A large number of results of a similar character have been obtained from other radio-active products, separated from the radio-elements, but the cases of thorium and uranium will suffice for the present to form a basis for the discussion of the processes that are taking place in radio-active bodies.
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 q_{0} 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 Kq_{0}dt, 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
dI = Kq_{0}e^{-λ(T - t)}dt,
where λ is the constant of decay of activity of the active matter. The activity I_{T} due to the whole matter produced in the time T is thus given by
I_{T} = [integral]_{0}^T Kq_{0}e^{-λ(T - t)}dt
= (Kq_{0}/λ)(1 - e^{-λT}).
