1922
Anal. Chem. 1982, 5 4 , 1922-1926
Quantitative Gas Chromatography/Fourier Transform Infrared Spectrometry with Integrated Gram-Schmidt Reconstruction Intensities D. 1.Sparks, R. B. Lam,' and 1. L. Isenhour* Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27514
An Interferogram-based callbratlon curve approach to quantltatlve GCIFTIR using Integrated Gram-Schmidt reconstructlon lntensltles Is presented. The expected quantltatlve response Is derived from theory and supported wlth emplrlcal evidence. Accurate on-the-fly quantltatlon Is demonstrated over a concentration range correspondlng to absolute analyte amounts ranging from 1 pg to greater than 100 pg using pentyl propionate as the analyte.
The combination of the separation ability of a gas chromatograph with the identification ability of an infrared spectrometer (GCIR) became practical with the introduction of rapid scanning Fourier transform infrared (FTIR) spectrometers in the 1960s. Subsequent development in the early 1970s has made GC/FTIR useful for a variety of qualitative applications which have been reviewed by Erikson ( I ) . The quantitative capabilities of the technique, however, have been little explored to date. The myriad difficulties associated with any attempt at absolute quantitation using FTIR have been outlined by Hirschfeld (2). Anderson and Julian have proposed using library absorptivity values for rough (=t25%) absolute quantitation of GC/FTIR data (3). While the dramatic improvement in spectral accuracy obtainable with Fourier transform instruments offers hope for the future development of accurate absolute quantitative methodology, researchers are currently limited to a calibration curve approach for handling the effects of large interlaboratory variance in instrumental and mathematical operating parameters. Erikson used a calibration curve approach for quantifying carbonyl sulfide and ammonia concentrations in coal gas, using one analytical wavelength for each component (4). Mamantov and co-workers also used a single wavelength calibration curve approach for quantifying matrix isolation GC/FTIR data (5). Greater sensitivity in quantitative trace gas analysis was demonstrated by Haaland, who used a least-squares approach employing the entire IR spectral range (6). The quantitative on-the-fly detection of GC/FTIR eluents involves several additional considerations. Upon exit from the chromatograph, absorbing species are in the vapor state diluted by a substantial volume of inert carrier gas. The variance of absorptivity with concentration due to intermolecular bonding effects (7)is thus minimized, extending the linear range of absorbance vs. concentration profiles. However, several complicating factors arise at the interface between the chromatograph and the spectrometer. The effective pathlength of the light pipe interface as it relates to factors such as the temperature and reflectivity of the gold surface in the light pipe must be determined to allow absolute quantitation using measured absorptivities (3). Additionally, the light pipe Foxboro Analytical, 140 Water St., P.O.B.5449, Norwalk, CT
06856.
can hardly be considered a zero dead volume chromatographic detector. The quantitative GCIR response must therefore be corrected for the effects of flow rate and light pipe volume. On the other hand, if conditions at the interface are held constant and a calibration curve approach is used, these problems can be circumvented. This paper presents a calibration curve approach to GC/ FTIR quantitation using the integrated peak areas of a reconstructed chromatogram as a quantitative metric. The Gram-Schmidt reconstructiontechnique, originally developed by de Haseth and Isenhour (8), constructs a gas chromatographic trace directly from the raw interferometric data. The Gram-Schmidt reconstruction is thus computationally efficient as there is no need to Fourier transform each interferogram in the GC/FTIR experiment prior to data analysis. In addition, comparison studies have shown Gram-Schmidt reconstructions to be more sensitive than either Euclidean distance or fast Fourier transform (FFT)reconstructions (9, 10). A theory is developed which shows that quantitative GC/FTIR results can be extracted directly from the interferometric data and empirical evidence supporting the validity of the theoretical predictions is presented.
