Decay and Growth Tables for Naturally Occurring Radioactive Series

H. W. Kirby. Anal. Chem. , 1954, 26 (6), pp 1063–1071. DOI: 10.1021/ac60090a035. Publication Date: June 1954. ACS Legacy Archive. Cite this:Anal. Ch...
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Decay and Growth Tables for the Naturally Occurring Radioactive Series H. W. KIRBY M o u n d Laboratory, Monsanto Chemical Co., Miamisburg, Ohio

T

HE tables which follow are intended primarily for use by the analytical chemist. While in no sense complete, they have been compiled in an effort t o reduce or eliminate much of the tedious and duplirative labor involved in the analysis of radioactive materials. The tables are divided into three sections devoted to the three naturallv occurring radioactive series. The nuclear data are those compiled by the Sational Bureau of Standards ( 2 ) . The decay of each radiosotope is given in one or more appropriate time units except \There the half life of a particular nuclide is either too lone - or too short to be of analvtical importance. The growth of families is given in relative disintegration rates, which are readily determined by standard counting methods, rather than in 'numbers of atoms. The criterion throughout has been utility to the analyst. Thus, where the growth of a family is either too slow to be useful, or is effectively complete in a f e n hours, it has not been computed. hlthough the uncertainty of the half lives and the precision of

Table 11. Thorium Series (4n) I

Min. 0 1

2 3 4

5 A 7 8 9 10 20 30 40 50 Hours 1 2 3 4

5 A

present-day counting techniques do not warrant more than three significant figures in most cases, values have been computed to four or five significant figures to permit reasonably accurate interpolation. The equations used for calculating growth and decay were adapted from those originally proposed by Rutherford (3)and Bateman ( 1 ) .

7 8 9 10 11 12 13 14 15

SYMBOLS

18

-1-0

T,, T 2 , T,,

(Decay in minutes and hours)

1 ime, .

16 17 I9 20 21 22 23

= number of atoms of the parent (first member) of a family at an arbitrary time, 1 = 0 = half life of the first, second, nth member of a

family

24

Thz28

Ac228

1

1 .oooo

.oooo

0,9981 0.9962 0.9943 0,9925 0.9906 0.9888 0.9869 0.9851 0.9831 0.9814 0.9630 0.9451 0.9274 0.9101

Rn220 1 .oooo 0,9999 0.4662 0.9997 0.2174 0.9996 0.1013 0,9995 0.0472 0.9993 0.0220 0.9992 0.0103 0.9991 0,0048 0.9989 0,0022 0.9988 0,0010 0.9987 0,0005 0.0474 0,0000 0 . 9960 0.9947 Ra224

1 .oooo

0,9934

0.8931 0,7976 0,7123 0,6362 0.5681 0.5074 0.4532 0.4047 0.3614 0.3228 0,2883 0,2575 0,2299 0.2058 0.1834 0,1638 0.1463

0.1306 0.1167 0.1042 0.0931

0.0831

0.0752 0.0603

0,9999 0.9999 0,9999 0.9998 0.9998 0.9998 0.9997 0.9997 0,9996 0,9996 0.9995 0.9995 0.9995 0.9994 0.9994 0.9993 0.9993 0.9993 0.9992 0.9992 0 9991 0.9991 0.9990 0.9990

0 9921

0.9843 0.9765 0.9688 0.9611 0.9535 0.9460 0.9385 0.9311 0.9237 0.91B4 0.9092

0.90'20 0.8948 0.8878 0.8807 0.8738 0 8(i69 0 86300 0.8533 0.8465

0.8398 0.8332 0.82flO

PbZlz 1.0000

biz12 1 .oooo

0,9989 0.9978 0.9967 0,9956 0.9946 0.9935 0.9924 0.9913 0.9902 0,9892 0.9784 0.9678 0.9573 0.9470

0,9888 0,9774 0.9662 0.9552 0.9443 0.9336 0,9229 0 9124 0.9020 0.8917 0.7932 0.7091 0.6324 0,5639

0.9367 0.8774 0.8219 0.7698 0.7211 0.6755 0.6327 0.5927 0.5552 0.5200 0.4871 0.4663 0.4274 0.4003 0.3750 0.3512 0.3290 0.3082 0.2887 0.2704 0.2833 0.2373 0,2222 0,2082

0.5024 0.2529 0.1272 0.0639 0.0322 0.0162 0.0081 0.0041 0.0021 0.0010

TlZOS 1 . 0000 0.7996 0,0394 0.5113 0,4089 0.3269 0.2614 0.2090 0.1672 0.1337 0.1069 0.0114 0.0012 0.0001 0.0000

0,0005

0.0003 0.0001 0.0001 0.0000

Table 1. Thorium Series (4n) Synonym

Mode of Decay

Energies in Rlev.

A,, Half Life

x

...

a

4.0

1.39

1IsThr

8-

0.002

6.7 T

1IsThz

8-

1 6

613H

HdTh

Cl

TIiX

a

Tn

a

Th.4

[00 ..80604180Y-1 0D-~

93% 5 . 7 a%5 4

3.64 D

1904 D - I [00 0079 H-1

...

a

TliR

P-

6.3

54.59

6 8 (1)

0.1586

AI-' 0.0127 S-I [0.7631 2 6 3 . 2 11 [4,3870S-1

7.8

0.0003S

2300 s - 1

88'3 0 3

[1 2 ' 3 0 6

a

8 8

3 04 X 10-7s

TIiC"

P-

1 8

3 I11

ThD

81

GROWTH AND DECAY EQUATIONS

...

Relative activity of a single radioactive species:

1 2694 D - l 0 0654H-l

10 6 TI

ThC'

...

(disintegration rate) of the parent of a family a t t = 0 S,,h = activities of the first and nth members of a family at any time = alpha activity of the parent at t = 0 = total alpha activity of thc chain a t any time = total beta activity of the chain nt any time

-I

TIiC

Stable

010 Oif

1 SOY

[ -

1*,)

[

2 7138D-1 0 1131 H -

(lecav constant of the nth memher of a family (A,< = 0.69315/

= activity

0.1035T-I 0 000283 D - '

r

=

m \

5.oXlo-"Y-'

72q5.4

128% 5 . 3

[(0.64% 8 3

101"Y

Decay Constnnts

2 3 X 106S-1

r

13 416H-1 0 2236 31-1

...

Activity of the first decay product relative to the initial activity of a parent which was pure a t t = 0:

ANALYTICAL CHEMISTRY

1064

Time. Days

Table 111. Thorium Series (4n)

Table IV. Thorium Series (4n)

(Decay in days)

(Growth of decay products from initially pure pareut.' Parent, thorium-228)

Time, Days RaZ"

Th2"

Raza

AcZB

Th'n

Razz'

PbZl'

2 3 4 5

1,0000 0.9997 0.9994 0.9992 0.9989 0,9986

1.0000 0,0663 0,0044 0.0003 0,0000

1.0000 0,9990 0.9980 0.9970 0.9960 0.9950

1 ,0000 0.8266 0.6833 0,5648 0.4669 0 3859

1,0000 0.2082 0.0433 0,0090 0.0019 0.0004

50 51 52 53 54 55

6 7 8 9 10

0 9983 0.9980 0 9977 0.9975 0.9972

0.9940 0.9930 0.9920 0.9910 0.9901

0.3190 0.2637 0.2180 0.1802 0.1489

0.0001 0.0000

56 0.9843 0.9456 57 0.9840 0.9446 58 0.9837 0.9437 59 0.9834 0.9427 60 0.9831 0.9418

11 12 13 14 15

0.9969 0.9966 0.9963 0.9960 0.9958

0.9891 0,9881 0.9871 0,9861 0.9851

0.1231 0.1018 0.0841 0.0695 0.0575

61 62 63 64 65

0.9829 0.9826 0.9823 0.9820 0.9817

0.9409 0.9399 0.9390 0.9380 0.9371

16 17 18 19 20

0.9955 0.9952 0.9949 0.9946 0.9943

0.9841 0.9832 0.9822 0.9812 0.9802

0.0475 0.0393 0.0325 0.0268 0,0222

66 67 68 69 70

0.9815 0.9812 0.9809 0.9806 0.9804

0.9362 0.9352 0.9343 0.9334 0.9324

0

1

0.9859 0.9856 0.9854 0.9851 0.9848 0.9845

0.9513 0.9503 0.9494 0.9484 0.9475 0.9465

0.9315 0.9306 0.9296 0.9287 0.9278

??

