ANALYTICAL CHEMISTRY
918 gives a large "mixing time" (1). This is of no disadvantage, because the mixing time map be measured accurately by using small amounts of ascorbic acid. Results in Table I show data for evperiments a t one enzyme concentration, with different amounts of ascorbic acid. The reproducibility of replicates is seen to be very good, showing the adequacy of the automatic end-point method.
method is advantageous, as a is large, the slope is small, and the intercept is easy t o determine. Figure 5 shows the expected straight line relationship. The intercept b/a is 18.8 seconds per mg., which corresponds to an enzyme activity of 20.2 catecholase units or 0.018 millimole (of ascorbic acid oxidized) per minute per milliliter as shown in Table I. REFERENCES
TREATMENT OF DATA
The treatment of the data in Table I is presented here to show a modified method of calculating the enzyme activity. The initial rate, KO& = a / b , is evaluated from the data by means of a plot of t / Q us. t according to Equation 4 (which is another form of Equation 3) rather than from a plot of 1/Q us. l / t ( 4 ) .
t
b
t
Asimov, I., and Dawson, C. R., J . A m . Chern SOC.,72, 820 (1950). Gould, B. S., Enzymologia, 7, 292 (1939). (3) Ludwig, B. J., and Xelson, J. AI., J . -4m. Chem. SOC.,61, 2601
(1) (2)
(1939). (4) Miller, W. H., Alallette, 11.F., Roth, Ibid., 6 6 , 5 1 4 (1944). (5) Ponting, J. D., Ibid., 76, 662 (1954).
L.J., and Dswson, C. R.,
RECEIVED for review November 26, 1954. Accepted February
Q=a+u
It is evident that t / Q is equal to b/a when t = 0. This plotting
14, 1935.
Mention of specific products does not imply t h a t they are endorsed or recornmended by the Department of Agriculture over others of a slmllar nature not mentioned.
Determination of Small and large Amounts of Fluorine in Rocks F. S. GRIMALDI, BLANCH INGRAM, and FRANK CUTTITTA
U. S.
Geological Survey, Washington 25,
D. C.
Gelatinous silica and aluminum ions retard the distillation of fluorine in the Willard and Winter distillation method. A generally applicable, simple method for the determination of fluorine in rocks containing aluminum or silicon or both as major constituents was desired. In the procedure developed, the sample is fused with a mixture of sodium carbonate and zinc oxide, leached with water, and filtered. The residue is granular and retains nearly all of the silica. The fluorine in the filtrate is distilled directly from a perchloric acid-phosphoric acid mixture. Phosphoric acid permits the quantitative distillation of fluorine in the presence of much aluminum at the usual distillation temperature and without the collection of large volumes of distillate. The fluorine is determined either by microtitration with thorium nitrate or colorimetrically with thoron. The procedure is rapid and has yielded excellent results on silicate rocks and on samples from the aluminum phosphate (leached) zone of the Florida phosphate deposits.
I
N T H E determination of fluorine it is common practice to isolate fluorine as fluosilicic acid by some modification or adaptation of the Willard and Winter distillation method ( 1 1 ) . It is known that gelatinous silica or large amounts of aluminum, or both, as well as other elements of the ammonium hydroxide group, retard the distillation of fluorine. For example, with the usual distillation temperature of 135" to 140" C. and with the collection of about 150 ml. of distillate, fluorine is not completely recovered in the presence of more than about 50 mg. of silicon dioxide in the gelatinous form ( 7 , 9, 11) and about 20 mg. of aluminum oxide. The interference from these sources is especially serious for silicate rocks, because silicon and aluminum are major constituents, and large samples are required for determination of small quantities of fluorine, about 0.0301,. The analyst is confronted with these same difficulties in the determination of fluorine in samples from the aluminum phosphate (leached) zone of the Florida phosphate deposits. It has been reported ( 2 , 11) that better recovery of fluorine is obtained by distilling a t higher temperatures (160" to 165' C.)
