Corrections for Matrix Effects in X-Ray Fluorescence Analysis Using Multiple Regression Methods BERNARD J. ALLEY Propulsion laboratory,
U. S.
Army Missile Command, Redstone Arsenal, Ala.
RAYMOND H. MYERS Department of Statistics, Virginia Polytechnic Institufe, Blacksburg, Va.
b Multiple regression analysis techniques for relating characteristic x-ray emission line intensities and ingredient percentages have been used to devise a method for estimating ingredient percentages in multicomponent mixtures. Based on a linear statistical model, a set of simultaneous regression equations and the corresponding set of inverse working equations were developed for the direct analyses of four ingredients in propellant slurries. The derived pa rtia I regression coefficients for matrix components provide corrections for measured matrix effects within samples. Calibration mixtures were systematically formulated according to a half fraction of a 2 4 factorial experiment. The advantages of multiple linear regression analysis over commonly used simple linear regression analysis are demonstrated by comparing the correlation indexes and standard errors for the regression of intensity ratios on the four ingredient percentages. Application of the multiple regression method i s limited to ingredient percentage ranges where the true relationships among intensities and percentages can be satisfactorily represented by the linear model chosen.
X
fluorescence methods are widely used for the compositional analyses of various materials, and new applications are increasing a t a rapid rate. The high speed and degree of precision attainable are especially attractive features of these methods. Unfortunately, the combination of speed and accurate analyses commensurate with the high precision is often difficult to achieve in practice because of the existence of sample matrix effects. The existence of significant matrix effects means that the intensity of fluorescent radiation from the analytical element is a function of the percentages of the matrix elements as well as its own percentage. The theoretical causes of matrix effects and problems arising therefrom are well known; Mitchell (11, 18) has described in detail the qualitative and quantitative aspects of -RAY
these effects and the associated problems. Because of matrix effects, the simple linear regression of x-ray intensity, R, from an element on its percentage, X , in a multicomponent mixturenamely, R = bo bX (1)
premixes. Similar mathematical methods for the x-ray fluorescence analyses of multicomponent mixtures have been reported by Sugimoto (16, IS) and Lucas-Tooth and Pyne (IO).
is usually an inadequate representation of the true relationship. Nevertheless, this simple equation can sometimes be used to give acceptable analyses when the matrix effects are small. Ingredient percentages in unknown samples are estimated either from calibration curves or the inverse of Equation 1:
The general techniques of multiple regression analysis are described, for example, by Ostle (14). On this basis, and assuming the functional relationship between the dependent and independent variables to be linear over the percentage ranges of the calibration ingredients, the statistical model for the analyses of four ingredients in a mixture is
+
X = ( R - b,)/b
(2) Even when matrix effects are large, Equation 2 can give percentage estimates of sufficient accuracy if the matrix effects are first minimized experimentally. Such experimental procedures classically involve either the preparation of families of curves or the alteration of samples prior to analysis; however, the preparation of families of curves requires a large number of standards, and the alteration of samples detracts from the speed of analysis. Because of these disadvantages, a number of mathematical methods of correcting for matrix effects have been developed from both theoretical and empirical considerations. Recently published methods have been reviewed by Campbell and Brown (4). This paper describes the use of multiple regression methods for the direct compositional analyses of multicomponent mixtures with special application to the analyses of four ingredients in propellant slurries. Since the methods have general applicability, the actual ingredients are denoted in the text by subscripts I, 2, 3, and 4. Multiple regression analysis is a logical extension of simple regression analysis; the influences of all analyzed elements on the x-ray intensity from the analytical element are evaluated simultaneously. Lamborn and Sorenson (9) have demonstrated the usefulness of multiple regression analysis for determining iron(II1) oxide in propellant
THEORETICAL
