Supplement to '"Nature" October ii, 1906 



SUPPLEMENT TO "NATURE." 



lUK EXTROPy Of h'ADIATlOS. 

 ]'orlcsi(iigcii liber die Theorie dcr Wiinucslraldiiiix- 

 Hv Dr. Max Planck. Pp. vili + 222. (Leipzig: 

 johann .\nibrosius Barth, igo6.) Price 7 marks. 



A QUANTITY of heat Q is transferred by radi- 

 ation from a bod)' the surface of which is at 

 temperature T, to a body the surface of which is at 

 a lower temperature T^. From this cause alone the 

 former body loses a quantity of entropy Q/T,, while 

 the latter gains a quantity of entropy Q/T.^. The net 

 g.iin of entropy of the bodies arising from the trans- 

 .•iction is 0(i/T,— i/T,). Where and how does this 

 gain take place ? 



Slime people have expressed the view thai the notion 

 of entropy has no place in radiation phenomena, and 

 that it is only a convenient symbol introduced for the 

 purpose of representing a certain class of heat pheno- 

 mena occurring in material bodies. But the entropy 

 gained or lost by a body measures the gain or loss 

 of unavailable energy on the supposition that energy 

 c.in be converted into work by means of ideal re- 

 versible engines working between the body and an 

 indefinite medium at unit absolute temperature. It 

 will thus be seen that the change of entropy above 

 considered represents a definite amount of what Mr. 

 .Swinburne calls " incurred waste," a change which 

 c.innot be undone, which leaves an indelible imprint 

 on the state of the universe, which represents a loss 

 of availability, or, from an engineering point of view, 

 a Uiss of value. \\'e might say that, though the 

 energy Q has not been altered in amount, it has be- 

 conif a less marketable commodity by the change. It 

 thus becomes important to examine exactly where and 

 how the change of entropy has taken place, that is, 

 to extend the notion of entropy to the ether. 



II we begin b\ attempting to applv reversible 

 thciniodynamics to the ether, we arrive ,it a single 

 result only, namely, Boltzmann's differential equation 

 connecting Maxwell's formula for the radiation 

 pressure with .Stefan's law for the intensity of radi- 

 ation inside a black cavity. For this particular kind 

 of radiation entropy is fully defined, and the energy 

 per unit volume being proportional to the fourth 

 power of the absolute temperature, the entropy is pro- 

 portional to the cube, being 4/3 of the energy divided 

 by ihe temperature. 



In this case there is no violation of the relation 

 between unavailable energy and entropy which forms 

 the basis of the thermodynamics of a material body. 

 At the temperature of the cavity the unavailable 

 energy represented by the entropy is 4/3 of the total 

 energy, but the discrepancy is accounted for by the 

 work of expansion against the radiation pressure. 

 Further, as the author shows, the gain of entropy 

 when communication is established between two black 

 cavities at unequal temperatures is calculable by 

 ordinary thermodynamic methods, just as is the gain 

 of entropy produced by diffusion of two portions of 

 gas at unequal pressure or temperature or both. 

 NO. 1928, VOL. 74] 



Irreversible changes will necessarily occur at the 

 surface of a body unless eilher the surface is per- 

 fectly reflecting or the incident radiation in the ether 

 is of (he character of black-cavity radiation ; for the 

 radiation emitted by a body is necessarily distributed 

 in all directions, while it can .-ibsorb radiation f.iUing 

 on it in particular directions. 



If, on the other hand, a uniformly healed black 

 body is radiating heat into space, the radiation re- 

 ceived at an external point will be limited in direc- 

 tion by the solid angle which the body subtends at 

 that point, and this will decrease as the distance from 

 the body increases, but no passage of heat from a 

 hotter to a colder body is necessarily associated with 

 the outward propagation of the radiation. We may 

 imagine an ideal perfectly reversible burning glass 

 capable of concentrating the radiation on a receiving 

 body in such a way that it converges from all direc- 

 tions on the body, the solid angle formed by the 

 directions being thus increased to 2t. If the radi- 

 ating and receiving bodies are perfectly black, the 

 latter will be in a state of thermal equilibrium with 

 the ether if its temperature is equal to that of the 

 radiating body, and the radiation may thus be 

 absorbed at the temper.alure of emission by perfectly 

 reversible methods. 



This does not mean that the outward propagation 

 of radiation from a finite body is reversible, for if a 

 body, say a sphere, commences to radiate into infinite 

 space previously devoid of radiation, available energy 

 is lost in consequence of the radiation pressure set 

 up. If, now, we imagine the sphere surrounded by 

 a concentric perfectly reflecting sphere, and suppose 

 that at the surface of this latter the energy of radi- 

 ation per unit volume is ^ and radiation pressure /, 

 then, if the volume of the sphere is decreased by dX , 

 the sphere will have to absorb heat-energy '^tfV 

 which is unavailable at the temperature T of the 

 sphere, and. moreo\er, available energy fdX will have 

 to be supplied in order to overcome the radiation pres- 

 sure. Hence it appears that even in this case the 

 entrop\- per unit volume at any point of the ether 

 assumes the form {\jf + f):T, where T is the black- 

 body temperature corresponding to the same intensity 

 of radiation per unit solid angle. .And as the radi- 

 ation proceeds outwards the quantity /dV/T repre- 

 sents the gain of entropy over and above the quantity 

 of entropy taken from the radiating sphere which is 

 given by the rfO/T formula. 



These introductory statements will give some idea 

 of the difficult task which Dr. Planck has under- 

 taken in his endeavour to trace the connection 

 between radiation |)henomena and the assumed prin- 

 ciples of irreversible thermodynamics. .So many 

 physicists have given up this task as hopeless that 

 Dr. Planck has had to rely, to a large extent, on his 

 own investigations; and the list of original papers, 

 published between iScjO and 1902, affords an insight 

 into the amount ot lime and thought the author 

 has given lo the subject in its many and varied 

 aspects. The present book, ba.sed as it is on the 

 courses of lectures delivered by Dr. Planck at Berlin 

 during the session 1905-6, is intended to place the 