and by collecting larger volumes of distillate. It is the authorq' experience, however, that such procedures fail to give quantitntive recovery of fluorine on such samples as described above. Up t o the present time the best approach has been to separate fluorine from silica and from the elements of the R203group before the distillation step. Some adaptation of the Berzeliua method is usually used for this separation, silica and aluminum being precipitated by zinc oxide from a nearly neutral or slightli ammoniacal solution ( 4 , 11). This procedure is time-consuming. and there is danger of loss of fluorine by coprecipitation on the large gelatinous precipitates obtained. Recently Shell and Craig (9) described a procedure primarily designed for the simultaneous determination of fluorine and silica in JThich a flu.: mixture of sodium carbonate and zinc oxide is used for decomposing silicate samples. This feature represents an important advance, because a granular precipitate retaining most of the silica is obtained when the melt is leached with water. The behavior of aluminum in the fusion depends on the composition of the sample being analyzed, and frequently as much as 95% of the aluminum will be found accompanying fluorine in the IT ater leach. Where the aluminum cannot be tolerated, it must be separated. This is ordinarily taken care of in Shell and Craig's procedure, where ammoniacal zinc oxide is added directly and prior t o the filtration of the water leach of the melt, in order to recover most of the dissolved silica. This investigation was undertaken to develop a generally applicable, simple method for the determination of both small and large amounts of fluorine in rocks, especially those containing aluminum and silicon as major constituents. Independent of Shell and Craig, the authors arrived a t a similar flux mixture for the decomposition of silicate rocks, and this feature of the procedure is the same as that of Shell and Craig. After the sample is fused, leached with water, and filtered, the fluorine in the filtrate is distilled directly from a perchloric acid-phosphoric acid mi.:ture. Phosphoric acid permits the quantitative distillation of fluorine in the presence of much aluminum a t the usual distillation temperature and viithout the collection of large volumes of distillate. The fluorine in the distillate may be determined either by microtitration with thorium nitrate or colorimetrically by a modification of the thoron [2-(2-hydroxy-3,6-disulfo-l-naphthylazo)-
V O L U M E 2 7 , NO. 6, J U N E 1 9 5 5
919
benzenearsonic acid] method of Horton and others ( 6 ) . -41though phosphoric acid has been used for the distillation of fluorine ( 3 , 6, 7 ) , it has not been generally adopted in the past because of the possibility of entraining sufficient phosphate with fluorine during the distillation to interfere subsequently with the determination of fluorine. This difficulty is of little consequence. Just prior to the completion of this work, Brunishob and Michod ( 1 ) used phosphoric acid to prevent the interference of aluminuni and iron( 111)in the distillation of fluorine.
aluminum oxide can be tolerated a t the distillation temperature of 140" =k 3" C. Distillation experiments a t 140" zt 3 " C. from perchloricphosphoric acid medium a t three levels of fluorine (0.4, 5, and 15 mg. of fluorine) and six levels of aluminum (20, 50, 75, 100, 150, and 200 mg. of aluminum oxide) for each level of fluorine showed that quantitative recoveries of fluorine were obtained in each instance. Thus, a t least 200 mg. of aluminum oxide can be tolerated in the distillation.
EXPERIMENTAL
Preliminary experiments sought to establish the following: The amount of silica that may be expected in the filtrate from a water leach of a zinc oxide-sodium carbonate fusion of a sample. The maximum amount of soluble aluminum that can be tolerated in the distillation of fluorine from a perchloric acid medium. The effect of a t least 200 mg. of soluble aluminum in the distillation of fluorine from a perchloric-phosphoric acid medium. The amount of phosphate that can be tolerated in the microtitration and colorimetric method for determining fluorine. The amount of phosphate that accompanies fluorine in the distillation from a perchloric-phosphoric acid medium. The minimum amount of distillate from the phosphoricperchloric acid medium I equired for quantitative recovery of fluorine.