(Thn28)

=

( ~ ~ ~ 2 4 = ) 1,00528 (,--it NoAi (RnZZO)

.VoAi

(PbP12)

e

\I

2v6k6 (Bi21Z) -.VOX,

-

e-M)

1.005?5

+

,--2i

0.00017 ,--ha'

E: h'di =

=

-

1.00592 ,-'It

NOXI

(T1208)

"1 =

.vlA7

-= A'oAI

5.02182

71 0.9801 72 0.9798 73 0.9795 74 0.9792 75 0.9790

26 27 28 29 30

0.9927 0.9924 0.9921 0.9918 0.9915

0.9743 0.9734 0.9724 0.9714 0.9705

0,0071 0.0058 0.0048 0.0040 0.0033

76 0.9787 0.9269 77 0.9784 0.9259 78 0.9781 0.9250 79 0.9779 0.9241 80 0.9776 0.9232

31 32 33 34 35

0.9913 0.9910 0.9907 0.9904 0.9901

0.9695 0.9685 0.9676 0.9666 0.9656

0.0027 0.0023 0.0019 0.0015 0.0013

81 82 83 84 85

0.9773 0.9770 0.9767 0.9765 0.9762

0.9222 0.9213 0.9204 0.9195 0.9186

36 37 38 39 40

0.9898 0.9896 0.9893 0.9890 0,9887

0.9647 0.9637 0.9627 0.9618 0.9608

0.0011 0.0009 0,0007 0.0006 0.0005

86 87 88 89 90

0.9759 0.9756 0.9754 0.9751 0 9748

0.9176 0.9167 0.9158 0.9149 0 9140

41 42 43 44 45

0.9884 0,9882 0.9879 0.9876 0.9873

0.9598 0.9589 0,9579 0.9570 0.9560

0.0004 0.0003 0.0003 0.0002 0.0002

91 92 93 94 95

0.9745 0.9743 0.9740 0.9737 0.9734

0.9131 0.9121 0.9112 0.9103 0.9094

2 3 4 5

46 47 48 49 50

0.9870 0.9868 0.9865 0.9862 0.9859

0.9551 0.9541 0.9532 0.9522 0.9513

0.0002 0.0001 0.0001 0.0001 0.0001

96 97 98 99 100

0.9732 0.9729 0.9726 0 9723 0 9721

0.9085 0.9076 0.9067 0.9058 0.9049

49

1.00598 e - A l t

-

(po211)A-7nA;a

0.0183 0.0152 0.0125 0.0104 0.0086

Fraction Remaining After a Given Time. EXAMPLE 1. What fraction of radium-223 remains after 15 days?

-

-k

1.14424

0.13833 e--hat

0.9792 0.9783 0.9773 0.9763 0.9753

USE OF THE TABLES

1.00528 g - A l t

0x1

0 9941 0.9938 0.9935 0.9932 0,9929

Activity of the lath member of the decay chain relative to the initial activity of a parent which was pure a t t = 0:

=

(pozls) E 4 Nohi

21 22 23 24 25

Activity of the second decay product relative to the initial activity of a parent which was pure a t t = 0:

(--Xif

NOXI

-

+

1.1575Y e - A z t 0.15287

e-Aet

-

0,00125

' E a Nohi

1.00598

- 1.15827 e - A 2 t

-t

0.163W ,-'st e--Xlt

a0

La = 2.67886 ,--If NoXi

-

4.17377 e - A 2 L t 0.00034 e - A a t 0.51287

-

- 0 00132 ,--hat

+

3,06999 e - A z L f 0,39330

- 0.00125 e F A 5 f 0.00215

Valid 1 hour after purification, assuming 100% retention of radon-220. b Assumes 100% counting yield of all betas.

0

Table V.

Thorium Series (4n)

(Growth of decay products from initially pure parent. Parent, thorium-228) Time, Day0 0 1

6

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 3% 40 41 42 43 44 45 46 47 48 49 50

Razz4 0.0000 0.1733 0.3165 0,4345 0.5319 0.6123 0.6786 0.7332 0.7781 0.8151 0.8456 0,8705 0.8910 0.9078 0.9214 0.9325 0.9416 0.9488 0.9547 0.9594 0.9631 0,9660 0.9682 0.9698 0.9710 0.9719 0.9724 0.9726 0.9727 0.9725 0.9723 0.9719 0.9714 0.9708 0.9701 0.9694 0.9687 0.9679 0.9671 0.9663 0.9654 0.9645 0.9636 0.9627 0.9618 0.9609 0.9899 0.9590 0.9581 0.9571 0.9562

Pb212

at/NoAl

,t?t/.Y,Al

0.0000 0.0878 0.2280 0.3578 0.4679 0.5593 0.6349 0.6972 0.7485 0.1907 0,8256 0.8540 0.8775 0.8967 0.9124 0.9252 0.9356 0.9440 0.9508 0.9563 0.9606 0.9640 0.9667 0.9687 0.9702 0.9713 0.9720 0.9724 0.9726 0.9726 0.9724 0.9721 0.9717 0.9711 0.9705 0.9699 0.9692 0.9684 0.9676 0.9668 0.9659 0.9651 0.9642 0.9633 0.9624 0.9615 0.9605 0.9596 0.9587 0.9577 0.9568

1 0000 1.5986 2.1666 2.6508 3.0535 3.3861 3.6603 3.8861 4.0718 4.2245 4.3503 4.4533 4.5372 4.6060 4.6619 4,7072 4.7437 4.7729 4.7968 4.8156 4.829Y 4.8410 4.8493 4.8554 4.8596 4.8622 4,8035 4,8637 4.8631 4.8617 4.8597 4.8572 4.8543 4.8511 4.8476: 4.8439 4.8399 4.8359 4.8317 4.8274 4.8230 4.8185 4.8140 4.8094 4.8048 4.8002 4.7955 4.7908 4.7861 4.7814 4.7767

0.0000 0.2204 0,5929 0.9404 1.2356 1 ,4809 1.6835 1 ,8506 1 ,9882 2.1015 2.1950 2.2717 2.3345 2.3861 2.4283 2.4626 2,4905 2.5130 2.5314 2.5462 2.5577 2.5669 2.5741 2.5795 2.5836 2.5865 2.5884 2.5896 2.5901 2,5901 2.5896 2.5888 2.5876 2.5862 2.5846 2.5829 2.5810 2.5789 2.5768 2,5746 2.5724 2.5701 2.5677 2.5653 2.5629 2.5605 2.5580 2.5556 2.5530 2.5506 2,5481

.

V O L U M E 26, N O . 6, J U N E 1 9 5 4

1065 -

Table VI.

EXAMPLE 3. What fraction of thorium-228 will remain after 1 year? Since the tables do not give decay values beyond 100 days, it is necessary either to make use of the decay constant given in Table I or to use Equation 5, thus:

Thorium Series (4n)

(Growth of decay products from initially pure parent.a Parent, radium- 224) ( R a m ) 2!3. = e-Xd &VOX,

(~~220)

=

1.00017 ( e - ~ l t

(P021')

=

1.00017 e-X1t

'* .vox1

Nohi

e4 1.13823 KOA1

-

1.00308 e - - X 2 t

+ 0.00291 e--XBt

+ 0.00143 e - x x z t - 1.13966

=

(Pb212)

- ,--~zt)

( ~ i z i 2 )'$3 = 1.15151 e - ~ i l A OX1

\I

s, Nflv,xi

0x1

=

+ 0.10798

+ 0,00001 e - - x 2 L -

1 .I8219

1.26564 2.00327

=

4.15185 e - & l t

-

-

3 05387

+ 0.00143 e - X z t

+- 0.11381 e - X s t - 0.00037 ~

+ 0.00291 e - X 3 t

1.25947 e - - d t

-

3.24033 e - - X 4 t

-

-

+ 0.10798 +

e-Xbt

- 0.0003i

0.18540

0 Valid 1 hour after purification, assuming 100% retention of radon-220 and its decay products. b Assumes 100% counting yield of all betas.

