Quantification of Hydrofluoric Acid Species by Chemical-Modeling

Anal. Chem. 1997, 69, 25-35

Quantification of Hydrofluoric Acid Species by Chemical-Modeling Regression of Near-Infrared Spectra Christopher J. Thompson,† J. D. Sheldon Danielson, and James B. Callis*

Center for Process Analytical Chemistry, Department of Chemistry, BG-10, University of Washington, Seattle, Washington 98195

Near-infrared spectroscopy in the wavelength region 8001100 nm has been applied to the measurement of aqueous solutions of hydrofluoric acid over the concentration range 0.01-2.65 M and the pH range 2-13. The analysis is based on the detection of subtle perturbations of the water spectrum which result from the presence of the acid and other species in solution. Spectral characterization of the hydrofluoric acid system was performed using a spectrophotometric titration and multivariate model-based regression. Excellent agreement was observed between the spectra and a simple equilibrium model describing the acid’s dissociation. Five species, including hydrofluoric acid, fluoride, bifluoride, hydronium, and hydroxide ions, were quantitated with estimation errors below 10 mM using this procedure. In addition, the species’ spectral profiles were recovered, and optimum estimates of the system’s equilibrium constants were determined. Many industrial processes require careful monitoring and control of hydrofluoric acid levels, due to HF’s corrosive nature, complexing ability, and high toxicity. Frequently, these measurements must be made over wide concentration and pH ranges as may be found in etching baths and dissolver solutions.1-3 Traditional methods for measuring HF are either electrochemical in nature or based on a titration with visual or electrochemical detection of the endpoint. Conventional titrimetric-based analyses can be labor intensive and time consuming and are not well suited for process measurements. Furthermore, they would be inappropriate for solutions with a varying background of another acid. Ion-selective electrodes for measuring fluoride and hydronium ions have been employed in acidic solutions.1,4 However, these latter measurements may not be useful in process situations since they require a constant ionic strength for comparison with standard solutions. Drift in response and a limited pH range are additional problems associated with ion-selective electrodes. Christian et al.2 reported a titanium-based electrode whose current density is linearly related to the HF concentration. In this case, the basis of the response is etching of the electrode’s surface. Although †

Present address: Batelle, Pacific Northwest Division, P.O. Box 999, MSIN K6-96, Richland, WA 99352. (1) Entwistle, J. R.; Weedon, C. J.; Hayes, T. J. Chem. Ind. (London) 1973, 433-434. (2) Christian, J. D.; Illum, D. B.; Murphy, J. A. Talanta 1990, 37, 651-654. (3) Nishimura, S.; Moriyama, J.; Kushima, I. Trans. Jpn. Inst. Met. 1964, 5, 79-85. (4) Eriksson, T. Anal. Chim. Acta 1973, 65, 417-424. S0003-2700(96)00455-6 CCC: $14.00

© 1996 American Chemical Society

the electrode is sensitive at low HF concentrations (0.01-0.07 M), its stability at much higher concentrations has not been reported and may prove to be problematic due to an accelerated dissolution of the electrode surface. Near-infrared spectroscopy (NIRS) has a number of features that make it attractive as a process measurement technique.5 Most of the absorption bands observed in this wavelength region (7002500 nm) are due to vibrational overtones and combinations of C-H, N-H, and O-H stretches. In the short wavelength (SW) region (700-1200 nm), the extinction coefficients for these transitions are relatively low, which results in a high degree of linearity and the ability to employ long path lengths, simplifying sampling requirements. Moreover, instrumentation for NIRS is inexpensive and rugged, and signal-to-noise ratios of 1 × 104 are readily obtained.6 Another benefit is the potential to perform noninvasive and multicomponent analyses.5,7 However, there are drawbacks to NIRS, including its low sensitivity and the characteristic broadness of absorption bands, which often result in overlapped spectra. For such spectra, multivariate calibration methods, such as multiple linear regression (MLR), principal component regression (PCR), and partial least squares (PLS), are usually employed to relate spectral variance to chemical concentrations. The application of NIRS to the measurement of caustic brine in solution8 illustrates the utility of NIRS combined with either MLR or PLS for process-type measurements. Allan and Gahler first reported the use of NIRS for determining HF in aqueous solutions.9 In their work, the free HF concentration was shown to be strongly correlated to the absorbance at 1835 nm. However, as they pointed out, the analysis is prone to interference by other acids and electrolytes. Furthermore, the large extinction coefficient of water at 1835 nm requires the use of thin cells, which are impractical for process measurements. In this study, we have explored the use of the SW near-IR region and multivariate calibration methods for developing an assay that can be adapted to process measurements. To extend the utility of the method, we have considered both wide concentration and pH ranges, as well as the presence of other ions in solution. (5) Callis, J. B.; Illman, D. L.; Kowalski, B. R. Anal. Chem. 1987, 59, 624A637A. (6) Nadler, T. K.; McDaniel, S. T.; Westerhaus, M. O.; Shenk, J. S. Appl. Spectrosc. 1989, 43, 1354-1358. (7) Kelly, J. J.; Barlow, C. H.; Jinguji, T. M.; Callis, J. B. Anal. Chem. 1989, 61, 313-320. (8) Phelan, M. K.; Barlow, C. H.; Kelly, J. J.; Jinguji, T. M.; Callis, J. B. Anal. Chem. 1989, 61, 1419-1424. (9) Allan, W. J.; Gahler, A. R. Anal. Chem. 1959, 31, 1778-1783.

Analytical Chemistry, Vol. 69, No. 1, January 1, 1997 25

EXPERIMENTAL SECTION Reagents. Concentrated hydrofluoric acid (48%-51%) was reagent grade (Baker). Stock solutions of approximately 3 M were prepared by diluting weighed aliquots of the concentrated acid in polyethylene containers. The concentrations were determined by titration with a 1.000 ( 0.002 M standard sodium hydroxide solution (VWR). A 4.500 M potassium hydroxide titrant was prepared using Baker reagent-grade reagent. Standardization of the titrant was performed by titrating aqueous solutions of potassium hydrogen phthalate (Aldrich). Samples. Fifteen dilution-series samples spanning 0-2.004 M in uniform increments were prepared by weight from a stock solution of HF. A second calibration set consisting of 45 titration samples was prepared by adding variable amounts of the titrant solution to aliquots of a 2.915 M HF stock solution. The relative amounts of acid and base were chosen such that the samples represented the complete titration of a 50-mL aliquot of the HF stock solution. The corresponding pH range of the titration samples was approximately 2-13. This same procedure was also used to prepare a set of 25 validation samples, except the amount of titrant added to each validation sample was randomly selected to fall within the same range as the calibration set. Instrumentation. SW near-IR spectra (800-1100 nm) were measured with a Hewlett Packard 8452A diode-array spectrophotometer which was controlled by a 386-based personal computer. Data acquisition was carried out using in-house written C-language programs10 which were called from the MATLAB11 software environment. This arrangement enabled real-time calculations to be performed on the spectra as they were acquired. Initial experiments with HF solutions indicated that drift in the instrument response was a significant factor in limiting the measurement reproducibility. Accordingly, a number of modifications were made to improve the long-term stability of the spectrophotometer. To reduce the temperature drift of the silicon detector, a baffle was installed to thermally isolate the power supply from the cavity housing the optics. In addition, a thermostating system, including a resistive heater, a circulating fan, and a temperature sensor (Part No. AD590, Analog Devices, Norwood, MA), was mounted inside the optics cavity. Fluctuations in the source intensity were reduced by monitoring the lamp’s output with a photodiode and adjusting the lamp voltage with an electronic feedback circuit. Repetitive measurements of the baseline showed that these modifications greatly stabilized the detector’s response. Over a 5-h period, offsets in the baseline were reduced by a factor of 20 at 850 nm and by a factor of 50 at 1050 nm. The greater improvement at longer wavelengths clearly demonstrates the advantage of thermostating the silicon detector. Sample thermostating was performed by a custom-built, thermostated cell holder, shown in Figure 1. Two independently controlled thermostating systems were built into the holder to regulate the sample’s temperature. The primary system used a thin heating foil (Thermofoil, Minco Products, Minneapolis, MN) to heat the copper block that holds the cuvette. Subambient operating temperatures were made possible by the secondary thermostating system, which employed a thermoelectric module (Part No. CP-1.4-127-06L, Melcor, Trenton, NJ) to cool an outer (10) Aldridge, P. K.; Kelly, J. J.; Callis, J. B. Anal. Chem. 1993, 65, 3581-3585. (11) The MathWorks, Inc. AT-MATLAB for MS-DOS Computers, Natick, MA, 1989.

26 Analytical Chemistry, Vol. 69, No. 1, January 1, 1997

Figure 1. Thermostated sample holder.

Figure 2. Automated sample-handling system.

heat shield. For both systems, temperatures were regulated with feedback circuitry, based on the response of a temperature sensor (Part No. AD590, Analog Devices). Sample equilibration was monitored with a thermistor (Part No. 112-103FAJ-BO1, Fenwall Electric, Milford, MA), which was interfaced to an A/D converter (Model VF900, Real Time Devices, State College, PA) operating at a resolution of 20 bits. The high-resolution A/D converter was also multiplexed to a platinum resistance thermometer (Part No. S9689PL5X12, Minco Products, Minneapolis, MN) for more accurate measurement of the cell holder’s temperature. Automated sample-handling was accomplished by a computercontrolled flow system (Figure 2). An electronically actuated fourposition valve (Part No. DCSD4PHC, Vici Valco Instruments Co., Houston, TX), constructed from Hastelloy C, was used to select between the system components that deliver samples to and from the cuvette. Automatic, sequential introduction of sample solutions into the cuvette was performed by an ISIS autosampler (Isco, Lincoln, NE) and a peristaltic pump. Removal of solutions from the cuvette was accomplished with a vacuum line. To isolate the vacuum and prevent liquid from being drawn through the fourposition valve at inappropriate times (i.e., when the valve is moving), an electronically actuated, two-position Teflon valve was placed in the vacuum line. This latter valve was opened only after the main valve was in the vacuum position. Thermal equilibration times were reduced by passing sample solutions through a temperature preconditioner, which contained a 5-ft coil of 1/16-in. PEEK tubing (Item No. 1531, Upchurch Scientific, Oak Harbor, WA) mounted in a thermostated copper block. Like the heat shield of the thermostated cell holder, the preconditioner was monitored with a temperature sensor (Part No. AD590, Analog

Devices) and regulated by a thermoelectric module (Part No. CP1.4-127-06L, Melcor) and feedback circuitry. The sample-handling system’s valves, pumps, and autosampler were interfaced to an 80386-based PC using a 48-line digital I/O board (Model DIO48-2, Real Time Devices). Software for controlling the system components was written in C and was interfaced to MATLAB using the “mex-file” utility. Measurement Protocol. A complete sample-measurement cycle consisted of four steps: (1) rinsing the cuvette twice with sample solution, (2) pumping sample solution into the cuvette and waiting for thermal equilibration to occur, (3) acquiring the sample’s spectrum, and (4) removing the sample from the cuvette. The cell holder was thermostated at 25 °C and was shown to be stable to (0.002 °C during the analysis of each sample set. Sample temperature equilibration was monitored by measuring the sample’s temperature every 2 s with the thermistor until the standard deviation of the previous 10 temperature readings dropped below 0.0004 °C. Each sample’s spectrum was immediately measured after the sample reached thermal equilibrium. All of the spectral measurements were summed to memory for 15 s (the maximum allowed by the HP 8452A) to maximize the signal-to-noise ratio. Sample spectra were referenced to a scan of the empty 1-cm polystyrene cuvette. To partially compensate for any remaining drift in the instrument, a new blank was acquired after every five sample measurements. In addition, accidental correlations to instrument and temperature drifts were minimized by analyzing samples in a randomized sequence. Data Analysis Software. Computations were performed using MATLAB. Nonlinear optimization routines, based on the Levenberg-Marquardt method12 and the Nelder-Mead simplex algorithm,13 were from the MATLAB Optimization Toolbox. Modules for spectral preprocessing and multivariate analysis were developed on a 486-based PC. THEORY Speciation in Aqueous HF Solutions. The nature and speciation of hydrofluoric acid have been actively investigated for many years. Two equilibria have generally been accepted for describing the HF system: Ka

HF \ y z H+ + FK2

HF + F- y\z HF2-

(1) (2)

to the formation of dimers or higher-order complexes involving neutral HF molecules.23 More recently, Giguere and Turrell24-26 have inferred from the infrared spectra of the HX acids that HF is, indeed, largely dissociated, with the apparent acid strength rationalized by the formation of ion pairs between H3O+ and F-:

H2O + HF h [H3O+‚F-] h H3O+ + F-

According to Giguere, the strongly hydrogen-bonded complex accounts for both the nearly complete ionization of HF and the observed weakness of the dilute acid. Giguere’s theory also includes provision for the formation of the bifluoride ion:

[H3O+‚F-] + HF h H3O+ + HF2-

(12) Marquardt, D. SIAM J. Appl. Math. 1963, 11, 431-441. (13) Nelder, J. A.; Mead, R. Comput. J. 1965, 7, 308-313. (14) Davies, C. W.; Hudleston, L. J. J. Chem Soc. 1924, 125, 260-268. (15) Broene, H. H.; DeVries, T. J. Am. Chem. Soc. 1947, 69, 1644-1646. (16) Hepler, L. G.; Jolly, W. L.; Latimer, W. M. J. Am. Chem. Soc. 1953, 75, 2809-2810. (17) Connick, R. E.; Hepler, L. G.; Hugus, Z. Z.; Kury, J. W.; Latimer, W. M.; Tsao, M. J. Am. Chem. Soc. 1956, 78, 1827-1829. (18) Ahrland, S. Helv. Chim. Acta 1967, 50, 306-318. (19) Mesmer, R. E. Anal. Chem. 1968, 40, 443-444. (20) Srinivasan, K.; Rechnitz, G. A. Anal. Chem. 1968, 40, 509-512. (21) Mesmer, R. E.; Baes, C. F., Jr. Inorg. Chem. 1969, 8, 618-626. (22) Baumann, E. J. Inorg. Nucl. Chem. 1969, 31, 3155-3162.

(4)

In the limit where virtually no free HF exists in solution (i.e., the first equilibrium in eq 3 lies far to the right), the equilibrium behavior of eqs 1 and 2 is essentially the same as that in eqs 3 and 4, except H+ becomes H3O+, and HF becomes [H3O+‚F-]. Iterative Model-Based Regression. Because HF exists in equilibrium with several other species in solution, developing a suitable calibration model based solely on spectroscopic data requires knowledge of the equilibria involved as well as accurate estimates of the equilibrium constants. Furthermore, identification of the spectral features associated with each species is desirable to ensure that the calibration model is based on chemically meaningful variance in the spectra. For these reasons, an iterative, model-based regression method,27 hereafter referred to as chemical-modeling regression (CMR), was used to characterize the titration spectra and develop a calibration for the HF system. In principle, this approach can be used to estimate equilbrium constants, species’ concentration profiles, and pure-component spectra. The utility of CMR for characterizing chemical equilibria has previously been illustrated in a study of the stepwise formation of zinc bromide complexes using Raman spectroscopy28 and in the elucidation of species in alcohol-water mixtures using NIRS.29 We begin by defining a matrix of measured spectra, R, as p samples by q wavelengths. Assuming linear additivity of components, the matrix of spectra R is equal to the product of a concentration matrix, C, and a matrix of standard spectra of the pure components, S:

R ) CST + ER Although the published values for both Ka and K2 show considerable variation,14-22 values for the pKa are on the order of 3, indicating weak acid behavior. This unexpected behavior has been attributed both to the high strength of the H-F bond and

(3)

(5)

where C is dimensioned p × r constituents, S is q × r, and ER is a p × q residuals matrix. In favorable situations (simple kinetics or equilibria), estimates of C may be obtained with a kinetic or thermodynamic model that describes the concentrations of each of the sample’s constituents. For example, if the spectra were acquired over the course of a titration experiment, an equilibrium (23) Warren, L. J. Anal. Chim. Acta 1971, 53, 199-202. (24) Giguere, P. A.; Turrell, S. Can. J. Chem. 1976, 54, 3477-3482. (25) Giguere, P. A. Chem. Phys. Lett. 1976, 41, 598-600. (26) Giguere, P. A.; Turrell, S. J. Am. Chem. Soc. 1980, 102, 5473-5477. (27) Sylvestre, E. A.; Lawton, W. H.; Maggio, M. S. Technometrics 1974, 16, 353-368. (28) Heuman, J.; Ozeki, T.; Irish, D. E. Can. J. Chem. 1989, 67, 2030-2036. (29) Alam, M. K.; Callis J. B. Anal. Chem. 1994, 66, 2293-2301.

Analytical Chemistry, Vol. 69, No. 1, January 1, 1997

27

model may be used to assess the concentration profiles as a function of the pH. Because model parameters (i.e., rate constants or equilibrium constants) are usually dependent on the experimental conditions, the calibration problem consists of determining the model parameters and obtaining estimates of the purecomponent spectra. The first step in chemical-modeling regression is to decompose the response matrix, R, by the singular value decomposition:30,31

where C ˆ + is the pseudoinverse of C ˆ and is defined as (C ˆ TC ˆ )-1C ˆ T. At this point, the initial values of the model parameters are evaluated by first calculating an estimate of U h using the rotation matrix,

R ) UΣVT

Z)U hˆ - U h

(7)

U h ) CHT

(8)

The rotation matrix H gives a nonorthogonal rotation of U h to C. To obtain an estimate of H, the chemical model must be employed to generate estimates of the concentrations. Generally, this involves specifying initial values for the model parameters. These values are then used to calculate an initial value for C. Given an estimate for C, eq 8 can be solved for the rotation matrix: T

H ˆ )C ˆ U h +

(11)

Optimum values for the model parameters may be obtained by varying the parameters and minimizing the norm of Z in a systematic fashion, such as by the use of the LevenbergMarquardt refinement algorithm.12 As noted by Shrager,34 the model parameters may be determined more accurately if the scores are weighted by the square of their singular values (i.e., the eigenvalues). In this case, the weighted residual matrix, Zw, is given by

Zw ) (U hˆ - U h )Σ h2

(12)

Once the model parameters have been determined, the pure component spectra may be found with

Sˆ ) V hΣ hH ˆ where the bars over the matrices indicate that rank reduction has been performed. The truncated scores matrix, U h , is now dimensioned p × s, where s is the number of factors retained (i.e., the pseudorank), Σ h is s × s, and V h is q × s. A useful method for assessing the pseudorank is evolving factor analysis (EFA).32,33 In EFA, the data matrix R is ordered according to some parameter, such as pH or concentration. Submatrices of the data matrix, formed by excluding successive spectra, are decomposed by the singular value decomposition. The logarithms of the singular values resulting from each decomposition are then plotted against the chosen parameter. In some cases, such a plot provides a clear indication of the number of significant components. As described earlier, the scores are a basis set for the concentrations and are related to them through a rotation matrix H:

(10)

and then determining a residual matrix, Z, by

(6)

where the columns of U (p × p) are unit length, orthogonal eigenvectors of RRT, the columns of V (q × q) are unit length, orthogonal eigenvectors of RTR, and Σ is a p × q diagonal matrix that contains the singular values (square roots of the eigenvalues of both RRT and RTR). The columns of U (“score” vectors) constitute a basis for the concentrations, and the columns of V (“loading” vectors) span the spectral space of S. Normally, the factors that are associated with small singular values are attributed to random noise and are discarded. This data reduction step results in a reduced-noise, biased representation of R which may be written as

R h )U hΣ hV hT

U hˆ ) C hH hT

(13)

and estimates of the pure-component concentrations may be obtained from

ˆˆ ) U C h (H ˆ T)+

(14)

ˆ is an estimate of the concentration profiles which where C includes the measurement errors. Prediction of an unknown sample’s concentrations, cˆ un, can be made using

cˆ un ) run Sˆ (Sˆ TSˆ )-1

(15)

where run is a 1 × q vector containing the unknown sample’s spectrum. An alternative approach, which takes into account the model constraints, is to iteratively vary the elements of cˆ un until a minimum in

||cˆ unSˆ T - run||

(16)

is obtained, subject to the concentration relations specified by the model. For example, in the chemical equilibrium process K

A y\z B + C

(17)

K ) [B][C]/[A]

(18)

(9) the relation

(30) Golub, G. H.; Vanloan, C. F. Matrix Computations, 2nd ed.; Johns Hopkins University Press: Baltimore, MD, 1989; pp 71-74. (31) Malinowski, E. R. Factor Analysis in Chemistry, 2nd ed.; Wiley Interscience: New York, 1991. (32) Gampp, H.; Maeder, M.; Meyer, C. J.; Zuberbu ¨ hler, A. D. Talanta 1985, 32, 1133-1139. (33) Cartwright, H. J. Chemom. 1987, 1, 111-120.

28


holds, and the elements of cˆ un in eq 16 could be constrained (34) Shrager, R. I. Chemom. Intell. Lab. Syst. 1986, 1, 59-70.

of the O-H stretch.37 Somewhat surprisingly, no positive features are apparent near 880 nm, the expected location of the second overtone of the H-F stretch (calculated from constants listed by Herzberg38). Calibration Using Dilution Standards. In the traditional NIRS calibration experiment, the experimental design is based on a dilution series of standard solutions. This approach was initially adopted for HF using a set of 15 samples that uniformly spanned the concentration range of 0-2 M. To account for the weak acid behavior of HF, the equilibrium concentrations of HF, H+, F-, HF2-, and OH- were estimated from the following system of five equations:

Figure 3. Short-wave near-infrared spectra of water and a 2 M solution of hydrofluoric acid.

Figure 4. Result of subtracting the spectrum of water from the spectrum of the 2 M solution of hydrofluoric acid.

according to

cˆ un 3 (K + δK) g cBcC/cA g (K - δK)

(19)

where δK represents the uncertainty in the equilibrium constant, and cA, cB, and cC are the concentrations of the species A, B, and C. RESULTS AND DISCUSSION Spectral Features of Water and Hydrofluoric Acid Solutions. Figure 3 shows the spectrum of water along with the spectrum of a 2 M HF solution. Both spectra are dominated by a broad absorbance band centered at 970 nm, which has been assigned to the 2ν1 + ν3 combination of symmetric and antisymmetric stretching motions of the O-H bond.35,36 Also indicated by Figure 3 is the relatively small influence HF has on the water spectrum. The relative changes are better illustrated in Figure 4, which displays the result of subtracting the water spectrum from the spectrum of the HF solution. Presumably, the large negative feature near 1008 nm is associated with the reduced concentration of water due to dilution by HF in addition to changes in the bulk water structure. Another interesting property of the difference spectrum is the slightly enhanced absorbance at 965 nm, which corresponds with the location of the second overtone (35) Bayly, J. G.; Kartha, V. B.; Stevens, W. H. Infrared Phys. 1963, 3, 211-223. (36) Buijs, K.; Choppin, G. R. J. Chem. Phys. 1963, 39, 2035-2041.

Ka ) [H+][F-]/[HF]

(20)

K2 ) [HF2-]/[HF][F-]

(21)

Kw ) [H+][OH-]

(22)

[H+] ) [F-] + [HF2-] + [OH-]

(23)

[HF]0 ) [F-] + [HF] + 2[HF2-]

(24)

where [HF]0 is the nominal concentration of HF and the values for Ka and K2 were taken from Baumann.22 Equations 20 and 21 are the mass-action expressions associated with the HF equilibria (eqs 1 and 2), eq 22 is the corresponding expression for the autoprotolysis of water, and eqs 23 and 24 are the charge balance and mass balance expressions, respectively. This system of equations contains nine variables: four of these (Kw, Ka, K2, and [HF]0) are known for each sample, while the five species’ concentrations are unknowns. The equations were solved for each of the 15 calibration samples to give the expected equilibrium concentrations (Figure 5). According to the model, HF is the dominant species in the solutions, with a maximum concentration of 1.79 M. Furthermore, the species’ concentration profiles are nearly linear, except at very low levels of nominal HF. Note that the model actually estimates the equilibrium activities, which we have assumed to be identical to the concentrations. This assumption may not be valid at higher concentrations, but it appears to be justified by the very strong correlation that was observed between the model and the spectra of the calibration solutions, as described below. The SW near-IR spectra of the calibration solutions were preprocessed with a moving-average smoothing filter (22-nm window) and the second-derivative transformation to minimize baseline offsets and slope changes. Figure 6 displays the resulting data set. A direct consequence of the derivative operation is the splitting of the spectra into regions of positive and negative absorbance. To better visualize the intensity changes associated with concentration, the second derivative water spectrum was subtracted from all of the spectra in the set (Figure 7). The resulting difference spectra show that the relative changes in absorbance appear to be linearly spaced, with the largest variations centered at 972 and 1004 nm. (37) Wheeler, O. H. Chem. Rev. 1959, 59, 642-645. (38) Herzberg, G. Molecular Spectra and Molecular Structure; Prentice Hall: New York, 1939; Vol. I.


29

Figure 7. Second-derivative difference spectra of the 15 dilutionseries solutions. Table 1. Intercorrelations for the Dilution-Series Experimental Design species H+ FHF2HF

Figure 5. Model equilibrium concentrations of the 15 calibration solutions.

Figure 6. Second-derivative spectra of the 15 HF solutions (0-2 M).

Multiple linear regression39 (MLR) with step-forward wavelength selection was used to relate the second derivative spectra to the model concentrations. Although the strongest correlation to concentration was observed at 996 nm, most regions of the spectra showed a linear relation to [HF]. To maximize the sensitivity of the measurement, the calibration was based on the second derivative absorbance at 1002 nm, where the spectral differences were larger. This approach resulted in a standard error of estimation of 0.005 M and an R2 (squared correlation coefficient) value of 0.999 94. In addition, the detection limit for (39) Martens, H.; Næs, T. Multivariate Calibration; John Wiley and Sons: New York, 1989; pp 63-64.

30 Analytical Chemistry, Vol. 69, No. 1, January 1, 1997

H+

F-

HF2-

HF

1.000 0.709 0.998 0.999

0.709 1.000 0.664 0.685

0.998 0.664 1.000 0.999

0.999 0.685 0.999 1.000

HF was estimated at 0.009 M, based on three times the standard deviation of the second derivative absorbances of five deionized water spectra. Model validation was performed using cross validation;40 the low standard error of prediction (0.008 M) suggests that the model is reliable for pure solutions of HF. The above calibration results demonstrate the feasibility of determining HF by SW-NIRS. However, there are two problems associated with a single-wavelength calibration. The first is that the variance in the spectra appears to be attributed in part to a lowering of the water concentration as [HF] increased. This would imply an indirect correlation to the HF concentration, which would not hold in process situations where the water concentration might be changed by the addition of other solutes to the solution. The second problem with the calibration concerns the experimental design and the rank of the spectral data matrix. Assuming the equilibrium model is valid, a high degree of correlation exists between the four significant species in solution. This is evident from the nearly linear concentration profiles in Figure 5 and the species’ intercorrelation coefficients, which are shown in Table 1. Because the values approach unity, the effective rank of the spectra is expected to be less than the number of species in solution. In fact, the rank appeared to be only 2, based on the apparent structure and reproducibility of the eigenvectors obtained from singular value decomposition of the matrix of second derivative spectra. As a result, HF cannot be measured independently using a dilution-series approach. In the regime of a titration experiment, the various ionic forms of HF would be expected to vary more independently. Consequently, a titration experiment was chosen as a preferred method for carrying out the process of system identification and the development of a multicomponent calibration for the HF system. Spectra of the Titration Samples. The spectra of the titration samples were preprocessed in the same manner as the dilution (40) Stone, M.; Roy, J. Statist. Soc. B 1974, 36, 111-148.

Figure 8. Second-derivative difference spectra of the titration samples. For clarity, only half of the spectra are displayed. The approximate pH range of the samples was 2-13.

Figure 9. Evolving factor analysis of the second derivative spectra.

series spectra. To the unaided eye, the resulting second derivative spectra were similar to those in Figure 6, except the relative variance between 960 and 985 nm appeared to be approximately twice as large. To better visualize trends in the data, the second derivative spectrum of the first titration sample (2.915 M HF with no added KOH) was subtracted from all of the spectra in the set. Half of the resulting difference spectra are shown in Figure 8. As the amount of added base increased, the region of maximum absorbance (972 nm) increased steadily until the endpoint of the titration. This behavior is consistent with the trend in the difference spectra of the series-dilution samples (Figure 7). Following the endpoint, the peak in Figure 8 shifts to longer wavelengths and decreases in intensity. Also apparent is the growth of new bands at 944 and 956 nm. These latter features appear to be associated with KOH and KF, because peaks at the same positions were observed in the difference spectra of pure KOH and KF solutions. Rank Determination. Evolving factor analysis was performed on the matrix of second derivative spectra; the results are shown in Figure 9. As indicated from the graph, five eigenvectors seem to be significant over the course of the titration. In addition, the rank appears to increase near 5, 12, and 32 mL of added base, suggesting a change in the species’ distribution as a function of pH. The appearance of the last component coincides well with the endpoint, which occurred at 32.39 mL. As an additional diagnostic, singular value decomposition was performed, and the resulting score and loading vectors were examined. Systematic patterns were evident in the first five

Figure 10. Expected equilibrium concentrations for the 45 titration samples.

eigenvectors, while the remaining factors seemed to be dominated by random noise. Indeed, an analysis of subsequent sample sets indicated that only the first five factors were reproducible. The rank of the data was therefore determined to be 5. Model Formulation. In the conventional equilibrium description of HF (eqs 1 and 2), five species are predicted to be significant at the beginning of the titration: HF, F-, HF2-, H+ (or H3O+), and H2O. With the addition of KOH, K+ and OH- are added to the system, eventually bringing the total number of species up to seven. Because the titration spectra were determined to be rank 5, it would appear that either the data are inconsistent with this model or some of the system components are not detectable in the spectra. However, the constraint of charge balance reduces the number of linearly independent species by one. Furthermore, if we assume that the hydration numbers of the species remain constant, then the water concentration is completely dependent on the other solution components, and the expected rank of the data (5) is in agreement with the experimental estimate. Therefore, estimation of the samples’ concentrations was based on the equilibria in eqs 1 and 2. As with the dilution standards, a quantitative expression for the equilibrium concentrations was obtained by solving the system of equations that describes the equilibrium behavior of hydrofluoric acid. In this case, the relevant equations include the massaction expressions (eqs 20-22), the mass-balance relation (eq 24), and a modified charge-balance equation which now includes the potassium ion:

[H+] + [K+] ) [F-] + [HF2-] + [OH-]

(25)

Dilution by the addition of base was accounted for by setting

[HF]0 ) VHFFHF/(VHF + VKOH)

(26)

[K+] ) VKOHFKOH/(VHF + VKOH)

(27)

and

where VHF is the initial volume of HF, VKOH is the total volume of added KOH, and FHF and FKOH are the formal concentrations of Analytical Chemistry, Vol. 69, No. 1, January 1, 1997

31

Figure 11. Optimized score vectors’ fit. