231 



imagine that the energy transferred to the gas in the first process 

 is remitted to the radiation by another way, e.g. with another 

 frequency. In this case energy would, however, move continuously 

 in a cycle. This one will not feel inclined to accept. 



In the way indicated above, say by the application of a sufficien 

 number of similar pistons, the equilibrium with the radiation can 

 be brought about for all principal frequencies of the gas. If there 

 are still other ways in which energy can be transferred from radiation 

 to the gas molecules or vice versa, the nature of the equilibrium 

 will presumably not be changed thereby. 



The result of these considerations is that we shall admit — in so 

 far as the above argument is not considered sufficiently cogent we 

 will put as a hypothesis which may be justified or not in its con- 

 sequences — that in the equilibrium between the molecular trans- 

 latory motion in a gas and radiation enei-gy elements of a magnitude 

 \ hv play a part, if r is a principal frequency of the gas. 



§ 3. The energy and the entropy of an ideal monatomic gas. In 

 calculating the mean ^) energy and entropy to be ascribed to a definite 

 mode of vibration with frequency v of the gas, we follow the 

 reasoning which Planck, Warmestrahlung 2^° Aufl. § 135 — 143 

 follows for ideal linear electrical oscillators. Considering that for the 

 gas according to § 2 we have to do with energy elements ^ hv we 

 obtain (cf. Planck I.e. equation (22)) : 



where s-, and z«v represent the mean entropy and energy for the 

 mode of vibration considered. 



The temperature T^ is determined by 1 — 1 = — . In this differen- 

 tiation at constant volume the wave-length A remains constant; v is 

 connected with A by the relation v = 7;., c representing the velocity 

 of propagation. In ^ 4 it will appear that in the gas, when in 

 thermodynamic equilibrium, on the suppositions for which the simple 

 laws of propagation of sound hold, c is proportional to f/Va, also 

 when equipartition does not hold, where U is the total energy of 

 molecular translatory motion of the gas. We will now assume that 

 Ibr each mode of vibration we may put c -^ u^l-i. This hypothesis is 

 inconsistent with observations for vibrations which we can observe 

 as sound, e.g. for vibrations with small amplitude, if they are 



1) Numerical mean if we think the gas to be repeated many times. 



