Page:Radio-active substances.djvu/99

and consequently, in the state of equilibrium, the radium would contain a certain quantity of emanation, $Q$, such that—

2,

$\Delta ={\frac {Q}{\theta }}$ . . . . .

and the radiation of the radium would then be proportional to $Q$.

Let us suppose the radium placed in the circumstances under which it gives off the emanation to the exterior; this is obtained by dissolving the radium compound or by heating it. The equilibrium will be disturbed, and the activity of the radium diminished. But as soon as the cause of the loss of emanation has been abolished (the body being restored to the solid state or the heating having ceased), the emanation is accumulated afresh in the radium and we have a period during which the evolution, $\Delta$, surpasses the velocity of destruction, ${\frac {q}{\theta }}$. We then have—

${\frac {dq}{dt}}=\Delta -{\frac {q}{\theta }}={\frac {Q-q}{\theta }}$,

from which—

${\frac {d}{dt}}\left(Q-q\right)=-{\frac {Q-q}{\theta }}$,

3,

$Q-q=\left(Q-q_{0}\right)e^{-{\frac {t}{\theta }}}$ . . . . .

$q_{0}$ being the amount of emanation present in the radium at time $t=0$.

According to Formula 3, the excess of the quantity of emanation, $Q$, contained by the radium in a state of equilibrium above the quantity, $q$, contained at a given moment, decreases as a function of the time according to an exponential law, which is also the law of the spontaneous disappearance of the emanation. The radiation of radium being proportional to the amount of emanation, the excess of the intensity of the limiting radiation above the actual intensity should decrease as a function of the time by the same law; the excess should thus diminish to one-half in about four days.

The preceding theory is incomplete, since the loss of emanation to the exterior has been neglected. It is also difficult to determine the manner in which this acts as a function of the time. In comparing the results of experiment with those of this incomplete theory, there is found to be no satisfactory agreement; the conviction is, however, retained that the theory in question is partially true. The