Method for Calculating Cross-Contamination in Column Chromatographic Separation of Radioactive Parent-Daughter Pairs P. J. Karol Department of’Chemistry, Carnegie-Mellon Uniuersity, Pittsburgh, Pa. 15213 I n the chromatographic separation of two species where one is the radioactive precursor of the other, radioactive transformation while on the column modifies the parent-daughter elution curve from that expected of stable isotopes of the same elements. Precise determination of the amount of daughter species produced by other mechanisms such as nuclear reactions or decay of an alternate (independent) precursor necessitates an accurate calculation of hereditary contamination in the daughter fraction. The extent of this contamination, which can be quite considerable, is a function of the column elution rate and the half-lives of the parent and daughter isotopes. The theoretical plate concept of column chromatography is applied to radioactive systems and used to derive a general expression for the modified elution curve from which the amount of contamination may be determined.
WITH r m CONTINUING improvements in high-resolution chromatography and their efficacious application to rapid radiochemical separations comes the ability to study shortlived isotopes previously inaccessible. For example, among the lanthanides there are presumably many short-lived radioisotopes or isomers which are undiscovered or incompletely analyzed owing to the general inability to achieve satisfactory separation from parent activities in a short amount of time. Rapid column chromatography is doing much to alleviate this deficiency. An analytical complication occurs, however, when shortlived nuclei are studied using chromatographic separations. This is the distorting effect on the elution curves, especially in the region of cross contamination, caused by radioactive transformation from a species of one element to a species of another element at a rate which is comparable to the separation rate of the two elements on the column. The usual recipe for analyzing cross contamination ( 1 ) is no longer applicable. A procedure will be derived in detail for calculating the proportionation of the effluent daughter isotope between that produced by parent decay and that produced by alternate parent or independent modes. For a list of important symbols, see Appendix 3. THEORY
The basis for the following derivation was suggested by a work of Kailen and Heilbronner ( 2 ) . These authors formulated the solution of a n analogous problem-the effect that thermal dissociation of a substance has on its gas chromatogram. I. Theoretical Plate Concept: Stable Species. The theorztical plate concept (3) is based on the successful application of a parameter, the plate height, in describing the spreading of an elution peak by the chromatographic process. The plate height, in turn, is a n outgrowth of the theoretical plate (1) E. Cilueckauf, Trans. Faraduy SOC.,51, 34 (1955). (2) J. Kallen and E. Heilbronner Help. Chim.Acta, 43, 489 (1960). (3) J. C. Giddings, “Dynamics of Chromatography,” Part I, Marcel Dekker, New York, N. Y., 1965, p 20.
model of Martin and Synge ( 4 ) which was invented to describe the dynamic equilibrium mechanism of column chromatography. These authors define the plate height as “the thickness of the layer such that the solution issuing from it is in equilibrium with the mean concentration of solute in the non-mobile phase throughout the layer.” A chromatographic column can then be looked upon as a succession of such plates in which the chromatographic partition process is repeated over and over. If the plate height is H and one has a column of length L, we define another parameter by L/H = n 1 where n 1 is the number of theoretical plates in the column. For a rigorous discussion of the merits and justifications of the theoretical plate concept and its relation to the Martin and Synge model, see Reference 3. The elution curve may be derived by considering the massbalance equation (net change = gain - loss),
+
+
dN, = (KN,-i
- KN,)dV
(1)
