Diffusion of the Thorium Emanation.
163. On account of the rapid decay of the activity of the thorium emanation, it is not possible to determine the value of K its coefficient of diffusion into air by the methods employed for the radium emanation. The value of K has been determined by the writer in the following way. A plate C, [Fig. 57], covered with thorium hydroxide, was placed horizontally near the base of a long vertical brass cylinder P. The emanation released from the thorium compound diffuses upwards in the cylinder.
Fig. 57.
Let p be the partial pressure of the emanation at a distance x from the source C. This will be approximately uniform over the cross section of the cylinder. From the general principles of diffusion we get the equation
d2p dp
K ---- = – ---- .
dx2 dt
The emanation is continuously breaking up and expelling α particles. The emanation-residue gains a positive charge, and, in an electric field, is removed at once from the gas to the negative electrode.
Since the activity of the emanation at any time is always proportional to the number of particles which have not broken up, and since the activity decays with the time according to an exponential law,
where p1 is the value of p when t = 0 and λ is the radio-active constant of the emanation.
Then
dp
---- = -λp,
dt
and
d2p
K ---- = λp.
dx2
Thus
Since p = 0 when x = ∞. B = 0. If p = p₀ when x = 0, A = p₀.
Thus
It was not found convenient in the experiments to determine the activity of the emanation along the cylinder, but an equivalent method was used which depends upon measuring the distribution of “excited activity,” produced along a central rod AB, which is charged negatively.
It will be shown later ([section 177]) that the amount of excited activity at any point is always proportional to the amount of emanation at that point. The distribution of “excited activity” along the central rod from the plate C upwards thus gives the variation of p for the emanation along the tube.
In the experiments, the cylinder was filled with dry air at atmospheric pressure and was kept at a constant temperature. The central rod was charged negatively and exposed from one to two days in the presence of the emanation. The rod was then removed, and the distribution of the excited activity along it determined by the electric method. It was found that the amount of excited activity fell off with the distance x according to an exponential law, falling to half value in about 1·9 cms. This is in agreement with the above theory.
Since the activity of the emanation falls to half value in 1 minute, λ = ·0115. The value K = ·09 was deduced from the average of a number of experiments. This is a slightly greater value than K = ·07, obtained for the radium emanation, but the results show that the two emanations do not differ much from one another in molecular weight.
Makower (loc. cit.) compared the rates of diffusion of the thorium and radium emanation through a porous plate, and concluded that the two emanations were of about the same molecular weight, thus confirming the results obtained by the above method.