webster. — planck's radiatiox formula. 133 



explaining chemical affinities, and which has other advantages over 

 Planck's oscillator and most other models of the quantum. 



Planck's Assumptions. The basis of the derivation of Planck's 

 formula is Boltzmann's relation between entropy and probably, which 

 Planck puts in the form S = k log IF where S is the entropy and k 

 is the gas constant reckoned for a single molecule, and IP is the "prob- 

 abilitv," considered as the number of ways in which the svstem can 

 be arranged so as to be indistinguishable from its present form. 



In applying this law to a system of similar molecular oscillators, two 

 arrangements are considered indistinguishable if they give equal num- 

 bers of oscillators in each of certain groups, any group, the n th , being- 

 defined by the condition that the energy of every oscillator in it must 

 lie between (n — 1) h v, and n h v. For this rule to be a real criterion of 

 indistinguishability, we must have the density of their representative 

 points of the energy scale constant through each one of these inter- 

 vals and changing abruptly on going from one interval to the next. 



These assumptions, as Planck shows, give for the mean energy of 



the oscillators in the n th group the value f n — - J h v. By setting the 



entropy equal to its maximum value consistent with a given total 

 energy, he finds for the probability that a given oscillator will be in the 



n th group the value w„ = 017" where a = r-= ^-f— and 



2 t, — J\ h v 



2 E — N h v „ . 



7 = 9 pi at l ~ ' E being the total energy of N oscillators. 4 



Now to obtain a law of distribution of energy in the black body 

 spectrum, it is necessary to have some law of emission and absorption 

 for the oscillators. To keep in touch with the classical electrodyna- 

 mics, he assumes that an oscillator absorbs energy as that system 

 would require if one might neglect reradiation entirely; but to ob- 

 tain the discontinuities required, he assumed, not the classical law of 

 emission, but that the oscillator can emit only when the representative 

 point reaches the boundary of one of these intervals, and that, if it 

 does emit, it must emit its whole energy, n h v. 



With these assumptions, he proves that the energy of such an 

 oscillator, exposed to radiation of continuously distributed frequencies, 

 will increase at a constant rate, thus insuring the constancy of the 

 density of points in each interval ; and he obtains the proper reduction 

 of their density from one interval to the next by assuming that the 



4 Planck, 1. c, § 139. 