where N, is the number of nuclei (or related variable such as weight or disintegration rate) of a given type in plate i, dN, represents the change in the number of nuclei in plate i due to repartition by flow of a n eluent volume dV through plate i after volume Y has already passed, and K represents an effective partition coefficient equal to the fraction of nuclei which is in the solution (mobile) phase per unit volume of solution phase. The first term on the right of Equation 1 is the number of nuclei gained from the effluence of the preceding plate i - 1 and the second term represents the loss of nuclei from plate i carried by the advancing eluent fluid. The function which describes the dependence on effluent volume of the number of nuclei emerging from the column is called the elution curve and is given by KNntl(V), the number of nuclei per unit volume that issues from the last plate, n 1. (The volume V which has emerged from any plate, the last plate in particular, is related to the volume V’ of eluting solution which has entered the column by V = V’ - E where e is the column void volume.) When eluent flow has not yet started, V‘ = V = 0, and the number of nuclei in the first plate, N1, is equal to N(O), the original number of nuclei loaded onto the column. Also when V’ = V = 0, N , = 0 for i > 1. By using this information as boundary conditions, Equation l may be solved to give
+
Details of the derivation are contpined in Appendix 1. For large values of n, the Poisson distribution in Equation 2 may be approximated by a Gaussian distribution yielding ,-(KV
-
n)*/2n
(3) Multiplication of Equations 2 and 3 by K gives the desired (4) A. J. P. Martin and R. L. M. Synge, Biochern. J . , 35, 91 (1941).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
1383
differing from Equation 2 only by the presence of the damping factor e-"" which accounts for decay. Equation 7 is approximated by the analog of Equation 3
If X P is small with respect to the flow rate, i.e., if A P = b/+ l = 0. The solution to Equation 9 contains several terms which will be discussed according to their various origins by use of the following definition:
Figure 1. Sample elution peak where the parameter V, is the volume a t which the maximum concentration of eluting species appears and /3 is the full width of the elution peak at e-l x height of the peak maximum
D,(V) = D,'(V)
elution curve function. A sample elution curve is illustrated in Figure 1. For both the Poisson and Gaussian forms, Vo, the volume corresponding to the peak maximum is given by Vo = n/K
(4)
From Equations 3 and 4 the width, 8, of the Gaussian elution curve at e-1 times the maximum is
+
(10)
D' represents the number of daughter nuclei which were present in the initial sample loading at the commencement of elution; the number of daughter nuclei which originate as decay products generated during the course of the chromatographic process is denoted by C. From the above definitions, D(0) = D'(0). As shown in Appendix 1, the D'(0) nuclei, hereafter referred to as primary daughter nuclei, are chromatogrammed in a manner analogous to P(O), that is D'n+l(V) = D(0)
Since the parameters Vo and p are known from experiment, the values of n 1 and K can be obtained from Equations 4 and 5. 11. Theoretical Plate Concept: Labile Species. PARENT SPECIES. Consider now the elution of P radioactive nuclei with decay constant X p . The mass-balance Equation 1 must now contain a n additional term, - XpP,dt, which accounts for the loss of P by radioactive transformation,
+ CdV)
(KOV)ne - ( K D -t A D )V n!
(1 1)
The other daughter nuclei, C(V) will be referred to as secondary daughter nuclei. In deriving a n expression for Cn+l(V),the only initial condition is that C,(O) = 0 for all i. The derivation of this expression is described in Appendix 1. Definition of several new parameters proves to be convenient at this time: t p = the time after elution commences at which the peak in the parent species elution curve emerges = VPAP/XP
(1 2)
a = parent-daughter separation factor = VPIVD
where K p is the parent partition coefficient and AP = Ap(dt/dV) = Ap/@, where @ = dV/dt is the constant flow rate of eluent through the column. The initial conditions used in solving Equation 6 are as follows: At the initiation of the elution process Pl(V) = P(O), [P(O) is not, however, equal to the original number of P loaded onto the column at t < 0 but may be obtained from the latter in a straightforward manner]; Pi(0) = 0 for i > 1. The details of the solution of Equation 6 are given in Appendix 1. The result is
1384
= tp/tD =
c = (Kp - K D
KD/Kp
+ A P - A D ) / ( K P- KD) =
The resulting solution is expressible as
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
(1 3)
+ 1 , u ) is defined
where the incomplete gamma function y(n through
SECONDARY CONTAMINATION
At some time T, fractions from vlowerto vupper are combined to form a consolidated daughter sample which will contain D* daughter nuclei as defined below. From a total of C* Equation 19, the total number of secondary daughter nuclei, denoted by the symbol C*, included in the consolidated sample may be derived yielding the expression
+
(1 6)
Equation 15 can be simplified by noting that, for the large values of n routinely encountered in column chromatographic separations, the following asymptotic relationship holds :
>> 1 >> hptP(11 -- aX D / X P ) .
