236. Variation of the activity over long periods of time. We are now in a position to calculate the variation of the α and β ray activity of the active deposit over long periods of time. If it is supposed that the matter initially deposited consists only of D, the amounts P, Q and R of radium D, E and F existing at any later time are given by the equations 3, 4, 5, section 197.
Since, however, the intermediate product E has a much more rapid rate of change than D or F, the equations can be simplified, without much loss of accuracy, by disregarding the change E, and by supposing that D gives out β rays and changes directly into the α ray product F.
Let λ_{1}, λ_{2} be the constants of change D and F respectively. Let n_{0} be the number of particles of D present initially. Then using the notation of section 197, the amount P of radium D at any time t is given by P = n_{0}e^{-λ_{1}t}. The amount Q of radium F is given by
Q = (n_{0}λ_{1}/(λ_{1} - λ_{2}))(e^{-λ_{2}t} - e^{-λ_{1}t}).
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Fig. 96.
The number of β particles emitted by D + E per second, some
