Page:Radio-activity.djvu/378

 It is, at first sight, a somewhat unexpected result that the final rate of decay of the active deposit from thorium gives the rate of change not of the last product itself, but of the preceding product, which does not give rise to rays at all.

A similar peculiarity is observed in the decay of the excited activity of actinium, which is discussed in section 212.

For a long exposure in the presence of a constant supply of thorium emanation, the equation expressing the variation of activity with time is found from equation (8), section 198,

I_{t}/I_{0} = Q/Q_{0} = (λ_{2}/(λ_{2} - λ_{1}))e^{-λ_{1}t} - (λ_{1}/(λ_{1} - λ_{2}))e^{-λ_{2}t} = ((λ_{2}e^{-λ_{1}t})/(λ_{2} - λ_{1}))(1 - ·083e^{-1·90 × 10^{-4}t}).

About 5 hours after removal the second term in the brackets becomes very small, and the activity after that time will decay nearly according to an exponential law with the time, falling to half value in 11 hours. For any time of exposure T, the activity at time t after the removal (see equation 11, section 199) is given by

I_{t}/I_{0} = Q/Q_{T} = (ae^{-λ_{2}t} - be^{-λ_{1}t})/(a - b),

where I_{0} is the initial value of the activity, immediately after removal, and

a = (1 - e^{-λ_{2}T})/λ_{2}, b = (1 - e^{-λ_{1}T})/λ_{1}.

By variation of T the curves of variation of activity for any time of exposure can be accurately deduced from the equation, when the values of the two constants λ_{1}, λ_{2} are substituted. Miss Brooks has examined the decay curves of excited activity for thorium for different times of exposure and has observed a substantial agreement between experiment and theory.

The results are shown graphically in Fig. 78. The maximum