--~ .~

0.2721 0.3155 0.3557 0.3928

3 . 085fi 8.1098 3.1315 3.1505

0.6044 0.7262 0.8402 0.9464

18 21 24

0.4271 0.4588 0.4880 0.5149 0.5832 0.6355 0.6749 0.7036

3.1667 3.1803 3.1914 3.2002 3.2137 3.2111 3.1956 3.1698

1 ,0449 1.1361 1,2203 1.2981 1.4962 1 ,6488 1 ,7644 1 ,8498

30 36 42 48 54 60 66 72

0.7369 0.7472 0.7425 0.7284 0.7082 0.6846 0.6590 0.6326

3.0953 3.0006 2.8944 2,7823 2.6681 2.5543 2.4425 2.3336

1.9514 1 ,9873 1 ,9805 1.9462 1 ,8948 1,8331 1.7657 1 ,6966

78 84 90 96 102 108 114 120

0.6061 0.5798 0.5542 0.5293 0.5053 0.4822 0,4600 0.4388

2.2283 2.1268 2,0294 1 ,9360 1 ,8467 1.7613 1.6797 1.6018

1,6249 1.5549 1,4863 1.4197 1.3554 1.2935 1 ,2341 1,1773

126 132 138 144

0.4185 0.3992 0.3807 0.3630

1 ,5275

1,1229 1 ,0709 1.0213 0.9739

n

10 11 12

15

Fraction remaining after 100 days

=

Fraction remaining after 65 days

= 0.9371 (Table

0.9049 (Table 111)

111)

~

Differential Decay. Mixtures of nuclides are sometimes encountered which cannot conveniently be resolved by chemical methods. Various means have been employed to determine the e ~ amount of each species present, but one of the most generally useful is the technique of differential decay.

Exmpm 4. A mixture is known to contain only radium-223 and radium-224 in equilibrium with their decay products. When counted, the activity of the mixture is found to have decayed to half its original value in 5 days. What fraction of the original activity was due to radium-224 and its decay produrts? Table VIII.

(Growth of decay products from initially pure parent. Parent, radium-224) Time, BIINoXI Hours PbQ12 Ut/.VOXl 0.0000 1 . 0000 0 0,0000 0.0878 2.9936 1 0.0617 0.2096 3.0087 0.1204 2 0.3425 3.0328 0.1748 3 0,4758 3.0594 0.2253 4

7 8

= 0.6943, or,

. - ~ - ~ - ~ _ _ _

~

Table VII. Thorium Series (4n)

5 ti

=

Fraction remaining after 365 days = (0.9049)3(0.9371) = 0.6043

- o.oooo2 ,-M 1.25947

NsXs

(T1108)

Fraction remaining after 1 year

1 ,4565 1.3889 1.3244

Synonym UI

Uranium-Radium Series (4n

Mode of Decay a

Energies, MeV. 4 2

56% 0 . 2

UXl

[44%

uxz [(O. I:%

IT)

0.1

90% 0 6 10% 1 2 (0 4)

[

0.02876 D -1

24.10 D

0.0012 H-1 1 14x1

0.6080 11 - 1

UZ

8-

2.3

6.7H

a

4.8

267,OOOY

0,0000026 Y -1

a

75% 4 . 7 [25% 4 . 6

80,000 Y

0,0000087 Y

1,620 Y

0.000428

Io

93% 4 . 8

... E 111

7% 4 . 6 5.5

a

3.825 D

0 . 1035 €1-1

T-1

0.1812 D -1 0.00755H-1 0,2273 M

0.7

26.811

6.7

c

-1

0.02586 M - 1 1,5518 H-1

2.0s

0 3466 S-1

0.019s

36 5 s - 1

(1)

7.1

U

23% 3 . 2 [77% 1 . 7 5.5

2.1111 H-1

0.000164S

4,227 S - 1

1.8

1.32 M

0.5251 31-1

22Y

0.03151

0.03

ponintials

4.85 D

(5.0)

(5)

5.3 138.4 D

the decay of bismuth-210 for the desired time is the product of its decay in 7 days (0.3677 from Table X ) , in 3 hours (0.9823 from Table IX), in 10 minutes (0.9990 from Table IX), and in 4 minutes (0.9996 from Table 1X)-i.e.,

Stable a

(0.3677) (0.9823) (0,9990) (0.9996) = 0.3607

1.5

B-

Isomer.

...

0.0351911-1

19.7M

7.7

92y0 1.1 8% 1 . 2 eaeb

Decay Constants 1.54 x 10-1OY-1

UII

a

,(a+b)

Half Life 4 . 5 1 X lO9Y

4- 2)

4.23M

...

T-1

0 1429 D-1

[

0,00595 Hi-

0 005008 D -1

[1 8279 Y-1 0.1639 M

...

-1

-1

A N A L Y T I C A L CHEMISTRY

1066 Table IX. Time,

Min. 0

Uranium-Radium Series (4n

1.oOoo

1 2 3 4 5 6 7 8 9 10 20 30 40 50

0,9999

0,9998 0.9996 0.9994 0.9992 0,9990

Hours 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1.0000 0.5444 0.2964 0.1614 0.0879 0.0478 0.0260 0.0142 0.0077 0.0042 0.0023 0.0000

~

1.0000 0.9999 0.9997 0.9996 0.9995 0.9994 0.9992 0.9991 0.9990 0.9989 0.9987 0.9975 0.9962 0.9950 0.9937

1.0000 0.7967 0,6348 0.5057 0.4029 0.3210 0.2557 0.2038 0.1623 0.1293 0.1030 0.0106 0.0011 0,0001 0,0000

0.9924 0,9850 0,9776 0,9703 0.9629 0.9557 0.9485 0.9414 0.9343 0.9273 0.9203 0.9134 0.9065 0.8997 0.8929 0.8862 0.8795 0,8729 0.8663 0.8598 0.8533 0.8470 0.8405 0.8343

0,9988 0.9976 0.9964 0.9952 0,9940 0.9928 0.9916 0.9904 0.9893 0.9881 0.9869 0.9857 0.9845 0,9833 0.9822 0,9810 0.9798 0,9786 0.9775 0.9763 0.9731 0.9739 0.9728 0.9716 ~~

1.0000 0.9744 0,9496 0.9253 0.9017 0.8787 0.8562 0.8344 0.8131 0.7923 0.7721 0.5962 0.4603 0.3554 0.2744

1.0000 0.9654 0.9320 0.8998 0.8687 0.8387 0.8097 0.7817 0,7547 0.7285 0,7034 0.4948 0,3480 0.2448 0.1722

0,2119 0,0449 0.0095 0.0020 0.0004 0.0001 0.0000

0.1211 0.0148 0.0018 0,0002 0,0000

~

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Th'ak 1.0000 0.9716 0.9441 0.9173 0.8913 0.8661 0.8415 0.8176 0.7945 0.7719 0,7501 0.7288 0.7081 0.6880 0.6685 0.6496 0.6312 0.6133 0.5959 0.5790 0.5626 0.5466 0.5311 0.5161 0,5014 0.4872 0,4734 0,4600 0.4469 0.4343 0.4220 0.4100 0.3984 0 3871 0.3761 0,3654 0.3551 0.3450 0.3352 0.3257 0.3165 0.3075 0.2988 0 2903 0.2821 0.2741 0.2663 0.2588 0.2514 0 2443 0.2374

1.0000 0.5915 0.3499 0.2069 0.1224 0.0724 0.0428 0.0253 0.0150 0,0089 0.0052 0.0000

1~0000 0.9999 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 0.9992 0,9991 0.9990 0.9980 0.9970 0.9960 0.9950 0.9940 0,9882 0,9823 0.9765 0.9706 0.9649 0.9692 0.9535 0.9478 0.9422 0.9366 0.9310 0.9255 0.9200 0.9146 0.9091 0.9038 0.8983 0.8931 0.8877 0.8824 0.8772 0.8720 0.8668

~

Table X. Lranium-Radium Series (4n Time, Days 0

+ 2)

Bill4

Th214

1,0000 0.9999 0.9998 0.9997 0.9997 0,9996 0,9995 0,9994 0.9993 0.9992 0.9991 0.9991 0.9990 0 9989 0,9988 0.9987 0.9986 0.9985 0,9984 0.9984 0.9983 0.9982 0.9981 0.9980 0,9979 0,9978 0.9978 0,9977 0,9976 0.9975 0.9974 0.9973 0.9972 0.9972 0.9971 0,9970 0.9969 0.9968 0.9967 0.9966 0.9966 0.9965 0.9964 0.9963 0.9962 0.9961 0.9960 0.9960 0 9959 0.0001 0 9958 0.0001 0,9957

1.0000 0.8343 0.6960 0.5806 0.4844 0.4041 0.3371 0,2813 0.2346 0.1967 0.1633 0.1362 0.1137 0.0948 0.0791 0,0660 0.0551 0.0459 0.0383 0,0320 0,0267 0.0222 0.0186 0.0155 0.0129 0.0108 0.0090 0.0075 0.0063 0.0052 0.0044 0,0036 0.0030 0,0025 0.0021 0,0018 0.0015 0.0012 0.0010 0.0009 0 0007 0.0006 0 0005 0.0004 0.0003 0 0003 0 . 0,002 0.0002 0.0002

