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Use of the spectral trace length in determining the concentration of known compounds in complex mixtures
125
/4n&ica chimica Acta, 284 W93) 125-130 Elsevier Science Publishers B.V., Amsterdam
Use of the spectral trace length in determining the concentration of known compounds in complex mixtures Frank R. Burden Chemistry Department, Monash University, Clayton, Victoria 3168 (Australia) (Received 6th April 1993; revised manuscript received 27th July 1993
Abstract The principal of minimum length of a spectral trace has been used to determine the concentration of a known compound in a mixture of other unknown compounds when the unknown compounds are not available for study without the known compound mixed in with them. This method works where the spectrum of the unknown compounds approximates a quadratic curve and does not have spectral peaks in a region where there are spectral peaks of the known compound. Keywor& Chemometrics; Spectral trace length; Mixtures
A common problem for a chemical analyst is to estimate the concentration of a known, and available, compound in a mixture of other unknown compounds, where the unknown compounds are never available without some of the known compound mixed with them. Unless there are particular chemical or physical methods whereby the known compound can be readily separated from the mixture, the problem is not easily resolved. In this paper a new procedure is proposed which makes use of the Zength of the signal trace of a spectrum rather than the peak heights or areas under the signal. The method is not confined to infrared (IR) spectra but it is most easily demonstrated by this technique. Such mixture resolutions have been tackled by a variety of methods [1,2] the most well-known of which is that of factor analysis [3,4]. Of course many such problems may be solved by the astute analyst picking an appropriate IR absorbance of
Corresw&cnce to: F.R. Burden, Chemistry Department, Monash University, Clayton, Victoria 3168 (Australia).
the known compound where it can be readily ascertained by direct examination of the mixture spectrum that the unknown compounds are unlikely to be producing confounding absorbances. Factor analysis, when producing concentrations rather than elucidating pure component spectra, relies on the boundary conditions of non-negative concentrations and extinction coefficients. ‘Ihe method presented here, in making use of the length of the signal trace as well as assuming the applicability of Beer’s law, is relying on producing a minimum in the signal length to produce the concentration of the known component. It is a variant of baseline correction methods [5,6] adapted to situations where the data is stored digitally but avoids the need to fit polynomials and other functions to the observed signals, a procedure which can easily lead to spurious results [7]. The method is presented algebraically together with some operating rules which set the boundaries of the methods’ usefulness. This method is then illustrated by using a theoretical example similar to that used in reference [2].
0003~2670/93/$06.00 Q 1993 - Elsevier Science Publishers B.V. All rights reserved
126
F.R. Burden /AnaL Chim. Acta 284 (1993) 125-130
METHOD
and these two terms can be represented
General Consider a mixture, M, of two components, K and U, where K represents a known and available compound and where U is an unknown compound or mixture of unknown compounds. It is desired to ascertain the concentration of K in the mixture using only the spectra of K and M with no knowledge of the spectrum of U in the absence of K. Let A(h) be the absorbance at wavelength A so that (1)
A,(A) =&(A) +Ax(A) which can be written as
(2) A,(A) = {C”U(A)) + CxK(A) where Co and Cx are the concentrations and, U(A) and K(A) are the absorbances at unit concentration of U and K The term {C,U(A)} in Eqn. 2 can be written as U(h) since the concentration is not of interest, or is even undefined, here. Eqn. 2 becomes A,(A)
=m+
C,K(A)
(3) with respect
Equation 3 may be differentiated to A to give A&( A) = U’(A) + C,K’( A)
(4) Using the formula for the length of a curve the length of the spectral trace L,(A,, A,) from A, to A, is given by
&(A,,
A,) =LI’[l
A, &(AW’(A)
A2) dCK
= / A,
=
, dA
+c
(6)
(1 +A’M(A)2)1
A, U’(A)K’(A) /
dLl(h
A21
= I
+ I 1
2
(8)
dCK
If the condition
dLM(AlfA21=.
can
be
dC, found, by numerically subcacting increasing amounts of K(A) from M(A), and the first term, I,, of Eqn. 8 is zero then Cx is the amount of K subtracted from M. This condition will occur when U’(A) = a constant, i.e., U(A) is linear, and ],:zK’(A)dA = 0.
