2 PERKS ARE PRESENT
estimation is frequently signaled by a failure to converge after iteration, while overestimation gives rise to small “noise” peaks. SPECSOLV is available for public user distribution through the Triangle Universities Computation Center network. Inquires regarding availability of source listings or decks may be addressed to: Program Librarian, University of North Carolina Computation Center, Phillips Hall, Chapel Hill, N.C. 27514.
ACKNOWLEDGMENT
-0
9.00
‘11.5
14.1
9.00
11.5
19.1
The authors acknowledge the work of W. L. Bowden and M. J. Powers, whose research problems have provided applications tests for SPECSOLV. The various user oriented suggestions of J. A. Baumann have been quite valuable. One of us (HSG) thanks W. E. Hatfield for his initial inspiration.
FREQUENCY (KILOKAYSERS)
Figure 9. The near infrared spectrum of the PF6 salt of [(NH&%Oyz)-
. R ~ C I ( b i p y ) ~ showing ]~+ a weak intervalence transfer band as a shoulder
to a much stronger absorption. The larger component is only partially shown
assuming purely Gaussian or bi-Gaussian line shape, the resulting ease and simplicity of use is significant. SPECSOLV has been used successfully by numerous researchers in this department who lack any knowledge of decomposition schemes or computerized methods. We feel that the ability to require as input only information immediately available from the experiment is a significant advance in this field. Many of the possible input parameters to SPECSOLV have default values. In any event, only the choice of the number of peaks to be found affects the search outcome; other parameters largely deal only with output and input forms. While the selection of the number of component peaks may seem arbitrary, under-
LITERATURE CITED R. D. B. Fraser and E. Suzuki, Anal. Chem., 41, 37 (1969). K. S. Seshadri and R. N. Jones, Spectrochlm. Acta., Part A, 19, 1013 (1963). R. P. Young and R. N. Jones, Chem. Rev., 71, 219 (1971). A. M. Kabiel and C. H. Boutros, Appl. Spectrosc., 22, 121 (1968). F. C. Strong, Appl. Specfrosc., 23, 593 (1963). J. Pitha and R. N. Jones, Can. J. Chem., 44,3031 (1966). B. G. M. Vandeginste and L. De Galan, Anal. Chem., 47, 2124 (1975). J. R. Beachamand K. L. Andrew, J. Opt. SOC.Am., 61, 231 (1971). J. W. Perram, J. Chem. Phys., 49, 4245 (1968). (IO) A. R. Davis et al.. Appl. Spectrosc., 26, 384 (1972). (1 1) L. M. Schwartz, Anal. Chem., 43, 1336 (1971). (12) R. N. Jones, Pure Appl. Chem., 18, 303 (1969). (13) J. Pitha and R. N. Jones, Can. J. Chem., 45, 2347 (1967). (14) W. L. Bowden, Doctoral Thesis, University of North Carolina, Chapel Hill, N.C.. 1975. (15) M. J. Powers, R. W. Callahan, D. J. Salmon, and T. J. Meyer, in press, 1976. (16) W. Kuhn and E. Braun, Z.Phys. Chem., Abt. B, 8, 281 (1930); 9, 426 (1930). (1) (2) (3) (4) (5) (6) (7) (8) (9)
RECEIVEDfor review March 1,1976. Accepted May 13,1976. This work was supported by a grant from the National Science Foundation, MPS75-00970.
Determination of Relative pK Values of Dibasic Protolytes by Regression Analysis of Absorbance Diagrams Friedrich Gobber” and Jurgen Polster Institute for Physical and Theoretical Chemistry, University of Tubingen, 0-7400Tubingen 1, Federal Republic of Germany
The absorbances of a dibasic acid measured at two distinct wavelengths with varying pH as the parameter form a conic section which can be determined by an appropriate leastsquares method. The conic section and the absorbances of the acid in its nondissociatedform and its completely dissociated form allow for the determination of the difference of the pK values ( ApK) and the absorbances of the ampholyte without using pH values. If data from more than two wavelengths are available, one ApK value can be computed for each pair of wavelengths. Because the reliabilities of the particular results are considerably different, an extensive error analysis is necessary to obtain an averaged ApK value together with an estimated standard deviation. The results for o-phthalic acid are given as an example.