for n
n
Substituting this expression into Equation 15 gives
n ! cn(1
X
- a)
A complete description of the elution curve is now possible in terms of its representative components: KpPn+,(V ) de) scribes the elution peak for parent species; K D D ’ , + ~ ( V describes the peak due to primary daughter, i.e., daughter in the initial load; K D C , + ~ ( Vis) the function depicting the distribution of secondary daughter nuclei which are generated during passage through the column. The latter will emerge at elution volumes lying between those of the Gaussian peaks corresponding to the parent and daughter atoms in the initial load. One may prefer to express the result in collected fraction number (or drop number) v rather than volume, in which case V / V p becomes v/vp, u p being the fraction number at which the parent elution curve peaks. The volume per fraction is equal to V p / v p and the number of secondary daughter nuclei in a given fraction may be expressed as
The number of original parent species which has subsequently decayed and been eluted as (secondary) daughter species into a given collected fraction is thus equal to --!!L
na naXptpe’ - a - C“+l = P(0) vpn![n (1 VP
[ ( ~ A P - AD)
-(AP YP
- a) + tp(Xp n
- AD) AD)]
+ l,nacI)]
1
X
(19)
VP
Likewise, from Equation 8 the number of undecayed parent nuclei in the given collected fraction is given by
[The sample consolidation time, T , will have been eliminated from the final expression. See Equation 27.1 Tailing of the parent species elution curve into the daughter fraction will also contribute to the number of secondary daughter nuclei in the daughter sample. The amount of tailing may be determined from Equation 8 using the usual radioactive growth and decay expression for parent-daughter pairs and generates a n additional term which rigorously should be included in Equation 22. To simplify the ensuing discussion, however, it will be assumed that such normal cross contamination is of much lesser consequence as is usually the case and will be ignored. The amount of secondary daughter, C*, must be subtracted from the experimental total number of daughter nuclei in the above sample if we want to obtain D*, the amount of (primary) daughter sample in the initial column loading. As is clear from Equation 11, we have
XDtP
vpd2*n e
= UP
Equations 22 and 23 are the desired results, expressing the proportionation of daughter nuclei in the elution sample between those present in the original parent-daughter mixture loaded onto the chromatograph column (i.e.,primary) and those produced by parent decay subsequent to the initiation of the separation process (i.e.,secondary). Equation 22 contains a factor which involves incomplete gamma functions. These are not amenable to routine calculation without access to either computer facilities or uncommon mathematical tabulations. However, since the number of theoretical plates present in chromatographic columns used for efficacious separations is generally greater than one hundred, the exact bracketed term in Equation 22 may be replaced by one involving the normal probability function which is tabulated in most basic mathematical handbooks (5). Justification for the above substitution is detailed in Appendix 2. Equation 22 becomes
and from Equation 1 1 , one obtains a similar expression for the number of undecayed primary daughter nuclei: (5) “Handbook of Chemistry and Physics,” Chemical Rubber Publishing Co.,Cleveland, Ohio. ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
1385
0.0 0 5
0.004 0
Y
L \
c 0
Figure 2. Elution curve calculated for a mixture of 166Tm(7.7 hr) and L66Yb (57 hr) -,166Tm where at the initiation of the elution process there is 40 times as much Yb present as Tm on the column. The flow rate is such that the Yb elution peak occurs after 1 hr of flow. Solid curves represent undecayed 166Yband ls6Tm which were originally part of the sample load. The dotted curve represents 166Tmproduced from 166Yb decay during the course of the elution
‘F 0 0
0.003
t L
d .0)
Ti 0.002 a
z
+ 0 L 0)
n
5 0.001
z
0.ooc I
I
IO
Equation 24 contains only decay constants, flow rate, peak positions, and peak widths as parameters, all of which are fixed by the particular separation under investigation. EFFECTIVE TIME OF SEPARATION