where R,, is the measured x-ray intensity ratio for the ith analytical ingredient in the j t h calibration mixture, the pds are true (population) partial regression coefficients, X I , is the percentage by weight of the lth compdnent in the j t h calibration mixture, and E , , is the random error associated with R,]. The percentages of all measurable components appear in each equation in contrast to the simple regression model where only the percentage of the analytical ingredient appears. The XI,’S are determined with negligible error compared with the R,,’s, and necessarily selected as independent variables. The XI,’s may be chosen deliberately during the preparation of calibration standards. For the purpose of making certain statistical inferences, it is further assumed that the R,,’s are normally and independently distributed for a given set of XI,’s with a mean
The variance of Rij for any set of X t j l s is ( ~ 3 3 . The regression equations required are VOL. 37, NO. 13, DECEMBER 1965
1685
Table 1. Factors and Factor Levels for Preparation of Calibration Standards Low High Factor Symbol level, g. level, g. Ingredient 1 A 4.5 5.5 700.0 Ingredient 2 B 660.0 145.0 Ingredient 3 C 125.0 150.0 170.0 Ingredient 4 D
kij
=
+
+
bio bilX1,j bi,Xz, j bi3X3,j
+
+ biaX4,
j
(4)
where &j is an estimate of Rij, and the bit’s are estimates of the Pil’s. The bit’s are usually derived by a least squares analysis which involves minimizing the sums of squares of the residual errors:
- &j)’
(5)
~ i i
The normal equations that must be solved to get the b d s are
i = (1, . . . , 4) k = (1,
.
I
where a$kR is the corrected sums of cross products of the E,, with xkj from the original data, and ukt represents the corrected sum of cross products (or sum of squares in the case where 1 = k ) . The Equations 6 can be written in matrix form as follows: (7)
where A is the coefficient matrix, b, is the vector of partial regression coefficients, and G, is the vector of corrected cross products of ingredient percentages with x-ray intensity ratios. Inverting the Equations 7 gives
b, = A-’ G, (8) where the b,l element of b, is the coefficient of XL, in the regression equation of the ith analytical ingredient. iilthough determination of the set of regression Equations 4 from calibration data involves a number of calculations, the use of x-ray intensity ratios eliminates the necessity for frequent recalibration. Simplified computational procedures for multiple regression have been described by Kramer (8). Equations 4 can now be used to develop a set of working equations for estimating the Xt,’s of unknown mixtures. Using matrix notation, then R=b+BX (9) where R represents the vector of x-ray intensity ratios, b the vector of intercept 1686
ANALYTICAL CHEMISTRY
-B-
b
(10)
Letting B-1 = D and B-lb = -Do, Equation 10 can be expressed in the desired algebraic form for estimating ingredient percentages :
2%= dtO + d,lR1 +
+ dt3R3 + d24R4
(11)
where d,i is the il element of matrix D, and the j subscript has been omitted. This is a case of using a set of simultaneous multiple linear regression equations in reverse. Williams (17) has discussed this more general problem. The multiple regression calculations described in this paper were made with an IBbI-7094 computer using an available program developed by Baer et ul. (I). The program will fit a simple linear regression model, as well as the multiple regression model represented by Equations 3, to the data for all dependent variables in a single run. EXPERIMENTAL
., 4)
Ab, = G,
x=
i = (1, . . ., 4)
j = (1, . . ., n)
n
+ B-’R
I
d&z
i = (1, . . .,4)
(ss,)~= j = 1 (
terms, and B the array of b J s . X is the vector of unknown ingredient percentages that are estimated in practice. If B is a nonsingular matrix, Equation 9 can be inverted to give
Instrumentation. h universal vacuum x-ray spectrometer marketed by Philips Electronic Instruments was used. The spectrometer has inverted optics and four sample compartments. Spectrometer components consist of a 4-inch b y 0.020-inch entrance collimator, sodium chloride and pentaerythritol analyzing crystals, and a gas-flow proportional detector. The associated electronic circuit panel (Type 12206/0) has a decade scaler with an electronic timer, and a pulse height analyzer. X-rays were provided by a chromium target Machlett OEG-50s x-ray tube operated a t 40-kv. constant potential and 30 ma. Voltage to the generator was stabilized with a 5-kv.amp. line voltage stabilizer. Procedure. Slurry samples were placed against 0.00015-inch Mylar films and t h e surfaces against t h e films were irradiated a n d analyzed. Samples were analyzed in duplicate in conjunction with a stable synthetic reference standard. T h e number of seconds required t o collect a preselected fixed count for t h e reference standard, t,, and slurry samples, t,, were measured in rapid succession at the peak analytical goniometer angle of each ingredient. Two pairs of samples denoted by T1 and T2, were analyzed from each calibration mixture on the same day. The calibration mixtures, denoted by 91,were prepared and analyzed over a period of several weeks. The response variable used for regression analysis was A
where Rij is the x-ray intensity ratio averaged over samples for the i t h ingredient in the j t h mixture.