THEORY Quantitative absorption spectrometry hinges on the use of the Beer's law relation
A ( $ = e(o)bC
(1)
where A($ is the absorbance at a certain frequency, e(o) is the absorptivity at that frequency, b is the effective pathlength of the sample cell, and C is the sample concentration. For concentration ranges where e($ is independent of concentration, A($ is linear with C. The wide range of linearity available in a GCIR experiment can be readily demonstrated. When the pentyl propionate absorbance maximum at 1200 cm-' is plotted vs. concentration for the series of standard mixtures described in the Experimental Section, linearity is maintained over the entire concentration range encompassed by the standards. The wide range of linearity for absorbance vs. concentration profiles in GCIR indicates that a total integrated absorbance (TIA) reconstruction could prove to be a useful quantitative metric. As stated previously, however, TIA reconstructions are not only computationallyinefficient but less sensitive than Gram-Schmidt reconstructions. This is clearly illustrated in Figure 1which compares a TIA reconstruction generated using full 2048 point FFT's on the upper trace with a 30 basis vector Gram-Schmidt reconstruction. Both reconstructions use the same data set, representing a portion of the multiple injection GC/FTIR run described in the Experimental Section. The Gram-Schmidt reconstruction is clearly superior in terms of signal-to-noise and sensitivity. Defining narrow absorbance windows containing only significant analyte absorbances for integration results in higher S I N spectral domain GC reconstructions. Integrating only over the spectral window from 1740 to 1800 cm-l, corresponding to the intense pentyl pro-
0003-2700/82/0354-1922$01.25/00 1982 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 54,
NO. 12, OCTOBER 1982
Is*m(d)= F’( 10-A(”Bref(o) )
1923
(9)
is now made. For low absorAn approximation to may be approximated by the first two terms bances, of its Maclaurin series expansion:
= 1 - A(o) In 10
(10) Substituting for A @ ) using the Beer’s law expression (eq 1) and using the result to replace in eq 9 gives eq 11.
Isa(d) = F1(Bref(o)- e(o)bC(ln 10)Bref(D)) (11) The linearity property of Fourier transforms is used to separate the right-hand side of eq 11 giving
m
-ll
Ixm-
una
Figure 1. (A) 30 basis veictor Gram-Schmicfl reconstruction and (6) 2048-point FFT total integrated absorbance reconstructlon of a 1900 interferogram segment 01 a quantitative GC/FTIR experiment.
pionate carbonyl band, results in reconstruction S I N values for the analyte that are considerably better than those obtained by integration of the entire spectral range. However, these “windowgram”S / N values are still a factor of 5 or more worse than the corresponding Gram-Schmidt values and the generality of the GC pieak detection has been sacrificed. A4s low sensitivity is the critical drawback in GCIR compared to associated techniques such as gas chromatography/mass spectrometry (GCMS), the sensitivity of a GCIR reconstruction technique is its most vital attribute. The interferogram is a transmittancle measurement and transmittance is generally not linear with concentration. As the Gram-Schmidt reconstruction deals directly with transmittance data in the interferogram or time domain, GramSchmidt intensities would not be expected to exhibit a linear dependence on concentiration. On the other hand, for the low absorbances generally encountered in a GCIR experiment, it will be shown that the Gram-Schmidt intensity actually is linear with concentration. The Gram-Schmidt intensity (GS) for a given sample interferogram is defined as
GS = 1I.I
- ... - (I*B,)211/2
-
(2)
