IONISATION BY RADON I N SPHERICAL VESSELS BY W. MUND
In quantitative work on the chemical effectsof a rays, it is found very convenient to mix a given amount of radon with the homogeneous system to be studied. Very often one wishes to know the number of pairs of ions which are formed in the reaction vessel. In a liquid the range of a rays is so small that adsorption by the walls of the vessel may be disregarded. In a gaseous mixture the loss of ionisation that occurs through many particles encountering the glass has to be taken into account. This problem was solved by S. C. Lind’ for small spherical bulbs having diameters about half the range of the a particles from radon. Ionisation per unit length of path varies for an a ray with the distance r - x, from the end point of the range. According to Geiger’s law it is expressed by i = k(r - x)! From this relation it may be shown that for values of x not exceeding 2 or 3 cms the a particle leaves on its track a number of ions very nearly proportional to 2. Assuming this simplification S. C. Lind has been able to calculate the ionisation in small spheres. The following results may be 1ooked.upon as a further step in the direction opened by Lind. Two assumptions have been found necessary: I. The validity of Geiger’s law. 2 . That all the Ra A and Ra C as soon as formed in the bulb is deposited on the walls. If these points are granted the merely mathematical problem is solved in two steps without any further simplification. I. Let No atoms of radon be uniformly spread in a sphere of radius R ; let r be the range of every a particle emitted by the radon; and k rt the number of pairs of ions which the particle would produce if not stopped by the wall. The total number of pairs of ions formed during the time t is given by:
’[
11=N,(~-ee-xt)kr3I
---+2:
3::0(,
(--)
+(I
32 r 1or2 --)2R 5 (------
r
3 R 3 R 2 R 3
In this formula the difference r - 2R is supposed to be positive; if zR a much simpler formula is obtained:
I1 = N0(r - e-”)kr“jr
- --
S. C. Lind: “The Chemical Effects of Particles and Electrons”, 82 (1921).
> r,
IONISATION BY RADON I N SPHERICAL VESSELS
891
11. Let equilibria amounts of Ra A be spread on the walls of the vessel corresponding to the initial number No of atoms of radon, Let r’ be the range of a! particles from Ra A. The total ionisation is given by:
A similar formula applies to ionisation, Is from Ra C, r ” beiny the range of particles, from this source. Again r’ - zR and r ” - zR are supposed to be positive. If they are not the formula becomes:
The mathematical argument leading to formulas (I), ( z ) , (3) and (4) is fully developed in an earlier (French) paper1. It may be briefly outlined thus: Consider a surface element of area ds on the wall of the spherical vessel. Using Geiger’s law the loss of ionisation is calculated which occurs through a! rays striking ds. The double integration takes into account the 01 rays of every direction and from every distance compatible with the radius R, the ranges r, r’ and r ” and the distribution of the radiating source. The formulas referring to radon Ra A and Ra C must be combined in different ways according to the size of the vessel: I.
< r)
In small spheres (zR
I
= I1
the total ionisation is expressed by
+ Iz + Is = N,(I
Putting r’/r = a, r”/r = b and r/R
+ 3p3 + (I f1 {
= p
5/3
--)
520
P
- e-Xt)krgF
(5)
the function F is given by:
32 (-p
IO
--p2
3
-
P39)
3
Assuming for a and b the numerical values a
=
4 - 0 = 1.142; b = 6.57 2 3.94
= 1.667
3.94
Formula (6) may be written: 5/3
F
= 2.249
-
0 . 9 8 9 ~f
0.1;~
(1.142
- -) P
+
0.02301
i
p3
+ (I --)
2
P
32 (-p 3
IO
--pz 3
+ (1.667
P
-
* Ann. Soc. Scient. Bruxelles, 44, 336. The expression (1) given in the present text is identical with formula (16) on p. 344 of my former paper. Being B somewhat simpler form of the same relation it should be preferred in numerical calculations. ilnother identical expression has been kindly brought t o my notice by Dr. D. C. Bardwell.
W. MUND
892 2.
If
2
>p >
2/1.142 =
1.75
+ 0 . 1 5 ~(1.142 --) + (1.667- :)'''}+ 5/3
F = 2.249-0.989p 3. If 1.75
>p >
i 2/1.667 = 1.2
0.02301p3 (b)
P
P
2 5/3
F = 4. If 1.2
2.249 -
0.989~
>p F
= 2.249
0.15p(1.667
- -P)
f 0.023p3
(c)
- 0.989~4- 0.023p3
The calculations may be avoided by plotting a curve from the following tables : Formula (a) Formula (e) From p = 1.75 t o p = 1.2 Fromp = cc t o p = 2.0 P F PF P F PF 00 0.000 1.75 0.731 1.279 10.0 0.111 1.7 0,759 I.290 I.I 1 0 5.0 0.226 I.130 1.6 0.817 1.307 I. 148 1.5 0* 880 I ,320 4.0 0.287 1.173 3.0 0.391 1.4 0.946 1.324 I .326 I .200 1.3 1.020 0.480 2.5 2.4 0.504 I.209 .... 2.3 0.530 1.218 Formula (d) 2.2 0.556 I , 224 From p = 1.2 to p = o 2.1 0,587 I .232 P F PF 1.2 I ,I O 1 I .321 Formula (b) 1.1 I . 191 1.310 From p = 2 to p = 1.75 1.0 1.283 I.283 P F PF 0.9 1.376 1.238 2 .o 0.621 I.241 0.8 1.470 I.176 1.9 0.659 1.252 0.7 1 565 1.095 1.8 0.706 1.270 0.6 I .661 0.997 0.5 1.757 0.878 0.4 1 .a55 0.742 0.3 1.953 0.586 0.2 2.051 0.410 '
0.I
2 . I50
0.215
0.0
2.249
0.000
In using these tables the ratio p = r/R must be known, r is easily found when the specific ionisation s of the gaseous mixture is given. It is then expressed by the relation.
The product p F tabulated in the third column becomes nearly constant for values of p increasing beyond 3.
IONISATION BY RADON I N SPHERICAL VESSELS
Between p
893
and p = 1.75 the error that would be made on assuming 1.10, never exceeds zoyo and vanishes again for p = 1.75. This approximative constancy of p F for large values of p means that in small bulbs the ionisation is almost proportional to the pressure P I = KP (8) = 3
p F to be constant and equal to its limiting value of about
For large diameters or pressures, on the contrary, p3 becomes negligible and the ionisation formula (d) may be further simplified
I
- e-xt)kr?'$(z.z49 - 0.989 r / R )
= N,(I
or
I
=
I, ( I - A / R P )
Where
I, = N,(I
(9)
- e-Xt)kpMX
2.249
and
For air a t 76 cm and I 5°C A = 138.84 and I, = K"(I - e - X t ) ~ . 6 9X IO^ X 2.249 Let C be the quantity of radon in curies. The ionisation for the duration of one second will be:
C X 3.72 X 10lO X 1.69 X IO^ X 2.249 ( I - 138.8/RP) dI/dt = C X 14.7X IO~;(I - 138.8/RP) (10) This allows a comparison with the empirical formula of Duane and Laborde' for cylindrical vessels a t I 5°C : dI/dt
=
I,,,
=
X X
13.j(1
-
0.572
S/V)
I,,, is the saturation current in electrostatic units. If 4.77 X is the charge of every ion I,,, = dI/dt x 4.77 x 10-lO X is the quantity of radon expressed in gram-seconds of Ra
10-l~
E.S.U.
S is the surface and V is the volume of the vessel, If the formula of Duane and Laborde were to be tested in the case of a spherical vessel, we would write. S V
-=-
and
4nR2 4/3 n
Ra
- _3 R
dI/dt X 4.77 X 10-l~ = C X 4.75 x 10; X 13.j(1 - 1.716/R For a pressure P = 76 dI/dt = C X 13.6 X 1 0 ~ (; I - 130.4/RP) Le Radium, 7, 162 (1910)
894
W. MUND
So there is found a fair agreement between the theoretical and the empirical ionisation formula though the latter originated from measurements on cylindrical vessels. This supports the statement made by Lind that the ionisation in a cylindrical tube may be put approximately equal to the ionisation in a spherical bulb of the same volume. Though nothing very definite is known on the mechanism of radiochemical reactions, it is quite possible that in several cases the intensity of chemical effects varies along the path of the a particle according to the same law as intensity of ionisation. If this could be proved to be true, the function F which expresses from the stand-point of ionisation the utilisation coefficient of the radon, would hold for chemical action as well. As a matter of fact, the simplified formula 8 has worked quite satisfactorily in expressing Lind’s results1 on the radiochemical water synthesis as well as one single result of those of Scheuer2,Formula (9) is identical with the empirical law that Wourtzel3 has deduced from his very careful measurements on the decomposition of HzS by means of radon4. Recently the formulas (d), (c) and (b) have yielded a constant ratio M/I in studying the radiochemical condensation of acetylene5. Kevertheless it must be admitted that the available data are scarcely accurate enough to settle the question.
Summary Assuming Geiger’s law, formulas and numerical tables are given for the calculation of the number of ions which are produced in a spherical vessel by the a rays of a given amount of radon in equilibrium with Ra A and Ra C wholly deposited on the walls. Loirvain March, 1926 0
S. C. Lind: J. Am. Chem. SOC.,41, 531 (1919). 0.Scheuer: Compt. rend. 159, 423 (1914). a E. Wourtzel: Le Radium, 2, 289 (1919). Wourtzel’s formula: M = M,(1 - A/RP) was used by him to calculate by extrapohtion the amount of chemical reaction per curie of radon in a bulb of infinite radius. He assumed that, at infinite radius, RaA and RaC would not reach the walls and that therefore the alpha radiation would be utilized completely. Thus I, = N,,(I- e-xt) (1.69 1.84 z.37)105.By equation above I, = N0(1 - e-xt)r.69X 106 x 2.249, The value taken by Wourtzel for I, is therefore 1.j5times that arrived at by equation 9. 6 W. Mund and PT7. Koch: Bull SOC.chim. belg. 34, 241 (1925):J. P*hys. Chem., 30, 289 (1926).
+
+