Linear background The main variable that is available is to choose the range A, to A,. This choice can be made by examining the spectrum of the known compound, K, to pick out some clearly defined absorbances. The integral /,:;?K’(A)dA should be zero which means in practice that the absorbances should effectively reduce to zero, or be equal, at the boundaries A, and A,. If the background is linear, i.e., u(A) = a + bh and U’(A) = b then both terms of Eqn. 8 will approach zero as the amount of K subtracted from M approaches C,, i.e., the shortest distance ,between two points is a straight line.
+A’,(A)2]idA
and the concentration C, is found by searching for a minimum in L&A,, A2). Differentiating Eqn. 5 with respect to C, gives %dA,,
by I, and
I2
dA
*I (1 +&(A)‘) A2
/ K4
K’(A)K’(A)
, dh
( 1 +Ah( A)2)2
(7)
Non-linear background If the background is quadratic, i.e., U(A) = a + bh + CA’ and IJ’(A) = b + 2cA the first term of Eqn. 8, I,, will not become zero unless c can be set to zero. Higher order traces in U(A) would prove less suitable though a L.orentz& trace of Au2 can be expanded in a the type (A -A”12+ Au2 Taylor series wi;h the coefficients of A” being Au2 when Au -SZ(A -A’) which will be the ( A”j2” case for the tail of a Lorentzian. The cubic term Au is then likely to be sufficiently small. (A”)6
127
F.R Burden /Anal. Chim. Acta 284 (1993) 125-130 IMPLEMENTATION
Using the above arguments, the following strategy may be utilised on instruments where the spectral trace is held in a computer memory in digitisied form. It is assumed that Beer’s law applies to K(A) at low concentrations. (A) The spectrum, M(A), of the mixture M must be recorded and stored. (B) The spectrum, K(h), of the known compound K must be recorded at a measured concentration. This may be achieved by adding some K to the mixture M and then subtracting the spectrum of M as recorded under A above. (C) Definitive peak absorbances of K must be identified in the mixture M in a region A, to A, where there are no identifiable spectral maxima due to U(A). (D) For the chosen peak absorbances of K the integral j/;K’(A)dA should be numerically evaluated. The range A, to A, should be adjusted until the integral differs insignificantly from zero. (E) The spectrum of K, in known quantity C, (estimated), should be numerically subtracted form M until the resultant trace has a minimum length, i.e., d&l(A) dC,( estimated)
=
0
If the resultant trace is linear then U(A) must be linear and therefore the concentration, C,, of K in the mixture M will equal C, (estimated). The linearity of the resultant trace is easily tested by using the initial slope at A, and continuing the line to A,. Variance from linearity is then ascribed to the quadratic term, c, in U(A) = a + bh + CA’. (F) If the quadratic coefficient, c, is non-zero then the spectral trace M(A) may be initially modified by numerically subtracting CA’ from it. This may be represented by M,(A) = M,(A) - c,A2
are then re-applied until ci is close enough to zero. If ci is sufficiently close to zero then the amount of K subtracted from M in the last cycle will equal the unknown concentration. If the final value of ci in the last cycle differs significantly from zero then the trace of U(A) in the range from A, to A, cannot be closely fitted to a quadratic. In this circumstance either the range A, to A, must be varied or another peak absorbance of K(A) tried. This procedure will work best where the chosen spectral peak of K(A) lies on a linear or gently curved portion of U(A). It will be at its worst, as will a straight application of Beer’s law, where the chosen spectral peak of K(A) ‘lies directly above a similar peak of U(A). In this case the concentration will be estimated as being too high to the extent of the absorbance of U(A). Recourse to more than one peak spectral absorbance of K(A) is the appropriate safeguard.
