Evaluation of experimental designs for multicomponent determinations by spectrophotometry

Analytica ChimicaActa, 207 (1988) 125-135

125

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

E V A L U A T I O N OF E X P E R I M E N T A L D E S I G N S F O R MULTICOMPONENT DETERMINATIONS BY SPECTROPHOTOMETRY L.L. JUHL and J.H. KALIVAS* Department o/Chemistry, Idaho State University, Pocatello, ID 83209 (U.S.A.) (Received 14th October 1987) SUMMARY Criteria are described and evaluated for choosing an optimal range of wavelengths in multicomponent determinations by spectrophotometry. By using Gaussian curves,results from a systematic study of the effects of resolution and intensity ratios on the performance of these criteria are given. A two-component mixture of 9-methylanthracene and pyrene was used for investigating optimal wavelength ranges and derivative orders. Favorable results were obtained with restricted wavelength ranges offeringbetter spectral characterization for each component. The limit of detection appears to be the most useful criterion. Until the recent past, multicomponent determinations by spectrophotometry have been done by using a single wavelength for each component. Recently, multicomponent determinations have been improved by using multiple wave~ lengths and matrix least-squares data processing [1-4]. The earliest methods used the same wavelength ranges for all components present in a mixture. However, Rossi and Pardue [3] have shown that accuracy can be improved if wavelength ranges are chosen which highlight the spectral features of each component [3 ]. Their study included normal and first-, and second-derivative absorption spectra. The problem encountered is knowing which wavelength ranges to use for optimal results. The selection of the best wavelengths is usually an empirical choice. Allowing for computer selection is preferable but necessitates the development of decision criteria. Various criteria have been used to predict the optimal combination of wavelengths [2,5-17]. The purpose of this paper is to evaluate various selection criteria for wavelengths which will allow for optimal precision and accuracy. In addition, criteria are evaluated for their effectiveness in selecting the best derivative order for a given set of absorption spectra. THEORY Beer's law can be expressed in matrix notation as R - - C K where R is a m × p matrix of responses for p sensors and m standard samples, C is a m × n matrix

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126 of concentrations for n components and m samples, and K is the n × p calibration matrix. Solving for the concentrations of the sample components requires inversion of K. The condition number of the calibration matrix K, cond (K), represents the numerical accuracy that can be lost when the matrix is inverted [18,19]. The larger cond (K) becomes, the larger the concentration errors can become. Ideally, cond(K) should be unity for no loss of accuracy. This will occur when the system under investigation has equal intensity ratios and no spectral overlap [20]. It has been shown that a minimum in cond(K) can be used as a criterion for determining the optimal set of sensors (wavelengths) [ 6,14,15 ], parameters [ 15,21 ], and derivative order [22 ]. Cond (K) is usually calculated from the ratio of the largest eigenvalue of K to the smallest eigenvalue of K. The possibilities of using cond(K) as a criterion for selection of optimal wavelength ranges are further investigated in this paper. Other criteria explored are the figures of merit recently derived by Lorber and co-workers [23,24 ], namely the limit of detection (LOD), accuracy (ACC), and total error propagation ( T E P ) . Because the background and theory were discussed in the earlier papers, only the key equations are presented here. The selectivity for component j is the ratio of that part of the signal orthogonal to the spectra of the other components SELj = Ifa~ll/frai[I

(1)

with aj representing the calibration response vector for component j and aj representing that part of aj orthogonal to the multicomponent spectrum. The II Ir signify vector norm. The sensitivity has been defined as SENt = [Ia* II/co

(2)

where c° is the calibration concentration of component j. With these definitions of selectivity and sensitivity, the limit of detection, LODj, for the j t h component can be estimated by LOD 2= 3Q/SENj, where Q is the error in measurements. This definition of LODj can be rewritten as LOD] = 3 {~dcO/SELj 1]a t 1]

(3)

The accuracy will be a function of two principal factors: the precision (noise) in the sample responses and the precision in the calibration measurements. Lorber and co-workers proposed that accuracy could be defined as the sum of the contributing precisions AC Cj = z~ca/c 0 --1-Ac d/cj

(4)

where ACCj is the total accuracy for component j while Ace/C° and ACd/Cj are the relative precisions for calibration and sample, respectively. Equation 4 can be expanded to

