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/dt = q - (λ + α)n.
The same equation is obtained when no emanation escapes, with the difference that the constant λ + α is replaced by λ. When a steady state is reached, dn/dt is zero, and the maximum value of n is equal to q/(λ + α).
If no escape takes place, the maximum value of n is equal to q/λ. The escape of emanation will thus lower the amount of activity recovered in the proportion λ/(λ + α). If n_{0} is the final number of emanation particles stored up in the compound, the integration of the above equation gives n/n_{0} = 1 - e^{-(λ + α)t}.
The curve of recovery of activity is thus of the same general form as the curve when no emanation escapes, but the constant λ is replaced by λ + α.