1.0000 0.8668 0.7514 0.6513 0.5646 0,4894 0,4242 0.3677 0.3188 0.2763 0.2395 0.2076 0.1800 0.1560 0 1352 0.1172 0 1016 0.0881 0,0763 0,0662 0.0574 0 0497 0.0431 0.0374 0,0324 0 0281 0.0243 0.0211 0.0183 0.0158 0.0137 0.0119 0.0103 0,0089 0.0078 0,0067 0.0058 0 0051 0.0044 0.0038 0.0033 0.0029 0.0025 0 0021 0,0019 0 0016 0.0014 0 0012 0.0010 0.0009 0,0008

1.0000 0.9950 0.9900 0.9851 0.9802 0.9753 0,9704 0,9655 0.9607 0,9559 0.9512 0.9464 0.9417 0.9370 0.9323 0.9276 0,9230 0.9184 0.9138 0.9092 0.9047 0,9002 0,8957 0.8912 0.8867 0.8823 0.8779 0.8735 0.8692 0.8648 0.8605 0.8562 0.8519 0 8477 0.8434 0 8392 0.8350 0.8308 0.8267 0.8226 0.8185 0.8144 0.8103 0.8063 0.8022 0.7982 0.7942 0,7603 0.7863 0,7824 0.7785

Time, Days 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91

92 93 94 95 96 97 98 99 100

+ 2)

Thlak 0.2374 0.2307 0.2241 0.2178 0.2116 0,2056 0.1998 0.1941 0.1886 0.1832 0.1781 0.1730 0.1681 0.1633 0.1587 0,1542 0.1498 0.1456 0.1415 0.1374 0.1335 0.1298 0.1261 0.1225 0.1190 0.1157 0.1124 0.1092 0.1061 0 . IO31 0.1002 0.0973 0 0946 0.0919 0 0893 0.0868 0 0843 0,0819 0.0796 0.0773 0.0751 0.0730 0.0709 0.0689 0.0670 0.0651 0.0632 0.0614 0.0697 0.0680 0.0564

Pb210

0.9957 0.9956 0.9955 0.9954 0.9953 0.9953 0.9952 0.9951 0,9950 0.9949 0.9948 0.9947 0.9947 0.9946 0.9945 0,9944 0,9943 0.9942 0.9941 0.9941 0.9940 0,9939 0.9938 0.9937 0.9936 0.9935 0.9935 0.9934 0.9933 0.9932 0.9931 0.9930 0.9929 0.9929 0.9928 0,9927 0.9926 0.9925 0.9924 0.9923 0.9923 0.9922 0.9921 0,9920 0.9919 0.9918 0.9917 0.9917 0.9916 0.9915 0.9914

Bill0

p0210

0.0008 0,0007 0.0006 0,0005 0,0004 0,0004 0.0003 0.0003 0,0003 0.0002 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0000

0.7785 0.7746 0.7707 0.7669 0.7630 0,7593 0.7554 0.7517 0.7479 0.7442 0.7404 0.7368 0.7331 0.7294 0.7258 0.7221 0.7185 0.7149 0.7114 0.7078 0.7043 0,7008 0.6973 0.6938 0.6903 0.6869 0.6834 0.6800 0,6766 0.6732 0.6699 0.6665 0.6632 0.6599 0.6566 0,6533 0.6600

0.6468 0.6436 0 6404 0,6372 0.6340 0.6308 0.6277 0,6245 0.6214 0.6183 0.6152 0.6121 0.6091 0.6060

V O L U M E 2 6 , N O . 6, J U N E 1 9 5 4

1067 __

Table XI.

Uranium-Radium Series (4n

(Growth of decay products from initially pure parent." (Ra22B) >>kl =

+ 2)

= .vox1

1,00001 e - x ? t

-

1.00056

-

1.00545 e - X 2 t

(BiZLO)

1.00001

e--X2t

f 0,00055 e-xx3f (1'0210)

-

0.00007 e--Xat

. E6 = I.OOOOI SOX, =

o\'.h1

(Bizlo)

-m sox, SOX,

(po?in)

e--Xlt

&! .\-OX1

.vox1 1.01378 ,--Ii

+ 0.00048 e--Zt

-

1.01428

=

1.01379 e P x l t

-

-

1,01489

=

1,01402

+ 0.00005

-

1,03267 e - x d

0.00179 e - x 2 t

0.00286 e -

4.02759

-

3.00958 e - x 2 t

-

+ 0.00056 e--13f

1.03267 e--Xit 0.01179 e--Xsf

-

f 0.02084

f 0.01868e-X9t

- 0.00010 e - x B t

- 2,01582 e - - h 2 t - 0.00006 e--X3t

-

+ 0.01868

O.OOOIO e--hBt

0,01179 =

+

=

,j.01405 e - - X l t

- a' =

+

O.OOOO~ 0.02084 e--X4t - 0.01179 e--Xat

- i.00906 e - x z f

ssxs

.\'OX1

sox1

+ 0.02636

- 2.02917 e - x 7 f

-

f 0.00286 e - x h s t

yalid 1 hour after purification, assuming 100% retention of radon-222 and its decay products. b Assumes 100% counting yield of all betas. a

-

Table XII.

Uranium-Radium Series

(Growth of decay products from initially pure parent Time Days

Rn22?

Pb2Io

Biz10

Po210

0.0000

0.0000

5

0.0000 0.1657 0.3040 0.4193 0.5156 0,5959

0.0000

6 7 8 9 10

0.6628 0.7187 0.7653 0.8042 0.8367

0,0002 0.0003 0.0003 0.0004 0.0005

11 12 13

0.8637 0.8863 0,9052 0,9209 0.9340

Parent, radium-226) W'\'O'oXI

.

&/.\'OX

1 0000 1.4892 1.9054 2.2525 2.5422 2.7838

0.0000 0.3194 0.5979 0.8304 1 ,0243 1,1861

0.0000 0.0001 0.0001 0.0002 0.0002

2,9853 3.1536 3.2939 3.4110 3.5086

1.3211 1 ,4337 1.5277 1 ,6062 1.6710

0.0005 0.0006 0.0007 0.0008 0.0009

0.0002 0.0002 0.0003 0.0004 0.0004

3,5901 3.6580 3.7147 3.7619 3.8014

1 ,7263 1.7719 1.8100 1.8417 1 ,8683

20

0,9449 0,9541 0.9617 0.9680 0.9733

0.0010 0.0010 0.0011 0.0011 0.0012

0.0005 0.0006 0.0007 0.0007 0.0007

3.8344 3.8618 3.8847 3.9038 3.9197

1 ,8905 1 ,9090 1 ,9245 1 ,9374 1 ,9482

21 22 23 24 25

0,9777 0.9814 0.9845 0.9871 0,9892

0.0013 0.0014 0.0015 0.0016 0.0017

0.0008 0.0009 0.0010 0.0011 0.0011

0.0000 0.0001 0.0001 0,0001

3.9331 3.9442 3.9535 3.9612 3.9675

1,9573 1 ,9649 1.9712 1.9766 1.9810

2G 27 28 29 30

0.9910 0.9926 0.9937 0.9948 0.9956

0.0018 0,0018 0.0019 0.0020 0,0021

0.0012 0.0013 0.0014 0.0015 0.0015

0.0001 0.0001 0.0001 0.0001 0,0001

3.9728 3.9774 3.9811 3.9842 3.9868

1 ,9848

31 32 33 34 35

0.9963 0.9969 0.9974 0.9979 0,9982

0,0022 0.0023 0 002.4 0.0024 0,0025

0.0016 0.0017 0.0018 0.0019 0.0019

0.0001 0.0001 0.0001 0.0001 0.0001

3.9890 3.9908 3.9923 3.9936 3.9947

1.9964 1 ,9978 1.9990 2.0000 2.0009

36 37 38 39 40

0,9985 0.9987 0,9989 0.9991 0.9993

0.0026 0.0027 0,0028 0.0029 0.0030

0,0020 0.0021 0.0022 0.0023 0.0024

0,0001 0.0001 0.0001 0.0001 0.0002

3.9956 3.9963 3.9969 3.9974 3.9978

2.0016 2.0023 2.0028 2,0034 2.0038

45 50 60 70 80 90 100

0.9997 0.9998 0,9999 0 I9999 0,9999 0,9999 0.9999

0.0034 0.0038 0.0047 0.0055 0.0064 0.0072 0.0081

0,0028 0,0032 0.0041 0.0049 0.0058 0.0067 0,0075

0,0002 0,0003 0.0005 0.0006 0,0009 0.0012 0.0015

3.9992 3.9997 4,0001 4.0003 4.0006 4.0007 4.0010

2.0055 2.0067 2.0086 2.0103 2.0120 2.0137 2.0154

0 1

2 3 4

14

15 16

17 18 19

_

_

~

Uranium-Radium Series (4n

+ 2)

Parent, lead-210)

= e-hlt

- E 2

= 1,00060 ( , - A d

-

e-X?t)

0.0000 0.0001

z53= 1,01815 e - x l t + 0.03034 e - x z t sox1

+

0.005.i~e--4t

(Pb?10)

_

x o x 1

* 'k? = 1,00001 e - - l t SOXI

(POZ,()

~

A\TOToxl

-

sox1

( ~ i 2 1 1 )

_

( C r o n t h of decay products froin initially pure parent. (I'hZ10) slxl

.\-ox1

(Pb21')

_

Table XIII.