Calibration Standards. Working equations for the x-ray fluorescence analyses of mixtures are established by analyzing accurately known calibration standards and fitting a suitable model to t h e d a t a as already described. The standards might be selected at random from a n available source or prepared deliberately by t h e analyst. I n either case they must be representative of the unknown mixtures to be analyzed. Reliable and efficient estimates of the partial regression coefficients in a model such as that represented by Equations 3 can be made with a reasonable number of standards if the standard compositions are chosen properly. Confusing the effect of one ingredient percentage on an x-ray intensity with the effect of another ingredient percentage (confounding), and high degrees of correlation among ingredient percentages must clearly be avoided. It is difficult to avoid confounding and correlations among ingredient percentages without either resorting to a statistically designed experiment for preparing standards or analyzing a large number of standards ( I O , 16). d number of experimental designs might be used. Complete 2 p factorials (5), where p is the number of factors, are satisfactory for linear regression models containing only main effects (ingredient percentages) and first-order interactions. Fractional 2 p factorials (6) are suitable for linear models having only main effects, and they require fewer standards than the complete factorials. Composite designs described by Box and Hunter (2) can be used for models having higher degree terms by adding to the appropriate factorials. A half fraction of a 24 factorial was used in this work to prepare standards for the model represented by Equations 3. The factors and factor levels are given in Table I. The factor levels were arbitrarily chosen to result in practical ingredient percentage ranges that were still sufficiently narrow to permit a good fit of the linear model to the data. The treatment combinations for the principal block of the factorial and the corresponding compositions of the standards are given in Table 11. The percentages of all standard mixture ingredients are accurate to 0.1% relative. The defining contrasts (6) are I, ABCD. This is to denote thc exact treatment combinations of the possible 16 that are to be run in the half fraction. In the resulting experiment whose factor levels appear in Table I the four-factor interaction ABCD is confounded with &, the constant term; hence, no information is obtained about
IQ
u -98ADJ &a“ 8-1.7738 ’0.08311 Xa *0.02197 R-, r\.
2
ta .90
I-
-z .78
/
, .’
//
Y
5.94 P
/
.78.
2 2.74
A
13
12
14
15
WEIGHT-PERCENT OF INGREDIENT 3, x3
Figure 2.
this interaction. This confounding is satisfactory, however, since the fourfactor interaction is assumed to be negligible. When working equations are to be derived for the control analysis of a single type of mixture whose nominal composition is known beforehand, it should be placed a t the midpoint of the design as shown in Table I1 to increase the precision. Following convention, the high level of each factor in the treatment combination is denoted by the presence of the lower case factor symbol, and the low level by absence of the symbol. The combination [ I ] denotes all factors at their low levels. Each mixture in Table I1 also contains a fifth ingredient which cannot be analyzed by the x-ray fluorescence method and was not included in the design. Each effect in a half fraction of a factorial experiment has an “alias,” and it should be clearly defined to ensure that effects of interest can be accurately evaluated. The aliases in this experiment are given by the following: A=BCD; B=ACD; C=ABD; D = ABC; AB=CD; AC=BD; a n d A D = BC. Each main effect (A,B,C,D) is confused lvith a three-factor interaction, and each two-factor interaction with another two-factor interaction. Each partial regression coefficient in Equations 4 will reflect the sum of the contributions of the main effect and its alias. The main effects in this experiment can be properly evaluated, because threefactor interactions are generally not significantly different from the experimental error. RESULTS
Working Equations. The x-ray intensity ratios, R,,,resulting from analyses of the calibration standards are recorded in Table 111. The regression equations for estimating the R,’s with all the XZ’Spresent are given below. They were derived using the
Table II.
