IONIZATION PRODUCED BY RADON I N SPHERICAL VESSELS BY GEORGE GLOCKLER
I n the study of the chemical effects of alpha particles produced by gaseous radon and its decomposition products R a A and Ra C, it has been customary to calculate the ionization by means of the average path law, as developed by Lindl and Lunn*. However, Mund' has lately deduced expressions for the total ionization by considering the losses of ionization caused by the fact that some of the alpha particles are intercepted by the wall of the reaction vessel before they have completed their range. Mund makes two assumptions in his derivation: first he assumes the validity of Geiger's law and second, that all of the Ra A and Ra C reach the wall before they decompose. I n this paper we shall develop the average path law in such a form that it can be compared with the equations of Mund. We shall also indicate a new derivation of his equations, but in our method we shall follow closely the geometrical picture used in the derivation of the average path law. Finally we shall present arguments for a correction to be applied to Mund's equations. The correction has to do with the assumption stated above that all of the Ra A and Ra C decompose on the wall. It will be shown that experimental data exist which indicate that only about 70% of Ra A and nearly all of R a C (9370) decompose on the wall. The following notation is used: N, = number of Rn atoms present initially t = time r = range of a p t . from Rn = 3.94 cm; (air N.T.P.) r' = do.fromRaA = 4.5ocm.;r' = a r ; a = 1.142 r" = do.fromRaC = 6,57cm.;r" = b r ; b = 1.667 II = Krx - K ( r - x)" = ionization produced by an a pt. from Rn in the distance x (Geiger's Law). 12, I3 = do for Ra A and Ra C. K = 6.74 X 1 0 ~ a t N . T . p . X = decay constant for Rn. R = radius of the spherical reaction vessel r p =R B = N0(x - e-X') Kr';N = K0(1 - e-")dV/V p = average path of an a particle p = K R for Rn and M R for Ra X and C on the wall. I, = Intensity of Ionization (Ionization per cm. path) V = Volume of reaction vessel n = fraction of Ra A decomposing in gas phase m = doof R a C
' J. Phys. Chem., 16, 564-613 (1912).
Lind: "Chemical Effect of Alpha Particles and Electrons " 82 (1921). JAnn. SOC.Scient. Bruxelles, 44,336 (1925); J. Phys. Chek., 30, 890 (1926).
IONIZATION BY RADON I N SPHERICAL VESSELS
‘3 23
I. The Average Path of Alpha Particles in a Spherical Vessel filled with Radon Consider in Fig. I N radon atoms decomposing in time t at the point P. Then the number of alpha particles leaving P in all directions (all directions are equally likely) in time t will be the same. The number dN that will leave in direction 6 and describe paths of length x is given by
The total path that all the alpha particles will describe which leave P will be
and the average path will be -
E SrxsinBdB
P=
2
0
N
klG. I
By the help of the cosine law x can be expressed in terms of O and the integration yields the result obtained by Lunn.1 B further integration over the whole spherical volume gives the average path p = .7j R (3) If it is assumed that all the Ra h and Ra C decompose on the wall, then the average path of the resulting alpha particles is given by the same method p = .5 R (4) and for all three sets of alpha particles combined = .j833 R (5) We have here sketched the method of deriving the expression for the average path so that it can be compared with the method of deriving Mund’s equations in Section IV. It will then be seen that the simple proportion indicated in equation I is the fundamental thought underlying both methods of treating the problem of calculating the ionization produced in spherical vessels. 11. The Average Path Law (Theoretical Expression) We may now write the average path law as follows: The ionization of N,,(I - e-Xt) alpha particles from radon in time t is
if we assume Geiger’slaw and since t,heaverage alpha particle will of course traverse the average path. Similar expressions are obtained for the ionization due to
* Lind: “Chemical Effect of Alpha Particles and Electrons,”
83
(1921).
