Kq₀
I₀ = ----
λ
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
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 or Ur X does not, of itself, prove directly that a material constituent responsible for the activity has been chemically separated. For example, in the case of the separation of Th X from thorium, it might be supposed that the non-thorium part of the solution is rendered temporarily active by its association with thorium, and that this property is retained through the processes of precipitation, evaporation, and ignition, and finally manifests itself in the residue remaining. According to this view it is to be expected that any precipitate capable of removing the thorium completely from its solution should yield active residues similar to those obtained from ammonia. No such case has, however, been observed. For example, when thorium nitrate is precipitated by sodium or ammonium carbonate, the residue from the filtrate after evaporation and ignition is free from activity and the thorium carbonate obtained has the normal amount of activity. In fact, ammonia is the only reagent yet found capable of completely separating Th X from thorium. A partial separation of the Th X can be made by shaking thorium oxide with water owing to the greater solubility of Th X in water.