Table I.
Retarding Effect of Aluniinum on Distillation of Fluorine"
F Taken, Mg. 5.00 5,OO
a
5.00 5.00 5.00 5.00 5.00 5.00 5.00 150 ml.
DistillationoTemperature, C. 140 3 1-10 3 140 i. 3 140 i.3 140 f 3 140 f 3 148 i.3 148 i 3 3 148 of distillate at piescribed temperature A1203
Taken,
Mg. 20 50 75 100 150 200 20 50 75
** *
______
+
F Recovered, Alp.
4 92
4.76 4.51 3.82 3,64 3 15 4.99 4.90 4 72 -
-_
Table TI. Effect of Phosphate on Spectrophotometric Determination of Fluorine with Thoron 11g. None 0 050 0.100 0,200 0.400 0.800
1'206,
1 .ti0
20 y 17 0.453 0.453 0.454 0,455 0.447 0.442 0,443
Absorbance 40 y F 60 y F 0.417 0.370 0,419 0.372 0.417 0.371 0.415 0,369 0.412 0.366 0.407 0.361 0.398 0.354
80 y E' 0.336 0.336 0.334 0,33.5 0.332 0.328 0.323
\Tith mixtures containing varying quantities of phosphate and 1 mg. of fluorine, it was found that no more than 50 y of phosphorus pentoxide could be tolerated in the microtitration with thorium nitrate (50 y of phosphorus pentoxide corresponds to 20 y of fluorine). At least 200 y of phosphorus pentoxide could be tolerated in the thorium-thoron method in the range from 0 to 100 y of fluorine. The data on the effect of phosphate in the thorium-thoron procedure are given in Table 11. To determine the amount of phosphorus carried over in the distillation with perchloric-phosphoric acid, separate 25-ml. portions of water were distilled with the mixed acids, and 150-ml. distillates were collected. The phosphate in the distillates was determined by the molybdenum blue method. The distillates contained from 0.2 to 0.4 Y of phosphorus pentoxide per milliliter. These correspond to from 30 to 60 */ of phosphorus pentoxide in the total distillates. As only an aliquot portion of the distillate is used for the determination of fluorine, no interference should be expected from this source. This is confirmed by the ewellent results obtained with the procedure on samples containing known amounts of fluorine (Tables I11 and IV). In several experiments the distillation temperature was raised t o 150" C. and no greater amounts of phosphate were found in the distillates. As the amount of phosphate carried over into the distillate may depend on the type of still used, it is recommended that each analyst test this for himself. Although no trap was necessary in the still, some other design may require a trap to prevent the entrainment of phosphate. The data in Table 1- indicate that essentially quantitative re-
111 the experiments 6 grams of a (1 5) zinc oxide-sodium carbonate mixture were used in the silica studies; aluminum was added as a solution of aluniinum perchlorate and fluoride as a standard solution of pure sodium fluoride. Either 20 nil. of perchloric acid alone or mixed with 2 ml. of phosphoric acid was used in the distillation experiments. A standard phosphate solution made by dissolving pure potassium dihydrogen phosphate was used where required. The techniques used in the experiments are given in the section on procedure. The following results were obtnined: Using- pure _ silica in amounts mrying from 0.01 to 0.70 giam, from 1 to 16 nig. of silica n ere Table 111. Test of Procedure in Determining Small Amounts of Fluorine found in the water leach of the Fluorine, % sodium carbonate-zinc oxide Thorium-Alizarin Thorium-Thoron Microtitration (Colorimetric) (Colorimetric) melt; the greatest amount of SairiEile Naterial Addeda residueb In filtrate Totald In filtrate Totald In filtrate Totald PresentC silica 71-as obtained from the Granite G - l . e Fluorspar 79 0.01 0.25 0.26 0.25 0.26 0.25 0.26 0.26 test on 0.70 gram of silica. S o (O.Ofil% F) Opalglass91 0.003 0.25 00.26 0 25 0.26 0.26 0.27 0.26 greater amounts of silica were ... 