EXAMPLES
The method was tested with a theoretical model similar to that of reference [2] using the programming language Mathematics [8] on an IBM style personal computer as well as with a practical illustration using a mixture of toluene and benzene. For the theoretical model three spectral peaks
2.5 B2 1.5 f
1 0.5
(9)
where the subscript 1 represents the first iterative attempt to remove the quadratic term from M(A). Steps (E) and (F) may now be applied to M,(A) so producing M,(A) and ci. These steps
0 1900
1950
2000
20%
2100
WevenUmber Fig. 1. The three Lorentzian peaks derived from the data in Eqn. 10 and Table 1.
F.R Burden /Anal. Chim. Acta 284 (1993) 125-130
128 TABLE 1 Lmentzian peak parameters for Fig. 1
A
AU A0
Peak 1
Peak 2
Peak 3
0.57 6 1990
0.67 5 2000
0.55 7 2023
1. 0.8 . 0.6 .
shown in Fig. 1 were derived using the data in Table 1 and a Lorentzian spectral shape given by 1100
1200
1300
1400
1500
16w
WOVOIlw Fig. 3. Toluene.
where, for peak i, A,, is the relative absorbance at unit concentration, AL+ is the peak width at half height and A$’is the location of the peak maximum. Figure 2 shows the three peaks of K(h) superimposed on a background function U(A) = O.OOl(lo00 - A) + 0.00012( 1900 - A)2 (10) with an arbitrary concentration C, of K at 3.34 mol dmm3. Equation 10 was chosen to represent a strongly curved baseline which would demand more than one iteration to produce C, which itself was deliberately specified to 3 significant figures for testing purposes. After a single minimization of the spectral trace length the value of Cx (estimated) was found to be 3.30 and the quadratic term of the remaining curve, c, calculated to be 0.000118. A second pass gave Cx
(estimated) at 3.34 with the quadratic term at 2 x 10e6, which is effectively zero. Although this example represents a single instance in a noise free environment it is sufficient to show a universal applicability. Noise will be partially cancelled by virtue of the integration of Eqn. 5. For the practical example a mixture of toluene and benzene was made up to a mole fraction of benzene of 0.43, X,,,,,, = 0.43. IR spectra were taken of the two pure compounds and of the mixture on a Fourier transform IR spectrometer and the files converted to a form suitable for being read by Mathematics. Although toluene was taken to be the unknown compound, with benzene as the known one, the spectrum of each, and the mixture, are shown in Figs. 3-5 in the
1
0.6
I 1900
2060
1950
2100
1100
1200
1400
WaVenwnber
W_
Fig. 2. The Lorentzian peaks plus background.
1300
Fig. 4. Benzene.
1500
1600
F.R Burden/Anal.
Chim. Acta 284 (1993) 125-130
0.6 fl 0.6 . I 20.4. 0.2
J
0: 1100
1200
1300
1400
1600
1800
1362
1394
1406
WaVenUlllber
Fig. 7. The mixture spectrum minus the quadratic function.
Fig. 5. Mixture.
1000 cm-’ to 1600 cm-‘. The peaks near 1400 cm-’ were chosen to illustrate the method and the spectrum from 1360 cm-’ to 1420 cm-’ is shown in Fig. 6 where the benzene peak is the smaller of the two. The resolution of the two peaks is not as good as in the theoretical example and therefore constitutes a more severe test of the method; the range 1382 cm-’ to 1408 cm-’ was selected to meet the integration requirement, D, of the implementation section. To ensure convergence a damping factor of 0.2 was used and convergence was then achieved to X be_ne = 0.43, (the exact answer) at the 10th iteration. Figure 7 shows the mixture spectrum with the final quadratic function subtracted and the fact that it deviates from a straight line implies that the toluene trace in the chosen range is range
0.45 . 0.4 0.36 . 0.3 . 0.25 1362
1364
1406
W-
Fig. 6. The toluene and benzene analysis.