The evaluation of the measurements from multistep acidbase equilibria requires the pH values in order to obtain the 15.46
e
pK values and the absorbances of intermediates (1-3). Because the pH measurements are relatively inaccurate compared to spectrophotometric data, we investigated what information can be extracted from the more reliable spectrophotometric data alone. We found that, in the case of dibasic acids, the difference between the pK values and the absorbance of the ampholyte can be obtained without using pH measurements ( 4 , 5 ) . Our former method involves arbitrariness to some extent. It needs a \angent to a curve which is only represented by some data points. This can be done by hand or by fitting a low order polynomial to a subset of the data points. In either case, certain decisions cannot be derived from exact criteria. In the present paper, we describe how the results can be obtained from the data by unique numerical procedures. These are executed by means of a computer program which allows for the evaluation of the statistics of the data, thereby providing estimates for the reliability of the results. This FORTRAN program is available from the authors.
ANALYTICAL CHEMISTRY, VOL. 48, NO. 11, SEPTEMBER 1976
4 4
0.5
0
I
b b0
Figure 1. Some curves resulting from Equation 5 with different values
Of
Figure 2. Some absorbance diagrams (A, vs. Ak) of @phthalicacid as constructed from a part of the measurements
The pairs (k,l) designate wavelengths according to Table I. For (6,3) the intersection points of the tangents are constructed which give the absorbance for (bh) = bo. The arrows indicate increasing pH
K
Physically admissible points lie between the points (1,O) and (0,l) and under the line (bhp) = bo - b. All types of conic sections occur (0 < K < 4: ellipse, K = 4: parabola, and K > 4: hyperbola) but in their physical part they are quite similar
The two points (bo,O)and (0,bo) represent simply limiting conditions of the pH: very high pH: b = bo, (bhz) = (bh) = 0 very low pH: b = (bh) = 0, (bhs) = bo
MATHEMATICAL DERIVATIONS Description of the Chemical System. In this section we will state some mathematical relations which are valid in a closed isothermal chemical system containing the three forms of a dibasic acid, B, at equilibrium. B stands for any one of the three components:.BH2, BH-, or B2-. There are only two reactions: K1
+ H+
(la)
3B2- + H +
(Ib)
BH2 BH-
BH-
The concentrations of these substances are designated by (bh2),(bh), and b, respectively. We introduce two equilibrium constants in the usual manner:
-
K1 = h (bh)/(bh2)
(24
-
(2b)
K2 = b h/(bh) The ratio K
= KI/K2
(3)
is one of the quantities we wish to estimate. Its logarithm is the difference between the pK values of the two equilibria. Because there is no exchange of B between the system and its environment, the sum of the concentrations must be a,constant:
b
+ (bh) + (bh2) = bo = constant
(4)
The combination of Equations 2, 3, and 4 allows for the elimination of (bh), leading to an implicit relation between the two concentrations b and (bhz): b2
+ (bhz)' + (2 - K)b(bhz) - 2bo(b + (bh2)) + bo2 = 0
(5)
A plot of (bh?) vs. b at different pH values is a conic section of a type depending only on K. The following properties are shared by all of these curves: 1)The straight line (bhz) = b is an axis of symmetry; 2) Each curve passes through the points (bo,O) and (0,bo); 3) The tangents to the curves at these two points are the axes (provided K # 0).