Mixture
Compositions of Calibration Standards in Weight Per Cent
Treatment combination
1
2 3 4 5
6
7
8 9
XI 0.5514 0.4426 0.5631 0.5624 0.4505 0.4425 0.5290 0.4702 0.5001
ab bc ad ac cd bd abcd 111
Midpoint
model represented by Equations 3, the data in Table I1 for the independent variables, and the data in Table I11 for the dependent variables.
kl k2
k3
kd
+ 0.03193X2 + 0.04375X3 + = -3.1237
f 1.905x1
+
x3
X*
12.53 14.26 12.79 14.83 14.52 12,30 13.95 13.07 13.51
15.04 14.75 17.39 15.34 17.03 16.72 16.35 15.68 16.02
Table 111. X-Ray Intensity Ratios from Analyses of Calibration Standards
Mixture
2 3
0.02657X4 0.04046x1 f 0.00294X3 0.02781X4
= -1.0253 - 0.07955X1 0.01481X2 f 0.07550X3 0.00736X4
x2
70.18 68.84 67.51 67.52 66.10 68.86 67.34 69.00 68.07
1
4
+ O.OO952X2 + = 0.5875
Adjusted calibration curve for ingredient 3
5 6 7 8
9
R1
R2
1.1240 0.9285 1.1214 1.1635 0.9415 0.9039 1.0712 0.9561 1.0186
0.8980 0.8872 0.8030 0.8706 0.8064 0.8314 0.8404 0.8731 0.8431
RI 0.8219 0.9308 0.7668 0.9272 0.9026 0.7596 0.8662 0.8206 0.8346
€24
0.9906 0.9944 1.1221 0.9832 1.1127 1.0994 1.0836 1.0290 1.0591
(12)
= 5.6792 - 0.2212X1 0.05461x2 - 0.05978X3 f 0.000938X4
A particular coefficient is interpreted as the amount that R, will vary per unit change in the corresponding X I if the other X’s are fixed. A positive coefficient indicates an over-all enhancement effect of X I on R,, and conversely, a negative coefficient indicates an overall absorption effect. Constant radiation background corrections are included in the intercept terms. All measurable effects on R,, instead of strictly the mutual absorption and enhancement of radiation among elements,
are evaluated. This simultaneous evaluation of effects by regression analysis is considerably more efficient than intuitive experimental approaches whereby the influences of sample matrix ingredients on each analytical ingredient are evaluated one a t a time. The inverse working equations for estimating ingredient percentages from measured x-ray intensity ratios are:
21= 22
-0.9779 f 0.5615R1 f 0.5543R2 - 0.01339Rs O .4213R4 = 113.37 - 3.687Rl 0.3575R2 - 17.87R3 25.25R4
+
+
VOL. 37, NO. 13, DECEMBER 1965
1687
Table IV.
Degrees
Ingredient 1 Sum of Mean squares square
of Source of variation freedom Among M’s 8 Regression (4) 3031.996 13.560 Residual (4) 1 Between T’s 0.453 8 12.420 MXT 12.635 Sampling error 18 3071.064 Total 35 All values multiplied by lo4.