absorbances
used in the
of higher order than a quadratic. This must occur because the absorbances of toluene and benzene are insufficiently resolved though it should be noted that the correct result was, nevertheless, produced. Conclusion A method has been presented whereby the concentration of a known substance in a mixture with unknown substance can, in many circumstances, be extracted from knowledge of the spectra of the known substance and the mixture alone. It has the strength that it can be automated when coupled with a few rules regarding the selection of the known spectral peak absorbance and the range of integration about this peak. A weakness of the method relates to overlapping peak absorbances of the known and unknown compounds. If these have exactly the same central wavelength then the problem is not resolvable. However even if they do not have exactly the same central wavelength there is a dividing line at which the problem is potentially resolvable. This line occurs when the underlying part of the unknown spectrum can be reasonably fitted to a quadratic curve in terms of the wavelength, though it would be possible to allow for higher order terms in principle. Whether or not this occurs is testable and to some extent manipulable by defining an appropriate range for the integration about the peak of the known compound.
130 REFERENCES 1 Y.-C. Ling, T.J. Vickers and C.K. Mann, Appl. Spectrosc., 39 (1985) 463. 2 J. Liu and J.L. Koenig, Appl. Spectrosc., 41 (1987) 447. 3 E.R. Malinowski, Factor Analysis in Chemistry, Wiley, New York, 1991. 4 P.J. Gemperline, S.E. Royette and K. Tyndall, Appl. Spectrosc., 41 (1987) 454.
F.R. Burden /Anal. Chim. Acta 284 (1993) 125-130 5 M.A. Raso, J. Tortajada, D. Escolar and F. Action, Comput. Chem., 11 (1987) 125. 6 M.A. Raso, J. Tortajada and F. Action, Comput. Chem., 15 (1991) 29. 7 W.F. Maddams, Appl. Spectrosc., 34 (1980) 245. 8 Mathematics Program Version 2.0, Wolfram Research, 1991.
/4n&ica chimica Acta, 284 W93) 125-130 Elsevier Science Publishers B.V., Amsterdam
Use of the spectral trace length in determining the concentration of known compounds in complex mixtures Frank R. Burden Chemistry Department, Monash University, Clayton, Victoria 3168 (Australia) (Received 6th April 1993; revised manuscript received 27th July 1993
Abstract The principal of minimum length of a spectral trace has been used to determine the concentration of a known compound in a mixture of other unknown compounds when the unknown compounds are not available for study without the known compound mixed in with them. This method works where the spectrum of the unknown compounds approximates a quadratic curve and does not have spectral peaks in a region where there are spectral peaks of the known compound. Keywor& Chemometrics; Spectral trace length; Mixtures
A common problem for a chemical analyst is to estimate the concentration of a known, and available, compound in a mixture of other unknown compounds, where the unknown compounds are never available without some of the known compound mixed with them. Unless there are particular chemical or physical methods whereby the known compound can be readily separated from the mixture, the problem is not easily resolved. In this paper a new procedure is proposed which makes use of the Zength of the signal trace of a spectrum rather than the peak heights or areas under the signal. The method is not confined to infrared (IR) spectra but it is most easily demonstrated by this technique. Such mixture resolutions have been tackled by a variety of methods [1,2] the most well-known of which is that of factor analysis [3,4]. Of course many such problems may be solved by the astute analyst picking an appropriate IR absorbance of
Corresw&cnce to: F.R. Burden, Chemistry Department, Monash University, Clayton, Victoria 3168 (Australia).
the known compound where it can be readily ascertained by direct examination of the mixture spectrum that the unknown compounds are unlikely to be producing confounding absorbances. Factor analysis, when producing concentrations rather than elucidating pure component spectra, relies on the boundary conditions of non-negative concentrations and extinction coefficients. ‘Ihe method presented here, in making use of the length of the signal trace as well as assuming the applicability of Beer’s law, is relying on producing a minimum in the signal length to produce the concentration of the known component. It is a variant of baseline correction methods [5,6] adapted to situations where the data is stored digitally but avoids the need to fit polynomials and other functions to the observed signals, a procedure which can easily lead to spurious results [7]. The method is presented algebraically together with some operating rules which set the boundaries of the methods’ usefulness. This method is then illustrated by using a theoretical example similar to that used in reference [2].
