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.
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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
K(d^2p/dx^2) = - dp/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, p = p_{1}e^{-λt}, where p_{1} is the value of p when t = 0 and λ is the radio-active constant of the emanation.
Then dp/dt = -λp,
and K(d^2p/dx^2) = λp.
Thus p = Ae^{-[sqrt](λ/K)x} + Be^{[sqrt](λ/K)x}.
Since p = 0 when x = [infinity], B = 0.
If p = p_{0} when x = 0, A = p_{0}.
Thus p = p_{0}e^{-[sqrt](λ/K)x}.