Parent, radiuin-226)

,-Alf

(RnZ22) s?x2 = 1,00001 e - x d (1'0218)

_

1 ,9880 1.9906 1.9929 1 ,9948

-

1.05449

Time. Days

Biz10

Po210

Time, Days

Bizlo

0 1 2 3

0.0000 0.0003 0.0013 0.0027 0.0047 0.0071

40 41 42 43

?

0.0000 0.1332 0.2485 0,3484 0.4364 0.5105

44 45

0.9939 0.9942 0.9946 0.9947 0.9950 0.99.51

6 7 8 9 10

0.5755 0 6320 0.6809 0 7233 0 7601

0.0098 0.0128 0 0160 0.0194 0.0230

46 47 48 49 50

0.9952 0.9953 0.9954 0.9956 0.9955

0,1889 0,1929

11 12 13 14 15

0.7919 0.8195 0.8434 0.8640 0,8820

0,0268 0.0306 0.0346 0.0387 0,0429

51 52 53 54 65

0.9956 0,9955 0.9955 0.9955 0.9955

0.1969 0.2009 0.2049 0.2088 0.2127

I6 17 18 19 20

0.8975 0,9109 0,9226 0.9328 0.9415

0.0471 0.0514 0,0557 0.0601 0.0645

56 57 58 59 60

0.9954 0.9954 0.9953 0.9953 0.9952

0.2167 0.2205 0.2244 0.2283 0.2321

21 22 23 24 25

0.9491 0.9556 0.9612 0.9661 0.9704

0.0089 0.0733 0.0777 0.0821 0.0866

61 62 63 64 65

0.9952 0.9951 0.9950 0.9950 0.9949

0.2359 0.2397 0.2435 0.2472 0.2510

26 27 28 29 30

0.9740 0.9772 0.9799 0.9822 0.9843

0.0910 0.0984 0,0998 0.1042 0.1086

6f1

67 68 69 70

0.9948 0,9948 0.9947 0.9946 0.9945

0.2547 0.2584 0.2621 0.2657 0.2694

31 32 33 34 35

0,9860 0.9875 0.9888 0.9899 0.9909

0.1130 0.1174 0.1217 0.1261 0.1304

71 72 73 74 75

0 9944 0,9944 0.9943 0.9942 0,9941

0.2730 0.2766 0.2802 0.2837 0,2873

36 37 38 39 40

0.9916 0.9924 0.9929 0,9934 0.9939

0.1347 0.1390 0.1432 0,1475 0.1517

76 77 78 79 80

0.9940 0.9940 0.9939 0.9938 0.9937

0,2908 0.2943 0,2978 0.3013 0.3049

Po2lQ

0.1537 0.1559 0.1601

0.1643 0.1684 0.1726 0.1767 0.1808

0.1848

Let Corepresent the oiiginal count obtained, let R1 be the fraction of Co due to radium-223 and its decay products, and let R2 be the fraction due to radium-224 and its decay products. In 5 days, radium-223 decays to 0.7339 of its original value (Table SVI)and radium-224 decays to 0.3859 of its original value (Table 111). Therefore,

+ I ,0000 Rr 0 . 5 CO = 0 . i 3 3 9 RI + 0 3859 Rt CO = 1,0000 R,

The two equations are solved simultaneously, and it is found that R, = 0.3279 COand R, = 0.6721 CO. The results give a measure of the relative activity of each isotope and its decay products in the mixture a t the time of the first measurement. rl determination of the absolute amount of each isotope depends upon the radioactive equilibrium constant (see below), the nature and energy of the radiation counted, the self-absorption of the sample. and the geometry of the counter, subjects which are beyond the scope of this discussion. Radioactive Equilibrium. \\-hen a radioisotope decays to a radioactive daughter whose half life is shorter than that of its parent, a condition of radioactive equilibrium will eventually be attained in which the daughter product is decaying as rapidly as it is being produced. The mixture then decays with the half life of the parent. The ratio of daughter activity to parent activity is given by the expression

ANALYTICAL CHEMISTRY

1068 Table XIV. Uranium-Actinium Series (4n Synonyni

Mode of Decay

BCU

a

Energies, MeV. 20% 4 . 6

80% 4 . 4 CY

...

...

5?% 20?& 3% 13%

[(1.2ia)

a

-4n

a

AcB

x

10-10Y-1

25.65H

0.0000202 Y -1

34,300y

0.0315 Y - 1

22.0 Y

0.0000863 D -1 0.0373D-1

[

18.6D

0 00155 H-1

1.9804 H-1

[

41% 35% 17% 7%

5.7 5.6 5.5 5.4

11.2D

69% 15% 12% 4%

6.8 6.6 6.4 6.2

3'92S

10.61 hl-1

0.00183 S

22.800M-1

7.4

p-)

0 0330 XI -1 0.0619 D-1 0 00258 H-1

:::

0,0192 M-1

8.0

1O-dS

416,000M-1

7.4

0.529

79 98 M-1

(?)

[i-:

8-

...

7.9

0.6486 D-1

21 M

r a L(0.0005%

88XlOeY

1.2

8-

.4OX

5.0 4.9 4.8 4.7 0.02

(4.9) 24% 6 . 1 22% 6 . 0 7% 5 9 25% 5 . 8 22% 5 . 7

a

ACK

Decay Constants

0.0270H-1

a

KdAc

AC.4

0.2

8-

Half Life

+ 3)

a

1.1520H-1

36.lhI

ACC

AcC"

8-

ACC'

a

AcD

...

...

Stable

...

Table XV. Uranium-Actinium Series (4n TI1227

FrZza

Time, Min.

Thlal

0 1 2 3 4 5 6 7 8 9 10 20 30 40 50

1.0000 0.9995 0,9991 0.9986 0.9982 0.9977 0.9973 0.9968 0.9964 0.9959 0.9955 0.9910 0.9866 0.9822 0.9778

1.0000 1.0000 0.9675 0.9361 0.9057 0,8763 0.9999 0.8479 0.8204 0.7937 0.7679 0.7430 0.9997 0.7189 0.9995 0.5168 0,9992 0,3715 0.9990 0.2671 0.9987 0.1920

0.9734

0.9984 0.9969 0.9954 0.9938 0.9923

Hours 1 2

3 4 5

0 9474

0.9221 0.8975 0.8736

jy--=e Xp

X2

- XI

(Decay in minutes and hours) Razz3 Pbz" Bill1 Time, Hours 1.0000 1.0000 0.9810 0.9623 0,9440 0.9261 0.9998 0.9085 0.8912 0.8742 0.8576 0.8413 0.9996 0.8253 0.9991 0.6811 0.9987 0.5621 0.9983 0.4639 0.9979 0.3829

0.1380 0.9974 0,0190 0.9949 0.0026 0.9923 0.0004 0.9897 0.0001 0.9872

Ti

Ti - Tz

1.0000 0.7255 0.5263 0,3819 0.2770 0.2010 0.1458 0.1058 0.0767 0.0557 0.0404 0.0016 0,0001 0.0000

0.3160 0.0998 0.0316 0.0100 0.0032

(6)

or, for a three-member chain, the ratio of the activity of the third member to that of the first member is (sa)

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

+ 3)

Ti1931 0.8504 0.8276 0.8056 0.7841 0.7632 0.7428 0.7230 0.7038 0.6850 0.6668 0.6489 0.6317 0.6148 0.5985 0.5825 0.5689 0.5518 0.5371 0.5228

Tti221

Frn:

Ra22:

0.9907 0.0000 0.9846 0.9892 0.9821 0.9877 0.9796 0.9861 0.9771 0.9846 0.9745 0,9831 0,9720 0,9815 0.9695 0.9800 0.9670 0.9785 0.9645 0.9170 0.9621 0.9755 0.9596 0,9739 0.9571 0.9724 0.9546 0.9709 0.9622 0.9694 0.9497 0.9679 0.9473 0.9664 0.9448 0.9649 0,9424 0.9634 0,9400

Pb211

0.0010 0.0003 0.0001 0,0000

If the half lives of the decay products are very much smaller than that of the parent, K , is very nearly equal to unity. The time required for the first decay product to reach 99.9% of equilibrium is approximately 10 KST2. EXAMPLE 5 . How long does it take for thorium-234 to reach 99.9% of equilibrium in a sample of uranium-238? How much thorium-234 is present in a microcurie of uranium-238 30 dayr after purification?

