258. Amount of the products. By application of the theory of successive changes, the probable amount of each of the products present in radium and the other radio-elements can readily be estimated.
Since each radio-atom expels one α particle of atomic weight about that of hydrogen or helium, the atoms of the intermediate products will not differ much in weight from the parent atom.
The approximate weight of each product present in a gram of radium can be readily deduced. Let NA, NB, NC be the number of atoms of the products A, B, C present per gram in radio-active equilibrium. Let λA, λB, λC be the corresponding constants of change. Then if q is the number of the parent atoms breaking up per second, per gram,
q = λANA = λBNB = λCNC.
Consider the case of the radium products, where the value of q is 6·2 × 1010 ([section 93]). Knowing the value of λ and q, the value of N can at once be calculated. The corresponding weight can be deduced, since in one gram of matter of atomic weight about 200, there are about 4 × 1021 atoms (section 39). The results are shown in the following table:—
| Product | Value of λ (sec)-1 | Number of atoms, N, present per gram | Weight of product gram of radium |
|---|---|---|---|
| Radium emanation | 2·0 × 10-6 | 3·2 × 1016 | 8 × 10-3 |
| Radium A | 3·8 × 10-3 | 1·7 × 1013 | 4 × 10-6 |
| Radium B | 5·4 × 10-4 | 1·3 × 1014 | 3 × 10-5 |
| Radium C | 4·1 × 10-4 | 1·6 × 1014 | 4 × 10-5 |
With the small quantities of radium available, the amounts of the products radium A, B and C are too small to weigh. It may be possible, however, to detect their presence by means of the spectroscope.
In the case of thorium, the weight of the product Th X, which is present in greatest quantity, is far too small to be detected. Since the value of λ for Th X is about the same as for the radium emanation, the maximum weight present per gram is about 4 × 10-12 of a gram, remembering that q for radium is about 2 × 106 times the value for thorium. Even with a kilogram of thorium, the amount of Th X is far too small to be detected by its weight.
This method can be used generally to calculate the relative amounts of any successive products in radio-active equilibrium, provided the value of λ for each product is known. For example, it will be shown later that uranium is the parent of radium and is half transformed in about 6 × 108 years, while radium and radium D are half transformed in 1300 and 40 years respectively. The weight of radium present in one gram of uranium, when equilibrium is established, is thus 2 × 10-6 grams, and the weight of radium D is 7 × 10-8 grams. In a mineral containing a ton of uranium there should be about 1·8 grams of radium and ·063 grams of radium D. Some recent experiments described in [section 262] show that these theoretical estimates are about twice too great.
259. Rayless Changes. The existence of well-marked changes in radium, thorium, and actinium, which are not accompanied by the expulsion of α or β particles, is of great interest and importance.