where 1is the 100 dimensional sample vector formed by taking 100 consecutive points of the sample interferogram starting 60 points past the light burst, and B,is the ith orthogonal basis vector, calculated as dericribed previously (8). The following set of definitions is needed before beginning the derivation of the concentration dependence of GS. Let Isa(d) and Iref(d]lrepresent the sample and reference interferogram arrays evaluated a t 2048 discreet values of the interferometeroptical displacement, d (cm). B-(s) and B,&) represent the sample and reference power spectrum arrays which each consist of 1024 points evaluated at 8-cm-l intervals of frequency (u). Let Fc ) denote the Fourier transform and P’() denote the inverse Fourier transform. If phase errors are assumed negligible, the following exipressions hold. Bsamlo)
Brefr:@
= p(Isam(4)
(3)
= J’(Iref(4)
(4)
Llm(d = JvLm; Iref(4 = p’
)
(5) (6)
The transmittance specti*um,T($,and absorbance spectrum, A($, can then be expressed as
T(o) = lo-/’@)= B,,(o)/Bref(D)
(7)
Rearrangement of eq 7 ‘ ,1wives
Bsa(ij) = 1O-A(@Bref(o) which upon inverse transforming both sides gives
(8)
Isam(d)
= F ’ ( B r e f ( U ) ) - F’(e(WJC(1n 10)Bref(o))
(12)
which simplifies to Iaam(4
= Iref(d) - C ( F l ( e ( o ) W n 1O)&edo))) (13)
The sample interferogram is now expressed as a function of the reference interferogram, the sample concentration, and another term, JL1(e(@b(lnlO)B,&)). This last term is a signal in the interferogram domain which depends on the sample absorptivity and the reference power spectrum. It can be considered an interferogram which does not change with concentration but does change if a different analyte with a different absorptivity is chosen and therefore may be considered constant for a given analyte. For the remainder of the derivation, the quantities in eq 13 are treated as vectors to enable substitution into eq 2. This includes reduction of the dimensionality of the interferogram vectors from 2048 points to 100 points. There has been some disagreementon which portion of the interferogramto extract the 100 point segment from in order to maximize the SIN in the reconstructed chromatogram (8,9,11). This parameter is partially a function of the specific analyte and preliminary work in our laboratory indicates that it may also be a function of the particular instrument used to collect the data. We used a 100-point segment displaced 60 points from the light burst as this region has been determined optimal for our GC/FTIR instrument. The notation will change as follows:
Ism@) = I Iredd) = R
(14)
F’(e(o)b(ln 10)Bref($) = X
(16)
(15)
Equation 13 may now by rewritten as
I=R-CX
(17)
and substituted into eq 2 to give
GS =
I(R, - C X ) * ( R ,- C X ) - ((R,- CX)-B# ((E1-
-
CX).B# - ... - ((E1 - CX)-B,}211/2(18)
Equation 18 can be simplified to give the result of interest
GS = CIX-X -
- ( B ~ S X- ... ) ~- (B,*X)211/2(19)
The mechanics of this simplification and can be found in the Appendix. It can be seen from eq 19 that if the conditions of the derivation are met, the Gram-Schmidt intensity should be linear with concentration as all terms in the parentheses are constant for a given analyte. The four conditions that must be met are as follows: (1)the sample absorbance must obey Beer’s law (e(a) does not vary with C), (2) the effective pathlength, b, must not vary during the experiment, (3) phase errors in the interferogram must be negligible, and (4)the two-term Maclaurin series approximation to must hold. The first two conditions are requirements for calling X a constant. As stated previously, the first condition is of little concern for concentrations generally encountered in a GCIR
1924
ANALYTICAL CHEMISTRY, VOL. 54, NO. 12, OCTOBER 1982 -.
i'
a
0
a a w 8
Table I. Quantitative Data for Standard Solutions final concn CJc/c, CJ r( CP PI mix comp molar pg/3 p L % CAC), % dl
d2 d3 d4 d5 d6 d7 d8 d9 all
PP PP PP PP PP PP PP PP PP AC
0.319 0.213 0.107 0.0426 0.0213 0.01065 0.00426 0.00213 0.00107 0.132
138 92.1 46.1 18.4 9.21 4.61 1.84 0.921 0.461 47.67
0.40 0.55
0.74 0.80 0.91 1.04 1.08 1.17 1.27
0.84 0.92 1.04 1.09 1.17 1.28 1.31 1.38 1.47
0.74
ABSORBANCE
Figure 2. Plot of the relative error in the Maclaurin series approximation to vs. A(P) for (A) a two-term and (B) a three-term substitution.
