Enzymatic Enthalpimetry, A New Approach to Clinical Analysis: Glucose Determination by Hexokinase Catalyzed Phosphorylation C. David McGlothlin and Joseph Jordan’ Department of Chemistry, The Pennsylvania State University, 752 Davey Laboratory, University Park, PA 76802
Direct hjection Enthalpimetry (DIE) has been adapted to enzyme catalyzed determinations. Advantage was taken of the specificity inherent in enzymatic processes. Using thermistor instrumented adiabatic calorimetry, temperature increments on the order of O . O 0 l o were measured with a precision and accuracy of 2 % . Capabilities of enzymatic enthalpimetry are documented by the quantitative analysis of glucose. Five-hundred-microliter samples, containing glucose in a range between 0.5 and 50 pmol, were phosphorylated in the presence of hexokinase under judiciously optimized experimental conditions including invariant pH and excess Mg-ATP reagent. Empirical calibrations were obviated by normalization in terms of a “conditional AH” parameter which represents the appropriate thermochemical summation of the phosphorylation process proper plus ancillary proton transfer. The methodological principles outlined in this paper are expected to have broad applicability in clinical and biological analysis.
where the reduced coenzyme NADPH is eventually measured by fluorescence or absorption spectrometry. Obviously, it would be preferable to make direct use of a primary enzymatic reaction (i.e., a process in which glucose itself is a reactant) and still employ a n instrumental method t o parametrize results. Because of the highly selective nature of enzyme catalysis, even a property as universal as the heat of reaction is appropriate. This has recently been demonstrated (3, 4 ) by the successful determination of glucose via heat conduction (Tian-Calvet) calorimetry, utilizing the heat evolved in the phosphorylation of glucose under specified experimental conditions. Glucose analysis by Direct Injection Enthalpimetry (DIE) via Reaction 2 is reported in this paper. Results are presented and discussed which indicate that DIE may be ideally suited for rapid and convenient determinations in clinical practice. A singularly attractive feature is that findings can be related to a thermodynamic parameter, viz., an appropriate AH assignment.
The determination of glucose is one of the most routinely performed of clinical analyses. As such, numerous procedures have, from time t o time, found their way into clinical laboratories. Henry ( 1 ) and Martinek (2) have reviewed the various methodologies for glucose, which include oxidation by copper(I1) or ferricyanide, condensation with anthrone or o-toluidine, and, more recently, enzymatic procedures. The enzymatic approach has unmatched advantages of specificity. However, its conventional implementation suffers generally from the drawback of relying on secondary reactions in the determinative step which entails pitfalls of error propagation inherent in indirect analysis. Thus, the usual determination of glucose with the aid of glucose oxidase
DIE was developed in the mid-sixties by Wasilewski e t al. ( 5 ) ,who used it to monitor heats evolved or absorbed in reactions which had very fast kinetics. The crux of the method was a quasi “instantaneous thermometric titration”: a “temperature pulse”, AT, in a n adiabatic cell was recorded upon injection of a reagent and rapid mixing. T h e reagent was added in sufficient excess t o drive the relevant equilibrium thermodynamically and kinetically to virtual completion, which was attained in 0.01 second. The “working equation” for DIE is an expression of the form:
METHODOLOGICAL PRINCIPLES
C,H,,O,
+
O2
+
glucose
H,O
H,O,
+
C6Hj2O7 (1)
oxidase
(P-D-glucose)
(gluconic acid)
depends on the absorptiometric (colorimetric) analysis of the product H202 by coupling with an oxygen acceptor dye which requires the presence of a second enzyme, viz., peroxidase. An alternative method of glucose analysis takes advantage of a phosphorylation reaction, e.g.,
hex0 kinase
glucose + Mg(ATP)’glucose-6-phosphate2~ + Mg(ADP)-
+
H’
(2)
Again the conventional determinative step requires another enzymatic process, viz.: glucose-6-phosphate
+
~Iu-6-P
NADP
dehydro g e n a e
6-phosphoglucono-5-lactone
+
NADPH
+
H’
(3)
To whom correspondence should be addressed. 786
ANALYTICAL CHEMISTRY, VOL. 47, NO. 6, MAY 1975
Q AT=-=k
-n*AH
k
where k and Q denote heat capacity (maintained invariant by adding always the same amount of reagent solution t o a given volume of “sample”) and total heat evolved, respectively. Either n (number of moles) or AH (effective heat of the reaction) can be evaluated from Equation 4, provided the other is known. The applicability of Equation 4 is contingent on negligible heats of mixing (Le., sample and reagent solutions must be isothermal) and negligible heats of dilution. T h e classical DIE procedure was appropriately modified for the determination of glucose by enzymatic enthalpimetry. The main change consisted of interchanging reagent and sample solutions: discrete glucose samples were injected sequentially into the adiabatic cell which contained a large excess of the reagent Mg(ATP)2-, a buffer of p H 8, and the enzyme hexokinase. Under these experimental conditions, the phosphorylation of glucose (Reaction 2) proceeded to virtual completion in a finite time period which ranged from 30 to 300 seconds. The progress of the reaction was recorded vs. time by monitoring A T as the unbalance potential of a sensitive thermistor bridge. A “kinetic DIE curve” of this type is illustrated in Figure 1, where the fol-
//
WheatstoM\ Bridge
fl n P;
to calibration heater circuitry
/injection
port
E post -reaction domain
I
0.01 caI
T i m , t (seconds)
to
Figure 1. Typical injection enthalpogram for glucose phosphorylation.
