Page:Radio-activity.djvu/359

 This must obviously be the case, for otherwise there would be a destruction or creation of matter by the mere process of separation of the source from its products; but, by hypothesis, neither the rate of supply from the source, nor the law of change of the products, has been in any way altered by removal.

Substituting the values of P, Q, R from equations (7), (8), and (9), we obtain

P_{1}/P_{0} = 1 - e^{-λ_{1}t}, Q_{1}/Q_{0} = 1 - (λ_{1}e^{-λ_{2}t} - λ_{2}e^{-λ_{1}t}) / (λ_{1} - λ_{2}), R_{1}/R_{0} = 1 - λ_{3}(ae^{-λ_{1}t} + be^{-λ_{2}t} + ce^{-λ_{3}t}),

where a, b, and c have the values given after equation (9). The curves representing the increase of P, Q, R, are thus, in all cases, complementary to the curves shown in Fig. 73. The sum of the ordinates of the two curves of rise and decay at any time is equal to 100. We have already seen examples of this in the case of the decay and recovery curves of Ur X and Th X.

201. Activity of a mixture of products. In the previous calculations we have seen how the number of particles of each of the successive products varies with the time under different conditions. It is now necessary to consider how this number is connected with the activity of the mixture of products.

If N is the number of particles of a product, the number of particles breaking up per second is λN, where λ is the constant of change. If each particle of each product, in breaking up, emits one α particle, we see that the number of α particles expelled per second from the mixture of products at any time is equal to λ_{1}P + λ_{2}Q + λ_{3}R +, where P, Q, R, are the numbers of particles of the successive products A, B, C, Substituting the values of P, Q, R already found from any one of the four cases previously considered, the variation of the number of α particles expelled per second with the time can be determined.

The ideal method of measuring the activity of any mixture of radio-active products would be to determine the number of α