experiment. The effective pathlength through the light pipe is dependent on the quality of the inner gold coating and the light pipe temperature but may be easily held constant for a given GCIR experiment. The third condition is necessary for the validity of eq 3-6. Assuming negligible phase errors in the interferogram is a stringent condition and is clearly not the case for current FTIR instrumentation. However, as assumed in the Mertz phase correction method (121,the phase angle due to signal, e(@, varies gradually with frequency in the spectral domain. The distribution of the phase spectrum in the interferogram domain is therefore confined to a narrow region (less than 50 points) centered at the light burst. If the assumptions in the Mertz phase correction are valid, phase errors should exhibit a negligible effect on the portion of the interferogram used in the Gram-Schmidt calculation for this work. The last condition holds only for low sample absorbances. Figure 2 is a plot of relative error vs. absorbance for the twoand three-term Maclaurin series approximations to The two-term error curve (curve A) shows that the maximum absorbance must be kept under 0.12 to keep the error in the approximation under 5%. Curve B shows that higher absorbances may be tolerated using the three-term approximation. A derivation similar to that shown for the two-term approximation results in an approximately second-order Gram-Schmidt dependency on concentration when a threeterm approximation is used. This allows for the use of a second-order fit to the quantitative data when the concentration range involved requires a three-term Maclaurin series in order to maintain accuracy. approximation to
EXPERIMENTAL SECTION The GC/FTIR quantitative analysis was performed on a series of nine standard solutions prepared with pentyl propionate (PPI as the analyte, acetophenone (AC) as the internal standard, and benzene as the solvent. The analyte and internal standard concentrations for each of the mixtures are listed in Table I. The molar concentrations have been converted to absolute analyte amounts for a 3.O-pL GC injection. These absolute amounts are directly proportional to their correspondingmolar concentrations but are more amenable to literature comparison. The relative error in the analyte concentration due to the dilution scheme, uC/C, has been calculated for each mixture according to the tables of standard volumetric glassware tolerances (13). The relative error values were calculated by using the following propagation of error equation (20) where uw/ W is the relative error in the initial weighing, q J F g is the relative error in the initial volumetric flask volume, and
upi/Pi and UF,/F~ are the relative errors in each subsequent dilution step for the pipet and volumetric flask volumes, respectively. The final column in Table I gives the relative dilution error in the quantitative analytical measurement using the internal standard, O,(CP~/CAC),and was calculated as follows:
These relative error calculations provide a lower limit of approximately 1% on our ability to determine the quantitative accuracy of the technique. The GC/FTIR system used to collect the data consisted of a Hewlett-Packard402B gas chromatograph interfaced to a Digilab FTS-14 infrared spectrometer. The interface, a gold-coatedlight pipe (14) of inner diameter 1.5 mm and length 54 cm (volume = 0.95 mL), and its associated l/ls in. glass lined, stainless steel, capillary transfer lines were wrapped with insulated heating tape and provided with thermocouple temperature monitoring and variac temperature control. A high sensitivity HgCdTe detector was used for GC/FTIR detection. The spectrometer and the optical bench housing the interface and detector were kept under dry nitrogen purge during data acquisition. The separations were performed on a 6 f t X 1/4 in. 0.d. glass column packed with 15%SE-30 on 80-100 mesh Chromosorb P AW-DMCS. A 3.0-wL portion of each standard mixture was injected sequentially, allowing 5.5 min between injections. The injection port, column oven, transfer line, and light pipe temperatures were held at 202 "C, 140 "C, 220 "C, and 193 "C, respectively. The flow rate, measured with a soap bubble flowmeter,was 55.5 mL/min. The Digilab FTS-14 collects 8 cm-' resolution data at the rate of one 2048-pointinterferogram every 0.8 s. These data were written onto magnetic tape and transferred to a Nova 3/12 minicomputer for further data analysis. A portion of the reconstructed chromatogram for the quantitative GC/FTIR experiment is shown in Figure 1. The reconstruction intensities for interferograms 1800-3700 are plotted, corresponding to the second half of mixture d7 and all of mixtures d6-d3, respectively. The four largest peaks correspond to the solvent peak for mixtures d 6 4 3 with each solvent peak followed by a peak for pentyl propionate and a peak for acetophenone, respectively. The 30-basisvector Gram-Schmidt reconstruction was performed with the GIFTS software package described elsewhere (15). The GIFTS package calculates the basis vectors using double precision arithmetic while performing all other calculations in the integer mode, thus speeding up the computation. The basis vectors were taken from positions corresponding to base line prior to the solvent peak for each of the injections. It was empirically determined that spreading the basis vectors throughout the chromatogram results in a consistent signal-tonoise ratio throughout the reconstruction. Selecting all basis vectors from the beginning of the chromatogram results in a signal-to-noiseratio that gradually decreases from the beginning to the end of the reconstruction. Base line correction was performed on all reconstruction peaks prior to area calculation. A linear least-squares routine was used to fit the best line to an array of 50 points consisting of 25 reconstruction points taken from both sides of the peak. The base line value at each point of the peak was then calculated and subtracted from the reconstruction intensity at that point. The area of the base-line-corrected peak was computed with a New-
ANALYTICAL CHEMISTRY, VOL. 54, NO. 12, OCTOBER 1982
1925
Table 11. Least-Squares Fits of First-, Second-, and Third-Order Polynomial Functions to the Quantitative Data goodness of fit values concn range, p g / 3 fiL order = 1 order = 2 order = 3 0.461-9.21 0.461-18.4 0.461-46.1 0.461-92.1 0.461-138
1.28 5.93 30.4 89.2
1.56 1.94 1.70 8.99 30.0
1.96 1.81 2.63
uG PPI3 uL
Flgure 3. Quantitativecalibration curve plots over (A) the second-order range spanned by mixtures d2-d9 and (E) the linear range spanned by mixtures d5-d9.