0003~2670/93/$06.00 Q 1993 - Elsevier Science Publishers B.V. All rights reserved
126
F.R. Burden /AnaL Chim. Acta 284 (1993) 125-130
METHOD
and these two terms can be represented
General Consider a mixture, M, of two components, K and U, where K represents a known and available compound and where U is an unknown compound or mixture of unknown compounds. It is desired to ascertain the concentration of K in the mixture using only the spectra of K and M with no knowledge of the spectrum of U in the absence of K. Let A(h) be the absorbance at wavelength A so that (1)
A,(A) =&(A) +Ax(A) which can be written as
(2) A,(A) = {C”U(A)) + CxK(A) where Co and Cx are the concentrations and, U(A) and K(A) are the absorbances at unit concentration of U and K The term {C,U(A)} in Eqn. 2 can be written as U(h) since the concentration is not of interest, or is even undefined, here. Eqn. 2 becomes A,(A)
=m+
C,K(A)
(3) with respect
Equation 3 may be differentiated to A to give A&( A) = U’(A) + C,K’( A)
(4) Using the formula for the length of a curve the length of the spectral trace L,(A,, A,) from A, to A, is given by
&(A,,
A,) =LI’[l
A, &(AW’(A)
A2) dCK
= / A,
=
, dA
+c
(6)
(1 +A’M(A)2)1
A, U’(A)K’(A) /
dLl(h
A21
= I
+ I 1
2
(8)
dCK
If the condition
dLM(AlfA21=.
can
be
dC, found, by numerically subcacting increasing amounts of K(A) from M(A), and the first term, I,, of Eqn. 8 is zero then Cx is the amount of K subtracted from M. This condition will occur when U’(A) = a constant, i.e., U(A) is linear, and ],:zK’(A)dA = 0.
Linear background The main variable that is available is to choose the range A, to A,. This choice can be made by examining the spectrum of the known compound, K, to pick out some clearly defined absorbances. The integral /,:;?K’(A)dA should be zero which means in practice that the absorbances should effectively reduce to zero, or be equal, at the boundaries A, and A,. If the background is linear, i.e., u(A) = a + bh and U’(A) = b then both terms of Eqn. 8 will approach zero as the amount of K subtracted from M approaches C,, i.e., the shortest distance ,between two points is a straight line.
+A’,(A)2]idA
and the concentration C, is found by searching for a minimum in L&A,, A2). Differentiating Eqn. 5 with respect to C, gives %dA,,
by I, and
I2
dA
*I (1 +&(A)‘) A2
/ K4
K’(A)K’(A)
, dh
( 1 +Ah( A)2)2
(7)
Non-linear background If the background is quadratic, i.e., U(A) = a + bh + CA’ and IJ’(A) = b + 2cA the first term of Eqn. 8, I,, will not become zero unless c can be set to zero. Higher order traces in U(A) would prove less suitable though a L.orentz& trace of Au2 can be expanded in a the type (A -A”12+ Au2 Taylor series wi;h the coefficients of A” being Au2 when Au -SZ(A -A’) which will be the ( A”j2” case for the tail of a Lorentzian. The cubic term Au is then likely to be sufficiently small. (A”)6
127
F.R Burden /Anal. Chim. Acta 284 (1993) 125-130 IMPLEMENTATION
Using the above arguments, the following strategy may be utilised on instruments where the spectral trace is held in a computer memory in digitisied form. It is assumed that Beer’s law applies to K(A) at low concentrations. (A) The spectrum, M(A), of the mixture M must be recorded and stored. (B) The spectrum, K(h), of the known compound K must be recorded at a measured concentration. This may be achieved by adding some K to the mixture M and then subtracting the spectrum of M as recorded under A above. (C) Definitive peak absorbances of K must be identified in the mixture M in a region A, to A, where there are no identifiable spectral maxima due to U(A). (D) For the chosen peak absorbances of K the integral j/;K’(A)dA should be numerically evaluated. The range A, to A, should be adjusted until the integral differs insignificantly from zero. (E) The spectrum of K, in known quantity C, (estimated), should be numerically subtracted form M until the resultant trace has a minimum length, i.e., d&l(A) dC,( estimated)
=
0