“marker spike” identifying beginning of reaction period; C“ heat evolved in phosphorylation
a
0,total
lowing features are apparent: a) An isothermal base line, corresponding to the initial temperature of the buffered Mg(ATP)2--enzyme mixture in the adiabatic cell; b) A rapid temperature drop, AB, following injection of the glucose sample which was advisedly cooler, to serve -as a convenient “marker spike”; c) A “reaction curve”, CD, reflecting heat evolved by the phosphorylation of glucose which followed a first order rate law; d ) A “post-reaction’’ temperature-time t r a c e , m . As can be seen in Figure 1, selfexplanatory extrapolations to C‘ and D’ readily yielded the quantity Q = h e AT which is required for evaluating n or A H from Equation 8 2 .
t-
3.3cm. -4
Figure 2. Adiabatic cell
Procedure. Seven (8) ml of phosphorylation reagent were placed in the dewar, which was immersed in a water bath maintained at 25.00 f 0.01 OC. Attainment of thermal equilibrium in the adiabatic cell yielded an isothermal base line (see Figure 1): this was facilitated by starting out with a slightly colder phosphorylation reagent and adjusting its temperature by in situ joule heating in the dewm. Five-hundred-microliter “samples” (containing various amounts of glucose, in a range of concentrations between 30 and 1000 mgidl) were then injected and corresponding “reaction curves” recorded. The recorder chart ordinate was calibrated calorimetrically by joule heating subsequent to each injection using the correlation:
EXPERIMENTAL Chemicals. Reagent or primary standard grade chemicals and doubly distilled water were used throughout. Calibration standards for glucose were prepared from 99+% 8-D-glucose or dextrose. Hexokinase, obtained as a suspension in ammonium sulfate (Type C-130, Sigma Chemical Co., St. Louis, MO) had a quoted activity of 190 Units/mg protein (1 Unit corresponds to the amount of enzyme required to catalyze the conversion of 1 pmol of substrate per minute) and was essentially free of other enzyme activity. The adenosine 5’-triphosphate (ATP), disodium salt, was crystalline and 99% in purity. The phosphorylation reagent used in most of the work was prepared by dissolving one gram of ATP and one gram of MgC126H20 in a small volume of 0.5M Tris[tris(hydroxymethyl)aminomethane]/Tris-HC1 buffer, p H 8.0. After addition of 300 Units of hexokinase, the pH was readjusted to 8.0 with 1M sodium hydroxide and the solution diluted to ’70 ml with buffer. This quantity was sufficient for fifty sequential determinations. Preparation of phosphorylation reagents with other buffers (in lieu of Tris) was effected by a similar procedure. Apparatus. The instrumentation for monitoring temperature has been described previously (6, 7 ) . T h e temperature sensor was a thermistor which had the following characteristics: resistance a t 25 oC-15,000 ohms; temperature coefficient in the 24-26 “C interval-0.039 ohmlohmPC. The thermistor was wired in a dc Wheatstone bridge, whose unbalance potential provided a linear measure of temperature which was recorded vs. time as the ordinate deflection of a strip chart millivoltmeter. An approximately 100-ohm resistor (in series with a lOO-ohm, 1%standard resistor) was used as a “calibration heater” to convert AT(OC) into Q(ca1oriesj under conditions of constant heat capacity ( k = constant; see Equation 4). The adiabatic cell consisted of a custom-made dewar (supplied by KonteslMartin, Evanston, IL) shown in Figure 2. Glucose sample solutions were injected with the aid of precision syringes (Hamilton Co., Reno, NV) via capillary stainless steel needles or Teflon tubing.
