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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 λ1, λ2, λ3 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=ne1t

(1)

dQ/dt=λ1P-λ2Q

(2)

dR/dt=λ2Q-λ3R

Substituting the value of P in terms of n in (1), dQ/dt = λ1ne1t2Q; the solution of which is of the form

Q=n(ae1t+be2t),

where a and b are constants. By substitution it is seen that a=λ1/(λ21). Since Q=0 when t=0, b= -λ1/(λ21)

(3)Thus Q = 1λ12(e2t-e1t)

Similarly it can be shown that

(4)

R=n(ae1t+be2t+ce3t)

where a=λ1λ212)(λ13), b=λ1λ221)(λ23), c=λ1λ231)(λ32)

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

n01P0=y2Q03R0

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=n0λ1e1t,

Q=n0λ12${\displaystyle {\big (}}$λ1λ2e2t-e1t${\displaystyle {\big )}}$,

R=n0(ae1t+be2t+ce3t),

where a=λ212)(λ13), b=112)(λ23), c=λ1λ2λ313)(λ23)

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