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
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 n₀ is the final number of emanation particles stored up in the compound, the integration of the above equation gives
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.
For example, if α = λ = ¹⁄₄₆₃₀₀₀, the equation of rise of activity is given by
and, in consequence, the increase of activity to the maximum will be far more rapid than in the case of no escape of emanation.
A very slight escape of emanation will thus produce large alterations both in the final maximum and in the curve of recovery of activity.
A number of experiments have been described by Mme Curie in her Thèse présentée à la Faculté des Sciences de Paris on the effect of solution and of heat in diminishing the activity of radium. The results obtained are in general agreement with the above view, that 75 per cent. of the activity of radium is due to the emanation and the excited activity it produces. If the emanation is wholly or partly removed by solution or heating, the activity of the radium is correspondingly diminished, but the activity of the radium compound is spontaneously recovered owing to the production of fresh emanation. A state of radio-active equilibrium is reached, when the rate of production of fresh emanation balances the rate of change in the emanation stored up in the compound. The differences observed in the rate of recovery of radium under different conditions were probably due to variations in the rate of escape of the emanation.
217. It has been shown in section 152 that the emanation is produced at the same rate in the solid as in the solution, and all the results obtained point to the conclusion that the emanation is produced from radium at a constant rate, which is independent of physical conditions. Radium, like thorium, shows a non-separable activity of 25 per cent. of the maximum activity, and consisting entirely of α rays.