127 ACCi - (l[dll/lla~l] + •d )e~ IIdll

(5)

where d is the sample response vector and ~d is the error propagation for the j t h component which can be calculated from K~= []a* [l" I]dU/dTa *. A more interpretable form of this is ACCi = (1/SEL/" [lai I[+ 1/SENj-c i )E~

(6)

with ci representing the concentration of the j t h component in the sample. From Eqn. 6, it can be seen that improving the accuracy for component j requires that both SEL~ and SENj be as large as possible. The TEPj can be described by the term ( eld [I/I[a* l[ + ~:~) from Eqn. 5. More insight can be obtained if T E P i is rewritten as TEPi = (1/SELi" Hai I[+ 1/SENj-.c~ ) lld[[

(7)

which differs from ACCi by one term. Some of these figures of merit have recently been used by Carey and Kowalski [25 ] for characterization of quartz microbalance and surface acoustic wave sensors. The figures of merit were applied to evaluate the interference correction procedure for inductively-coupled plasma atomic emission spectrometry [ 24] as well as being used as objective functions for simplex optimization of such a system [26 ]. EXPERIMENTAL

Apparatus, computations and reagents The simulated absorption spectra were modeled by Gaussian curves generated over 100 wavelengths. Amplitudes, one-half bandwidths, and peak-to-peak resolution were 10 units unless specified otherwise. Normal random noise with a zero mean and a relative standard deviation of 2 % was added to the synthetic spectra. FORTRAN was used for the programs on the Idaho State University Hewlett-Packard 1000 computer system. A Model 2300 Varian spectrophotometer was used for ultraviolet absorption measurements. An Apple II + was used to control the spectrophotometer in addition to providing data storage and graphics output. Derivatives were calculated with stored spectra using a two-point slope procedure developed by Varian. All absorption and derivative values were transferred to the HP-1000 computer for subsequent data processing. Multiple linear regression was used to calculate the calibration matrix and analyte concentrations. Stock solutions of 9-methylanthracene and pyrene were prepared with reagent-grade chemicals. Concentrations were 1.002 and 1.001 mg 1-1, respectively, using a 75/25 (v/v) solvent mixture of spectroscopic grade acetonitrile and distilled water.

128

Procedure Each Gaussian curve simulating a spectrum for a component present in a synthetic sample was based on one unit each. Gaussian absorption spectra representing pure standards were based on concentrations of two and three units. This gave two standard spectra for each component. The wavelength ranges studied are listed in Table i and illustrated in Fig. 1. A mixture of 9-methylanthracene and pyrene at 1 mg l - 1 each was used as the sample. Three pure standards for each component at concentrations of 2, 3 and 4 mg l - 1were used for calibration, evaluation of criteria, and quantitative determinations. TABLE 1 Wavelengthranges studied Wavelengthrange (nm) Representation

I

c

I

I

229-391 a

260.0

295.0

229-265 c

295-391 d

229-245 e

d

if!,,

225.0

229-278 b

A

330.0

WAVELENGTH

365.0

400.0

(rim)

Fig. 1. Spectra of 9-methylanthracene (---) and pyrene (--): (A) zero-order derivative (absorbance ); (B) first-orderderivative; (C) second-orderderivative.Wavelengthranges correspondto those in Table 1.

129 RESULTS AND DISCUSSION

A simulated systematic study was conducted for two and five components with varying spectral effects to assess general trends in the figures of merit and cond (K). Resolution and intensity ratios were the spectral effects varied. This part of the study was patterned after a similar investigation by Otto and Wegscheider [20]. The results presented here will be directly comparable with theirs. Listed in Table 2 are the results for two components with varying resolution. As the resolution decreases, the values for cond (K) increase indicating a deterioration in concentration estimates as would be expected. This is confirmed by the relative errors tabulated. The values for cond (K) are very close to those of Otto and Wegscheider [ 20 ]. The sensitivity for the two-component system is seen to degrade with a decrease in resolution. Similar conclusions can be made for SELj, LODj, TEPj, and ACCj. It appears at this stage that cond(K) and the other criteria are in agreement and can be used to describe the system adequately. Table 3 presents results for different intensity ratios for two components. Component 2 has increasing intensity while component 1 remains constant. Cond (K) is only indicative for the complete system while SELj, SEN j, LODj, TEPj, and ACCj contain information about individual components. Specifically, component 2 can still be determined with an intensity ratio of 100 while component 1 cannot. Composite values could be reported for the figures of TABLE2 Trends of the figures-of-merit for two components with varying resolution Resolution a C o n d ( K )