I324
GEORGE GLOCKLER
the alpha particles from Ra A and Ra C if we assume that they reach the wall before decomposing. The total ionization due to all three sets of alpha particles in time t is given by -
I
=
B[I +a'+
b'
- ((I -
$)'+
(a
- $)?:+( b -
;)')](7)
For large values of p this expression may be expanded binomially:
We will call equation 7 the theoretical average path law because it is based on a calculated value of the average path.
111. The Average Path Law (experimental expression) Before we proceed to evaluate expression 7, we shall restate the average path law on the basis of the experimental work of Lind and Bardwe1l.l These investigators have tested expression 5 for the average path by studying water synthesis in a vessel containing a small thin-walled alpha ray bulb in the center. Under these conditions all alpha particles emerging from the central bulb have the same distance to travel through the gaseous mixture. In their second experiment they broke the alpha ray bulb, causing the radon to distribute itself uniformly throughout the gas-mixture. From these experiments they found on the basis of the known fact that chemical reaction is proportional to the path, that for all three sets of alpha particles the average path is =
.61 R
(9)
They2express the total ionization due to the three sets of alpha particles from radon in terms of the ionization per cm. path.
I . = -dI= * 3 (r - x ) x
dx which reduces for x = o to
Kr2$ r The ionization due to the alpha particles from Radon is
I,,,
=
2
3
-
I1 = N,(I - e=Xt)I,3o p
=
B
Similar expressions are obtained for I1 and It. However we shall make use of the experimental value of the average path of equation 9 (Section VI) and we shall find there that the assumption that 100% Ra A and 1 0 0 7 ~ Ra C decompose on the wall is not supported by experiment. We shall find that 3070 ( = n) of Ra A and 7y0 ( = m) of R a C decompose in the gas phase. We now use this result and obtain five equations of the type ( 1 2 ) which when added give for the total ionization
* J. Am. Chem. Soe., 45, 2585 (1923). 'Lind and Bardwell: J. Am. Chem. Soc., 47, 2675 (1925).
IOSIZATION BY RADON I N SPHERICAL VESSELS
'325
We shall refer to this equation as the experimental average path law for it is based on the experimentally determined average path. Instead of using the Ionization intensity in stating the average path law we might have corrected equation ( 7 ) for the fractions of Ra A and Ra C decomposing in the gas phase in the manner just outlined.
IV. Derivation of M u d ' s Equation We shall now briefly indicate a method of derivation of hIund's equations making the same assumptions that he does, but we shall follow the geometrical picture developed in Section I for the deduction of the average path. Consider N,(I - e-xt) radon atoms decomposing in time t in a spherical vessel of Volume V. The number K of radon atoms decomposing in the elementary Volume dV located at P (Fig. I ) is
3- =
S 0 ( r - e-xt)
V
dV =
X0(1- e-i') 4 3TR3
~ z ~ b ~ s i n y d y d (14) b
As in eection I we have the number d?r' of radon atoms given by equation ( I ) . From the geometry of figure I it follows that
Instead of calculating now the average path of an alpha particle leaving the point P as was done in section I we shall follow Mund and calculate the total ionization of all the alpha particlesdecomposingin time t in the whole volume V and subtract therefrom the loss in ionization occasioned by the fact that some of the alpha particles cannot complete their range because they are intercepted by the wall. The total possible ionization produced by So( I - e-&') alpha particles in time t is when all of them complete their range
B = S , ( I- e-Xt) KrZS (16) The loss in ionization is given similarly by Geiger's law so that the net ionization is I =B -K (r - x ) * d~ S (17) where the integration extends over the whole volume and over all the possible values of x. Equation (17) is the fundamental relation from which all the cases of interest may be computed. I
.) Zonzzatzon by gaseous radon.
The ionization in large vessels from ( I ) , (14), and ( I j ) .
(2
R > r) is given by (17) on substituting
1326
GEORGE GLOCKLER
which on integration yields II=B
20
3520
The ionization due to gaseous radon in small vessels (zR