0.16 .. 0.16 , . 0.15 .. 0.16 found in the filtrates from the ... 0.11 .. 0.11 .. 0.10 0.11 Phosphate silicate s a m p l e s a n a l y z e d . rock 120 0.03 0.23 0.20 0.23 0.26 0.23 0.26 0.26 ... 0.17 .. 0.15 .. 0.15 .. 0.16 These amounts do not retard ... 0.11 .. 0.12 .. 0.11 .. 0.11 the distillation of fluorine. Diabase IV-1 Opal glass 91 0.01 0.23 0.24 0.25 0.26 0 20 0.21 0.22 Table I gives data on the (O.OIG~?, F) . . . 0 . 0 7 . . 0 . 0 8 . . 0 . 0 7 . . 0.07 1 ecovery of fluorine in the presa h-ational R . -..- ..__ - -~-i-r-~-i of .i i Standards standard samples. ence of various quantities of Based on original sample. C Calculated from fluorine contents of sample and material added. aluminum when perchloric acid d Sum of fluorine in residue and in filtrate. alone is used for the distile See Schlecht (8) l a t i o n . .4bout 2 0 m g . of I .
ANALYTICAL CHEMISTRY
920 Table IV. Test of Procedure in Determining Larger Amounts of Fluorine by Titration Fluorine, % In filtrate I n residue b 5 72 0.03 5.73 ..
Sample' Opal glass 91
..
Phosphate rock 120
..
..
Fluorspar 79 a
b
..
.. ..
Total 5.75 , .
3.80 3.76 46.07 46 07
..
..
Fluorine, % ' (NBS Certificate Values) 5.72 3.76 40.15
National Bureau of Standards standard samples. Based on original sample.
Table V.
Completeness of Recovery of Fluorine with 3' C. Distillation at 140"
*
Portion of Distillate First 50 ml. Second 50 ml. Third 50 ml. Total found Total taken
Fluorine Recovered, MeL Sample 1 Sample 2 1.89 14.30 0.11 0.64 0 02 0.06 2.02 15.00 2.00 15.00
-
-
coveries of fluorine are obtained xyith the collection of less than 150 nil. of distillate a t 140" =t3 " C. REAGEVTS AND APPARATUS
The procedure for the titration of fluoride is well known, and except where specific instructions are desirable only a brief listing of the reagents is made. Reagent grade chemicals were used in all experiments. Sodium Fluoride. The primary standard sodium fluoride was prepared by treating sodium bicarbonate solution with an excess of hydrofluoric acid in a platinum dish, evaporating the excess acid, and heating the residue at 600" C. in air. (Reagent grade sodium fluoride is not a dependable primary standard.) Standard fluoride solution for microtitrimetry, 1 ml. contains 1 mg. of fluoride. Standard fluoride solution for spectrophotometry, 1 ml. contains 5 y of fluoride. Thorium nitrate solution for microtitrimetry, 0.02S. The method of standardization is given in the procedure. Thorium nitrate solution for spectrophotometry, 1 ml. contains 50 y of thorium dioxide Sodium alizarin sulfonate, 0.1% aqueous solution, Phenolphthalein, 0.1 70 alcoholic solution. Chloroacetate buffer, 0.2M in monochloroacetic acid and 0.2M in its sodium salt. Sodium carbonate wash solution, a 27, aqueous solution. Sodium hydroxide, a 27, aqueous solution. Hydroxylamine hydrochloride, a 10% aqueous solution. Thoron, 2-(2-hydroxv-3,6-disulfo-l-naphthylazo)benzenearsonic acid, a 0.10% aqueous solution. Other Reagents. Hydrochloric acid (1 99), phosphoric acid (857,), perchloric acid (70'%), anhydrous sodium carbonate, zinc oxide. Fluorine still, capacity 125 ml. The still used was manufactured by Ace Glass, Inc., Vineland, X. J. (Xo. 6431 in Catalog 50), in vhich a steam delivery tube was substituted for the dropping funnel. Spectrophotometer, Beckman DU, supplied with 2-em. Cores cells.