Component SELj

SENt

LOD i (10 -a)

10.0

1 2

0.626 0.626

26.4 26.4

0.28 0.28

2.84

TEPj 6.03 6.02

ACC/ (10 -3 )

Rel. error (%)

0.76 0.76

0.02 0.11

2.0

14.1

1 2

0.141 0.141

5.93 5.93

1.81 1.81

28.3 28.3

3.36 3.36

0.5 0.6

1.0

28.2

1 2

0.071 0.071

2.99 2.99

1.93 1.93

56.1 56.7

6.65 6.72

1.2 1.3

0.5

56.0

1 2

0.035 0.035

1.51 1.51

4.31 4.31

111.0 113.4

13.4 13.2

2.3 2.1

0.4

68.7

1 2

0.029 0.029

1.23 1.23

6.17 6.17

142.1 134.1

16.0 16.9

3.9 4.0

0.3

88.0

1 2

0.022 0.022

0.94 0.94

6.76 6.76

178.9 185.0

21.3 22.0

6.2 6.1

aIncrements from peak to peak.

130 TABLE 3 Trends of the figures of merit for two components with varying intensity ratios Intensity ratio

Cond(K)

Component

SELj

SENi

LOD i (10 -3 )

TEPj

ACCi (10 -6 )

Rel. error

(%) 1

2.84

1 2

0.626 0.626

26.4 26.4

0.128 0.28

6.03 6.02

75.6 75.8

0.02 0.1

10

16.0

1 2

0.629 0.629

26.4 264.5

2.14 0.21

34.6 3.43

75.9 7.47

1.1 0.2

100

155.9

1 2

0.639 0.639

26.9 2679

34.3 0.34

372.4 3.16

88.1 0.75

15.7 0.1

TABLE 4 Trends of the figures of merit for two components with varying amplitudes and complete separation Amplitudes Cond(K)

Component

SELj

SENj

LODj (10 -4 )

TEPj

ACCj (10 -2 )

Rel. error

(%) 1.0 a

1.01

1 2

1.00 1.00

1.02 1.02

1.98 1.98

2.83 2.83

1.96 1.96

0.5 0.01

5.0 b

1.01

1 2

1.00 1.00

10.52 10.54

1.43 1.43

2.83 2.83

0.19 0.18

0.01 0.1

~Bandwidth 0.5, resolution 30. bBandwidth, 2.5, resolution 40.

merit which would allow for a characterization of the complete system similar to cond(K) [26]. Using cond (K) as a criterion is not recommended because of the restriction mentioned above and recognized by Lorber [ 23 ]. Another disadvantage results when constant intensity ratios with complete resolution are present but the amplitudes are allowed to vary. Table 4 shows the results of such an investigation. It is seen that cond(K) remains the same for both sets of amplitudes while the SENj, LODj, and ACCj terms present more descriptive criteria. Apparently SELj and TEPj have restrictions similar to cond(K). In a previous comparison of cond(K) and the determinant of K, it was found that if the intensity ratios of a two-component system were allowed to vary at constant resolution, cond (K) was preferred [ 19 ]. Conversely, when the amplitudes are allowed to increase while a constant intensity ratio is maintained, the determinant of K is a better criterion. It appears that SENj, LODj, and ACCj represent the best criteria to use for describing potential quantitative capabilities.

131

Listed in Tables 5 and 6 are the results for the simulated study with five components. The same overall trends that resulted for two components with varying resolution and intensity ratios are revealed in the Tables. The figures of merit are in agreement among themselves and with cond (K). A degeneration of selectivity is seen with a decrease in resolution as would be expected. These results are promising for the utility of SENj, ACCi, and LODe-as useful criteria in selecting operating parameters and the study was extended to experimental data for two-component mixtures. Shown in Table 7 are experimental results for determination of 9-methylanthracene and pyrene with absorbance data. The relative errors indicate that the choice of wavelength range is not critical for this system. Wavelength range d has the greatest selectivity but the ACCi values show the poorest accuracy for this range. The large ACCj values are attributed to low sensitivity values (see Eqn. 6). Based on SELj, SENi, TEPj, and ACCj, the wavelength range b appears to be best; however, the LOD leads to a different conclusion. In particular, with range b, pyrene is poorly estimated and has a corresponding poor LODj. With LOD~ as the sole criterion, range a seems best. TABLE 5 T r e n d s of figures of merit for five c o m p o n e n t s with different levels of resolution Resolution

