activity rises, but at the same rate as the first rayless transformation. This will also be the case if the rayless product has a slower rate of change than the succeeding active product. Given an activity curve of the character of Fig. 76, we can deduce from it that the first change is not accompanied by rays and also the period of the two changes in question. We are, however, unable to determine from the curve which of the periods of change refers to the rayless product. It is seen that the activity curve is unaltered if the values of λ 1 , λ 2 , that is, if the periods of the products are interchanged, for the equation is symmetrical in λ 1 , λ 2 . For example, in the case of the active deposit of thorium, without further data it is impossible to decide whether the period of the first change has a value of 55 minutes or 11 hours. In such cases the question can only be settled by using some physical or chemical means in order to separate the product A from B , and then testing the rate of decay of their activity separately. In practice, this can often be effected by electrolysis or by utilizing the difference in volatility of the two products. If now a product is separated from the mixture of A and B which loses its activity according to an exponential law, falling to half value in 55 minutes (and such is experimentally observed), we can at once conclude that the active product B has the period of 55 minutes.
The characteristic features of the activity curve shown in Fig. 76 becomes less marked with increase of the time of exposure of a body to the emanation, that is, when more and more of B is mixed with A at the time of removal. For a long time of exposure, when the products A and B are in radio-active equilibrium, the activity after removal is proportional to Q, where
(see equation 8, section 198). The value of Q, in this case, does not increase after removal, but at once commences to diminish. The activity, in consequence, decreases from the moment of removal, but more slowly than would be given by an exponential law. The activity finally decays exponentially, as in the previous case, falling to half value in 11 hours.
In the previous case we have discussed the activity curve obtained when both the active and inactive product have comparatively rapid rates of transformation. In certain cases which arise in the analysis of the changes in actinium and radium, the rayless product has a rate of change extremely slow compared with that of the active product. This corresponds to the case where the active matter B is supplied from A at a constant rate. The activity curve will thus be identical in form with the recovery curves of Th X and Ur X, that is, the activity I at any time t will be represented by the equation
where I₀ is the maximum value of the activity and λ2 the constant of change of B.
204. In this chapter we have considered the variation with time, under different conditions, of the number of atoms of the successive products, when the period and number of the changes are given. It has been seen that the activity curves to be expected under various conditions can be readily deduced from the simple