Section 197, That the Number of Particles of the Matter _b_ Present At

any time is given by

$$ Q = \frac {nλ1} {λ1 − λ2} (e^{–λ2 t} - e^{λ_1 t}) $$ .

Differentiating and equating to zero, it is seen that the value of Q passes through a maximum at a time T given by the equation

$$ λ2 e^{–λ2 T} = λ1 e^{–λ1 T} $$ .

For the sake of illustration, we shall consider the variation of the activity of the active deposit of thorium, due to a very short exposure to the emanation. Thorium A gives out no rays, and thorium B gives out α, β, and γ rays, while thorium C is inactive.

The matter A is half transformed in 11 hours, and B is half transformed in 55 minutes. The value of λ₁ = 1·75 x 10⁻⁵(sec.)⁻¹ and λ₂ = 2·08 x 10⁻⁴(sec.)⁻¹.

The activity of the mixture of products A + B is due to B alone, and will, in consequence, be always proportional to the amount of B present, that is, to the value of Q.

The variation of activity with time is shown graphically in Fig. 76. The activity rises from zero to a maximum in 220 minutes and then decays, finally decreasing, according to an exponential law, with the time, falling to half value in 11 hours.

This theoretical curve is seen to agree closely in shape with the experimental curve (Fig. 65), which shows the variation of the activity of the active deposit of thorium, produced by a short exposure in presence of the emanation.

There are several points of interest in connection with an activity curve of this character. The activity, some hours after removal, decays according to an exponential law, not at the rate of the product B, from which the activity rises, but at the same rate as the first rayless transformation. This will also be the case if the rayless product has a slower rate of change than the succeeding active product. Given an activity curve of the character of Fig. 76, we can deduce from it that the first change is not accompanied by rays and also the period of the two changes in question. We are, however, unable to determine from the curve which of the periods of change refers to the rayless product. It is seen that the activity curve is unaltered if the values of λ₁, λ₂, that is, if the periods of the products are interchanged, for the equation is symmetrical in λ₁, λ₂. For example, in the case of the active deposit of thorium, without further data it is impossible to decide whether the period of the first change has a value of 55 minutes or 11 hours. In such cases the question can only be settled by using some physical or chemical means in order to separate the product A from B, and then testing the rate of decay of their activity separately. In practice, this can often be effected by electrolysis or by utilizing the difference in volatility of the two products. If now a product is separated from the mixture of A and B which loses its activity according to an exponential law, falling to half value in 55 minutes (and such is experimentally observed), we can at once conclude that the active product B has the period of 55 minutes.

The characteristic features of the activity curve shown in Fig. 76 becomes less marked with increase of the time of exposure of a body to the emanation, that is, when more and more of B is mixed with A at the time of removal. For a long time of exposure, when the products A and B are in radio-active equilibrium, the activity after removal is proportional to Q, where

$$ Q = \frac {n₀} {λ1 − λ2} (\frac {λ1} {λ2} e^{–λ2 t} − e^{λ1 t}) $$,

(see equation 8, section 198). The value of Q, in this case, does not increase after removal, but at once commences to diminish. The activity, in consequence, decreases from the moment of removal, but more slowly than would be given by an exponential law. The activity finally decays exponentially, as in the previous case, falling to half value in 11 hours.

In the previous case we have discussed the activity curve obtained when both the active and inactive product have comparatively rapid rates of transformation. In certain cases which arise in the analysis of the changes in actinium and radium, the rayless product has a rate of change extremely slow compared with that of the active product. This corresponds to the case where the active matter B is supplied from A at a constant rate. The activity curve will thus be identical in form with the recovery curves of Th X and Ur X, that is, the activity I at any time t will be represented by the equation

$$ \frac {It} {I₀} = 1 − e^{–λ2 t} $$,

where I₀ is the maximum value of the activity and λ₂ the constant of change of B.

=204.= In this chapter we have considered the variation with time, under different conditions, of the number of atoms of the successive products, when the period and number of the changes are given. It has been seen that the activity curves to be expected under various conditions can be readily deduced from the simple theory. In practice, however, the investigator has been faced with the much more difficult inverse problem of deducing the period, number, and character of the products, by analysis of the activity curves obtained under various conditions.

In the case of radium, where at least seven distinct changes occur, the problem has been one of considerable difficulty, and a solution has only been possible by devising special physical and chemical methods of isolation of some of the products.

We shall see later that two rayless changes occur in radium and actinium and one in thorium. It is at first sight a very striking fact that the presence of a substance which does not emit rays can be detected, and its properties investigated. This is only possible when the rayless product is transformed into another substance which emits rays; for the variation of the activity of the latter may be such as to determine not only the period but also the physical and chemical properties of the parent product. In the two following chapters the application of the theory of successive changes will be shown to account satisfactorily for the complicated processes occurring in the radio-elements.

Footnote 294:

Ramsay, Proc. Roy. Soc. p. 470, June, 1904; C. R. 138, June 6, 1904.

Footnote 295:

Phil. Mag. February, 1904.