Cond(K)

Component

SEL i

SEN i

LOD i (10 -3 )

TEP i

ACCj (10 -3 )

Rel. error

(%) 10

14.8

1 2 3 4 5

0.409 0.210 0.177 0.211 0.408

17.2 8.85 7.42 8.86 17.2

1.07 2.08 2.48 2.08 1.07

18.2 35.2 42.5 35.4 18.2

1.16 2.25 2.71 2.26 1.16

0.51 1.3 1.2 0.50 0.20

8

34.6

1 2 3 4 5

0.241 0.101 0.0791 0.100 0.205

10.1 4.22 3.31 4.21 10.1

2.02 4.85 6.18 4.86 2.12

33.7 77.2 108 77.8 33.6

2.00 4.59 6.44 4.02 1.99

2.4 8.1 9.7 6.2 1.9

7

57.8

1 2 3 4 5

0.167 0.063 0.048 0.063 0.168

7.06 2.66 2.03 2.67 7.07

3.35 8.89 11.7 8.921 3.28

49.1 143 173 144 49.1

2.81 8.16 9.90 8.23 2.80

2.6 7.9 12.1 9.9 2.8

1 2 3 4 5

0.106 0.036 0.027 0.037 0.108

4.48 1.54 1.14 1.55 4.54

4.37 12.7 17.2 12.3 4.35

83.2 315 492 313 82.9

4.57 2.70 1.73 1.48 4.55

2.0 7.8 13.1 11.9 4.3

6

105

132 TABLE 6 Trends of figures of merit for five components with different levels of resolution within and among runs

Resolution

Cond(K) Component SEL/

SEN1

LODj (10 -~)

TEPi

ACCj (10 -3 )

Rel. error

(%) 20,20,20,7

20,7,7,7

5.07

32.4

1 2 3 4 5

0.918 0.838 0.773 0.386 0.413

34.9 35.1 32.3 16.2 17.1

0.388 0.386 0.419 0.837 0.793

6.67 7.29 7.91 15.9 14.9

0.52 0.56 0.61 1.24 1.17

0.25 0.20 0.40 0.89 0.42

1 2 3 4 5

0.199 0.085 0.081 0.168 0.764

8.36 3.61 3.42 7.12 32.1

1.794.15 4.37 2.11 0.466

38.2 91.1 92.9 45.6 10.0

2.37 5.67 4.78 2.84 0.62

1.4 3.4 3.1 1.3 0.34

In a previous study [ 22 ], an attempt was made to use cond (K) as a measure for the selection of an optimal derivative order. It was concluded that cond (K) was insensitive to noise present in the derivatives and could not be used to predict the best order. The following results give an evaluation of the performance of the other criteria in finding an acceptable derivative order. Table 7 also includes results for the first- and second-derivative data. As a general trend, sensitivity is lower than with absorbance data; this is expected because the Hail] is smaller for the derivatives. Similarly, TEPj systematically decreases with each successive differentiation because of the decrease in ]]d ]1.In contrast, ACCt increases with each order which is primarily attributable to the decrease in SEN t. Selectivities increase with differentiation as a result of the creation of more spectral features. The individual component spectra become more orthogonal, allowing improved resolution. In fact, only the LODt is observed to show random behavior. According to Eqn. 3, LOD/will increase or decrease depending on the direction taken by SELj and Ed. For instance, LODt for wavelength range a is seen to increase continually in spite of the improvement in SELt. The errors in the measurements are seemingly degrading LODt faster than SELj can improve it. Besides the primary effects on SELt and e~ with an increase in the derivative order, the gradual decrease in a t will also cause the decrease in LOD/. Thus LOD/appears to be the better guide for the selection of the correct derivative order to use. For wavelength range c, the LODt continually improves and the concentration estimates are acceptable. Wavelength range b enhances the results for the first derivative but is less reliable for the second derivative. The noise level as well as IlatH appear to have become more significant factors in the second de-