The form in which results calculated directly from Equations 23 and 24 appear can be expressed in a more convenient alternative form, in terms of a n “effective time of separation” of parent and daughter components by which is meant the following: if one were to have a mixture containing P(0) and D(0) nuclei at time t = 0 and an hypothetical instantaneous separation were to occur at some subsequent time r = T ~ then , at time T $ the number of parent nuclei F ( T ~=) P(0)e-’pT8, the number of undecayed original daughter nuclei D / ( T ~=) D(0)-XDr8,and the number of decayproduced daughter nuclei C ( n ) = ApP(0)[e-XP‘r - e - x D T * ] / (AD - X p ) . The time T~ is defined by equating the ratio ~/(TJ/D(O : ~) ( T ~ ) / P (from O ) the above hypothetical partition to the equivalent ratio calculated from Equations 23 and 24 using data from the actual experimental chromatogram. This ratio is used in order to obviate the necessity of knowing the value for D(O)/P(O). “Effective time of separation” is thus the time at which a hypothetical instantaneous separation of parent and daughter species would yield the same parent-daughter proportionation in a sample as is obtained from the actual column procedure. From the above discussion
where
1386
I
I
I
I
20 30 Fraction Number
1
I
40
1
which yields the effective time of separation
Equations 23 and 24 can be used to evaluate C*/P(O) and D*/D(O) from data. Note that from Equations 23, 24, and 27, T, the time of consolidation of the numerous fractions into a single sample, cancels out when Expression 27 is used, provided normal tailing (cross contamination) is negligible in comparison to hereditary contamination in the sample, as is assumed. EXAMPLE
As an illustration of the phenomenon under discussion one can consider the following case: cation exchange separation of 166Yb(57 hr) parent from ls6Tm(7.7 hr) daughter where at the start of elution there are 40 times as many lesYb nuclei as ls6Tm nuclei, the column contains 200 theoretical plates as determined from the Yb elution curve and Equation 8, and the Yb and Tm maxima elute at fraction numbers 22.5 and 32, respectively. Figure 2 is the calculated elution curve for a flow rate of 22/3minutes per fraction, that is, the Yb peaks -1 hr after elution commences. The solid curves are for l66Yb and (primary) l“Tm. The tailing of Yb into Tm is minimal. The dotted curve delineates the amount of parent-produced (secondary) 166Tm in each fraction. If one were to consolidate fractions 28 through 38 into a l66Tm sample, 27% of the total 166Tmwould be contaminant ls6Trn. Alternatively, this result is expressible as an effective time of separation, T * of 0.13 hr. If instead one combined fractions 32 through 38, then 15 % of the total le6Tmin the sample would be due to hereditary contamination reducing T~ to 0.06 hr. This example serves to illustrate the magnitude of the hereditary contamination correction only for these specific circumstances.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
In the general case a rough estimate of C*/ D* can be made by an a priori argument. If the entire primary daughter elution peak were consolidated into the final sample then D* 5 D(0)e-XDr.and the number of secondary daughter nuclei due to parent decay can be n o greater than the number of decayed parent nuclei, i.e., C* < P(O)(1 - e--Xpr*). Taking t p as a very rough approximation to r8,the relative magnitudes of contaminant to indigenous (secondary to primary) species and hence the practicality of the foregoing method may be extracted from
This is our basic equation, from which all our solutions are obtained by specialization. Solution of Equation 1 of Section I and Equation 6 of Section 11. These two sets of equations differ only in that in Equation 1 the atoms are nonradioactive, while in Equation 6 they are radioactive. Therefore, the solutions of Equation 1 can be obtained from those of Equation 6 by setting the decay constant in the latter equal to zero. We write Equation 6 in the form dP1(o) - - KPPi-l(v) - u p , ( ~ ) du
This is consistent with the preceding rigorous calculation of a specific example.
where u =
CONCLUSION
KP
+
('44)
Ap
Explicitly, Equation A3 is the set of differential equations
A general method has been devised for calculating the
_-
dP: do
component elution curves in a system where one radioactive species is produced by transformation of a radioactive parent at a rate comparable to the separation rate of the parent and daughter species o n a chromatographic column. The result is expressible as a hypothetical instantaneous effective separation time which may be calculated from the half-lives and column elution characteristics of the nuclides.
-
--UP:
dPn+1 - KPP, do
APPENDIX 1
The differential equations we have to deal with are all of the form
-
UP,+,
I 1
I
These are to be solved for the case of the initial conditions P:(O) = P(O), and P,(O) = 0 for i
f
so that Equation A 1 can be written d (&'Ni) = e f Uri(u) do Integration of the latter between o = 0 and result that can be put in the form
Continuing in this way with i = 4 , 5 . . . ,n we obtain
=
N,(0)e-rr'
+ e-t'
L'
L: =
>1
('47)
In view of Equations A6 and A7, Equation A2 gives, for i = 1, P,(V) = P(O)e-"'. and for i
LLV s,;
P,(V) = Kpe-". For i
(AS)
> 1,
=
P,-l(c)e""dc
('49)
2, Equation A9 gives, with uw of Equation A8
P,(v)= KpP(0)e-''.
dc = K,.P(O)Ve-"'.
(A10)
For i = 3, Equation A9 gives, with use of Equation A10
+ 1 in succession,
n!