ton-Cotes quadrature algorithm.
RESULTS AND DISCIJSSION The quantitative results obtained frolm the Gram-Schmidt reconstruction of the multiple injection GC/FTIR experiment described in the Expei:imental Section support the predictions derived from theory, Figure 3B,A shows plots of the quantitative metric vs. concentration for the concentration ranges spanned by mixtures (1-9 and mixtures d 2 4 9 , respectively. The concentration values plotted on the abscissa are the absolute quantities of analyte injected for a 3.0-pL injection volume, with the corresponding molar concentrations listed in Table I. The ordinate value for each point was obtained by dividing the area of the pentyl propionate peak by that of the acetophenone iinternal standard. and multiplying the result by 1000. The w e of an internal standard compensates for quantitative variances due to variances in the actual injection volume. The data in Figure 31B were fit to a linear polynomial while a second-order polynoimial was used for the data in Figure 3A. In all cases an intercept term was included in the linear least-squares polynomial. The point (0,O) may not be included as an additional data point as there is always a negative y intercept due to noise considerations. The implications of this are 'discussed in the :following paper (16). The predicted linearity of the Gram-SSchmidt metric at low concentrations is clearly supported by the data presented in Figure 3B. This is the range of greatest interest as few components in the complex mixtures generally studied by GCIR exhibit concentrations greater than the upper limit of this range. Figure 3A illust#ratesthat even for situations involving much higher analyte concentrations, a second-order polynomial fit is sufficient to accurately describe the data. The upper limit of the range plotted in Figure 3A, corresponding to an analyte concentration of over 0.2 M, is well above the concentrations encountered in almost any conceivable GC/FTIR application. First-, second-, and third-order polyinomial fits were calculated for the range illustrated in Figure 3B and for each subsequent range obtained by adding one additional data point to its predecessor. The goodness of fit data for these calculations are summarized in Table 11. The goodness of fit (GF) for each data set was calculated as shown below:
GF
=
I(C(J/~ - y,)')/(n - d - 1)11/' i
(22)
where yi is the experimental and yc the expected or calculated ordinate value for each point, n the total number of points, and d the degree of the polynomial used in the least-squares fit. These GF values are thus a measure of the standard deviation of the ordinate value of each point from the best
0
6.0
1.0
0.0
i o
I
lo.
rrG PP/3 uL
Flgure 4. Plot of the relative error in concentration vs. concentration for the data shown in Flgure 38.