(5) where Q/l is the calorie-equivalent per unit ordinate deflection; Y is the total ordinate displacement during t seconds; lE and R denote, respectively, the relevant potential drop and the resistance at constant current (i.e., lE = iR);and the subscripts H and S identify the calibration heater and the standard resistor. This technique obviated any explicit use of T and k in Equation 4 , yielding assignments of the quantity nAH directly from Q. Thermodynamic Assignments. Where temperatures are consequential (especially for equilibrium constants and Gibbs Free Energies), all values reported in this paper imply a temperature of 298 K (25 “C).
RESULTS Reaction mixtures used in this investigation invariably contained an amine buffer of pH 8 because, a t lower pH, equilibrium 2 was appreciably shifted to the left. Under these circumstances, a realistic formulation of the relevant reaction would be: glucose Mg(ATP)2OH- = glucose-6-phosphate2- Mg(ADP)- + H20, which is an alternate version of Equation 2. For maintenance of pH, the phosphorylation proper was necessarily coupled with the process: Buffer-NHZ H20 = Buffer-NH3+ OH- which is equivalent to Reaction 6, viz.,
+
+
+
+
-
+
Buffer -NH, + H’ Buffer -NH,’ (6) The actual heat of reaction operative in Equation 4 corresponded to the summation: A H = AHz i- AH6
(7)
Extensive experimental findings obtained using Tris as the buffer system are summarized in Table I. The invariance of ANALYTICAL CHEMISTRY, VOL. 47, NO. 6, MAY 1975
787
Table I. Verification of the Relationshipa:
Q = - n l H . Phosphorylation of Glucose in Tris Buffer
Glucose sampleb Precision Concentration, mgldld
30 50 100 150 200 2 50 300 3 50
0.833 1.39 2.78 4.17 5.56 6.95 8.33 9.72
11.1
500
13.9 20.8 27.8
AH,
calories
mol x
400 750 1000
Heat evolved,
Total amount, n,
s
0.0150 0.0243 0.0493 0.0755 0.0981 0.1257 0.1488 0.1749 0.1979 0.2533 0.3702 0.5004
143
kcallmol'
RSD,%
0.0002 0.0002 0.0006 0.0003 0.0005 0.001 0.0007 0.0008 0.002 0.002 0.001 0.002
13
-18.0 -17.5 -17.7 -18.1 -17.6 -18.1 -17.8 -18.0 -17.8 -18.2 -17.8 -18.0
*1 i4 *0.9
*l *2 *2 i1
13 13 i0.8
i1
Average 12 -17.9 Results obtained as in Figure 1, based on Equation 4 . Volume = 500 p l . Total A H for Reactions 2 and 6 in 0.5M Tris buffer, pH 8.0. To facilitate clinical comparison, concentrations have been expressed in this column in units of mg/dl which is common usage in relevant literature. a
~.
- ___
. ..
.-
. .
AH in column 6 substantiates the applicability of Equation 4. I t is apparent that glucose can be analyzed enthalpimetrically with a precision of 2% if the AH assignment of -17.9 kcal/mol is substituted into that expression. The virtual completeness of phosphorylation was painstakingly verified by allowing the reaction to proceed for fifteen minutes and quenching. This was accomplished by removal of the hexokinase via either adsorption on bentonite (8) or precipitation with barium hydroxide and zinc sulfate. Residual glucose was tested for with the aid of Reaction 1, which is known to be highly specific for the free glucose molecule and not to proceed with glucose-6-phosphate (9). No measurable amount of unreacted glucose was detected. Selected enthalpimetric studies, similar to those reported in Table I in the presence of Tris, were also performed by carrying out the phosphorylation of glucose in the presence of other amine buffers. These yielded corresponding experimental estimates for the overall operative reaction heat which depended on the buffer. A synopsis of the data is presented in Table 11. It is apparent from column 4 that subtraction of the appropriate AH6 values (which accounted for the contribution of buffer protonation in each instance in accordance with Equation 6) yielded a remarkably invariant assignment of AH, = -6.6kcal/mol
(8)
for the phosphorylation step per se (Reaction 2). Based on evident considerations of equilibrium thermodynamics and enzyme kinetics, a series of ad-hoc experiments were carried out for elucidating optimum conditions for the quantitative analysis of glucose by DIE via Reaction 2. The following requirements transpired. a) The pH should be maintained between limits of 7.6 and 8.5 where the effective A H (as defined in Equation 7) was found to be invariant in any given buffer system within experimental uncertainty. (In acid solution, Reaction 2 may not proceed to completion.) b) Buffer capacity should be adequate for maintaining the requisite pH invariance while hydrogen ions are produced by the phosphorylation process. In this context, a Tris concentration of 0.5M was found to be ample in all situations. Several reference sera were assayed accordingly by DIE 788
ANALYTICAL CHEMISTRY, VOL. 47, NO. 6, MAY 1975
..