+
PROCEDURE
Isolation of Fluoride. Fuse a 1.000-gram sample with a mixture of 1.2 grams of zinc oxide and 6 grams of sodium carbonate in a platinum or nickel crucible. The fusion must be made under oxidizing conditions to prevent attack on platinuin when this container is used. Cool. Place the crucible and contents in a 100-ml. beaker and add 30 ml. of water. Cover and heat on the steam bath until the cake has softened. Remove the crucible, police, and wash it. The volume of solution at this point should be about 50 ml. Break up any lumps with a stirring rod flattened a t one end, and then bring the solution to boil, stirring to prevent bumping. Boil for several minutes. remove from the heat. and allow the residue to settle.
Filter the solution directly into the distillation flask, decanting as much of the liquid as possible. Wash the residue by decantation with several small portions of hot sodium carbonate wash solution. Transfer the residue to the filter paper and wash several more times. Reject residue. The total volume of solution a t this point should be about 70 ml. Add slowly 20 ml. of perchloric acid and 2 ml. of phosphoric acid to the flask, keeping the flask cool by immersing it in cold water . Begin the distillation, collecting the distillate in a 250-ml. volumetric flask. When the temperature reaches about 135' C., pass in steam from an auxiliary flask. Increase t,he temperature to 140' C. and collect 150 ml. of distillate a t this temperature. The distillate should be collected a t a rate of about 4 ml. per minute. Dilute the distillate to 250 ml. with water and mix. Determination of Fluoride by Microtitration. Transfer a 50ml. aliquot to a 125-ml. Erlenmeyer flask. Add 12 drops of sodium alizarin sulfonate, and then titrate rvith sodium hydroxide until the indicator becomes pink. Discharge the pink color b y 99) hydrochloric acid. Add 2.5 ml. of careful addition of (1 buffer, and then titrate with 0 . 0 2 5 thorium nitrate solution. A more detailed description is given by Reynolds and Hill ( 7 ) . The thorium nitrate is standardized against solutions of sodium fluoride, titrating aliquots of 0, 0.1, 0.2, 0.4, 0.6, 0.8, 1, 2, and 3 mg. of fluorine. A curve is drarvn plotting milliliters of thorium nitrate against milligrams of fluorine. I n this procedure no more than 3 mg. of fluoride is titrated. If it should be desirable t'o t,itrate more than 5 mg. of fluoride, a close control of p H within the limits 2.9 to 3.1 is recommended by Shell and Craig (9). Each sodium fluoride solution used in the standardization is obtained by taking a water leach from a blank fusion, adding fluoride equal t o five times each amount of fliiorine to he titrated, and distilling according to the procedure. The usual 50-ml. aliquot from 250 nil. of final solution is taken for the titration. The working curve is a straight line above 1 mg. of fluorine, but is slightly curved helow this amount necessitating several points in the l o m r region. h slightly lower fluorine equivalent, is obtained for the thorium nitrate if standardized against pure sodium fluoride solutions which have not been distilled. This difference does not appear to be due to the inclusion of phosphoric acid in the distillation medium. Determination of Fluoride Spectrophotometrically with Thoron. Transfer a 5- to 15-ml. aliquot to a 25-nil. volumetric flask. ildd a drop of phenolphthalein and neutralize xyith sodium hydroxide to a faint, pink rolor of the indicator. Add, by means of pipets, 0.2 ml. of perchloric acid (70%), 1 ml. of hydroxylamine hydrochloride, 2 ml. of standard thorium nitrate solut,ion (equivalent to 100 y of thorium dioxide), and 2 ml. of thoron reagent. ddjust the volume to 25 ml. with distilled water and mix. 1Ieasure the absorbance of the solution after 15 minutes on a Beckman DU spectrophotometer a t 545 mp: using a 2-cni. Cores cell and a slit width of 0.05 mm. The refrence cell should be a blank containing perchloric acid, hydroxylamine hydrochloride, and thoron a t the same concentrations as in the sample. The standard curve, absorbance us. concentration of fluoride, is obtained by plotting the absorhances for colors developed rvith known amounts of fluoride in the range of 0 to 100 y of fluoride. h straight line is obtained. The standard fluoride solution is prepared directly; previous distillation is unnecessary. One microgram of fluoride in 25 ml. decreases the absorbance of the thorium-thoron color by about 0.002. Although a p H of 1 is used in the shove thoron procedure, the t,horium-thoron system is fairly stable in the range of p H 0.65 to 1.7.
