Page:EB1911 - Volume 22.djvu/814

 appear, and these are quite distinct in chemical and physical properties from the parent matter. The radiating power is an atomic property, for it is unaffected by combination of the active element with inactive bodies, and is uninfluenced by the most powerful chemical and physical agencies at our command. In order to explain these results, Rutherford and Soddy (20) in 1903 put forward a simple but comprehensive theory. The atoms of radioactive matter are unstable, and each second a definite fraction of the number of atoms present break up with explosive violence, in most cases expelling an or  particle with great velocity. Taking as a simple illustration that an particle is expelled during the explosion, the resulting atom has decreased in mass and possesses chemical and physical properties entirely distinct from the parent atom. A new type of matter has thus appeared as a result of the transformation. The atoms of this new matter are again unstable and break up in turn, the process of successive disintegration of the atom continuing through a number of distinct stages. On this view, a substance like the radium emanation is derived from the transformation of radium. The atoms of the emanation are far more unstable than the atoms of radium, and break up at a much quicker rate. We shall now consider the law of radioactive transformation according to this theory. It is experimentally observed that in all simple radioactive substances, the tensity of the radiation decreases in a geometrical progression with the time, i.e. I/I0=e−t where I is the intensity of the radiation at any time t, I0 the initial intensity, and a constant. Now according to this theory, the intensity of the radiation is proportional to the number of atoms breaking up per second. From this it follows that the atoms of active matter present decrease in a geometrical progression with the time, i.e. N/N0＝e−λt where N is the number of atoms present at a time t, N0 the initial number, and λ the same constant as before. Differentiating, we have dN/dt＝−N, i.e. λ represents the fraction of the total number of atoms present which break up per second. The radioactive constant λ has a definite and characteristic value for each type of matter. Since λ is usually a very small fraction, it is convenient to distinguish the products by stating the time required for half the matter to be transformed. This will be called the period of the product, and is numerically equal to log Qe2/λ. As far as our observation has gone, the law of radioactive change is applicable to all radioactive matter without exception. It appears to be an expression of the law of probability, for the average number breaking up per second is proportional to the number present. Viewed from this point of view, the number of atoms breaking up per second should have a certain average value, but the number from second to second should vary within certain limits according to the theory of probability. The theory of this effect was first put forward by Schweidler, and has since been verified by a number of experimenters, including Kohlrausch, Meyer, and Begener and H. Geiger. This variation in the number of atoms breaking up from moment to moment becomes marked with weak radioactive matter, where only a few atoms break up per second. The variations observed are in good agreement with those to be expected from the theory of probability. This effect does not in any way invalidate the law of radioactive change. On an average the number of atoms of any simple kind of matter breaking up per second is proportional to the number present. We shall now consider how the amount of radioactive matter which is supplied at a constant rate from a source varies with the time. For clearness, we shall take the case of the production of emanation, by radium. The rate of transformation of radium is so slow compared with that of the emanation that we may assume without sensible error that the number of atoms of radium breaking up per second, i.e. the supply of fresh emanation, is on the average constant over the interval required. Suppose that initially radium is completely freed from emanation. In consequence of the steady supply, the amount of emanation present increases, but not at a constant rate, for the emanation is in turn breaking up. Let q be the number of atoms of emanation produced by the radium per second and N the number present after an interval t, then dN/dt＝q-N where is the radioactive constant of the emanation. It is obvious that a steady state will ultimately be reached when the number of atoms of emanation supplied per second are on the average to the atoms which break up per second. If N0 be the maximum number, q＝λN0. Integrating the above equation, it follows that N/N0＝1−e−t. If a curve be plotted with N as ordinates and time as abscissae, it is seen that the recovery curve is complementary to the decay curve. The two curves for the radium emanation period, 3·9 days, are shown in fig. 1, the maximum ordinate being in each case 100.

This process of production and disappearance of active matter holds for all the radioactive bodies. We shall now consider some special cases of the variation of the amount of active matter with time which have proved of great importance in the analysis of radioactive changes.

(a) Suppose that initially the matter A is present, and this changes into B and B into C, it is required to find the number of atoms P, Q and R of A, B and C present at any subsequent time t. .

Let 1, 2, 3 be the constants of transformation of A, B and C respectively. Suppose n be the number of atoms of A initially present. From the law of radioactive change it follows: P＝ne−1t

Substituting the value of P in terms of n in (1), dQ/dt = λ1ne−1t−λ2Q; the solution of which is of the form Q＝n(ae−λ1t+be−λ2t), where a and b are constants. By substitution it is seen that a＝λ1/(λ2−λ1). Since Q＝0 when t＝0, b= −λ1/(λ2−λ1)

Similarly it can be shown that

where a＝, b＝, c＝

It will be seen from (3), that the value of Q, initially zero, increases to a maximum and then decays; finally, according to an exponential law, with the period of the more slowly transformed product, whether A or B.

(b) A primary source supplies the matter A at a constant rate, and the process has continued so long that the amounts of the products A, B, C have reached a steady limiting value. The primary source is then suddenly removed. It is required to find the amounts of A, B and C remaining at any subsequent time t.

In this case of equilibrium, the number n0 of particles of A supplied per second from the source is equal to the number of particles which change into B per second, and also of B into C. This requires the relation n0=λ1P0=y2Q0=λ3R0 where P0, Q0, R0 are the initial number of particles of A, B, C present, and λ1, λ2, λ3 are their constants of transformation. Using the same quotations as in case (1), but remembering the new initial conditions, it can easily be shown that the number of particles P, Q and R of the matter A, B and C existing at the time I after removal are given by P=e-λ1t, Q=$\big($e-λ2t-e-λ1t$\big)$, R=n0(ae-λ1t+be-λ2t+ce-λ3t), where a＝, b＝, c＝

The curves expressing the rate of variation of P, Q, R with time are in these cases very different from case (1). (c) The matter A is supplied at a constant rate from a primary source. Required to find the number of particles of A, B and C present at any time t later, when initially A, B, and C were absent.

This is a converse case from case (2) and the solutions can be obtained from general considerations. Initially suppose A, B and C are in equilibrium with the primary source which supplied A at a constant rate. The source is then removed and the amounts of A, B and C vary according to the equation given in case (2). The source after removal continues to supply A at the same rate as before. Since initially the product A was in equilibrium with the source, and the radioactive processes are in no way changed by the removal of the source, it is clear that the amount of A present in the two parts in which the matter is distributed is unchanged. If P1, be the amount of A produced by the source in the time t, and P