1069

V O L U M E 26, NO. 6, J U N E 1 9 5 4 -

Table XYI.

Uranium-Actinium Series (4n f 3) (Decay in days)

Time, Days

Th2al

0 1 2 3 4 5

1.0000 0.5228 0.2733 0.1429 0.0747 0.0391

6

0.0204 0.0107 0.0056 0.0029 0.0015

0.9995 0.9994 0.9993 0.9992 0.9991

0.7996 0.7704 0.7422 0.7151 0.6889

0,0008 0.0004 0.0002 0.0001 0.0001

0.9991 0.9990 0.9989 0.9988 0.9987

0,0000

Th227

Razz1

50 51 52 53 54 55

0 0 0 0 0 0

9957 9956 9955 9954 9953 9953

0 1552 0 1495 0 1440 0 1387 0 1337 0 1288

0 0453 0 0426 0 0400 0 0376 0 0354 0 0332

0.6898 0.6484 0.6095 0.5729 0.5385

36

57 58 59 60

0 9952 0 9951 0 9950 0 9949 0 9948

0 1241 0 1195 0 1182 0 1109 0 1069

0 0312 0 0294 0 0276 0 0260 0 0244

0.6637 0.6394 0.6160 0.5935 0.5718

0.5062 0.4758 0.4473 0.4204 0.3952

61 62 63 64 65

0 0 0 0

9947 9947 9946 9945 0 9944

0 1030 0 0 0992 0 0 0956 0 0 0921 0 0 0887 0

0.9986 0.9985 0.9984 0.9984 0.9983

0.5509 0.5307 0.5113 0.4926 0.4746

0.3715 0.3492 0.3282 0.3085 0.2900

66 67 68 69 70

0 0 0 0 0

21 22 23 24 25

0 9982 0 9981 0 9980 0.9979 0 9978

0 4572 0 4405 0 4244 0 4089 0 3939

0 2726 0 2563 0 2409 0 2264 0 2128

71 72 73 74 75

0.9939 0,9938 0.9937 0.9936 0.9935

0.0709 0.0683 0.0658 0.0634 0.0611

26 27 28 29 30

0.9978 0 9977 0.9976 0.9975 0.9974

0.3795 0,3656 0.3522 0.3394 0.3269

0 2001 0 1881 0.1768 0 1662 0.1562

76 77 78 79

so

0 9935 0 9934 0 9933 0 9932 0 9931

0 0 0 0 0

31 32 33 34 35

0 9973 0 9972 0.9972 0.9971 0.9970

0.3150 0.1468 0.3035 0.1380 0.2924 0.1297 0.2817 0.1219 0.2714 0.1146

81 82 83 84 85

0 0 0 0 0

9930 0 0489 0 0067 9929 0 0471 0 0063 9929 0 0454 0 0059 9928 0 0437 0 0055 9927 0 0421 0 0052

36 37 38 39 40

0.9969 0.9968 0.9967 0.9966 0.9966

0.2614 0.2519 0.2427 0.2338 0,2252

0.1077 0.1013 0.0952 0.0896 0.0841

86 87 88 89 90

0 0 0 0 0

9926 9925 9924 9923 9923

0 0 0 0 0

41 42 43 44 45

0.9965 0.9964 0.9963 0.9962 0.9961

0.2170 0.2091 0.2014 0.1940 0.1869

0.0791 0.0743 0.0699 0.0657 0.0617

91 92 93 94 95

0 9922 0 9921 0 9920 0 9919 0 9918

0 0 0 0

46 47 48 49 50

0.9960 0.9960 0 9959 0 9958 0 9957

0,1801 0.1735 0.1672 0.1611 0.1552

0 0580 0.0545 0.0513 0 0482 0.0453

96 97 98 99 100

7 8 9 10 11

12 13 14 15 16 17 18 19 20

Ac22i

Table XVII. (A4c227j55

=

,-Ad

-\!x2

=

0,9QO20 ( e - k l t

-

0,012 i e - X i t

-

.YoA I

(T)iZ2i)

0 0856 0 0168 0 0823 0 0158 0 0793 0 0149 0 0764 0 0140 0 0736 0 0131

0589 0567 0547 0527 0507

0 0124 0 0116 0 0109 0 0103 0 0096

n

0091 0 0085 0 0080 0 0075 0 0071

0406 0 0049 0391 0 0046 0377 0 0043 0363 0 0041 0349 0 0038

0337 0324 0312 0301 0 0290

0 9917 0 0279 9917 0 0269 9916 0 0259 9915 0 0250 9914 0 0241

0 0 0 0

Uranium-Actinium Series (4n

(Growth of decay products from initially pure parent.a

9943 9942 9941 9941 9940

0229 0216 0203 0190 0179

0 0036 0 0034 0 0032 0 0030 0 0028

0026 0025 0023 0022 0 0021 0 0 0 0

+ 3)

Parent, actinium-227)

,-Ad)

.\TOXI

(Frz2?) ?-2aA2Q

SOA,

rd.3

(Razz9 NOXI

=

iRn"g) X,h, = NoAi (po21j)3 x 5= SOXI

=

1.00309

e-Azoi)

-

2.48912

f 0.00002

+

-

2.49248 e-'?'

-

0.00002

f 1.48875 e-Xat

+ 0.00006

f 1.48895 ,--st

+ 0,00007 ,--Et

SOXI

-53 9OAI

.VEX6 = 1.00370 e - A 1 ' (Pb2ll) XOXI , (Bi211)

%* = 1.00370 e - x 1 f

- 2,49268 e-Az'

-

0.00002

e-A2at

(T1207j

'xp = 1.00370 .\-ox,

-

-

0.00003

e

~~

.vox1

=

ut

.VOX1

2 U"

=

N

A, a

b

-

5.01707 e - - X ~ '

418.089 =

1.48541e-X31

.\-A3

2.01939

10.95034 , - A I f

- 912.529 ePAlt

2.49313 e - A z f

+ O . O O O Oe - ~x z a t + 6.94518

f 0.002 e-AnaL

- 4.98561 e - - x z '

f 1.48939 e - A 3 t

-

0.01205

f 0,00008 e--xflt

+ 0.00007 , - A d

+ 495.432 e - A d + 0,005 f 2.97813 e--Xat

+ 0,00014

,--Ef

Valid 1 hour after purification, assuming 100% retention of radon-219 and its decay products, Assumes 25 kev threshold and 100% counting yield for all countable betas.

__

ANALYTICAL CHEMISTRY

1070 Table XVIII.

+ 3)

Uranium-Actinium Series (4n

(Growth of decay products from initially pure parent.

Parent, actinium-227)

0 1 2 3 4

0.0000 0.0361 0.0710 0.1045 0.1368 0.1679

0.0000 0,0018 0.0057 0.0113 0.0186 0.0274

0.0120 0.0553 0.1054 0.1615 0.2229 0.2890

1.0000 4.6052 8.7808 13.457 18.577 24.087

0.0000 0.0154 0.0229 0.0341 0.0486 0.0660

50 51 52 53

0.8324 0.6804 0.8379 0.6905 0.8432 0.7002 0.8484 0.7096 0.8533 0.7188 0.8581 0.7278