If the resultant trace is linear then U(A) must be linear and therefore the concentration, C,, of K in the mixture M will equal C, (estimated). The linearity of the resultant trace is easily tested by using the initial slope at A, and continuing the line to A,. Variance from linearity is then ascribed to the quadratic term, c, in U(A) = a + bh + CA’. (F) If the quadratic coefficient, c, is non-zero then the spectral trace M(A) may be initially modified by numerically subtracting CA’ from it. This may be represented by M,(A) = M,(A) - c,A2
are then re-applied until ci is close enough to zero. If ci is sufficiently close to zero then the amount of K subtracted from M in the last cycle will equal the unknown concentration. If the final value of ci in the last cycle differs significantly from zero then the trace of U(A) in the range from A, to A, cannot be closely fitted to a quadratic. In this circumstance either the range A, to A, must be varied or another peak absorbance of K(A) tried. This procedure will work best where the chosen spectral peak of K(A) lies on a linear or gently curved portion of U(A). It will be at its worst, as will a straight application of Beer’s law, where the chosen spectral peak of K(A) ‘lies directly above a similar peak of U(A). In this case the concentration will be estimated as being too high to the extent of the absorbance of U(A). Recourse to more than one peak spectral absorbance of K(A) is the appropriate safeguard.
EXAMPLES
The method was tested with a theoretical model similar to that of reference [2] using the programming language Mathematics [8] on an IBM style personal computer as well as with a practical illustration using a mixture of toluene and benzene. For the theoretical model three spectral peaks
2.5 B2 1.5 f
1 0.5
(9)
where the subscript 1 represents the first iterative attempt to remove the quadratic term from M(A). Steps (E) and (F) may now be applied to M,(A) so producing M,(A) and ci. These steps
0 1900
1950
2000
20%
2100
WevenUmber Fig. 1. The three Lorentzian peaks derived from the data in Eqn. 10 and Table 1.
F.R Burden /Anal. Chim. Acta 284 (1993) 125-130
128 TABLE 1 Lmentzian peak parameters for Fig. 1
A
AU A0
Peak 1
Peak 2
Peak 3
0.57 6 1990
0.67 5 2000
0.55 7 2023
1. 0.8 . 0.6 .
shown in Fig. 1 were derived using the data in Table 1 and a Lorentzian spectral shape given by 1100
1200
1300
1400
1500
16w
WOVOIlw Fig. 3. Toluene.
where, for peak i, A,, is the relative absorbance at unit concentration, AL+ is the peak width at half height and A$’is the location of the peak maximum. Figure 2 shows the three peaks of K(h) superimposed on a background function U(A) = O.OOl(lo00 - A) + 0.00012( 1900 - A)2 (10) with an arbitrary concentration C, of K at 3.34 mol dmm3. Equation 10 was chosen to represent a strongly curved baseline which would demand more than one iteration to produce C, which itself was deliberately specified to 3 significant figures for testing purposes. After a single minimization of the spectral trace length the value of Cx (estimated) was found to be 3.30 and the quadratic term of the remaining curve, c, calculated to be 0.000118. A second pass gave Cx
(estimated) at 3.34 with the quadratic term at 2 x 10e6, which is effectively zero. Although this example represents a single instance in a noise free environment it is sufficient to show a universal applicability. Noise will be partially cancelled by virtue of the integration of Eqn. 5. For the practical example a mixture of toluene and benzene was made up to a mole fraction of benzene of 0.43, X,,,,,, = 0.43. IR spectra were taken of the two pure compounds and of the mixture on a Fourier transform IR spectrometer and the files converted to a form suitable for being read by Mathematics. Although toluene was taken to be the unknown compound, with benzene as the known one, the spectrum of each, and the mixture, are shown in Figs. 3-5 in the
1
0.6
I 1900
2060
1950
2100
1100
1200
1400
WaVenwnber
W_
Fig. 2. The Lorentzian peaks plus background.
