be without effect in the 5 . 1 X phosphoric acid system (Figure 1). The concentration of phosphoric acid in the system has EL very pronounced effect on the separation of iron(II1) from phosphoric arid. For example, at 5M HCl, with 12.1M &Pod, 99% of the iron and 2170 of the phosphate are extracted while with 5 . l M H3P04, 89% of the iron and less than 3% of the phosphate are removed. Therefore, the most efficient separations can be made from the most dilute phosphoric acid solutions and quantitative separations of ferric ion from phosphoric acid
LITERATURE CITED
Table 11. Molar Concentration of HCI Necessary to Initiate Extraction of Phosphate
[HsP041 Moles/liter
(1) Ihdson, R. W., Forney, G. J., Swift, E. H., J . Am. ('hem. SOC. 5 8 , 2573 (1936). (2) Gee, A . , Deitz, V . R., A s . ~ I , .CHEM. 25, 1320 (1953).
[HC!I Moles/liter 3.5
5.1
7.7 8.7
(3) Laurene, A. H., Campbell, I>. E., Wiberley, S. E., Clark, H . M., J . Phys. Chem. 6 0 , 901 (1956).
3.0 1.1
12.1
(4)Nachtreib, S . H., Conway, J. G., J . Am. Chem. SOC.70, 3547 (1948). ( 5 ) Sachtreib, N. H., Fryxell, R. E., Ibid., p . 3552. (6) Saldick, J., J . Phys. Chem. 6 0 , 500
0
(1956).
XoRroN H.~BERMAN
can be achieved with only a few extractions when the more dilute phosphoric acid solutions are used.
W. R. Grace & Co., Washington ~~~~~~~hCenter Clarkesville, Md.
Compositional Analysis of n Component Systems by X-Ray Absorption Method
SIR: Previous workers (1, 3) have used various absorption techniques such as calibration curves of some material immersed in a coniplex or absorption edge spectrometry with certain assumptions about the complex which contains t,he unknown. These methods are perfectly valid pro.vided that one unknown constituent and one complex are used. When large numbers of different elements in different complexes are studied, many calibration curves or certain constants must be obtained. The method proposed in this paper relies only on the linear absorption at n different wavelengtlis for an n component system. The wavelengths can be chosen completely a t random; with knowledge of the elements present, however, more accurate results can be obtained by a judicious choice of the n wavelengths. .4 simple method, into which matrix effects do not enter, for the analysis of foils and liquid materials has been t,he goal of laboratories wishing to obviate the need for calibrat.:ion curves generally used in fluorescence analysis. Since the mechanism of absorption involves either the absorption or the nonabsorption of a photon, freedom from the effect of other materials present in the unknown is achieved. Consider a material which consists of constituent .4, 30%; constituent jg, 20%; and const,ituent C, 50%, i:n which the three materials, A , B, and C, are thoroughly mixed in the unknown. The method of fluorescence analysis would necessarily derive calibrat,ion curves for each of the constituents immerscd in the other two, while on the other hand, in an absorption experiment' an x-ray photon would be either absorbed or not absorbed by
one of the three constituents. If it is absorbed by the unknown, then this would decrease the obseived count by one. Statistically speaking, if a large number of photons were to impinge on plzP1
+ PZZPZ+
p32P3
pllP1
+
+
p21P2
+...+
p31P3
pnlPn-
1 Io In - = 0 at X1 d 11 -
and similarly at x2 to A,,
+ . . . + pnzP,- d-1 In Io1 2 -
=
0 a t Xz
(3)
the sample and only 50YG of the beam was detected in the counter, then the 50% that was absorbed would divide in c , PA, , p the ratio ~ ~ A : ~ , u B : ~ Mwhere and pc are the linear absorption coefficients of A , B. and C, respectively. The absorption equation for a simple system of n components is
Pl+PZ+P3+ ~
Therefore, the algebra is
. . . +P n =1
solution
by
matrix
PI = Ai("+ I ) Pz = Azc" + 1)
(4)
Il
=
I o e-WmPn)d
(1)
where
ll lo
x-ray intensity a t detector; intensity of unabsorbed x-ray beam; p,, = linear absorption coefficients of nth component; P , = fractional parts of nth component where 2: P,, = 1 ; d = thickness of material. Rewriting Equation 1 for two components we would have pJ'1
= =
+
pz1P2
- -1 In 1-0 d
= 0
11
at
(2)
Extending t,his equation to n component.s gives
or, more generally,
P,
=
A m ( n + l )( V I
=
1,2,3 .
. . n)
where the,l,, are the elements of the last row of the inverse matrix of the coefficients. If the quantity 1 ' d is kno\\-n,then the number of measurements is reduced by one and Equation 3 takes a >lightly different form; the nth equation is deleted from the set VOL. 36, NO. 9, AUGUST 1964
1877
Table I.
Sn
Ag (looqp p=105 P/Pl.OW
86.0 627.8
146 1533 174 1827 217 2278,5
176 1284 8 209 1525.7 247 1803.1
/I
P/Pl
283
J ! P/P1.389
J !
P/ !J
(100cic)a p = 7 3
73.0 766 5
a
Table II. Experimental Absorption Data Cu, Zinc, 100yo 100% Yellow brass
Theoretical Solution of Silver Solder by Absorption Method
Cu
(lOOYc); p = 8 3 130 1105 260 2210 38.5 329.25 50.9 432.65
Zn (100%)"
Solder
p = 7 1
145 1029 5
In
39 276 9
In
I
7.7419
I 1
5 I2
14 5313
I
pllpl
+ p 2 1 p 2 + ' . ' + pnlPn
d ' 1
iLl(n-1)
+
P1
+
...
P1 =
P~ =
! J d n
=
Io
I, io -'
2 In I, '
+
kz(n-lP2
2-1 In 1
. . $-
!J22p2
=
(6)
IO + pn(n-l)P, = d1 In 1,-i -
2 (! "> 2 (A 5)
j=1
j=1
Ai,
d
ln
I,
il2, d In I ,
(m = 1, 2 , 3 . . . n) (8) where .Imn are t,he elements of the inverse matrix of the coefficients. .is can be seen from Equation 3, if n measurements are imide, the necessary computations are only in the deter1) column of the mination of the ( n inversc of an ( n 1) x in 1) matrix. Such a coniputation requires the evalua1 determinant:: of order n. tion of n If the thickne$s is known (Equation 6 ) , then n - 1 mcasurements are required, but the computational incrrase is the evaluation of n* detrrminants of order
+
+
+
1878
0
627 1284 1525 1803 1
8 8
7 1
1105 2210 329 25 432 65 1
1029 276 320 416 1
The inverse of this matrix is very tedious; therefore, a matrix inversion routine for the Royal hlcBee LGP-30 digital computer was used. The results of the inversion are
or, more generally,
+
n - 1. When n is small ( n 5 4), neither of the two is restrictive as far as comput'ation is concerned. For large n(n 2 5)) the method ending in Equation 5 is preferred and will yield a check on the t,hickness of the material. .is a n example of a complicated system, if one has a foil 0.01 cm. thick of a silver solder which ha. the composition h g (40961, Sn (4073, Cu (14961, and Zn (6y0), t,hen Table I is formed.
766 5 1533 1827 2278 5 1
ANALYTICAL CHEMISTRY
PA,
=
Ps,
=
Pcu =
Pen= d
=
39.22% 40.09% 13.89% 5.90% 0.01009 cm.
The discrepancy is caused by the computer program round off error. This shows that a four component system can be solved with measurements a t four different wavelengths, the four \vavclengths having been chosen at random. To demonstrate the feasibility of the method, a yellow bra+ foil (6774 copper,
405
380
In
9 I1
1.00
PWL.BI
2250
260
In L!
4 00
1,
3397, zinc, as determined by electron microprobe), 0 001 in. (0.00254 cm.) thick was used. Measurements of the linear absorption coefficient of pure copper and zinc foils were made a t t n o x-ray wavelengths ( T V L ~ ]and TVLpl). The yellon brass sample was also measured a t these wavelengths and Inlo/Il determined. Measurements 13 ere made a t least lo7 counts on each foil a t each wavelength. Table I1 gives the experimentally measured values. Setting u p the matrix equation as in Equation 3 n e obtain the solution Pa = 66 6% PB = 33 3% 1 - = 397 d
In 2 14.0645 13 I Pl.bZ9 a 58'6 In 2 17.187 416.06 14 Values obtained from Handbook of Chemistry and Physics, 41st Ed., p. 2664. 45'2 320.92
PwLal
5 9 92 06
-7 -14 -14 -17
7419 5313 0645 1S17 I 1
I
1
The statistical error does not alter the results significantly because of the large number of counts taken. Measurement of the thickness of the copper and zinc standard foils seems to be the main factor in the observed deviation. ACKNOWLEDGMENT
The author thanks Fred Plock for performing the x-ray intensity measurements used in the experimental portion of this paper. LITERATURE CITED
(1) Barieau, R. E., Norelco Reporter V, 5-6, 101-4, (1958). ( 2 ) Crout, P. D., A I E E Trans. 60, 1235 (1941). ( 3 ) Edeston, B. H., Whisman, M. L., -Torelco Reporter V, 2, 49-57 (1958).
ROBERT LEFKER
U. S. Army Electronics Research and
Development Laboratory Fort lIonmouth, 1;.J.