The tangents at these points intersect at the origin (0,O) where all B is in the form BH-. By Equation 4,it is easily verified that only the triangle (O,O), (bo,O), (0,bo) and its interior represents physically admissible points of the system because outside of it at least one of the concentrations is negative. Some parts of the curves shown in Figure 1 lie outside of this triangle, and therefore they have no physical meaning. Transforming to Absorbance Values. Our measurements are absorbance values at some appropriate wavelength Xi, measured a t different values of pH. If C ~ designates R the molar absorptivity of B2- at XL, etc., the absorbance A / is given by the Lambert-Beer-Bouguer law: AI = c/Bb + tiBH(bh)
tmH2(bhd
(6)
(The length of the absorbtion cell is assumed to be one.) Replacing (bh) according to Equation 4 yields A/ =
(CLB-
tiBH)b +
ti^^^ - tmH)(bhd
tmHbo
(7)
The same equation with different 2s and subscript h instead of 1 holds for another wavelength and the two independent variables, b and (bhz), are thus transformed into ( A / , A / ( ) by a linear transformation including translation. A plot of'A/ vs. A/,, which we call the absorbance diagram ( G ) , is again ii conic section (see Figure 2). The six parameters of the linear translormation are determined as follows. If the measurements start at very high pH, the curve starts at the image of (l),k,O),called ( A / I ( , A / ~ I ) ) and ends for very low pH at the image ol' (O,bl)), called (A/HH~,A/,BH?). Straight lines are transformed into straight lines and hence the point of intersection of the tangents to the curve at these two points, called ( A / I ~ H , A , is ~ Ittw ~ ~image ~ ) , of the origin (0,O). The second part of the subscript ol' A represents that one of the three forms of I3 which is present with concentration bo at the respective point. These t h r w pairs of original and image points determine the transt'orm;it ion from concentrations to absorbances and the inverse t r:i tisl'ormntion. From Equation 6, we see that ALB
=
t/lih)
ANALYTICAL CHEMISTRY, VOL. 48, NO. 11, SEPTEMBER 1976
(8) 1547
and so on, for the six possible combinations of subscripts. Determination of K. It is easily verified that the center C of the conic section Equation 5 is given by
C: bc
= (bh2)c = 2bo/(4
-
(9)
K),
or K
=4
- 2bo/bc
(10)
The center of the conic section in the absorbance diagram is the image of C, called (Alc,Akc). Thus, from Equation 7:
puting with correlates”. The term “correlate” is just another name for a Lagrange-multiplier which is introduced to satisfy one constraint. Equation 15 can be derived by Lagrange’s method too, but then there are three equations to be solved instead of the two in the straightforward derivation outlined above. We check in the final stage of the computation to determine whether all data points are really close to the resulting curve, since Equation 15 is a good approximation for the square of the shortest distance from (Alm,Ak,) to the curve only if (alm,akm)is small. We take the value of the function M
Inserting bc = (bh& and A ~ B H = qBHbO, etc., according to Equation 8 yields
Q(a,P,r,s,~)= C am2 + akm2
+
as an obvious criterion for the goodness of fit. The set (a,P,y,d,t), yielding the lowest value of Q, is chosen as the best fit of the resulting conic section (Equation 14) to the data points. At the minimum of Q, the partial derivatives of Q with respect to the coefficients must be zero. This condition yields the necessary five equations for the five unknown coefficients. But the derivatives of Q are sums of derivatives of Equation 15 with respect to a, etc. These contain fourth-order terms in their denominators, and therefore it seems necessary to use minimization methods for general functions. To avoid this technical complication, we tried the following algorithm: If the denominators in Equation 15 are constants, the coefficients minimizing Q can be computed from linear normal equations because F is a linear function of (a,P,y,G,t) (see Equation 14). This set of normal equations is solved several times, using constant denominators which are computed with the coefficientsfrom the previous step. For the first step, the denominators are set equal to 1in order to start the iteration. Typically, after 5 to 10 iterations, the first decimal places of the coefficients remain constant and the value of Q is not further decreased. Then the iteration is terminated and the last set of coefficients is used for the further computation. Inspection of the average and maximum magnitude of the individual terms in Q (Equation 16) will reveal the quality of the fit. After the coefficients have been determined, the remaining steps are simple: As described in the previous section, the absorbances of the ampholyte are the coordinates of the intersection point of the tangents to the conic section a t the beginning and the end of the curve. Both of these points must be measured because Equation 14 describes the whole conic section, whereas we are only interested in a part of it. For K , only the value of Alc is missing (see Equations 10 and 13). This component of the center of the conic section (Equation 14) is given by
Aic - A ~ B H=
( t l ~ CiBHz
-~
C ~ B H ) ~ C
(12)
and by multiplying by bo/bc and ordering, we get bo/bc = (Ais
+ A ~ B -H ~~A ~ B H ) / ( -A A~ c~ B H )
(13)
This is the unknown part of Equation 10, and thus K can be determined if the three points mentioned above and the center of the conic section in the A plane are known. Any one of the quantities A ~ B HAlc, , and K may be ill-determined by the input data. We shall show later how the reliability of the results can be estimated. It is therefore not necessary to investigate how errors are propagated in each individual step.