+
TEST OF PROCEDURE
The procedure was applied to the determination of microgram and milligram amounts of fluorine. For the small amounts of fluoiine, 1-gram portions of t n o silicate rocks, granite G-I, and diabase W-1 ( 8 ) , were used, to which various amounts of a standard sample such as fluorspar, phosphate rock, or opal glass \$ere added. The fluorine contributed by the additions varied from 0.05 to 0.2 mg. I n several tests the residues obtained from the fusion and water leach were re-fused to determine the amount of fluorine remaining in the residue. The fluorine contents of the granite and diabase were established by replicate analyses, the granite containing 0.61% fluorine and the diabase 0.016% fluorine. After distillation the fluorine was determined by microtitration and colorimetrically by the thoron procedure and by Talvitie's alizarin method (IO)on aliquots of the same distillates. Talvitie's procedure was included for comparison purposes. The results given by the various methods (Table 111) are in good agreement. The results indicate that the residue from the first
V O L U M E 2 7 , N O . 6, J U N E 1 9 5 5 fusion and leach need not be reworked, as the amount of fluorine retained is negligible for all practical purposes. The slightly greater amount of fluorine retained when phosphate rock was used as the addit'ion material is to be expected because of the presence of calcium and phosphate ions in the rock. Standard samples of phosphate rock, fluorspar, and opal glass were used in testing the procedure on milligram amounts of fluorine. Both the phosphate rock and fluorspar were directly distilled from the perchloric-phosphoric acid medium without preliminary treatment. A 0.25-gram sample m s taken for the phosphat,e-rock test, and 25 nig. n-ere used for the fluorspar. The fusion procedure cannot lie used on phosphate concentrates because of the retention of much fluorine by the residue obtained after leaching the melt. .I 0.5-gram sample for the opal glass was fused, leached nith water, and half of filtrate was taken for distillation. For all s:imples, the fluorine was determined by t,itration. The results obtained are compared to the Sational Bureau of Standards certificate values in Table IV and show good agreement. The retention of fluorine in the residue after leaching is again negligible for the opal glass. The direct ilistillntion procedure was applied also to samples
921 from the aluminum phosphate (leached) zone of the Florida phosphate deposits, and again excellent results were obtained. LITERATURE CITED
(1) Brunishola, G., and llichod, J., Helv. Chim. Acta, 37,874 (1954). (2) Dahle, D., and Wichmann, H. J., J . Assoc. Ofic. Agr. Chemists, 19,320 (1936). ( 3 ) Ibid.. 20. 297 (1937). (4j Hoffman', J . I:, and Lundell, G. E. F., J . Research S a t l . Bur. Standards, 3, 581 (1929).
( 5 ) Horton, A. D., Thomason, P. F., and lliller, F. J., ANAL. CHESS., 24, 548-51 (1952). (6) Reynolds, D. S., J . Assoc. Ofic.A g r . Chemists, 18, 108 (1935). (7) Reynolds, D. S., and Hill, W,L., IXD.ENG.CHEM., ~ A L ED., . 11, 21 (1939). (8) (9) (10) (11)
Schlecht, W.G., ASAL. CHEY.,23, 1568 (1951). Shell, H. R., and Craig, R. L., Ibid., 26, 996 (1954). Talvitie, N. A., IND.E m . CHEM.,Ax.4~.ED.,15, 620 (1913). Willard, H. H., and Winter, 0. B., Ibid.. 5 , 7 (1933).
