130. Theory of the phenomena. These processes of decay and recovery go on at exactly the same rate if the substances are removed from the neighbourhood of one another, or enclosed in lead, or placed in a vacuum tube. It is at first sight a remarkable phenomenon that the processes of decay and recovery should be so intimately connected, although there is no possibility of mutual interaction between them. These results, however, receive a complete explanation on the following hypotheses:
(1) That there is a constant rate of production of fresh radio-active matter by the radio-active body;
(2) That the activity of the matter so formed decreases according to an exponential law with the time from the moment of its formation.
Suppose that q₀ particles of new matter are produced per second from a given mass of matter. The rate of emission of energy due to the particles produced in the time dt, is, at the moment of their formation, equal to Kq₀dt, where K is a constant.
It is required to find the activity due to the whole matter produced after the process has continued for a time T.
The activity dI, due to the matter produced during the time dt at the time t, decays according to an exponential law during the time T – t that elapses before its activity is estimated, and in consequence is given by
where λ is the constant of decay of activity of the active matter. The activity IT due to the whole matter produced in the time T is thus given by
The activity reaches a maximum value I₀ when T is very great, and is then given by