NUMERICAL PROCEDURES The part of our data which we are now considering consists of two series of measurements, Ai, and Ahm with m = 1,2, . . . ,M, where each m corresponds to a different pH value. If we plot these M points, they lie near to a conic section in the A-plane, which we express implicitly by (compare to Equation 5): F(A1,Ak) Ai2
+ a A h 2 + PAlAk + ?Ai+
6Ak
+
t
=0
(14) Multiplying Equation 14 by a factor does not ch,ange the curve represented by this equation. Therefore, we impose a restriction on the possible six coefficients by setting the one of A12 equal to one. Then the remaining five coefficients ( a ,P, y, 6, t ) can be uniquely determined, except for the case where the coefficient of Ai2 is exactly equal to zero. By a proper choice of the coefficients in Equation 14, we wish to obtain a best fit to the given data points. We define the best fit as follows: Consider first one single data point, (Alm,Akm),for the best set of the coefficients. We add to this point a vector (alm,akm), which we suppose to be relatively small and to have the folalm,Akm+ akm)lies on lowing properties: 1) The point (Al, the curve (i.e., fulfils Equation 14) and 2) alm2 akm2is a minimum for all vectors with property 1) [(al,,ak,) is the shortest possible correction vector]. By inserting (Al, al,) in place of Al and (Akm ah,) in place of Ah into Equation 14 and neglecting terms of second order in al, and ahm,we get a linear relation between al, and ahm. Solving for the minimum distance (property 2)) then leads to:
+
+
+
+
am2 + akm2= F2(Alm,Akm)/[(2A~,+ PAkm + 7)’ + + PAim + S ) 2 ] (15) In (7), a similar procedure is described for the case of linear relations instead of Equation 14 and it is called there “com1548
a
ANALYTICAL CHEMISTRY, VOL. 48,
NO. 11,
SEPTEMBER
m= 1
(16)
ERROR ANALYSIS In previous sections, we have shown how the interesting quantities are derived from one pair of the series of measurements, (Al,,Ah,)(m = 1,2, . . . ,M). What we really have are series of measurements at wavelengths A&, . . . ,XL, with L about 10. Thus, we are able to compute as many results as there are pairs (&,Ah) with 1 5 1,k 5 L and 1 # k. The number of such pairs is L ( L - 1)/2. We now append subscripts 1 and k to the results, meaning k computed from the measurements at Xi and that, e.g., ~ l was Ak. In the example given in the application section, (0-phthalic acid), we found that the set of numbers Klk cannot simply be averaged in order to get the desired final result: an estimate 1976
of K. About one half of the values lie close together, but others differ from this cluster of results by at most two orders of magnitude, and even negative values of K occur which are impossible for physical reasons. We suppose that some of the results are affected by large numerical errors which may occur, e.g., in the solution of the normal equations, or in the determination of the intersection point of the two tangents, or the computation of Equations 13 and 17. If our measured data happen to yield values near to a singular case (singular matrix, parallel tangents, or vanishing denominators), we have to expect that the errors present in the measurements are amplified and the result is not useful. Because of the complicated dependence of the results on the input data, we did not try to investigate the sensitivity of each of those individual steps. Instead of this, we tested the sensitivity of the whole process by a MonteCarlo method (8). 1) Compute the quantities Klk,(m,P,Y,a,€)lk from a pair of measurements (Alm,Ak,), m = 1,.. . ,M. 2) An “errorless” data set (Alm’,Akm’)is constructed where this point lies on the conic section defined by (oI,P,Y,6,€)&(see Equation 14) and close to (Alm,Ak,). 