millicalories per micromole of glucose) listed in column 6 of Table I. We anticipate that the broad methodological principle, exemplified in this paper by the phosphorylative determination of glucose, may be applicable to a wide range of clinical and biological analyses. The salient general feature of the approach is the idea of utilizing a nonspecific property (viz., the heat of reaction) for quantitative measurement, while taking advantage of the inherent specificity of enzymatic or immunologic processes.
LITERATURE CITED (1) (2) (3) (4)
(5) (6) (7) (8)
V. J. Pileggi and C. P. Szustkiewicz, in "Clinical Chemistry, Principles and Technics," 2nd ed., R. J. Henry, D. C. Cannon, and J. W. Winkelman, Ed., Harper and Row, New York, NY, 1974, p 1272. R. G. Martinek, J. Am. Med. Techno/.,31, 530 (1969). R. N. Goldberg. E. J. Prosen, B. R. Staples, R. N. Boyd, and G. T. Armstrong, "NBS Report No. 73-178" (1973), 21 pages. R. N. Goldberg, E. J. Prosen, B. R. Staples, R. N. Boyd, G. T. Armstrong, R. L. Berger and D. S.Young, Anal. Blochem., in press. J. C. Wasilewski, P. T-S, Pei, and J. Jordan, Anal. Cbem., 36, 2131 (1964). J. Jordan and N. D. Jespersen, Colloq. lnt. CNRS, 201 (Thermochimie), 59 (1972). N. D. Jespersen, Thesis, The Pennsylvania State University, 1971. R. A. Darrow and S. P. Colowick, in "Methods in Enzymology", S. P. Colowick and N. 0. Kapian, Ed., Vol. V, Academic Press, New York, NY, 1962, p 233.
(9) A. Huggett and D. A. Nixon, Lancet, 2, 368 (1957). (10) E. Lange, in "The Structure of Electrolytic Solutions", W. J. Hamer, Ed., John Wiley & Sons, New York, NY, 1959, p 135. (1 1) J. J. Christensen and R. M. Izatt, J. Pbys. Cbem., 66, 1030 (1962). (12) R. N. Goldberg, personal communication. (13) K. Burton and H. A. Krebs, Biocbem. J., 54, 94 (1953). (14) R. C. Phillips, P. George, and R. J. Rutman, J. Am. Cbem. Soc., 68, 2631 (1966). (15) R. A . Robinson and R . H. Stokes, "Electrolyte Solutions", 2nd ed., Academic Press, New York, NY, 1959, pp 491-508. (16) G. Ojelund and I. Wadso, Acta Cbem. Scand., 22, 2691 (1968). (17) R. G. Bates and H. B. Hetzer, J. Pbys. Cbem., 65, 667 (1961). (18) S. P. Datta, A. K. Grzybowski, and B. A. Weston, J. Cbem. Soc., 1963, 792. (19) E. R. B. Smith and P. K. Smith, J. Biol. Cbem., 146, 187 (1942). (20) J. J. Christensen, R. M. Izatt, D. P. Wrathail, and L. D. Hansen, J. Cbem. SOC.A 1969, 1212. (21) S.P. Datta, A . K. Grzybowski, and R. G. Bates, J. Phys. Cbem., 68, 275 (1964). (22) E. Raabo and T. C. Terkildsen, Scand. J. Clin. Lab. Invest., 12, 402 (1960). (23) K. M. Dubowski. Clln. Chem., 8, 215 (1962). (24) N. Nelson, J. Biol. Cbem., 153, 375 (1944).
RECEIVEDfor review November 15, 1974. Accepted January 15, 1975. Supported by Research Grant GP-38478X from the National Science Foundation. Presented in part before the Symposium on Enthalpimetric Analysis, First National Meeting, Federation of Analytical Chemistry and Spectroscopy Societies (FACSS), Atlantic City, NJ, November 1974.
Theory of a General Method for Phase Analysis Aleksandar Bezjak Department of Chemistry, Faculty of Pharmacy and Biochemistry, University of Zagreb, Zagreb, Yugoslavia
A new theory of a general method for phase analysis of multiphase systems has been described. The method for the determination of the number, nature, and quantity of phases of the multiphase system, even if the phases are partly known or poorly defined, has been proposed. The method is based on X-ray diffraction patterns or other structural data of several samples containing various quantity of phases of the same multiphase system. The proposed method for the determination of the number of phases has been illustrated, using a three-phase model system.
