Substituting the values of P, Q, R from equations (7), (8), and (9), we obtain
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 λ1P + λ2Q + λ3R + ..., 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 or β particles expelled from it per second. In practice, however, this is inconvenient and also very difficult experimentally.
Certain practical difficulties arise in endeavouring to compare the activity of one product with another. We shall see later that, in many cases, all of the successive products do not emit α rays. Some give out β and γ rays alone, while there are several “rayless” products, that is, products which do not emit either α, β, or γ rays. In the case of radium, for example, radium A gives out only α rays, radium B no rays at all, while radium C gives out α, β, and γ rays.
In practice, the relative activity of any individual product at any time is usually determined by relative measurements of the saturation ionization current produced between the electrodes of a suitable testing vessel.
Let us consider, for example, the case of a product which gives out only α rays. The passage of the α particles through the gas produces a large number of ions in its path. Since the α particles from any individual product are projected with the same average velocity under all conditions, the relative amount of the ionization produced per second in the testing vessel serves as an accurate means of determining the variation of its activity. No two products, however, emit α particles with the same average velocity. We have seen that the rays from some products are more readily stopped in the gas than others. Thus the relative saturation current, due to two different products in a testing vessel, does not serve as an accurate method of comparing the relative number of α particles expelled per second. The ratio of the currents will in general depend upon the distance between the plates of the testing vessel, and, unless the relative ionization due to the average α particle from the two products is known from other data, the comparison of the currents can, at best, be only an approximate guide to the relative number of α particles escaping into the gas.
202. Some examples will now be considered to show how the factors, above considered, influence the character of the curves of activity obtained under different experimental conditions. For the purpose of illustration, we shall consider the variation after removal of the excited activity on a body exposed for different times to a constant supply of the radium emanation. The active deposit on removal consists in general of a mixture of the products radium A, B, and C. The nature of the rays from each product, the time for each product to be transformed, and the value of λ are tabulated below for convenience:—