The activity reaches a maximum value I_{0} when T is very great, and is then given by
I_{0} = Kq_{0}/λ;
thus I_{T}/I_{0} = 1 - e^{-λT}.
This equation agrees with the experimental results for the recovery of lost activity. Another method for obtaining this equation is given later in section 133.
A state of equilibrium is reached when the rate of loss of activity of the matter already produced is balanced by the activity supplied by the production of new active matter. According to this view, the radio-active bodies are undergoing change, but the activity remains constant owing to the action of two opposing processes. Now, if this active matter can at any time be separated from the substance in which it is produced, the decay of its activity, as a whole, should follow an exponential law with the time, since each portion of the matter decreases in activity according to an exponential law with the time, whatever its age may be. If I_{0} is the initial activity of the separated product, the activity I_{t} after an interval t is given by
I_{t}/I_{0} = e^{-λt}.
Thus, the two assumptions—of uniform production of active matter and of the decay of its activity in an exponential law from the moment of its formation—satisfactorily explain the relation between the curves of decay and recovery of activity.
131. Experimental evidence. It now remains to consider
further experimental evidence in support of these hypotheses.
The primary conception is that the radio-active bodies are able to
produce from themselves matter of chemical properties different
from those of the parent substance, and that this process goes
on at a constant rate. This new matter initially possesses
the property of activity, and loses it according to a definite law.
The fact that a proportion of the activity of radium and thorium
can be concentrated in small amounts of active matter like Th X