3.5657 3.6113 3.6556 3.6985 3.7403 3.7807

297.15 300.94 304.63 308.21 311.69 315.06

1.3720 1.3921 1.4116 1,4305 1.4489 1.4668

6 7

0,0374 0,0487 0,0609 0.0741 0.0880

0.3593 0.4330 0.5098 0.5892 0.6708

29.938 36.084 42.485 49.103 55.901

0.0860 0.1084 0.1328 0.1591 0.1869

56

9 10

0.1979 0.2268 0.2546 0.2814 0.3072

57 58 59 60

0.8627 0.7364 0.8671 0.7449 0.8713 0.7530 0.8754 0.7610 0.8793 0.7687

3.8201 3.8582 3.8951 3.9310 3.9657

318.34 321.51 324.59 327.58 330.48

1.6842 1,5010 1.5174 1.5333 1.5487

11 12 13 14 15

0,3321 0,3561 0.3791 0.4014 0,4228

0.1027 0.1179 0.1336 0.1497 0,1662

0.7542 0.8390 0.9249 1.0117 1.0990

62.849 0.2161 69.917 0.2465 77.077 0.2779 84.305 0.3102 91.580 0.3431

61 62 63 64 65

0.8831 0.8868 0.8903 0.8936 0.8969

0.7762 0.7834 0.7901 0.7972 0.8038

3.9994 4.0320 4.0637 4.0943 4.1240

333.28 336.00 338.64 341.19 343.66

1.5637 1.5782 1.5923 1,6059 1.6191

16 17 18 19 20

0,4434 0.4633 0,4824 0,5009 0.5186

0,1829 0,1999 0,2170 0,2342 0.2515

1.1866 1.2742 1.3618 1.4491 1.5359

98.880 106.19 113.48 120.76 127.99

0.3766 0.4105 0.4447 0.4791 0.5136

66 67 68 69 70

0.9000 0.8102 0.9030 0.8164 0.9059 0.8224 0.9087 0.8282 0.9114 0.8338

4.1527 4.1803 4.2074 4.2334 4.2586

346.06 348.37 350.62 352.79 354.89

1.6319 1.6443 1.6563 1.6679 1.6792

21 22 23 24 25

0,5357 0.5522 0,5681 0,5834 0,5981

0,2688 0.2860 0,3032 0.3203 0,3372

1.6221 1.7075 1.7921 1.8757 1.9583

135.17 142.29 149.34 156.31 163.19

0.5481

0.5826 0.6170 0.6512 0.6851

71 72 73 74 75

0.9140 0.8393 0.9165 0.8446 0.9189 0.8497 0.9212 0.8546 0.9234 0.8694

4.2830 4.3066 4.3294 4.3514 4.3728

356.92 358.88 360.78 362.62 364.40

1.6901 1.7007 1,7109 1,7208 1,7304

26 27 28 29 30

0,6123 0.6259 0,6391 0.6518 0.6640

0.3540 0.3706 0,3871 0.4033 0.4193

2.0397 2.1199 2.1988 2.2764 2.3526

169.97 176.66 183.23 189.70 196.05

0.7187 0.7520 0.7849 0.8174 0.8494

76 77 78 79 80

0.9255 0.9276 0.9295 0.9314 0.9332

0.8640 0.8685 0.8727 0.8770 0.8810

4.3934 4.4133 4.4326 3.4512 4.4691

366.11 367.77 369.38 370.93 372.43

1.7396 1.7485 1.7572 1.7656 1.7737

31 32 33 34 35

0,6757 0.4351 0,6871 0,4506 0,6980 0.4808 0.7086 0.4658 0.7186 0.4955

2.4274 2.5008 2.5727 2.6430 2.7119

202.28 0.8809 208.40 0.9119 214.39 220.25 0.9424 0.9724 225.99 1.0018

81 82 83 84 85

0.9350 0.9367 0.9383 0.9399 0.9414

0.8849 0.8887 0.8923 0.8959 0.8993

4.4866 4.5030 4.5195 4.5351 4.5502

373.87 375,25 376.62 377.93 379.18

1.7816 1,7890 1.7963 1,8034 1.8102

36 37 38 39

0,7283 0,7377 0.7468 0.7555 0,7638

0.5099 0,5240 0,5378 0,5513 0.5846

2.7793 2.8451 2.9095 2.9723 3.0336

231.61 237.10 242.46 247.69 252.80

1.0306 1.0688 1.0865 1.1136 1.1041

86 87 89 90

0.9428 0.9442 0.9455 0.9468 0.9480

0.9026 0.9057 0.9088 0.9117 0.9156

4,5648 4.5789 4.5925 4.6056 4.6182

380.40 381.57 382.71 383.80 384.85

1.8168 1,8232 1.8294 1,8352 1.8409

0,7719 0.7797 0,7872 0.7944 0.8013

0.5776 0,5901 0.8024 0,6144 0.8262

3.0933 3.1516 3.2084 3.2637 3.3176

257.78 262.63 267.37 271.98 276.46

1.1660 1.1912 1.2158 1.2399 1.2634

91 92 93 94 95

0.9492 0.9174 0.9503 0.9200 0.9514 0.9226 0.9525 0.9251 0.9535 0.9275

4.6304 4.6422 4.6536 4.6648 4.6751

385.87 386.85 387.80 388.71 389.59

1.8464 1.8517 1,8568 1.8618 1 8666

0,8080 0,8145 0.8207 0.8266 0,8324

0.6376 0,6487 0,6596 0,6702 0,6804

3.3700 3.4210 3.4706 3.5189 3.5657

280.83 1.2863 285.08 1.3086 289.22 1.3303 293.24 1.3514 2 9 7 , l 5 1.3720

96 97 98 99 100

0.9544 0.9554 0.9563 0.9511 0.9579

4.6853 4.6951 4.7046 4.7137 4,7225

390.44 391.26 392.05 392.81 393.54

1.8712 1.8757 1.8800 1.8841 1 8881

5

8

40

41 42 43 44

45 46 47 48

49 50

Table XIX. Uranium-Actinium Series (4n -k 3)

54 55

88

K, =

(Growth of decay products from initially pure parent Parent, thorium-227)

0.9298 0.9320 0.9341 0.9362 0.9382

4.51 X IO9 4.51 X 109

-

365

= 1.0000 (data from Table

VIII)

Time for 99.9% equilibrium = (IO) (1.0000)(24.1) = 241 days Because of its extremely long half life, the decay of uranium238 is negligible. Since Ke = 1.0000 and the decay of thorium234 in 30 days is 0.4220 (Table X), solution of Equation 2 yields

- 2.52024 e-XZt + 0.00276 e - X s t +

(TI207) % = 2.61756

.YOXI

2=

11.05767

-

10.06006 e--Zt

Noh

+

o,oo239 e - - X s t

@e 5.03448 e--X1t - 5.03941 'VOX1

O.OOOOI e-'& - 0.00008 e - - X 7 t 0.00001 e--Xst 4-

=

e-X2t

- oooool

+ 0.00601 e - X s t +

e--Xst

p,ooool e - ~ B t - o , ~ o ,o- -~~ ~ t Valid 1 hour after purification assuming 100% retention of radon-219 and its decay products. b Assumes 25-kev. threshold and 100% counting yield for all countable betas. 0

Growth of Decay Products. EXAMPLE 6. How much polonium-210 is present in a sample of lead-210 15 days after purification? The answer is read directly from Table XIII. The activity of polonium-210 will be 4.29% of the initial activity of the lend-210 . .. - - - - -. EXAMPLE 7. Assuming 100% retention of radon-219, how much will the alpha activity of the actinium-227 chain increase during the first 36 hours after purification? From Table S X I the ratio of the alpha activity a t 36 hours to the initial a1ph.a activity iS 0.0795/0.0120 = 6.625. EXAMPLE 8. Assuming 100% radon retention, what is the beta activity of the radium-226 chain relative to the activity of

V O L U M E 26, N O . 6, J U N E 1 9 5 4 Table XX.

Uranium-Actinium Series (4n

(Growth of decay products from initially pure parent. Time, Days

.

1071

+ 3)

Parent, thorium-227)

Time, Days

0 1 2 3 4 5

0.0000 0.0589 0.1121 0.1600 0.2031 0.2417

1.0000 1.1968 1.3746 1.5326 1.6724 1.7952

0.0000 0.1133 0.2202 0.3165 0.4030 0.4804

50 51 52 53

6 7

8 9 10

0.2760 0.3066 0.3335 0.3572 0.3779

1.9025 1.9955 2.0754 2.1432 2.1998

11 12 13 14 15

0.3958 0.4111 0.4241 0.4350 0.4438

10 17 18 19 20

Hours 0 1 2 3

1 0000 1.0160 1.0363 1.0580 1.0798 1.1021 1.1238

0.0000 0.0048 0.0167 0.0308 0,0454 0.0601 0.0747

0,0120 0.0136 0.0153 0.0169 0.0186 0.0203 0.0221

0.0000

6

0.0000 0.0003 0.0015 0.0037 0.0070 0.0113 0.0165

7 8 9 10 11 12

1.1688 1.1937 1.2187 1.2438 1.2690 1,2943

0.0228 0.0299 0.0378 0.0464 0.0558 0.0658

1.1454 1.1668 1.1881 1.2092 1.2302 1.2511

0,0892 0.1035 0.1178 0.1320 0.1460 0.1600

0.0238 0.0256 0.0273 0.0291 0.0309 0.0327

0.0125 0.0126 0.0127 0.0129 0,0130 0.0131

13 14 13 16 17 18

1.3197 0.0764 1.3451 0.0875 1.3706 0.0992 1.3960 0.1112 1.4214 0.1238 1.4469 0.1367