line or curve through the points. The GF values for the linear fit in Figure 3B and the second-order fit in Figure 3A were 1.28 and 8.99, respectively. Visual inspection indicates a good fit in both cases as it does for the fits to the other concentration ranges in Table I1 which have similar GF values, although plots of these other ranges have not been included to avoid redundancy. Using the GF data as a guide, it can be seen from Table I1 that a linear model adequately describes the empirical data out to about 20 pg of analyte absolute, while a second-ordermodel extends the range to about 100 fig. The maximum analyte absorbance for mixture d4, which at 18.4 pg of analyte is in the linear response range, is 0.186 absorbance unit. Referring to Figure 2, the error in the two-term Maclaurin series approximation to lo-* for this absorbance is greater than 10%. The two-term approximation was shown in the Theory section to lead to a linear Gram-Schmidt response; however, this example shows that the error in the approximation does not directly correspond to the quantitative error resulting from the assumption of a linear Gram-Schmidt response. This is reasonable as the absorbance values in the spectral domain vary from zero to a maximum and using the absorbance maximum therefore provides only an upper limit to the deviation from linearity. The third-order fit gives better GF values for the last two concentration ranges in Table 11. However, the use of a third-order polynomial is not statistically justified for the number of data points used without a theoretical prediction for such a response. With transmittance exponentially related to absorbance, one might assume an exponential dependence of the Gram-Schmidt intensity on concentration. However, the empirical data do not support such an assumption. The goodness of fit parameter is qualitatively useful but does not give a quantitative indication of the relative error in the Gram-Schmidt technique. However, as the GF value is a measure of the absolute error in the quantitative metric (q,), the relative error in concentration at a given concentration (uc/C) can be calculated according to eq 23, where 6y/6x is
ANALYTICAL CHEMISTRY, VOL. 54, NO. 12, OCTOBER 1982
1928
isting Fourier transform based methods of analysis.
APPENDIX The mechanics of simplifying eq 18 to give eq 19, both found in the theory section, are presented here. Theory Section. The distributive property of dot products over subtraction is used to expand eq 18 to give
GS =
IRl-Ri
- 2CR1.X
+ C2XaX - (fil*Bl- CB1*X)2- ... - (Rl-B, - CB,-X)211/2(24)
( R , . B 2 - CB2.X)'
The first basis vector is formed from an arbitrary reference interferogram, designated Rl, via eq 25. I
'
0.
10.
I
20.
30.
10.
SO.
60.
,
10
BO.
90.
I 100
uG P P I 3 uL
Flgure 5. Plot of the relative error in concentration vs. concentration for the data shown in Flgure 3A.
the first partial derivative of the best-fit line or curve to the data with respect to x . Figure 4 is a plot of crc/C vs. C for the linear fit illustrated in Figure 3B. At concentrations corresponding to less than 1 pg of analyte, the error is large due to the low S I N in the reconstructed chromatogram for such low concentrations. The relative error quickly approaches the dilution error level of approximately 1% which it reaches at an analyte level of about 5 pg. Figure 5 is a similar plot generated for the second-order fit of Figure 3A. Once again the error is S I N limited at lower concentrations and reaches a minimum of approximately 1 % corresponding to the dilution error. In this case, however, the relative error rises again at higher concentrations as a result of the roll off of the GramSchmidt response with a corresponding loss of sensitivity. The overall quantitative accuracy of the technique is quite good, surpassing the often quoted value of 5% for quantitative infrared spectrometry over a range from 1pg to greater than 100 pg. This range can be expected to shift for different analytes due to differing molar absorptivities and different instruments according to their sensitivity.
