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
thus
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 I₀ is the initial activity of the separated product, the activity It after an interval t is given by