V gives a
('42)
( 6 ) W. Rubinson, Chemistry Department, Brookhaven National
Laboratory, private communication, 1970.
rl(o) = 0; r,(c) = KpP,--1 for i
P,+,(V) = P(0) (KPVY p - b v
r,(u)ef"do
(7) W. Rubinson, J. Cl7em. Phys., 17,542 (1949).
646)
We note that Equations (A5) are of the form (AI) with = u and
where N,(u) is the number of atoms (or other entity) in the ith plate after volume o of eluent has been passed; r,(u) is the known rate (per unit volume of eluent) of addition of atoms to the ith plate; and {is a known constant. Equation A1 with t in place of u and X in place of { is the customary equation for a radioactive nuclide of decay constant X that is being created at rate r , ( f ) . There are a variety of ways of solving such equations. In the original version of this work, the author solved them by the method of the Laplace Transform. Rubinson (6), who was kind enough to read that version critically, then communicated to the author a set of solutions obtained in a more elementary way by the method he had used to derive the equations of radioactive transformation (7). The fact that the two methods gave identical results makes it fairly certain that the final equations are mathematically correct. We will follow Rubinson's simpler method in our exposition. In Equation A1 we recognize that
N,(v)
>1
which, in view of Equation A4, is Equation 6 of Section II. To prove Equation 2 of Section I, we note if the spccies is nonradioactive, then the A p in Equation A4 is q u a l to zero (see the remark following Equation 6 of Sci'LiOii !I), in which case Equation A3 reduces to dP,ldu
=
KrP,-i
-
KpP,
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
(A 13)
1387
and Equation A12 reduces to
Recalling that a = KD/KP
and defining Except for notation these are identical with, respectively, Equations 1 and 2 of Section I. Solution of Equation 9 of Section 11. We write these equations in the form dD,ldv = A p P ,
x = wo;
u =
wv
Then Equation A25 can be written
+ K D D ~ -- ~pDi (i = 1, 2,
. . ., n
+ 1)
(A15)
where p = KD
(A281
The integrals are elementary. Their values and those of subsequent integrals that will arise can be written down directly by noting that by successive partial integrations
+ AD
and solve them for the case of the initial conditions D,(O) = 0 for all i
(-417)
i.e., for the case where there is no daughter activity in the initial loading, D f = Ci. Any amount Dl'(0) = D'(0) of daughter activity in the initial loading will pass through the column according to an equation of the same form as that of the parent Equation, A12: ( K DV" D'"+l(V) = D'(0) -e - p v n! and the overall distribution of daughter in the column is obtained by adding this expression to the distribution C,,,,(V) of daughter generated from the parent o n the column [cf. Equation A36 below]. Equation A15 is of the form A1 with
+
ri(v) = K ~ D i - i ( u ) ApPi(v), and { = p
('419)
1
l'
+
[ K ~ D i - l ( o ) ApPi(o)]ePudv
(A20)
(KPu)'-' ApP,(v) = ApP(0) -e(i l)!
-
D1(V) = ApP(0)e-pv
Lv
e-(' -
xne-zb =
(A301
DdV)
=
KP A p P ( 0 ) x e-pv l a
(A31a)
W
where
- a)2(1 - e-")
+(1
- a)(. - e-')u (a3- e-')-
+ U2
2!
(A31b)
Equations A31 display a pattern that leads us to infer that KP" D n + W = APP(O)w"+' e-''
dv
L+1
(A32a)
where
or, with the abbreviation w = u - p
-
In the same way Equation A20 can be evaluated for the case i = 3 by use of Equations A24 and A30. The result can be put in the form
"
For the case i = 1, Equation A20 gives, with use of Equation A21 (note that Di(0) = 0 for i > l )
Xne-2d.x
-,+I +
Is = (1
where ApPi(v) is known from Equation A12:
Lm
This is the so-called "incomplete gamma function," conventionally denoted by 1, u) (cf. Equation 16). Evaluation of Equation A28 gives a result that can be manipulated into the form
In view of Equations A17 and A19, Equation A2 gives Di(V) = e - p v
=
xne-zh
(A22)
For the subsequent deductions, it will simplify matters greatly if we write Equation A21 in the form [cf. (A22)]
Rubinson (6) has proved Equation A32 by mathematical induction, but the proof will not be given here. We proceed to transform Equation A32 to a form convenient for numerical evaluation. Writing A32b as
Then with Equations A23 and A24, Equation A20 for the case i = 2 is and introducing the new variables a 1 u; z = -u = y 1 - a 1-a
y = -
1388
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
+u
(A331
where
we get
5 (e-?’ 9 -
-
(1
j!