-...__
using a phosphorylation reagent buffered at pH 8.0 with 0.5M Tris. Results were compared with three conventional glucose determinations, including condensation with 0-toluidine, oxidation-reduction with Cu(I1) and arsenomolybdate and an enzymatic glucose oxidase procedure. Results are listed in Table 111. Accuracy was computed with reference to the values obtained by the glucose oxidase method which is recognized as the most reliable. I t is apparent that the accuracy of the enthalpimetric determination of glucose (typical error, 2%) was comparable with its precision (see Table I), viz., f 2 % on the average. Errors obtained in the classical 0 - toluidine and arsenomolybdate analyses were of the same order of magnitude. The enthalpimetric procedure had the great advantage that protein exclusion was unnecessary; protein must be removed in the other methods because of interferences in spectrophotometric measurements in the determinative steps.
DISCUSSION As far as the relevant thermochemistry is concerned, the work described in this paper has yielded the first reported assignment for the heat of phosphorylation of glucose via Reaction 2, Le., with Mg(ATP)2- as the phosphorylating agent, viz. -6.6 kcal per mol (see Equations 2 and 8 above) with a precision of f 2 % . The prevailing conditions were: ionic strength, 0.3; 8 X 10-5M < [glucose] < 3 X 10-3M; [Mg(ATP)2-] [Mg(ADP)-] = 0.05M; pH 8. It transpires from the pertinent literature (IO) that overall corrections for heats of dilution (to concentrations and ionic strength = zero) are within the limits of our experimental uncertainty. Consequently, the appropriate thermodynamic assignment is
+
AH,. = (-6.6
*
0 . l ) k c a l p e r mol
(9)
where the standard states are 1 molal (which is indistinguishable from 1 molar within the applicable confidence interval) ideal aqueous solutions with respect to all reactants and products (including pH 0), and the reference state (activity coefficients = 1)is infinite dilution. In the context of Assignment 9, it should be noted that it derives from experimental measurements of actual reaction heats corresponding to the sum specified in Equation 7 and listed in Table 11. A related value for the conditional heat
Table 111. Comparison of the Enthalpimetric Method with Various Conventional Procedures for the Determination of Glucose in S e r u m Samplesa
Table 11. Thermochemistry of the Glucose Phosphorylation Reaction in the Presence of Various Buffers of pH 8.0 (1)
(qa*d AH = AH2 + A H 6
Buffer system
tris(Hydroxymethy1)aminomethane, pK 8.1 Glycylglycine, pK 8.25 Glycylglycylglycine, pK 7.84 N,N-bis(2-hydroxyethyl)glycine, pK 8.33
-17.9
-17.1
($Sd
(4iId
AH6
AH2
-11.3
-6.6
Reference
-15.8
-10.6 - 9
-6.5 -6.8
-12.9
- 6.3
-6.6
Average -6.6 Determined in present study, see Reactions 2 and 6 identified by subscripts; values listed represent averages of five replicates. Assignments taken from the literature (16-21). Calculated from columns 2 and 3 via Equation 7 . Heats of reaction are expressed in kilocalories/mol . of phosphorylation of glucose, in the presence of Tris buffer at pH I7.2, has been reported by a group at the National Bureau of Standards in an earlier pioneering investigation (Goldberg et al., reference 3, 4 ) . In that study, experimental conditions were at variance with ours, viz., the solution was more acidic and magnesium was added in substoichiometric amounts with respect to ATP (whereas our conditions were judiciously selected to bind virtually all ATP and ADP as the magnesium complexes specified in Equation 2). Relevant thermodynamic data available in the literature ( 1 1 ) reveal that under Goldberg’s conditions (3, 4 ) several ATP and ADP species in addition to those shown in Reaction 2 were involved. Taking this situation into account, Goldberg (12) has recently computed from his measured heats an assignment for the process: glucose
+
ATP4’ == glucose-6-phosphate2-
+
ADP3‘
+
H’
(10)