RECEIVEDfor review December 9, 1954. Accepted February 14, 10%. Work completed a s part of a program conducted b y the U.S. Geological Survey on behalf of the Division of Raw Materials of the Atomic Energy Commission, Publication authorized b y the Director, C. 9. Geological Survey.
Absolute Assay of Beta Radioactivity in Thick Solids Application to Naturally Radioactive Potassium ANDREW D. SUTTLE, JR.',
and W. F. LIBBY2
Department o f Chemistry and Institute for Nuclear Studies, University o f Chicago, Chicago, 111.
The method of absolute measurement of beta radioactivity in solids has been found reliable within about 5%. The simplicity of this method and the scarcity of other practical absolute assay techniques led to this study. A wide variety of simple beta emitters display exponential absorption curves when the sample, absorber, and counter are placed close in cylindrical geometry. The absorption coefficient for the material constituting the sample can be used to calculate the absolute specific radioactivity of the solid sample. The absorption coefficients for a given absorbing material vary smoothly with the energy of the beta radioactivity transition, so that reliable values of the energies of such radioactivities can be obtained from the absorption coefficients. The coefficients vary with the atomic weight of the absorber. This effect is only partially elucidated.
T
HE possibility of calculating the absolute quantities of radio-
activity from relatively simple measurements on solids has always been interesting. A prescription has been outlined (3, 7 , 8, 11). The possibility of converting the count rate obtained from the radiations from a solid sample to an absolute measure of the specific radioactivity of the sample seems t o exist in a general way. This is true, however, only in so far as the absorption of beta radiation is accurately exponential under the conditions of measurement. These conditions are that the source, absorber, and detector be in relatively intimate contact, as, for example, in the arrangement in which the absorber is wrapped directly around the wall of the Geiger counter and the source is placed directly over it (8). It is apparent that the contact can allow some appreciable distance between the Geiger counter and the source and 1 Present address, Research a n d Development Division, Humble O il and Refining Co., Baytown, Tex. 2 Present address, U. S. Atomic Energy Commission, Washington, D . C.
the absorber, providing that the cylindrical geometry is maintained. This is revealed by data given in this paper, in 11hich in the one instance the abeorptiori curve is measured with the sample as the wall of the counter, and in the other instance as a cylindrical sample surrounding a counter n i t h about 1 cm. between the sample and the n-all of the Cciger counter. In both instances the absorption curves are exponential. The absorber must lie close to the counter because the chance of recording the beta rays scattered in the absoiber dependq strongly on this juxtaposition (5). Under the condition that the absorption curve is accurately exponential, one can write that the fraction of the radiation coming from a n infinitesimal thickness, dx,of material a t a depth, 1 -2 2, from the surface of the sample, \vi11 be e - h where G is the ratio of 3.ir to the solid angle subtended by the sensitive counter volume a t the position of the tiny element of the sample being considered, and X is the reciprocal of the mass absorption coefficient for the beta radiation. It is of course equal t o the 1 half thickness times -. Therefore an expression can be n ritten ln2 for the total radioactivity, total count rate, expected from a sample for which it is assumed that G is not dependent seriously on the depth of the sample being considered and has been averaged over its area.
R (c.p.m.)
=
AuX/G
(1)
where A is the area of the sample in square centimeters, u is the absolute specific radioactivity of the sample in disintegrations per minute per gram of sample, A is the reciprocal of the a b s o r p tion coefficient of the radiation in units of grams per square centimeter, and G is defined above. This equation holds if the source is thick as compared with A. For finite thickness, it becomes :
