the equation (4), section 197, that the number of particles of the
matter B present at any time is given by
Q = (nλ_{1}/(λ_{1} - λ_{2})) (e^{-λ_{2}t} - e^{-λ_{1}t}).
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
λ_{2}e^{-λ_{2}T} = λ_{1}e^{-λ_{1}T}.
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 λ_{1} = 1·75 x 10^{-5}(sec.)^{-1} and λ_{2} = 2·08 x 10^{-4}(sec.)^{-1}.
The activity of the mixture of products A + B is due to B alone, and will, in consequence, be always proportional to the amount of B present, that is, to the value of Q.
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Fig. 76.
The variation of activity with time is shown graphically in Fig. 76. The activity rises from zero to a maximum in 220 minutes and then decays, finally decreasing, according to an exponential law, with the time, falling to half value in 11 hours.