:: D (active about 40 no rays Soluble in deposit of slow years strong acids change) and volatized below 1000° C. :: E (same) 6 days β (and γ) Non-volatile at 1000°C. :: F (same) 143 days α rays Volatile at 1000° C; deposited from solution on to bismuth plate. ? — — — =236. Variation of the activity over long periods of time.= We are now in a position to calculate the variation of the α and β ray activity of the active deposit over long periods of time. If it is supposed that the matter initially deposited consists only of D, the amounts _P_, _Q_ and _R_ of radium D, E and F existing at any later time are given by the equations 3, 4, 5, section 197. Since, however, the intermediate product E has a much more rapid rate of change than D or F, the equations can be simplified, without much loss of accuracy, by disregarding the change E, and by supposing that D gives out β rays and changes directly into the α ray product F. Let λ₁, λ₂ be the constants of change D and F respectively. Let _n₀_ be the number of particles of D present initially. Then using the notation of section 197, the amount _P_ of radium D at any time _t_ is given by $$ P = n₀ e^{–λ_1 t} $$ . The amount _Q_ of radium F is given by $$ Q = \frac {n₀ λ_1} {λ_1 − λ_2} (e^{–λ_2 t} - e^{–λ_1 t}) $$ . [Illustration: Fig. 96.] The number of β particles emitted by D + E per second, some months afterwards, is $$ λ_1 n₀ e^{–λ_1 t} $$, and the number of α particles emitted by radium F is $$ \frac {λ_1 λ_2 n₀} {λ_1 − λ_2} (e^{–λ_2 t} − e^{–λ_1 t} ) $$ . The results are shown graphically in Fig. 96, by the curves _EE_ and _FF_, in which the ordinates represent the number of β and α particles expelled per second by the products D and F respectively. The complete calculation for three changes shows that the number of β particles soon reaches a practical maximum, and then decays nearly exponentially with the time, falling to half value in 40 years. The number of α particles expelled per second increases for several years, but reaches a maximum after 2·6 years and then diminishes, finally falling off exponentially with the time to half value in 40 years. The experimental curve of the rise of α ray activity, shown in Fig. 93, as far as it has been determined, lies accurately on this curve, if the maximum is calculated from the above theory. The observed activity after a period of 250 days is marked by the point _X_ on the curve.