23 =
-4.028 2.932R2
+ 1.285Rl + 16.31R3 +
(13)
4.663R4
24 = 58.08 - 0.3090R1 35.34RZ
- 4.413B3 7.537R4
The best set of regression equations for estimating the R,’s, as determined by stepwise regression analysis using the IBM-7094 computer, is:
I&
=
+
+
-0.5987 l.882X1 O.OO658Xz 0.01715X3
+
& = 0.8878 + 0.00669X2 0.03078X4
&=
-1.7738
+ 0.02197X2 +
(14)
0.08311X3
24 = 5.7684 - 0.2220X1 0.05550X2 - 0.06072X3 At least one of the original independent variables in each equation of the set was found to have a n insignificant effect on estimating Ri and, therefore, was deleted. Even the percentage of the analytical ingredient may be deleted, as shown by the fourth equation of the set. All of the coefficients except that for the analytical ingredient, in each of the equations in set 12, can be interpreted as correction factors for the effects of sample matrix ingredients on the x-ray intensity ratio. This can be illustrated by considering the simple equation for R3 in Equations 14. If the are adjusted for the departure of the over individual X2’sfrom the mean, X2, all calibration standards, the equation becomes
A D J & = 68 -1.7738
- 0.02197 (X2 - XZ)=
+ 0.08311Xs +
0.0219722 (15) The simple linear regression of RS on X3 with no correction for matrix effects is shown in Figure 1. The calibration data are also plotted to show the scatter 1688
ANALYTICAL CHEMISTRY
757.999 3.390 0.453 1.553 0.702
Analysis of Variance”
Ingredient 2 Sumof Mean squares square 362.920 6.883 0.570 8.430 7.919 386.722
90.730 1.721 0.570 1.054 0.440
Ingredient 3 Sumof Mean squares square 1265.908 32.687 0.002 7.700 8.316 1314.613
about the regression line. Equation 15 and the adjusted calibration data are shown in Figure 2 to illustrate the reduction in scatter achieved by applying the matrix correction. The amount of improvement obtained by matrix correction depends on the magnitude of the matrix effect, and can be substantially greater than that illustrated by Figures 1 and 2. A greater improvement is shown, for example, by the data for ingredient 2 in Table V. Analysis of Variance. The sources of error i n t h e experiment and their magnitudes can be evaluated by making an analysis of variance. This analysis for the four slurry ingredients is shown in Table IV. T h e sums of squares and mean squares are for individual x-ray intensity ratios. The total sum of squares for variation among mixtures has been partitioned (14) into the residual sum of squares, (SSE),, and the sum of squares for regression, (SS Reg),. The sum of squares for regression is determined as follows:
The residual sums of squares, defined by Equations 5, represent the amount of deviation of the x-ray intensity ratio measurements about the regression plane-Le., the amount of error associated with the estimation of Ri. The standard error, ( S E ) ~is, obtained from the residual sum of squares: =
[(SS&/(n - p - 1 ) P
(17)
where n is the number of calibration standards analyzed, p is the number of degrees of freedom for regression, and n - p - 1 is the number of degrees of freedom for residual error. -4minimum of five calibration standards is needed to derive each equation in the set of Equations 12, because p = 4. It is desirable to analyze a larger number of calibration standards, so that there will be degrees of freedom for estimating the residual mean squares and the standard errors. The sources of variation listed in Table IV can be tested for statistical significance by calculating the ratios
316.477 8.177 0.002 0.962 0.462
Ingredient 4 Sum of Mean squares square 940.675 29.326 3.062 11.493 15.850
1000.406
235.169 7.337 3.062 1.437 0.881
( F ratios) of their mean squares to the experimental error mean square, and comparing them with critical values in a table of tabulated F values. The r l l X T interaction contains the largest random error and, therefore, is considered to be the best estimate of the true experimental error in this experiment, B u t this is likely to be an underestimate of the true error because it does not include the source of random error which would result from the replication of calibration standards. The sampling error is calculated from the 18 pairs of sample analyses, and represents sample repeatability with the instrumental operating conditions essentially fixed. The validity of the linearity assumption made in connection with the regression model can be checked by testing the significance of the residual mean squares. The relative merits of using simple or multiple regression methods for the analyses of multicomponent mixtures can be assessed by comparing the standard errors and correlation indexes for the two types of regression analysis. The values of these parameters for this experiment are listed in Table V. The correlation index, R,2, is the fraction of the total corrected sum of squares among calibration standards that is explained by regression, and