RADIOACTIVE EQUILIBRIUM 61 



law. For X substitute the value 2.097 X 10 -6 , and solve for t. 



. 2.097 X 10~ 6 * 



login 2 = 



6 2.3026 



t = 3.825 days 



The quantity 1/X is that average time in seconds in which the number 

 of original atoms is reduced to 1/e ( = 0.36788) of their original count. 

 This is called the average lije of a radioactive substance, even though 

 some of its atoms may exist for only a short time and others for a long 

 time. The average life of radon is 1/X = 1/(2.097 X 10~ 6 ) sec = 

 476,871 sec = 5.52 days. 



Radioactive Equilibrium 



The following example illustrates what is meant by an equilibrium 

 state. Suppose that you are given a closed vessel containing many 

 millions of microorganisms. At the close of each day you are asked to 

 investigate the number of deaths and the number of births. Suppose 

 that at the end of 30 days you find that the birth rate equals the death 

 rate. You have found that an equilibrium state has been established 

 if from then on the death rate equals the birth rate. 



Similarly the population content of 1 gram of freshly prepared radium 

 sulphate is about 26.8 X 10 20 molecules. In the first day (about one- 

 millionth of these) 3.2 X 10 15 of the radium atoms will disintegrate to 

 form the atoms of the radon gas. As soon as some radon atoms are 

 formed, however, some of them will disintegrate, so that by the end of 

 3.825 days half of them have disintegrated. In this process about 16 per 

 cent of the radon atoms disintegrate per day. Hence at the end of the 

 first day 16 per cent of 3.2 X 10 15 atoms have disintegrated, leaving 

 2.7 X 10 15 radon atoms. By the end of the second day 3.2 X 10 15 

 4- 2.7 X 10 15 , or 5.9 X 10 15 , are present, of which 0.9 X 10 15 disinte- 

 grate, leaving 5 X 10 15 atoms. By the end of the third day there are 

 8.2 X 10 15 atoms, of which 1.3 X 10 15 disintegrate. At the end of 

 about 30 days nearly as many radon atoms are disintegrating as are 

 supplied by the disintegrating radium atoms. Radioactive equilibrium 

 has then been established between radium and radon. The maximum 

 amount of radon that can accumulate from a given quantity of radium 

 under these circumstances is called its equilibrium amount. Table II— 1 

 shows the various amounts of the radioactive substances in equilibrium 

 with 1 mg of radium. 



Figure II— 3 shows the increase in number of radon atoms, despite 

 their decomposition in the presence of the more slowly decomposing 