and obtained ,lHolo= -5.7 kcal/mol. Reaction 2, whose heat has been measured in the present investigation (see Equation 9), represents the obvious thermodynamic summation of Reactions 10, 11,and 12. Mg(ATP)’- = Mg2’ + A T P 4 (11) Mg2’ t ADP3- = Mg(ADP)(12) Using known heat assignments for Reactions 11 and 12, Goldberg (12) has performed an alternative calculation of AHo10, based on the independent experimental measurements reported in this paper. The two calculations were in excellent agreement. This consistency is gratifying. However, as far as analytical applications are concerned, Reaction 10 is impractically slow in the absence of magnesium. This was indeed our rationale for relying on Reaction 2 rather than Reaction 10. Our experimental conditions were carefully selected to have Reaction 2 prevail to the virtual exclusion of all competing equilibria. In contradistinction Goldberg et al. (3, 4 ) , had a “mixture of reactions” occurring simultaneously. The equilibrium constant and Gibbs Free Energy of Reaction 2: K2 = 4.0
X
Glucose found byb
1 0 - 5 , A G 2 ” = + 6 . 0 kcal/mol
(13)
was estimated from appropriate thermodynamic parameters of Reactions 10, 11, and 12 which are available in the literature (13, 1 4 ) , viz. AGO10 = + 3.9 kcal/mol, Klo = 1.4 X K11 = 1.5 X and K l z = 1.9 X lo4.
’
( ’ ‘true’ )
glucose
Condensation
concen-
with
trationbSC
66 142 292 71 199
300
o-toluidined
66 13 8 277 72 198 290
Redox with
Cu(1I) and
This work
arsenomolybdate e
Concentration
62
68 144 296 68 198 305
139 3 04
70 193 316
h
r
Oaf
+3
+1 +1 -4 -0.5
+2 Standard reference sera (so-called “Calibrate” and “Versatol” series) supplied by Warner-Lambert Co., Morris Plains, N J . All concentrations are expressed in mg/dl, see footnote d to Table I. Determined by the glucose oxidase method of Raabo and Terkild~ e n ( 2 2 1Method .~ of D ~ b o w s k i ( 2 3 )Somogyi-h’elson .~ method(24). Accuracy with reference t o column 1. a
f
--
Combination of Equations 9 and 13 yields: AS,“ =
-
298
A G 2 ” = -42 c a l o r i e s / m o l - d e g r e e
(14) Assignment 13 makes it evident that Reaction 2 proceeded to satisfactory completion under the conditions selected in this study for the determination of glucose. Indeed, using reasonable estimates of pertinent activity coefficients ( 1 5 ) ,Equation 13 yields the following conditional equilibrium constant for pH 8 and ionic strength of 0.3: K2’ = 0.6 X 108K2 = [ glucose-6-phosphate2-][ Mg(ADP1-1 = [glucose][ Mg (A T P )’ -1
.4
103
(15) This corresponds to 99.5+% phosphorylation of the glucose present, taking into account the amount of MgATP added in all our experiments (at least a tenfold stoichiometric excess). Even though the conditional equilibrium constant of Reaction 10 exceeds that of Reaction 2 by a factor of thirty-five, a further gain in the completeness of the phosphorylation would evidently be inconsequential from an analytical point of view. On the other hand, the faster kinetics of Reaction 2 represent a significant asset, enabling the rapid determination of a wide range of glucose concentrations by direct injection enthalpimetry. A pH 1 7.6 is crucial for driving Reaction 2 to completion as written. In this context, the procedural detail of readjusting the pH (by adding sodium hydroxide after addition of ATP, see experimental section above) is not trivial; if this were omitted, the pH might drop to “dangerous” levels, buffer notwithstanding. From the viewpoint of clinical practice, the glucose analysis method described in this paper has several attractive features. Major advantages are outlined below. 1) A “single-step procedure” is feasible. Presence of protein and hemolysis do not interfere. Laborious and timeconsuming pretreatments (which “haunt” other methods) are consequently obviated. 2) Empirical standardization is unnecessary. Instead, results may be related directly to a fundamental property, viz., the appropriate “operational AH assignment” (-17.9 ANALYTICAL CHEMISTRY, VOL. 47, NO. 6, MAY 1975
789
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 , MAY 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 that 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
