The decay curve of the radium emanation is shown in the same figure. The curve of recovery of the lost activity of radium is thus analogous to the curves of recovery of uranium and thorium which have been freed from the active products Ur X and Th X respectively. The intensity It of the recovered activity at any time is given by
where I₀ is the final value, and λ is the radio-active constant of the emanation. The decay and recovery curves are complementary to one another.
Fig. 85.
Knowing the rate of decay of activity of the radium emanation, the recovery curve of the activity of radium can thus at once be deduced, provided all of the emanation formed is occluded in the radium compound.
When the emanation is removed from a radium compound by solution or heating, the activity measured by the β rays falls almost to zero, but increases in the course of a month to its original value. The curve showing the rise of β and γ rays with time is practically identical with the curve, [Fig. 85], showing the recovery of the lost activity of radium measured by the α rays. The explanation of this result lies in the fact that the β and γ rays from radium only arise from the active deposit, and that the non-separable activity of radium gives out only α rays. On removal of the emanation, the activity of the active deposit decays nearly to zero, and in consequence the β and γ rays almost disappear. When the radium is allowed to stand, the emanation begins to accumulate, and produces in turn the active deposit, which gives rise to β and γ rays. The amount of β and γ rays (allowing for a period of retardation of a few hours) will then increase at the same rate as the activity of the emanation, which is continuously produced from the radium.
216. Effect of escape of emanation. If the radium allows some of the emanation produced to escape into the air, the curve of recovery will be different from that shown in Fig. 85. For example, suppose that the radium compound allows a constant fraction α of the amount of emanation, present in the compound at any time, to escape per second. If n is the number of emanation particles present in the compound at the time t, the number of emanation particles changing in the time dt is λndt, where λ is the constant of decay of activity of the emanation. If q is the rate of production of emanation particles per second, the increase of the number dn in the time dt is given by
dn = qdt – λndt – αndt,
or dn