CONCLUSION Quantitative GC/FTIR using the Gram-Schmidt chromatographic reconstruction method has been shown to be accurate over a concentration range of greater than 2 orders of magnitude. The resulting calibration curves can be analyzed using easily computed first- or second-order least-squares polynomials. Extension of the approach to higher concentrations involves the analysis of more complex response curves but has little practical utility. Extension to lower concentrations is of much greater importance and requires increasing the S I N ratio of the reconstructed peaks. Various instrumental improvementssuch as faster interferometer scan rates, higher IR throughput, extended precision analog to digital converters, and more concentrated chromatographic peaks using capillary GC can be expected to dramatically increase the achievable S I N ratios and thus lower the detection limit substantially. An alternative approach to increasing S I N is the application of data processing techniques, an example of which is the cross-correlation S I N enhancement technique presented in the following paper (16). The importance of the fact that the quantitative method described in this paper deals with data in the interferogram or time domain should be stressed. With the recent advances in qualitative analysis of GC/FTIR interferogram domain data (17,18), it may become possible to perform a complete qualitative and quantitative analysis on a GC/FTIR experiment without computing a single Fourier transform. This offers a large advantage in computational efficiency over ex-
B, = Rl/(Rl.Rl)l/2
(25)
The basis vectors form an orthogonal set; therefore all basis vectors except B1 are orthogonal to R , and their dot products with Rl are zero. Substitution of eq 25 into eq 24 and simplifying the result using the orthogonal relation of R, to B2, B3, etc. gives
GS = IR,.R1- 2CR1.X
+
C'X-X - {R1*R1/(R1-RJ1/2C R , . X / ( R 1 . R l ) ~ ~-~(0 ) ~- CB2.X)2 - ... (0 - CB,-X)211/2(26)
Expansion of the squared terms gives
+
2CR1.X C'X*X - R,*R1+ 2CR1.8 C2(R1*X)z/(R,*RJ- C2(B2*X)2- ... - C2(B,-X)11/2 (27)
GS =
IR,*R1-
Collecting like terms gives
GS = lC2X*X- C2(R1.X)2/(R1-R1) - C2(B,*8)2- ... C2(B,*X)211/2(28)
Substituting the relation given in eq 29 into eq 28 gives eq 30.
(B,.X)2 = (R1/(R1.R,)1/2.X)2 = (R,.X)2/(Rl.R,)
(29)
GS = 1C'X.X - C'(B1-X)' C2(B2-X)2- ... - C2(B,-X)211/2(30) When C2 is factored out of eq 30, the result of interest, eq 21, is obtained.
LITERATURE CITED (1) Erikson, M. D. Appl. Spectrosc. Rev. 1979, 75, 261. (2) Hlrschfeld, T. I n "FTIR, Applications to Chemical Systems"; Ferraro, J. R.,Basiie, L. J., Eds.; Academic Press: New York, 1979; p 193. (3) Anderson, C. P.; Julian, R. L., presented at the 1980 Pittsburgh Conference, paper 362. (4) Erlkson, M. D.; Frazier, S. E.; Sparaclno, C. M. Fuel 1981, 60, 263. (5) Hembree, D. M.; Garrison, A. A,; Crocombe, R. A.; Yokley, R. A.; Wehry, E. L.; Mamantov, 0.Anal. Chem. 1981, 5 3 , 1783. (6) Haaland, D. M.; Easterllng, G. Appl. Spectrosc. 1980, 3 4 , 539. (7) Tomaselli, V. P.; Zarrabl, H.; Moiler, K. D. Appl. Spectrosc. 1980, 3 4 , 415. (8) de Haseth, J. A,; Isenhour, T. L. J. Chromatogr. Sci. 1977, 49, 1977. (9) White, R. L.; Glss, G. N.; Brissey, G. M.; Wllklns, C. L. Anal. Chem. 1981, 5 3 , 1778. (10) Hanna, D. A,; Hangac, 0.;Hohne, 8. A,; Small, G. W.; Wleboldt, R. C.; Isenhour, T. L. J. Chromatogr. Sci. 1979, 17, 423. (11) de Haseth, J. A,; Leclerc, D. F., presented at the 1982 Pittsburgh Conference, paper 038. (12) Mertz, L. InfraredPhys. 1987, 7 , 17. (13) Lam, R. 6.; Isenhour, T. L. Anal. Chem. 1980, 5 2 , 1158. (14) Azzaraga, L. V. Appl. Spectrosc. 1980, 3 4 , 224. (15) Hanna. A,: Marshall. J. C.: Isenhour, T. L. J. Chromatogr. Sci. 1979, 17, 434. (16) Lam, R. B.; Sparks, D. T.; Isenhour, T. L. Anal. Chem. 1982, 5 4 , .
I
0000
(17) Small, G. W.; Rasmussen, G. T.; Isenhour, T. L. Appl. Spectrosc. 1979, 3 3 , 444. (18) de Haseth, J. A,; Azzaraga, G. T. Anal. Chem. 1981, 5 3 , 2292.
RECEIVED for review January 11,1982. Accepted June 4,1982. This work was supported by National Science Foundation Grant No. CHE 8026747.