)=O
e-‘
g)
j!
(A34)
By the definition of the incomplete gamma function 1, u), (Equation 16)
r(n
+
From the above equations, the following approximation results
so Equation A34 can be written
which is the form used in Equation 24. Insertion of Equation A35 into A32a and reversion to our original notation by means of Equations A33, A27, A26, A22, A16, and A4 gives
APPENDIX 3. LIST OF IMPORTANT SYMBOLS
+
(Kp - K D A P - AD)/(Kp - K D ) see Equations 14 and A37 = number of daughter nuclei in plate i which have been generated during the course of elution by parent decay: secondary daughter nuclei in plate i = number of (secondary) daughter nuclei in a sample consolidated from the chromatographic effluent, which nuclei were originally parent species at the time the elution process commenced, i.e., “hereditary contaminant” in final sample; see Equation 22 = number of (primary) daughter nuclei in a sample consolidated from the chromatographic effluent, which nuclei have undergone no bansmutation during elution; see Equation 23 = total number of daughter nuclei in plate i = number of daughter nuclei in plate i which have not undergone radioactive transformation, Le., are indigenous to the column loading: primary daughter nuclei in plate i = number of daughter nuclei in column loading at initiation of elution process = index for plate number in column; 1 5 i 5
=
c=---
W
KP - KD
- AD KP - KD
AP -I----
Equation A36 is converted to Equation 15 of the text by means of the appropriate substitutions. As remarked below Equation A1 7, the solution, Equation A36, pertains to the case in which there is no daughter activity in the initial load on the column. If the initial load contains a n amount D’(0) of daughter activity, then Equation A18 must be added to Equation A36. See Equation 10. APPENDIX 2
Equation 21 is exact and may be used as such. However, the bracketed term contains the incomplete gamma function for whose evaluation one must resort to Pearson’s “Tables of the Incomplete Gamma Function” (8) which are useful over a limited range of n. Otherwise numerical determination cia computer is necessary. For large values of n such as those routinely encountered in column work, a n asymptotic approximation to r(n 1, u ) is possible. From the chapter o n “Probability Functions” in Reference (9) one finds that
+
r(n where q
=
2(n
+ 1, 4
= n![1
- Q(x2)d1
031)
+ l ) , x 2 = 2u and
is a chi-square probability function. For v isn 1 > 50
+
Q(x21d = QUI
>
100, that (B3)
(8) K. Pearson, “Tables of the Incomplete r-Function,” University Press, Cambridge, England, 1957. (9) M. Abramowitz and Irene A. Stegun, Ed., “Handbook of Mathematical Functions,” Dover Publications, New York, N. Y., 1965.
n + l
the fraction of all nuclei of given type in the mobile phase per unit volume mobile phase = the fraction of all daughter nuclei in the mobile phase per unit volume mobile phase. = the fraction of all parent nuclei in the mobile phase per unit volume mobile phase = number of nuclei of particular species on column at initiation of elution process = number of theoretical plates in the column = n factorial = n(n - l)(n - 2). . . 3 . 2 . 1 = total number of parent nuclei in plate i = number of parent nuclei in column loading at initiation of elution process = time elapsed since initiation of eluent flow = time at which the peak in the elution curve for parent species emerges from the column = time at which the fractions comprising the final sample of “daughter species” are combined = eluent volume =
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
1389
volume at which the peak in the general elution curve emerges; see Figure 1 VD = volume at which the peak in the elution curve for daughter species emerges VP = volume at which the peak in the elution curve for parent species emerges a = parent-daughter separation factor, i.e., Vp/Vu P = width of elution curve at times the maximum; see Figure 1 ~ (+ n 1, u ) = incomplete gamma function; see Equation 16 AD = decay constant of daughter species XP = decay constant of parent species AD = fraction of daughter nuclei per unit volume which decay in the time the unit volume flows, i.e., b d t l d V AP = fraction of parent nuclei per unit volume which decay in the time the unit volume flows, i.e., XpdtjdV V = fraction number YP = fraction number at which parent species elution curve peaks VO