133 TABLE 7 Results of wavelength selection for the determination of 9-methylanthracene and pyrene from absorbance data and from first- and second-derivative data Wavelength range a

Component

SELj

SEN i

LODj (10 -3 )

TEPi

ACCj (10 -2 )

Rel. error

RSD (%)

(%) Absorbance data

a

9-MA b Pyrene

0.907 0.907

0.912 0.544

1.49 1.03

3.01 5.04

2.19 3.68

0.3 0.9

1.7 1.5

b

9-MA Pyrene

0.914 0.914

1.58 0.843

2.68 1.47

2.86 5.40

1.26 2.39

0.1 1.6

1.4 2.2

c

9-MA Pyrene

0.899 0.899

1.56 0.753

2.65 1.22

2.86 5.92

1.29 2.66

0.2 0.9

2.2 2.1

d

9-MA Pyrene

0.966 0.9066

0.118 0.604

0.097 0.331

8.04 2.30

20.7 5.91

1.9 0.9

2.3 3.0

e

9-MA Pyrene

0.687 0.687

0.899 1.09

0.387 0.5/42

6.51 4.65

5.17 3.69

0.7 0.5

3.9 2.4

/

First-derivative data c

a

9-MA Pyrene

0.982 0.982

0.214 0.359

0.304 0.440

2.25 3.25

13.4 19.4

0.8 1.3

2.4 3.4

b

9-MA Pyrene

0.976 0.976

0.531 1.01

0.351 0.643

2.11 3.86

7.46 13.65

0.1 0.9

0.6 1.6

c

9-MA Pyrene

0.967 0.967

0.746 1.55

0.320 0.696

2.04 4.46

7.54 16.5

0.12 0.2

0.4 1.3

d

9-MA Pyrene

0.999 0.999

0.456 0.131

1.05 0.204

10.5 2.04

170.1 3.31

4.3 0.6

9.9 4.5

e

9-MA Pyrene

0.887 0.887

0.500 0.357

0.470 0.302

3.19 2.06

28.6 18.3

0.6 0.3

0.8 0.6

S e c o n d - d e r i v a t i v e data c

a

9-MA Pyrene

0.995 0.995

0.388 0.243

0.425 0.678

2.26 3.62

51.6 82.4

1.7 1.6

4.9 3.1

b

9-MA Pyrene

0.996 0.996

0.657 0.430

0.539 0.823

2.30 3.51

30.4 46.5

1.0 1.8

1.9 4.3

c

9-MA Pyrene

0.994 0.994

0.655 0.348

0.310 0.584

2.17 4.09

30.5 57.5

0.6 0.4

2.6 3.8

d

9-MA Pyrene

0.999 0.999

0.024 0.149

2.18 0.353

12.5 12.04

829.2 135.6

5.6 0.5

8.3 1.1

e

9-MA Pyrene

0.999 0.999

0.049 0.339

4.59 0.663

14.0 2.01

410.1 58.9

6.8 0.7

16.1 1.3

aSee Table 1. b9-Methylanthracene. CLOD/stated as 10-1 units not 10 -3.

134 rivative. Wavelength range e shows a slight improvement with the first derivative but with the second derivative the concentration estimates are unsatisfactory for 9-methylanthracene because the second-derivative spect r u m for this component approaches baseline. This behavior is correctly predicted by the LOD i. It is concluded t h a t if one is concerned only with selection of an optimal set of wavelengths to use for spectrophotometric determinations, the choice of a proper criterion among those presented is usually not critical. The criteria were found to correspond reasonably well with each other. Perhaps a combination of the figures of merit would be more satisfactory. Nonetheless, the LODj seems to be the criterion of choice when the selection of both the optimal wavelength range and derivative order are important. Narrow ranges of wavelengths representing highly selective regions for a given component were seen to offer better results for t h a t component. W h e n selectivity increases, the spectrum of interest is more orthogonal to the remaining spectral features, thus offering better results. This is in agreement with the results of Rossi and Pardue [3]. For systems with very similar spectra, data t r e a t m e n t would perhaps best be handled by factor analysis or partial least squares in latent variables rather t h a n multiple linear regression as used in the present study [20,27]. This research was supported by Grant 586 from the Faculty Research Committee, Idaho State University, Pocatello, ID.

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