There are several different techniques today for the direct analysis of multiphase systems, the best information being supplied by X-ray diffraction, infrared absorption, nuclear magnetic resonance, and differential thermal analysis. None of these methods, however, has yet been made applicable for qualitative and quantitative analysis of complicated multiphase systems with unknown or less known compounds and poorly crystallized or nearly amorphous phases. For such systems even the basic information, Le., the number and nature of phases, cannot be determined by the common "finger-print" and calibration procedures. Therefore, a theory with some new relations has been developed to deduce a general method for phase analysis regardless of the degree of knowledge of the system. The theory includes the criterion for the determination of the number of phases in different samples of a given multiphase system. The problem of the number of phases in problematic two-phase systems was first discussed by Teichgraber ( 1 ) . He established the criterion to determine 790
ANALYTICAL CHEMISTRY, VOL. 47, NO. 6 , M A Y 1975
whether or not a system consists of two phases. His theoretical consideration led him t o conclude that for any twophase system the following relation must be satisfied:
where F ( n ) are standardized elements of X-ray diffraction patterns for different samples (i, j , and k ) of the two-phase system under consideration. According to Teichgraber, the best way to define the X-ray diffraction patterns in particular 2 6 intervals is to represent them by Fourier series, so t h a t the elements defining the diffraction curves are the Fourier coefficients. Beside the accurate determination of the number of phases in the multiphase system, the other basic problem is how to obtain the spectral functions of individual phases when these phases are not available in pure state. So far this problem has not been considered theoretically.
THEORY The Criterion Equation for the Determination of the Number of Phases. In a detailed phase analysis of multiphase systems, the experimental curve obtained by applying a physicochemical method of structural investigation can be represented as a sum function composed of curves of each phases i
Such a sum function Q(x) is composed of i individual functions q ( x ) which have to be multiplied by weight fractions
wi of the respective phases. T h e individual functions pi(x) and the sum function @(x)are defined by a row of elements f r ( n )and F ( n ) ,respectively. For every element of the sum function and for every sample of the system, the equation analogous to Equation 2 has to be satisfied, Le.:
+
where K1 # K2 # . . . # Kj,for j = 1 , 2 . . . r 1. In the special case, when K1 = Kz = . . . = Kj = 1, Le.,
for
cuij,i= 1, we have i
Eai = i=l
r
1
(10)
i= 1
..........
(3)
where r = number of phases; j = number of samples; f i b ) = n t h element of the X-ray diffraction pattern, the IR spectra, or NMR spectra of i t h phase; wj,i = weight fraction of i t h phase in j t h sample of the system; F,(n) = n t h element of the X-ray diffraction pattern, the IR spectra, or NMR spectra for j t h sample of the system. By analyzing r + 1 samples with different weight fraction of r individual phases, i.e, the mutual interdependence of r 1 sum functions, it is possible to obtain a n equation which shows whether or not it is a system of r phases. If all r 1 samples contain identical phases, then the determinant Cr+l must be zero:
+
+
Combining Equation 6 and Equation 10, we obtain:
and t h a t for a two-phase system gives
Equation 12 and Equation 1 are identical, Le., the Teichgraber's equation is a special case of criterion equation. Evaluation of Individual P h a s e S p e c t r a from Multip h a s e S p e c t r a . In order to obtain spectral functions of individual phases when the phases are not available in pure state, it has been necessary to start on the problem with the series of Equations 3. If the system of r 1 samples is truly 1 equaan r-phase one, then we must consider r of the r tions. This series of r equations can be written in matrix form:
+
+
...