is determined from the regression and residual sum of squares in Table IV. It measures how well the model fits the data, and has a value of 1 for a perfect fit, The data in Table V show that multiple regression analysis is superior to simple regression analysis for all ingredients, with the greatest improvement noted for ingredients 2 and 3. Estimating Ingredient Percentages. T h e percentages of ingredients in the slurry mixtures are estimated by means of Equations 13. T h e precision and accuracy of these estimates are of primary concern. The absolute errors of estimating ingredient percentages in t h e calibration standards are given in Table VI. The rootmean-square errors using working equations derived from both simple and multiple regression analysis are given in Table VII. The working equations
derived from multiple regression analysis give smaller errors of estimation for all ingredients. Estimates of calibration ingredient percentages are likely to be more accurate than the estimates of unknown percentages in mixtures not included in the regression analysis. The accuracy of the method can be evaluated by establishing joint confidence interval estimates on the ingredient percentages. Williams (18) discusses the general problem of attaching confidence intervals to the solution of a set of simultaneous regression equations similar to those in 12. H e derives a n approximate expression for the variance of each of the estimates of the independent variables (the Xi in 12) and uses this to obtain a n approximate expression for a Confidence interval estimate on each X , which of course would be useful in the application considered here. However, this approximation is good only when the partial regression coefficients are large in comparison with their standard errors. Such was not the case for the particular mixture considered here. Box and Hunter (3) mention the problem of determining joint confidence interval estimates on the solution of a set of simultaneous equations when the coefficients are subject t o error. Their nork was actually a part of a more specific problem in statistical experimental design; however, it can be applied here. The results would be joint intervals on the percentages. This type of interval can, in general, give results which are unrealistic because the resulting confidence region is often very large. However, the confidence regions in this work were narrow and the results have much practical use. The details are given by Myers and Alley ( 1 3 ) . DISCUSSION
2(X$,- f,,)z
Ingredient 1
2 3 4
(18)
j=1
Eisenhart ( 7 ) has pointed out that when the values of X,,have been selected by the analyst and the corresponding R,, values observed, the equations obtained by minimizing the foregoing expression are meaningless. Accordingly, the only correct estimate of the postulated linear relationship between R,, and X,, is given by Equations 4, and Equations 11 are the correct ones to use for estimating the Xz’s.
Correlation Indexes and Standard Errors from the Regression of X-Ray Intensity Ratios on Ingredient Percentages
Simple regression SE R2 0.979 0.01502 0.02407 0,561 0.02472 0.868 0.01465 0.938
The actual relationships
Rlulti le regression, . alyvariables SI R2 0.00921 0.996 0.00656 0.981 0.01429 0.975 0.01354 0.970
tures may not be linear as assumed in the model represented by Equations 3. The ranges of percentages for which this linear model can be satisfactorily used depend mainly on the relative values of the mass absorption coefficients of mixture elements for the analytical wavelengths. The linear model represented by Equations 3 can be used in many cases to give sufficiently accurate analyses over reasonably wide percentage ranges, but these ranges are difficult to determine beforehand. Matrix effects generally present a more serious limitation to analyses over wide ranges than the assumption of linearity. Analyses using the simple linear relationship of Equation 1, for instance, are commonly restricted to ingredient percentages which vary by not more than 10 to 20% of the amount present. The narrow percentage ranges in Table I1 are acceptable for the analyses of propellants described here, but do not represent the maximum ranges that can be analyzed in practice. Ideally, one might select a more complex model which more closely approximates the underlying functional relationship among the variables. Consider, for example, the polynomial model =
Pi0
Multiple regression, best variables SE R2 0.995 0.00853 0.00581 0.978 0.01247 0.971 0.01211 0.970
between
R,, and X , , in multicomponent mix-
Rt,
The possibility of directly deriving working equations for estimating the X,’s-Le., without first deriving the set of regression Equations 4-might be considered. The least squares analysis would be made in the manner already described, and would involve minimizing the expression
SS, =
Table V.