3) The distances between the real and the “errorless” points are used to estimate the standard deviation of the measured data. 4) The “errorless” data points are changed by normally distributed pseudo-random numbers with the same standard deviation as estimated in step 3 and another value of K can be computed from these simulated measurements. Step 2 is done by a modified Newton method along the gradient of F (Equation 14). The measured point (Alm,Akn) is corrected by the vector (alm,akm)which is described in the sequel to Equation 14. A new correction vector is computed for the resulting point. This is repeated several times until the correction vector is negligible. If this is not the case after five corrections, we assume a failure of the Newton method. Because both the Newton method and Equation 15 rely on the same linearization, such an event indicates that the conditions for this linearization are not fulfilled. The Monte-Carlo method can be performed in a very short space of time on a fast computer and, by repeating step 4, a large number of simulated results can be attached to each Klk. The standard deviation of the simulated results is taken as an estimate of the standard deviation of Klk. We found that the results are fairly independent of the initialization of the random number generator and the number of simulations if 10 or more simulations are executed. This investigation enables us to distinguish between good and poor results. A combination (I,h)is assumed to yield poor results if one of the following errors is detected: 1)The matrix of the normal equations is almost singular. 2) The Newton method does not converge. 3) Negative values of Klk are computed from the real or some simulated data. For each remaining Klk a stand’ard deviation is computed from the results from the real and the simulated data. It may k a much greater standard deviation happen that some ~ l has than most of the others. This means that this result is very sensitive to random variations of the input data which must be assumed to be present in the measured data too. The decision whether such a result is used as a valid result or not is left to the user of our program. In our example, about one half (11 out of 21 possible combinations) of the results passed through these tests. The final result at the end of the following section is the pK difference (logarithm of K) obtained by taking the mean of the individual results which are considered to be valid by the criteria given above. From this set of valid results the standard error S and the confidence interval are computed by standard procedures (9).
Table I. Absorbance Measurementsa No.
1 2 3 4 5 6 7
Wavelength, nm
305 295 290 286 272 268 255
Absorbances of
BHZ 0.0843 0.0617 0.2383 0.5148 0.6979 0.6063 0.7050 s = 0.0005
BZ0.060 0.0120 0.0470 0.1600 0.4470 0.4320 0.7000 s = 0.0005
BH-
0.376 0.633 0.763 0.633 0.534 0.892 s = 0.001
Standard deviation a Absorbances of o-phthalic acid (designated by B). Those of BH, and BZ- are measured, that of BH- is a result of computations. One complete series of measurements consists of 113 measurements at intermediate pH values while these values occur at very low (BH,) and very high pH (B2-) (3, 4 ) . Measurements were performed at 20 “C with a total concentration of o-phthalic acid of 5.8 X mol/l. Standard deviations are estimated for the whole data sets (BH,, BZ-) or for some results from different combinations ( h k , hl) in the case of BH-.
