62 



APPLIED RADIOACTIVITY 



radium, during the course of 35 days. Note particularly how the growth 

 of radon conforms to an exponential rise in time, and that for all practi- 

 cal purposes it attains its maximum value in about 30 days. 



A mathematical analysis of radon equilibrating in the presence of 

 radium leads to the following general law 



N = N max (1 - e~^ ) 



where iV max = (Xi/X 2 )iVo- Here N is the number of radium atoms 

 originally present, Xi the decay constant of radium, and X 2 the decay 

 constant of radon. Since the radium atom has a very long life compared 

 to the radon atom (X 2 > Xi ) and no radon is present initially, the above 

 expression describes the results accurately. 



15 20 



Time in days 



Fig. II-3. The net increase in number of radon atoms in the presence of radium. 

 Despite the rapid decay, radon is formed at the same rate as it disintegrates at about 

 the thirtieth day. After this ./Vmax maintains its constant value. 



Radon, when used for therapeutic purposes, is pumped off the radium 

 and sealed in small capsules. Initially such a capsule contains only 

 radon gas. Radon progressively disintegrates, and the successive 

 products of decay are RaA, RaB, RaC, etc. The state of equilibrium 

 between radon and its products is reached in about 4 hours. The rate 

 of decay of radon cannot be neglected in calculating the equilibrium 

 of the end products despite its rapid decay. The relative number of 

 these atoms is therefore different when in equilibrium with radon than 

 when in equilibrium with radium. This type of equilibrium is referred 

 to as " transient." In transient equilibrium the products RaA, RaB, 