=
To
4J
=
effective time of separation, i.e., the time at which a hypothetical instantaneous separation of parent and daughter species would yield the same sample as i$ obtained from the column procedure; see E q u a t m 27 column flow rate, d Vldt ACKNOWLEDGMENT
The author expresses his considerable gratitude to Dr, William Rubinson of Brookhaven National Laboratory for his thorough examination of the mathematicai manipulations employed in this work and in particular for his valuable comments and suggestions on simplifying the finel f o r p in which these derivations are Dublished. RECEIVED for review February 2 5 , 1971. Accepted May 19, 1971. Presented in condensed form at the 159th ACS Meeting, Houston, Texas, February, 1970. The author gratefully acknowledges the support of the ACS Petroleum Research Fund, Grant Number 2085-C3, and also partial support by a University Scaife Gtant
~
Separation of Uranium from Seawater by Adsorbing Colloid Flotation Young S. Kim and Harry Zeitlin Department of Chernistrjs and Hawaii Institute of Geophysics, Unirersity of Hawaii, Honolulu, Huw’aii 96822 A procedure is described for the separation from seawater of uranium present as the stable tricarbonatouranyl anion by an adsorbing colloid flotation technique which utilizes a collector-surfactant-air system. At p H 6.7 i 0.1 the uranium is adsorbed effectively on the positively charged ferric hydroxide collector. Upon addition of the anionic surfactant, sodium dodecyl sulfate, and the bubbling of air through the seawater, the colloidal particulates of ferric hydroxide enriched with uranium by absorption a r e floated within 2-3 minutes to the surface as a stable froth which is easily removed. Uranium was analyzed spectrophotometrically using Rhodamine B. Average recovery of uranium from seawater by this method is 82%.
A m c h r PAPER ( I ) has described the first application of a bubble technique to seawater for the separation of a trace metallic constituent. Under optimal conditions, molybdenum as molybdate is floated to the surface quantitatively and reproducibly in less than five minutes as an easily removable froth by a positively charged iron(II1) hydroxide collector, an anionic surfactant (dodecyl sodium sulfate), and air. The behavior of the collector-surfactant-air system toward a metallic anionic species such as molybdate prompted an investigation to determine whether the flotation method could be applied successfully to other trace metals which exist in seawater as anions. This communication is concerned primarily with the extension of the separation process to uranium believed to be present in seawater as the very stable tricar( I ) Y. S. Kim and H . Zeitlin, Sepur. Sci.. in press. 1390
bonatouranylate ion, UO,(CO.,),~--( 2 ) = 1.7 ?< (3)]. Other methods for the separation of uranium in seawater which is present in the 2.9-3.3 c(g/l. range include coprecipitation with aluminum phosphate and ferric hydroxide and solvent extraction ( 4 ) . In order to determine the separated uranium spectrophotometrically, a modified procedure was worked out involving Rhodamine B (5, 6) which proves to be comparable to other spectrophotometric and fluorometric methods ( 7 , 8), eliminating the need for a fluorescence attachment. Rhodamine B has not been employed previously for the determination of uranium in natural waters. EXPERIMENTAL Apparatus and Equipment. A Beckman DU spectrophotometer was used for absorbance measurements. The absorbances were read in low volume matched quartz cells of 1.0-cm path length. The pH of the solutions was de(2) E. D. Goldberg, “The Sea,” M . N. Hill, Ed., Vol. 2, lnterscience, New York, N. Y.. 1966, p 5. (3) A. G. Klygin and I . D. Smirnova, R i m . J . Iuorg. Chem., 4, 42 (1959). (4) J. P. Riley and G. Skirrow, “Chemical Oceanography,” Vol. 2, Academic Press. London, New York, 196.5. pp 391-392. (5) Frausto da Silva and Legrand de Moura. 1/7r. Co/?f.P w c r f i t l Uses Ar. Euergy, 28, 537 (1958). ( 6 ) H. H. Ph. Moeken and W. A. H. Van Neste. A i d . Chirn. Acrci, 37, 480 (1967). (7) Academy of Sciences of the USSR, “Analytical Chemistry of Uranium,” Israel Program for Scientific Translations, 1963. (8) E. B. Sandell, “Colorimetric Determination of Traces of Metals,” 3rd ed., Interscience, New York, N . Y., 1965, p 903.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