APPLICATION The method as described in the preceding sections was applied to measured absorbances of o-phthalic acid which reacts according to Equatiqn 1. First, one has to decide which wavelengths should be used for the measurements. By recording absorbances of the system at some values of pH over a fairly wide range of wavelengths, it is possible to determine those ranges of wavelengths which show strong dependence of absorbance on the pH ( 4 ) .This is a necessary condition for proper wavelengths. Because data from two different wavelengths are needed for each evaluation, the absorbances at the two wavelengths must be different in order to avoid an almost singular transformation from concentrations to absorbances (see Equation 7). This is probably achieved by choosing values of the wavelengths which are not too close to each other. The first column of Table I gives the wavelengths Xi we used for our measurements. As mentioned in the previous section, 10 of the possible 21 combinations of wavelengths failed to yield acceptable results. Most of the failures are caused by the wavelength X1 which shows only weak dependence of absorbance on pH. The measurements at A7 caused some failures too. One combination, (A,,&,) was rejected because the absorbances are too similar. For each wavelength Xl, the absorbance is measured starting, e.g., a t very low pH and increasing pH by a small value (about 0.05 in our example) per step. The absorbance is stationary in the beginning (pH < 1;all B is in the form BH2) and becomes stationary at the end (pH > 8; only B2- present). The stationary values are the absorbances A ~ B and H ~A ~ B (columns 3 and 4 of Table I). In Figure 2, we give some examples of absorbance diagrams with measurements from two wavelengths plotted in the A plane. The pH is increased by adding small amounts of a strong base (NaOH). This causes a slight dilution which has to be accounted for by correcting the measured values. The last column of Table I gives the absorbances of the intermediate BH-. They are computed as coordinates of the point of intersection of the two tangents a t both ends of the physical part of the conic section (see Figure 2). The final estimate of the difference of pK values of ophthalic acid obtained from our measurements is ApK pK2 - pK1 = 2.43. Standard error: S = 0.01; 95% confidence interval: ApK f 2.2 s.
ANALYTICAL CHEMISTRY, VOL. 48,
NO. 11, SEPTEMBER 1976
1549
DISCUSSION
LITERATURE CITED
This work expands our formerly graphical methods for the determination of relative pK values without using pH measurements. One objective was the use of the proper function for the fit-the conic section-as suggested in the theory of the observed chemical system. Another problem became apparent in the course of this work: some combinations of series of measurements yield distinctively less reliable results than others. There are several possible reasons for this fact which cannot be investigated in advance. Therefore, we used the criteria of the error analysis section in order to exclude those meaningless results. This second part of the computations required a t least 90% of the total computing time which was about 1 min on a fast computer in our example (7 series of measurements, 113 measurements per series, 30 Monte-Carlo tests per combination).
B. J. Thamer, J. Phys. Chern., 59, 450-453 (1955). G. Heys, H. Kinns, and D. D. Perrin, Analyst (London), 97,52-54 (1972). R. Blume and J. Polster, Z.Naturforsch. 6,29, 734-741 (1974). R. Blume, H. Lachmann, and J. Polster, 2.Naturforsch., 6,30, 263-276 (1975). (5) J. Polster, Z.Phys. Chern. (FrankfurfamMain), 97, 113-126 (1975). (6) H. Mauser, "Formale Kinetik", 1st ed.. Bertelmann Univ.-Verlag, Cusseldorf, 1974, p 328, (7) E. Hardtwig, "Fehler- und Ausgleichsrechnung", 1st ed., Bibliographisches Institut, Mannheim, 1968, p 159. (8) Y. Bard, "Nonlinear Parameter Estimation", Academic Press, New York, 1974, p 46. (9) Anal. Chern., 46, 2258 (1974).
(1) (2) (3) (4)
RECEIVEDfor review December 23,1975. Accepted June 3, 1976.
