During this the centennial year of Albert Einstein’s epochal papers from his miraculous year, I was prompted to read again his paper on the quantum nature of radiation. 1 Surprisingly, I found a gap in his earliest argument for the existence of photons, a gap that apparently has remained unnoticed up to the present.
Einstein’s argument was based on his proof that for sufficiently large frequencies the entropy of thermal radiation varies logarithmically with the volume, in the same manner it does in an ideal gas. He concluded that “monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as if it consisted of mutually independent quanta [photons] of magnitude hv,”1 where v is the frequency of the radiation and h is Planck’s constant. In a footnote Einstein also gave a derivation, based on his entropy—volume relation, for the familiar pressure—volume formula of an ideal gas. But substituting in this formula Einstein’s relation for the number of photons—that is, the energy of the monochromatic thermal radiation divided by hv—results in an incorrect expression for the pressure. This pressure is one-third of the energy density of the radiation, as was originally shown by Maxwell for the case of electromagnetic waves. Moreover, a kinetic-theory calculation of the pressure shows that this relation is also valid for photons.
Maxwell’s relation also played a fundamental role in Boltzmann’s derivation of the dependence of thermal radiation on the fourth power of the temperature, and in Wien’s derivation of the general dependence of the thermal energy density on both frequency and temperature. I have not found any evidence, however, that Einstein’s contemporaries noticed the problem with calculating the pressure of monochromatic thermal radiation from Einstein’s entropy—volume relation, nor has this problem been pointed out in commentaries about this paper. 1,2
What is the solution to the pressure paradox? Einstein considered the volume dependence of the radiation entropy for fixed frequency v, but in an actual thermodynamic process, where the volume V is changed by moving a piston, the frequency does not remain fixed, because the wavelength is proportional to the linear dimensions of the cavity. Hence, the frequency v varies as V −1/3, giving the entropy an additional volume dependence that was not discussed by Einstein. It can be readily verified that this addition leads to a thermodynamic derivation of Maxwell’s relation for the pressure of isotropic radiation, filling in a longstanding gap in Einstein’s earliest thermodynamic argument for the existence of photons.