From Equation 4
and then transformed to: where Dl'sdenote co-factors of Cr+ldeterminant. Then dividing the j t h co-factor by the r 1st co-factor, Equation 5 takes the form:
+
r
a,F,(n) =
(6)
Frtl(d
1.1
where L Y ~ = -Dj/Dr+1. In this form, the equation is the criterion equation and has to be satisfied if the presumed number of phases in the system is correct. The best way to obtain aJ's coefficients is to compute them from experimental data for a large number of elements FJ(n)by the least square method:
a j F l ( l )+ a , F , ( l ) ajF,(2) + a2F,(2)
...
cr,F,(n)
+ .., +
a,.Fr(l) = Fr+l(l) -I- A(1)
-t
arFr(2) = Fr+,(2)
+
A(2)
arFr(n) = F,+,(n)
+
A(n)
+ a,F2(n) +
,
..+
...+
(7) where n >> r, and A(n) = elemental deviations. T h e reliability factor R which includes all deviations A from the criterion caquation shows the accuracy of analysis:
cI
I
Theoretically this factor has to be zero, but it is usually about 0.05 because of experimental errors. T h e criterion Equation 6 is satisfied even if data for sum functions are not normalized, Le., if r
X U . ' ~= ,K~j # i.1
1
(9)
This represents another series of linear equations which serve for the calculation of the individual phase function e ( x ) from sum functions aJ(x)'s (j= 1,r) = u,(x) =
€ll@j(x)
+
€12@*(X)
+
+
€2,@,(X)
.. + ..+
E,r@r(X)
E*l@l(X)
+ .
E2r@r(X)
qr(x) =
Erl@l(X)
+
Er2@2(X)
+ ... +
€m@r(X)
Cnl(X)
. . .
*
(15)
Conversion factors e's are related to the weight fractions, thus: ?L'11
*
. . 11'1,
€11
.
. Eir (161
where Z denotes unit matrix. Consequently, exact e-coefficients can be calculated when weight fractions are known. If weight fractions are not known, e's can be defined with more or less accuracy from special assumptions only. Several methods for the determination of the most probable t-values from series of special conditions will be published separately. Q u a n t i t a t i v e Analysis. Criterion Equation 6 can be transformed into a form convenient for quantitative phaseanalysis. When summary functions F l ( n ) ,F * ( n ) ,. . . , Fr(n) in the determinant Equation 4 and in Equation 6 are substituted by individual phase functions f l ( n ) ,f z ( n ) , . . . , ANALYTICAL CHEMISTRY, VOL. 47, NO. 6, M A Y 1975
791
Table I. Data on the Criterion Analysis for the Determination of the Number of Phases
3
I
Sample
R
a1
a2
e3
1 2 3 124 125 126 134 135 136 145 146 156 234 235 236 245 246 256 345 346 356 456 1234 1 23 5 1 236 1245 1 24 6 1256 1 34 5 1 34 6 1 35 6 1 456 2345 2346 2356 2456 3456
0.15 0.08 0.00 0.18 0.00 0.20 0.15 0.29 0.15 0.12 0.12 0.24 0.21 0.13 0.23 0.12 0.29 0.15 0.10 0.13 0.00 0.00 0.00 0.00 0.00 0.12 0.29 0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.11 0.55 -1.00 -0.24 0.50 -0.30 -0.53 -1.54 -2.04 0.10 1.20 0.33 2.22 1.43 0.20 1.54 2.04 0.42 0.24 0.50 -1.00 -0.90 -1.00 -1.75 0.79 -0.49 -0.91 -0.37 -1.22 2.00 1.05 -0.75 -2.45 1.22
0.91 0.46 2.00 1.26 0.50 1.24 1.50 2.49 3.01 0.91 0.58 -0.16 0.70 -1.19 -0.39 0.81 -0.60 -1.07 0.61 0.79 0.00 2.00 1.05 2.00 1.05 -1.37 1.05 1.12 0.85 1.70 1.00 1.75 0.85 1.70 -0.75
0.50 0.00 0.85 0.00 1.70 1.60 0.38 0.76 0.53 0.53 -2.00 -1.80 0.90 1.75 0.53
62 51 5341 6341
0.00 0.00 0.00
0.00 0.00 0.00
2.00 -1.00 -1.00
-1.00 2.00 2.00
combination
Figure 1. Idealized X-ray
Flgure 2.
diffraction patterns of individual phases
Sum functions belonging to samples 1-6
'"1r i g h t
Sample
10.
