Case 2. The activity curve for a long exposure to the emanation will now be considered. The activity after removal of A and C is proportional to λ1P + λ3R, where the values of P and R are graphically shown in [Fig. 75] by the curves AA, CC. Initially after removal, λ1P₀ = λ3R₀, since A and C are in radio-active equilibrium, and the same number of particles of each product break up per second. The activity due to A alone is shown in curve AA, [Fig. 75]. The activity decreases exponentially, falling to half value in 3 minutes. The activity due to C at any time is proportional to R, and is initially equal to that of A. The activity curve due to C is thus represented by the curve CC, which is the same curve as the upper curve CC of [Fig. 73]. The activity of A and C together is represented by the upper curve A + C ([Fig. 75]), where the ordinates are equal to the sum of the ordinates of the curves A and C. This theoretical curve is seen to be very similar in shape to the experimental curve ([Fig. 67]) showing the decay of activity of the active deposit from a long exposure measured by the α rays.
Fig. 75.
203. Effect of a rayless change on the activity curves. Certain important cases occur in the analysis of radio-active changes, when one of the products does not give rise to rays and so cannot be detected directly. The presence of this rayless change can, however, be readily observed by the variations which occur in the activity of the succeeding product.
Let us consider, for example, the case where the inactive matter A, initially all of one kind, changes into the matter B which gives out rays. The inactive matter A is supposed to be transformed according to the same law as the radio-active products. Let λ1, λ2 be the constants of the change of A and B respectively. If n is the number of particles of A, initially present, we see from the equation (4), [section 197], that the number of particles of the matter B present at any time is given by
Differentiating and equating to zero, it is seen that the value of Q passes through a maximum at a time T given by the equation
For the sake of illustration, we shall consider the variation of the activity of the active deposit of thorium, due to a very short exposure to the emanation. Thorium A gives out no rays, and thorium B gives out α, β, and γ rays, while thorium C is inactive.
The matter A is half transformed in 11 hours, and B is half transformed in 55 minutes. The value of λ1 = 1·75 x 10-5(sec.)-1 and λ2 = 2·08 x 10-4(sec.)-1.