ACTIVITIES OF RADIOACTIVE SUBSThXCES
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may be regarded as intermediate between dissolved but superficially adsorbed emulsifiers on the one hand and solid powder emulsifiers on the other. We may fairly conclude that the different theories are not essentially antagonistic but rather are different factors in a rather complex mechanism. This analysis is intended not as a denial of any of these factors but to emphasize and account for the survival of films of one liquid over films of the other during the process of emulsification, and to point out that this survival may be a principal factor in-determining which liquid shall become the outer phase. SUMMARY
The various factors which contribute to the stability of one type of emulsion rather than the inverse type are supplemental rather than rival. The direction of film curvature for minimum energy has probably been overemphasized a t the expense of the more mechanical forces operating during the emulsification process. The rupture of a film separating two droplets can be resisted by a larger rise in interfacial tension a t the threatened point if the reserve emulsifying agent is dissolved in the liquid forming the film, i.e., the external phase, owing to the lower rate of adsorption in that case. REFERENCES
(1) FINKLE, DRAPER, AND HILDEBR.4ND: J. Am. Chem. SOC. 46,2780 (19%). (2) GRIFFIN:J. Am. Chem. SOC.46, 1648 (1923). (3) HARKINB, DAVIS, AND CLARK:J. Am. Chem. SOC. 39, 354,541 (1917). (4) LANGMUIR: Chem. Met. Eng. 16,468 (1916);J Am. Chem. SOC.39, 1848 (1917). (5) SCARLETT, MORGAN, AND HILDEBRAND: J. Phys. Chem. 31, 1566 (1927).
CALCULATION OF ACTIVITIES OF RADIOACTIVE SUBSTANCES I N SERIES DISINTEGRATIONS D. E. HULL School of Chemistry, Institute of Technology, 0.niversity of Minnesota, Minneapolis, Minnesota Received April 18, 1941
I t is the object of this paper to. present a set of formulas which enable one to calculate quickly the activity of any member of the three naturally occurring radioactive series a t any time after any arbitrary initial conditions. The method of calculation has proved to be of convenience to the author, and the required formulas are given for the benefit of others who may find them of use.
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D. E. HULL
The general equations for calculating the quantity N of the nth member of a radioactive series a t any time t in terms of the initial quantity of the first member, Po, and of the respective decay constants, AI, A,, etc., have been given by Bateman (1). The equations for calculating the activity are obtained very simply from Bateman's equations by multiplying through by An/Ai. Thus,
(S)= cle-xlt
+ c,e-xzt + . . . +
cne-hnl
where (N) represents the activity of the nth member measured in, say, millicuries a t tirnc t, and
etc. I n applying these equations to actual cases, it is found that, because of the large differences in order of magnitude of the decay constants, many of the coefficients are very close to unity, while others are negligibly small. Only when the decay constants involved are of a similar order of magnitude are the coefficients much different from one of these values. By use of the above equations, the values of the coefficients necessary to calculate the activity of any member of the series formed from any preceding member have been determined for all cases of significance occurring in the three radioactive families. The values of the coefficients for the uranium, thorium, and actinium series are given in tables 1, 2, and 3, respectively. The accuracy with which the values are given is sufficient to keep the error in calculation to less than 0.1 per cent of the initial activity, which is all that is justified by our present knowledge of the decay constants. Exceptions to this occur in cases where two decay constants involved in the same formula have closely similar values (radium B and radium C ; radon and radium E), in which cases there is a larger error in the difference of the constants. Formulas are not included for cases where N is formed from P through an intermediate so long-lived that the activity of N follows that of the long-lived intermediate within 0.1 per cent, since this would involve needless duplication in the tables. For example, the formation of radium F from ionium is representeG with sufficient accuracy by the formula for the production of radium D from ionium, the corresponding coefficients in the two formulas differing by only one or two units in the fourth decimal. list of the half-lives and branching ratios which have been used in these calculations is appended to each table, together with the numeral or letter which is used as a subscript to the corresponding decay constant in the formulas.
TABLE 1 Cranium series
-
(USI) = (G)(1 - e--hzt) (US%) = 0.9985(USI)(1 - e-Ast) , (UZ) = 0.0015(US,)o(e-~zt- e--hzt) (Io) = (U)(I - e-isc) 1 (Ra) = l.0195(lo)G(e-h~c - e-hst) 1 (Ra) = ( U ) ( l - 1.0195e-Ast 0.0195e-h6s") (Rn) = ( R a ) ( l - ech7