t r a c t i a n of p u r phases
1
2
3
4
h
z(l
0.60
zt2
0.30 0.10
0.40 0.20 0.40
0.20 0.60 0.20
0.40 0.45 0.15
0.20 0.10 0.70
K~
6
0.05 0.45 0.50
f r ( n ) ,a-coefficients are identical to weight fractions in the r + 1 sample, Le., CY1
= ?('r+I,1
Or
=
and as for Equations 5 and 6, we obtain.
tcr+i,r
Thus by introducing individual phase functions into Equation 6, we obtain a system of equations first proposed by Copeland and Bragg ( 2 ) for quantitative determination of weight fractions:
fl(l)z~r+l,l + J2(1)2(r+l,2 + . f1(z)2f'r+i,i + . f ~ ( 2 ) 2 ( ' ~ + 1 , 2+ .
...
. f j ( ~ t ) z f ' ~ + 1 , 1+
J2(n)2cr+1,2 f
.. ..
..
f
fr(lhr+l,r
= Fr+i(l)
+ fr(2)z4,.ti,r = Fr+,(21
+ f r ( t ~ l ~ c r + i=, r Frti(t?)
(18) The theory of the general method for phase analysis also interrelates the weight fractions of individual phases from different samples of multiphase system. T h e relation is derived by substituting the last column in the determinant = Equation 4 by any of the weight fraction columns w,,~(-i 1, r 1). This gives another determinant C l r + l , which is also equal to zero,
+
1
C'r+i
792
2L'i,1
=I ' zc'2,1 . . .
zc1,2 w2,2
. . . zrl,r . . , ZC'?,,
0.40
?VI, i
'1'2, i
I = 0 (19)
ANALYTICAL CHEMISTRY, VOL. 47, NO. 6, M A Y 1975
r+l
1wj, i D j = 0
(20)
j- 1
2 j=l
cyjzej,i
= Z(' r + l , i
(21)
The equation shows that a-coefficients of the criterion Equation 6 can be used to determine weight fractions of 1 samples of the rcorresponding individual phase in r phase system.
+
DISCUSSION T o illustrate the course of analysis of a multiphase system, a three-phase model-system was set up consisting of six samples with different weight fractions of individual phases. Individual phase-curves, which are in fact idealized X-ray diffraction patterns of pure phases are shown in Figure 1. Every one of the six sum functions (Figure 2 ) has been represented by 80 amplitudes which were used to calculate sine and cosine coefficients of the respective Fourier series. The Fourier coefficients obtained were substituted into the criterion equation. Assuming a two-phase system, the R-factors and a-coefficients were calculated by the
least square method applying the following series of n equations:
~,A,(iz) + @ f i A , ( n= ) A,(u)
(22)
b u t assuming a three-phase system we applied n equations:
where n is 80, because 40 sine and 40 cosine coefficients were used. I , J,K , and L are indices of samples included in particular combinations. For the two-phase criterion analysis, we considered 20, i.e., all the possible combinations of three from the six different samples, while in case of the three-phase criterion analysis, 15 combinations of four from the six different samples were included. For 18 combinations of the two-phase criterion analysis, R factors were bigger than zero. They varied from 0.08 to 0.29. In every case of the three-phase criterion (except for combinations 1 3 4 5, 1 3 4 6, and 1 2 5 6) R factor was zero, thus confirming t h a t the system was really a three-phase one. T h e special case is represented by those combinations (1 2 5 and 1 3 4) of the two-phase criterion which give R factors zero, in spite of the system being a three-phase one (Table I). Such a paradoxical result arises either when the weight fractions of one phase in all the three considered samples are zero, or when the samples consist of pseudo-phases. Pseudo-phases are mixtures of two individual phases co-existing in the approximately same weight ratio in different samples. For example, in samples 1, 2, and 5 the weight ratio of phase 1 and phase 2 is constant and equal to 2:l. Such a mixture of two phases with constant weight ratio has a diffraction pattern which figures as an individual phase function for all the three samples mentioned. Therefore, the system including samples 1, 2, and 5 seems to be a two-phase one, Le., to consist of a pseudo-phase X and of the individual phase 3. An analogous situation is obtained when combining samples 1, 3, and 4, but in this case we have pseudophase Y containing pure phases 2 and 3 of weight ratio 3:l. T h e presence of pseudo-phase Y and pure phase 1 in samples 1, 3, and 4 results in R = 0 in the two-phase criterion analysis, although the system is in fact a three-phase one. Such a n incorrect conclusion is deduced only when all the samples included contain the same pseudo-phase. Pseudo-