+ P~IXI,,+
The first-order interaction terms in the model denote deviations from parallelism (19). The regression equations derived from such a model would perhaps be useful for estimating the Rz,’s from known X,,’s. They could not, however, be inverted to give equations for estimating unknown X2,’s, because the matrix of partial regression coefficients would not be square. The estimated coefficients in Equations 4 can be tested by standard statistical methods to ascertain whether they are significantly different from zero. Those coefficients which are not can be omitted from the equations during the least squares analysis to improve the precision of estimating the R,,’s. This is readily performed by
Table VI. Errors of Estimating Ingredient Percentages in Weight Per Cent
Mixture 8,- X I 8,- ~2 1 0.0059 -0.33 2 -0.0009 -0.32 3 -0.0038 -0.03 4 -0,0026 0.47 5 0.0039 -0.14 6 0.0009 0.13 7 0.0053 -0.47 8 -0.0048 0.50 9 -0.0037 0.18
8, - ~a 2,- X, 0.27 0.12 0.01 -0.22 0.20 -0.09 0.11 -0.25 -0.16
-0.13 0.09 0.13 0.11
-0.10 0.06 -0.29 -0.13 0.29
Table VII. Root-Mean-Square Errors of Estimating Ingredient Percentages Using Simple and Multiple Regression Working Equations
In-
gredient 1 2
3 4
Simple regression 0,0071 0.9970 0.3376 0.2193
Multiple regression 0.0039 0.3301 0.1775 0.1677
computer using Baer’s program (1). The resulting equations, giving the smallest calculated errors of estimation and containing only significant coefficients, are called the best set for estimating the Rij’s. This best set of regression equations cannot always be inverted, however, and even when they can, there is no assurance that they will produce a better set of working equations for estimating the X,’s than the set of Equations 4. ACKNOWLEDGMENT
The authors express appreciation to Boyd Harshbarger for consultative assistance; to J. L. Gill, J. T. Peeler, and E. L. Bombara for helpful suggestions and contributions; to R. Tankersley for the computer calculations; and to S. F. Schultz for the preparation of calibration standards. LITERATURE CITED
(1) Baer, R. M., John, P. W. M., Tornheim, L., California Research Corp., “Stepwise Regression,” Share Distribution 1333 SC BSlMR (PA) (1962). (2) Box, G. E. P., Hunter, J. S., Ann. Math. Statist. 28, 195 (1957). VOL. 37, NO. 13, DECEMBER 1965
1689
(3) Box, G. E. P., Hunter, J. S., Biometrika 41, 190 (1954). (4) Campbell, W. J., Brown, J. D., ANAL. CHEM.36, 312R (1964). ( 5 ) Davies, 0. L., ed., “Design and Analysis of Industrial Experiments,” 2nd ed., p. 247, Hafner Publishing Co., New York, 1956. (6) Ibid., p. 440. ( 7 ) Eisenhart, C., Ann. Math. Statist. 10, 162 (1939). (8) Kramer, C. Y., Ind. Quality Control 13, 8 (1957). (9) Lamborn, R. E., Sorenson, F. J., Advan. X-Ray Anal. 6 , 422 (1963).
(10) Lucas-Tooth, J., Pyne, C., Ibid., 7, 523 (1964). (11) Mitchell, B. J., “Encyclopedia of Spectroscopy,” p. 736, Reinhold, New 1960. York. ~.~~~ (12) Mitchell, B. J., “Encyclopedia of X-Rays and Gamma Rays,” G. L. Clark, ed., p. 531, Reinhold New York, 1963. (13) Myers, R. H., Alley, B. J., “Use of Regression Analysis for Correcting Matrix Effects in the X-Ray Fluorescence Analyses of Pyrotechnic Compositions,” Proceedings of Tenth Annual Conference on the Design of Experiments, 1964.
(14) Ostle, B., “Statistics in Research,” 2nd ed., p. 159, Iowa State University Press. Ames. 1963. (15) Sugimotd, AI., Bunseki-Kagaku 11, 1168 (1962); C.A. 58, 30769 (1963). (16) Sugimoto, M., Bunseki-Kagaku 12, 475 (1963); C.A. 59, 12152h (1963). (17) Williams, E. J.. “Regression Analvsis,” D. 162, Wilev “ , New York, 1959. (18jZbid.,-p. 166. (19) Youden. W. J.. “Statistical Methods y, New York, IYOl.