Determination of Trace Amounts of Beryllium in Rocks by Ion Exchange Chromatography and Spectrophotometry F. W. E. Strelow," R. G. Bohmer,' and C. H. S. W. Weinert National Chemical Research Laboratory, P.O. Box 395, Pretoria 000 1, Republic of South Africa
A method is described for the accurate determination of trace amounts of beryllium in rocks. Elution from a column of AG 50W-X8 cation exchanger with 2.0 M nitric acid in 70% methanol separates beryllium from Mg, Ca, Mn(ll), Fe(lll), AI, and many other elements which are retained. In a second cation exchange step, Ti( IV), Cu(ll), the alkalies, and several other elements are eluted with 0.3 M hydrochloric acid in 90% acetone, followed by 0.5 M sulfuric acid and then by 0.5 M nitric acid, all containing 0.015 % hydrogen peroxide. Beryllium is again eluted with 2.0 M nitric acid in 70% methanol. Determination is carried out by measuringthe absorbance of the complex of beryllium with chromearurol S in the presence of benryldimethylhexadecylammonium chloride at a wavelength of 610 nm. Additional purification steps are described for samples containing interferingelements and very low beryllium concentrations. The practical sensitivity of the method is about 0.02 fig of beryllium in 1 g of rock, or 0.02 ppm. The relative standard deviation for rocks containing more than 1 ppm beryllium is about 2 % . Relevant elution curves and results for several standard rocks are presented.
Beryllium occurs in rocks in concentrations of a few ppm or less. Published values for this element in international standard rocks show large amounts of scattering (1-3). Often only the estimated sensitivity limit of the method used is given, indicating that the berylliurn concentration should be less (1-3).Since the final accuracy of the determination of a t;ace component in a complex matrix is related to the purity of the material on which the final determination is carried out, it appears that an effective separation of beryllium from the major and minor elements occurring in rocks should be of some use. Many instrumental techniques can be used for the f i n d determination. A spectrophotometric approach will be Department of Inorganic and Analytical Chemistry of the University of Pretoria. 1550
described here because the instrumentation is relatively inexpensive and generally available. It has been shown that the sensitivity of the determination of beryllium by spectrophotometry of its complex with chromeazurol S can be considerably enhanced by the addition of micelle forming agents such as quaternary amines containing one long carbon chain, e.g. zephiramine (benzyldimethyltetradecylammonium chloride) ( 4 ) or benzyldimethylhexadecylammonium chloride (BDHA) ( 5 ) .The second reagent in the presence of 0.1 M ammonium chloride increases the molecular absorption of the complex of beryllium with chromeazurol S to t = 85 000 at 611 nm and to 90 000 at 620 nm, as compared with a value of about 20 000 for emax in the absence of a micelle forming reagent. Sommer and Kuban (6) have used polyvinyl alcohol for micelle formation and obtained a value for emax of about 52 000. Many multivalent elements interfere, but the addition of Ca-EDTA suppresses most interferences, provided the interfering elements are not present in excessive amounts. Zirconium and uranium(V1) interfere nevertheless (6).Furthermore, though the amount of aluminum causing less than f 2 % deviation from the true value increases from 0.2 times to 50 times the amount of beryllium when Ca-EDTA is added, this is not sufficient for rock samples with aluminum-beryllium ratios as large as lo5.The same argument applies to iron(II1) and titanium for which the limiting ratios to beryllium are 32 and 10, respectively, when Ca-EDTA is present (6). A recently developed method (7) using cation exchange chromatography in nitric acid-methanol for the separation of beryllipm seemed to be promising for the purpose. The method separates beryllium from all major rock forming elements with the exception of titanium and the alkalies. In addition some trace elements such as Cu(II),Bi(III), Pb(II),and Hg(I1) also accompany beryllium. Possibilities to include the separation of beryllium from these elements in an extended method were investigated. I t has been shown that titanium can be separated from beryllium by selective elution with 0.5 M sulfuric acid containing hydrogen peroxide (8). Cu(II),
ANALYTICAL CHEMISTRY, VOL. 48, NO. 11, SEPTEMBER 1976