phases can be detected by comparing the R values for various Combinations. For example, if samples 1, 2, and 5 and 1,3, and 4 were to contain two phases only, the system consisting of samples 2, 3, and 5 would have been a two-phase one as well. However, the R value for the 2 3 5 combination was 0.24 showing that the system of the mentioned samples was in fact a pseudo-two-phase one. For the three-phase criterion including combination of samples 1 3 4 5, 1 3 4 6, and 1 2 5 6, and R-factors were higher than zero, because samples 1 3 4 and 1 2 5 represent pseudo-two-phase systems, which in combination with the other samples cause the appearance of the pseudo-four-phase systems. Thus the system of samples 1 2 5 6 seems to be a four-phase one, because it includes pseudo-phase X and pure phases 1 2 and 3. The two groups of samples 1 3 4 5 and 1 3 4 6 with the pseudo-phase Y and pure phases 1 2 and 3 represent another pseudo-four-phase system. After the permutation which replaces sample 5 by sample 1 or sample 6 by sample 1 in Equation 23, c q = 0 was obtained by the least square method. R = 0 and CY^ = 0 for the permutations mentioned, further confirm that samples 3 4 l represent a pseudo-twophase system. Therefore replacement of the constant A L ( ~by) term A r ( n ) , A J ( ~or) A K ( ~is)recommended to avoid incorrect conclusion on the number of phases when pseudo-phases are present. The problem of pseudo-phases is of special importance in the study of some reaction mechanisms, where reaction products form simultaneously, a t constant weight ratio. The theory of the criterion for the determination of the number of phases has already been applied successfully in the analysis of the calcium silicate hydrate system ( 3 ) ,portland cement clinker ( 4 ) ,and polyethylene-styrene co-polymer system ( 5 ) .
LITERATURE CITED (1) M. Teichgraber, Faserforsch. Textiltech., 18, 359 (1967). (2) L. E. Copeiand and R. M. Bragg, Anal. Chem., 30, 196 (1956). (3) A. Bezjak and I. Jelenic, 2nd International Symposium on Autoclaved Calcium Silicate Building Materials, Hannover, 1969. (4) A . Bezjak and I. Jelenic, Proceedings of the VI International Congress on the Chemistry of Cement, Moscow. 1974, in press. ( 5 ) A. Bezjak, 2 . Veksli, V. Tomaskovic, M. Bari, J. Dobo, and I. Dvornik, "Proceedings of the Third Tihany Symposium on Radiation Chemistry, Tihany, Hungary," 1971, p 1013.
RECEIVEDfor review June 3, 1974. Accepted January 6, 1975.
Titration Errors Inherent in Using Gran Plots Stanley
L. Burden and David E.
Euler'
Chemistry Department, Taylor University, Upland, IN 46989
Titration end points from a set of strong acid-strong base titrations were obtained with Gran plots and compared to visual Indicator end points. The effects on titration error and end-point uncertainty which arise from measurement precision and the location on the titration curve of points used in making the Gran plots are discussed. Titration errors and end-point uncertainties comparable to visual indicator end points were obtained only by wisely selecting the points used In the analysis, carefully fitting the Gran plots to the points, and giving close attention to the precision of the measurements. Present address; D e p a r t m e n t of Physiology, S t r i t c h School of M e d i c i n e , L o y o l a U n i v e r s i t y , Chicago, IL.
In the past several years, interest has developed in using Gran plots to determine titration end points. Although this technique has been mentioned frequently in the literature and it is now beginning to appear in textbooks, little attention has been given to the titration errors which result from using it. McCallum and Midgley ( I ) have discussed errors which origidate in the chemistry of the particular titration reaction used and which cause non-linearities in Gran plots. For example, they describe a method to correct for the effects of the solubility of a precipitate in a precipitation titration, the autoprotolysis of water in an acid-base titration, and the effect of variation in activity coefficients. T h e authors make no statement, however, regarding the magniANALYTICAL CHEMISTRY, VOL. 4 7 , NO. 6, MAY 1975
793
