a dilute solution.