How did Planck use the concept of statistical entropy in trying to understand the meaning of his own law?

Question

I was reading Introducing Quantum Theory: A graphic guide (by J.P.McEvoy & Oscar Zarate) and came across Planck's predicament of understanding his very own law that accurately explained the blackbody radiation. It says (in the voice of Planck):

... From the very day I formulated the radiation law, I began to devote myself to the task of investigating it with true physical intuition. After trying every possible approach using traditional classical applications of laws of thermodynamics, I was desperate. But forcefully I was compelled to consider statistical version of the second law $$S = k .\log{W}$$ that I rejected and hated the most. After some of the most intense weeks of my life, the light began to appear to me ... The light was energy is not continuous but discrete in nature.

That was hard indeed! But why did he hate the Boltzmann's version of entropy??

Nevertheless, my question is: How did Mr. Planck use the entropy concept to come to the revolutionary conclusion that energy is discrete ? Please give me a math-free explanation.

Answer 1

Boltzmann's version of entropy require a finite number of states, and Planck had asserted that probability has no meaning without a “finite number of equally likely configurations.”{1} That is, in order to be able to use Boltzmann's equation and obtain finite results, he needed to use a discrete number of states, but light was supposed to be continuous. Some background: Einstein investigated the photoelectric effect and was able to resolve certain difficulties in the application of Planck’s theory by viewing quantum light bundles as particles (photons) of energy. However, other contradictions were created in the process. In Planck’s analysis, rays of light from a nonfocused system are emitted in all directions, which does not fit the definition of a particle. Bragg reflections of X-rays from a crystal exhibit wavelike properties, and particles have never been considered to have such a structure. Various particles, such as electrons, neutrons, and protons, also show wavelike properties. Electromagnetic radiation from an antenna can only have a narrow beam if the aperture is a large number of wavelengths and a corresponding number of elements are operating in synchronization. The size of atoms (10–10 m for hydrogen) is much smaller than the vast majority of the wavelengths of blackbody radiation. If short-wavelength radiation is electromagnetic in nature, the particle theory has another conflict.

Today’s scientists have settled for the “wave–particle duality” concept, in which particles sometimes act like waves and vice versa.

{1} Planck, M. Eight Lectures on Theoretical Physics; Wills, A. P., Translator; Columbia University: New York, 1915.