660 Dr. A. H. Gompton on Radioactivity 



thermodynamic reasoning predicts a change in the energy 

 evolved in radioactive disintegration when the potential of 

 the gravitational field is varied. The following analysis 

 shows, however, that this change is by no means large 

 enough to account for the lack of radioactivity of the earth's 

 interior. The cycle considered by Professor Donn.an consists 

 of the following four steps : — 



1. A system at gravitational potential Z changes from 

 state 1 to state 2, an amount of energy Q being liberated, 

 which results in u change of mass from m x to m 2 . 



2. The system in state 2 is raised from potential Z to 

 Z + oZ. 



3. At potential Z + £Z the system is changed back from 

 state 2 to state 1. 



4. The system in state 1 is lowered from potential Z-fSZ 

 to potential Z. Being then in its original condition, the 

 total energy evolved by the system is zero, and it possesses 

 its original mass. 



If the change SZ in the gravitational potential is small, the 

 total work done by the system in performing'this cycle is 



Q_ ?n2 8Z-(;Q + i3szVm 1 5Z = 0, 

 or e)Q 



which is the expression obtained by Donnan. If now we 

 consider Q as a function of Z only, we may write 



Putting R as the ratio between the energy evolved and the • 

 mass which disappears, we have 



m 1 — ?n 2 — Q/R, 



whence dQ/Q = dZjR. 



The difference in the gravitational potential b©tw r een the 

 surface and the centre of the earth is about o x 10 11 cm. 2 sec. -2 , 

 and the ratio R is of the order of the square of the velocity 

 of light, or 10 21 cm. 2 sec. -2 . Hence the decrease from this 

 cause in the energy of radioactive disintegration, being less 

 than one part in a billion, is wholly inad< quate to account 

 for the small amount of heat developed in the earth's 

 interior. 