RECEIVEDfor review May 20, 1965. Accepted September 15, 1965.
Spectrophotometric Determination of Bromine and Hydrogen Bromide E. C. CREITZ Fire Research Station, National Bureau o f Standards, Washington, D.
b A method is described for determination of Brz and HBr in concentrations ranging to 5 pg. of bromine per rnl. The analysis depends on oxidation of o-tolidine by Brz and subsequent colorimetric determination of the yellow oxidation product. Total bromine was determined, after reduction of Brz, by measurement of the optical absorption of AgBr suspensions, and HBr was obtained by difference. An overall reproducibility of about 0.002 absorbance units (approximately 0.07 pg. per rnl.) was obtained. Some oxidizing or reducing substances may produce interferences, as well as other halogens and S-’.
C.
powders as supports for perfluoro-oils and waxes as recommended by Lysyj and h’ewton (6) produced columns lacking in resolution. The use of conventional bubblers for absorption of HBr and Br2 in water is feasible if the concentration of the resulting solution can be kept low enough so that hydrolysis will help prevent loss of the absorbed compounds. Conventional titration methods (4) would, however, be impractical a t these low concentrations. The sensitivity of colorimetric methods and the availability of a spectrophotometer suggested the approach to be described. EXPERIMENTAL
S
of combustion products of flames burning mixtures of fuel, air or oxygen, and small quantities of brominated compounds required a sensitive method for the determination of 51’2 and HBr. Gas chromatographic methods, which have proved satisfactory for Clz and HC1 in combustion products ( 3 ) , have not been usable for the corresponding bromine compounds. Adsorption with incomplete elution, reaction with the column packing, and poor resolution are characteristic of the columns tried for separating Br2 and HBr from the other combustion products. I n the case of HBr, formation of the mono- and dihydrates add to the difficulties. While dry HBr could be separated with some degree of satisfaction, it was expected that only the hydrates would be found in combustion products containing water vapor. Attempts to dehydrate them invariably resulted in some free Brg. Our lack of success with chromatographic procedures was confirmed by Fish (a). The use of perfluorocarbon molding TUDIES
1690 *
ANALYTICAL CHEMISTRY
Method. It is generally believed t h a t Br2 hydrolyzes, in dilute solution, according t o the equation: Br2
+ HzO ,-
+
HBr
H+
HBrO
it H + +it BrO-
+ Br-
(1) 0.100
0075
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WAVELENGTH, rnp
Figure 1. Absorbance of a silver bromide suspension as a function of wavelength
followed by partial or complete ionization of the respective acids. The analyst is thereby provided with approaches based on the chemical properties of HBr and on the different chemical properties of HBrO. I n the absence of other ionizable bromine-containing compounds, either approach is adequate for the determination of Br2. However, in the presence of such compounds (which would add to the Br- concentration), only the properties of HBrO may be used for the determination of Br2. One may obtain a relationship between analyses based on the two approaches, so that, if additional Brion is present, its concentration may be determined by difference in the results obtained by the two approaches, one based on Br-, the other based on BrO-. The relationship between the two analytical approaches must be determined on solutions containing Br2 alone. However, very dilute Brz solutions of accurately-known concentrations cannot be prepared directly, but require standardization against a solution containing a known concentration of Brion. Potassium bromide was used for this purpose. Total bromine from such a dilute Br2 solution was determined by reduction of HBrO followed by addition of XgNO3-HKO8 solution and turbidimetric measurement of the resulting AgBr suspensions. The Brz was also determined colorimetrically by measurement of the transmittance of the yellow oxidation product of o-tolidine (4,4’diamino-3,3‘ - dimethylbiphenyl). I n acid solution, its absorbance maximum occurs a t a wavelength of 432 mp. A calibration curve for total bromine was produced by using a solution of KBr of k n o m concentration. These data were then used t o relate the two analytical methods and to standardize them. The absorbance of AgBr suspensions is related to both \vavelength and par-