1.2718 1.2924 1.3128 1.3330 1.3530 1.3729

0.1738 0.1876 0,2012 0.2147 0.2282 0.2415

19 20 21 22 23 24

1.4723 1.4977 1.5230 1.5482 1.5734 1.5986

0.1499 0.1635 0,1773 0,1915 0.2058 0.2204

1.3927 1.4124 1.4319 1.4512 1.4703 1.4892

0.2547 0.2679 0,2809 0,2938 0.3067 0.3194

0.0456 0.0143 0.0475 0.0145 0.0495 0.0147 0.0514 0.0149 0.0533 0.0152 0.0553 0.01 54

0.2508 0.2427 0.2348 0.2271 0.2197

26 28 30 32 34 36

1.6486 1.6982 1.7474 1.7961 1.8443 1.8921

0,2501 0.2803 0.3111 0.3422 0,3735 0.4050

1.5268 1.5639 1.6004 1.6364 1.6718 1.7067

0.3446 0.3694 0,3938 0.4179 0.4416 0.4650

0.0592 0,0159 0.0632 0,0164 0.0672 0.0169 0.0713 0.0175 0.0754 0,0181 0.0795 0.0187

0.2125 0.2055 0.1988 0.1922 0.1858

38 40 42

1.9393 1.9859 2.0319 2.0774 2.1223 2.1666

0.4365 0.4680 0.4995 0,5308 0,5619 0.5929

1.7411 1.7749 1.8083 1.8411 1.8735 1.9084

0.4880 0.5106 0.5330

0.0837 0.0880 0.0923 0,0966 0.1010 0.1054

04

68 7%

2.2533 2.3376 2.4195 2.4989 2.5761 2.6508

0.6541 0.7141 0.7728 0.8301 0.8860 0.9404

1.9677 0.6397 2,0281 0.6801 2.0868 0.7194 2.1437 0,7575 2.1990 0,7944 2.2525 0,8304

0.1143 0.0246 0.1234 0.0263 0.1327 0.0281 0.1422 0.0300 0.1518 0.0320 0.1615 0.0341

80 88 96

2.7936 2.9276 3.0535

1.0448 1.1481 1.2356

2.3550 2.4514 2.5422

0.8989 0.9635 1.0243

0.1814 0.2019 0,2229

0.0386 0.0434 0.0486

108 120

3.2278 3.3861

1.3641 1.4809

2.6685 2.7838

1.1088 1.1861

0.2564 0.2890

0.0569 0.0660

55

1.2600 1.2245 1.1898 1.1557 1.1223 1 ,0896

0.5529 0.5380 0.5233 0.5089 0.4947 0.4808

0.5495 0.6109 0.6651 0.7127 0.7543

56 57 58 59 60

0.2333 1.0576 0.2266 1.0262 0,2201 0.9956 0.2136 0.9657 0.2073 0.9365

0.4672 0.4538 0.4406 0.4278 0.4152

2.2463 2.2835 2.3121 2.3330 2.3467

0.7903 0.8212 0.8473 0.8691 0.8870

61 62 63 64 65

0,2012 0.1952 0.1893 0.1836 0.1780

0.9080 0.8802 0.8531 0.8266 0.8009

0.4029 0.3909 0.3791 0.3670 0.3564

0.4509 0.4562 0.4601 0.4626 0.4639

2.3540 2.3556 2.3517 2.3430 2.3300

0.9012 0’.9121 0.9200 0.9251 0.9277

66 67 68 69 70

0.1725 0.1672 0.1620 0.1570 0.1521

0.7758 0.7514 0.7276 0.7045 0.6821

0.3455 0.3349 0.3245 0.3144 0.3045

21 22 23 24 25

0.4640 0.4631 0.4612 0.4585 0.4551

2.3132 2.2928 2.2694 2.2431 2.2144

0.9280 0.9262 0.9226 0.9173 0.9105

71 72 73 74 75

0.1473 0.1426 0.1381 0.1337 0,1294

0.6602 0.2949 0.6390 0.2856 0.6183 0.2765 0.5983 0.2677 0.5788 0.2591

26 27 28 29 30

0.4510 0.4463 0.4410 0.4353 0.4292

2.1836 2.1509 2.1165 2.0808 2.0438

0.9023 0.8929 0.8825 0.8711 0.8588

76 77 78 79 80

0,1252 0,1212 0.1172 0.1134 0.1097

0.5599 0.5416 0.5238 0.5065 0.4897

31 32 33 34 35

0,4227 2.0059 0.4159 1.9671 0.4088 1.9277 0.4015 1.8878 0.3940 1.8475

0.8459 0.8323 0.8181 0.8035 0.7885

81 82 83 84 85

0.1061 0.1026 0,0992 0.0960 0.0928

0.4735 0.4577 0.4425 0.4277 0.4133

?

44 46

48 36 37 38 39 40

0.3863 0.3785 0.3706 0.3627 0.3547

1.8069 1.7662 1 ,7255 1.6848 1.6443

0.7732 0.7576 0,7419 0.7260 0.7100

86 87 88 89 90

0.0897 0.0867 0.0838 0.0810 0.0783

0.3994 0.3860 0.3729 0.3603 0.3481

0.1796 0.1736 0.1678 0.1622 0.1567

41 42 43 44 45

0.3467 0.3386 0.3306 0.3226 0.3147

1.6039 1.5639 1.5242 1.4849 1.4461

0.6940 0.6779 0.6619 0.6459 0.6301

91 92 93 94 95

0.0756 0.0731 0.0706 0.0682 0.0659

0.3363 0.3248 0.3137 0.3030 0.2926

0.1514 0.1463 0.1414 0.1366 0.1319

46 47 48 49 50

0.3068 1.4077 0.6143 0.2990 1.3699 0.5987 0.2913 1.3328 0.5832 0.2837 1.2960 0.5679 0.2761 1.2600 0.5529

96 97 98 99 100

0.0636

0.2826 0.1274 0.0615 0.2728 0.1231 0.0394 0.2634 0.1189 0.0573 0.2543 0.1148 0.05.54 0.2456 0,1109

pure radium-226 6 days, 6 hours after purification? Although the growth factors are not given for fractional days after the first 5 days, a linear interpolation will produce an answer which is correct to better than 0.5%. From Table XII, the beta growth factors of radium-226, 6 and 7 days after purification, are 1.3211 and 1.4337, respectively. The average increase between these two times is 0.0047 Der hour, so the increase in 6 hours is 0.0282. The beta growt,h factor for 6 days, 6 hours is 1.3211 0.0282 = 1.3493. E x A i r P m 9. A mixture of thorium-227 and thorium-228 is purified and a sample is alpha-counted periodically. At 24 and 120 hours the counts are 10,000 and 20,000 counts per minute, respectively. Assuming 100% retention of the radon isotopes, calculate the counts due to each thorium isotope a t the time of purification. This problem is treated in the same manner as in Example 4. The growth factors are found in Tables V and XX. Let A be the number of counts a t zero time from thorium-227 and B the nJmher of counts from thorium-228, a t zero time.

+

.

1.0000 1.0234 1.0472 1.0711 1.0952 1.1195 1.1441

0.2761 0.2687 0.2614 0.2542 0.2471 0,2401

54

Table XXI. Growth of Alpha and Beta Activities from Initially Pure Parents during First 120 Hours

52 56

60

0.5550

0.5766 0.5979

0.0104 0.0118 0.0121 0.0122 0.0123 0.0124

0.0193 0.0200 0.0207 0.0214 0.0222 0,0229

+ 1.5986 B 20,000 = 1.7952 A + 3.3861 B 10,000 = 1.1968 A

A = 1597 counts per minute from thorium-227 at zero time

B

=

5060 counts per minute from thorium-228 at zero time LITERATURE C I T E D

(1) Bateman, H., Proc. CamSridge Phil. Soc.. 15, 423 (1910). (2) Natl. Bur. Standards (E. S.),Circ. 499 (September 1950). (3) Rut.herford, E., Chadwick, J., and Ellis, C . D., “Radiations from Radioactive Substances,” p. 13, London, Cambridge

University Press, 1951.

RECEIVED f o r review September 17, 1953. Accepted December 24, 1953. hlound Laboratory is operated by Monsanto Chemical Co. for the United States Atomic Energy Commission under Contract Xumher AT-33-1-GES53.