1300
Fig. 4. Benzene.
1500
1600
F.R Burden/Anal.
Chim. Acta 284 (1993) 125-130
0.6 fl 0.6 . I 20.4. 0.2
J
0: 1100
1200
1300
1400
1600
1800
1362
1394
1406
WaVenUlllber
Fig. 7. The mixture spectrum minus the quadratic function.
Fig. 5. Mixture.
1000 cm-’ to 1600 cm-‘. The peaks near 1400 cm-’ were chosen to illustrate the method and the spectrum from 1360 cm-’ to 1420 cm-’ is shown in Fig. 6 where the benzene peak is the smaller of the two. The resolution of the two peaks is not as good as in the theoretical example and therefore constitutes a more severe test of the method; the range 1382 cm-’ to 1408 cm-’ was selected to meet the integration requirement, D, of the implementation section. To ensure convergence a damping factor of 0.2 was used and convergence was then achieved to X be_ne = 0.43, (the exact answer) at the 10th iteration. Figure 7 shows the mixture spectrum with the final quadratic function subtracted and the fact that it deviates from a straight line implies that the toluene trace in the chosen range is range
0.45 . 0.4 0.36 . 0.3 . 0.25 1362
1364
1406
W-
Fig. 6. The toluene and benzene analysis.
absorbances
used in the
of higher order than a quadratic. This must occur because the absorbances of toluene and benzene are insufficiently resolved though it should be noted that the correct result was, nevertheless, produced. Conclusion A method has been presented whereby the concentration of a known substance in a mixture with unknown substance can, in many circumstances, be extracted from knowledge of the spectra of the known substance and the mixture alone. It has the strength that it can be automated when coupled with a few rules regarding the selection of the known spectral peak absorbance and the range of integration about this peak. A weakness of the method relates to overlapping peak absorbances of the known and unknown compounds. If these have exactly the same central wavelength then the problem is not resolvable. However even if they do not have exactly the same central wavelength there is a dividing line at which the problem is potentially resolvable. This line occurs when the underlying part of the unknown spectrum can be reasonably fitted to a quadratic curve in terms of the wavelength, though it would be possible to allow for higher order terms in principle. Whether or not this occurs is testable and to some extent manipulable by defining an appropriate range for the integration about the peak of the known compound.
130 REFERENCES 1 Y.-C. Ling, T.J. Vickers and C.K. Mann, Appl. Spectrosc., 39 (1985) 463. 2 J. Liu and J.L. Koenig, Appl. Spectrosc., 41 (1987) 447. 3 E.R. Malinowski, Factor Analysis in Chemistry, Wiley, New York, 1991. 4 P.J. Gemperline, S.E. Royette and K. Tyndall, Appl. Spectrosc., 41 (1987) 454.
F.R. Burden /Anal. Chim. Acta 284 (1993) 125-130 5 M.A. Raso, J. Tortajada, D. Escolar and F. Action, Comput. Chem., 11 (1987) 125. 6 M.A. Raso, J. Tortajada and F. Action, Comput. Chem., 15 (1991) 29. 7 W.F. Maddams, Appl. Spectrosc., 34 (1980) 245. 8 Mathematics Program Version 2.0, Wolfram Research, 1991.