Boltzmann's constant appears in many distributions of statistical physics, and I have been left confused whether it is always the same constant with certainty. For example, suppose we define the Boltzmann's constant so that it is the constant that works with certainty for gases, i.e. it gives the Maxwell's speed distribution right. Could it be then that the same Boltzmann's constant would no longer give the Planck's distribution for black body radiation right, and some "other Boltzmann's constant" would be needed? The question is reasonable, since it is an empirical fact that the Planck's distribution usually does not approximate the real empirical radiation distributions very well. This issue has not been seen as a serious flaw, since it has been explained by the fact that real hot objects are not ideal black bodies. Based on this alone one might think it would be reasonable to speculate that the Planck's distribution for black body radiation might actually need a different Boltzmann's constant that the Maxwell's speed distribution for gases. The question is affected by a claim that Max Planck actually produced an estimate for the Boltzmann's constant by studying the black body radiation. I have the book Introductory Statistical Mechanics by Bowley and Sanchez, and it says this concerning Planck's achievements: If Planck estimated Boltzmann's constant accurately from data concerning black body radiation, that would imply that the Boltzmann's constant is the same for gases and black body radiation after all. But how is that possible, since at the same time the real radiation distributions are usually not very close to the ideal black body radiation? What was that data that Planck used really?