Page:Radio-activity.djvu/383

 (thorium X) all of one kind, which changes into B (thorium B), find the activity of A and B together at any subsequent time. This corresponds to Case I. (section 197). The amount Q of B at any time T is given by

Q = (λ_{1}n_{0}/(λ_{1} - λ_{2}))(e^{-λ_{2}t} - e^{-λ_{1}t}),

and the activity I at any time of the two together is proportional to λ_{1}P + Kλ_{2}Q, where K is the ratio of the ionization of B compared with that of A.

Then     I_{t}/I_{0} = (λ_{1}P + Kλ_{2}Q)/(λ_{1}n_{0}) = e^{-λ_{1}t}[1 + (Kλ_{2}/(λ_{2} - λ_{1}))(1 - e^{-(λ_{2} - λ_{1})t})],

where I_{0} is the initial activity due to n_{0} particles of Th X.

By comparison of this equation with the curve of variation of the activity of Th X with time, shown in Fig. 47, it is found that K is almost ·44. It must be remembered that the activity of the emanation and Th X are included together, so that the activity of thorium B is about half of the activity of the two preceding products.

The calculated values of I_{t}/I_{0} for different values of t are shown in the second column of the following table, and the observed values in the third column.

++-++ ++-++ ++-++
 * Time  |Theoretical|Observed|
 * |  value   | value  |
 * 0       |   1·00    | 1·00   |
 * ·25 days|  1·09    |  —    |
 * ·5  "  |   1·16    |  —    |
 * 1    "  |   1·15    | 1·17   |
 * 1·5  "  |   1·11    |  —    |
 * 2    "  |   1·04    |  —    |
 * 3    "  |    ·875   |  ·88   |
 * 4    "  |    ·75    |  ·72   |
 * 6    "  |    ·53    |  ·53   |
 * 9    "  |    ·315   |  ·295  |
 * 13    "  |    ·157   |  ·152  |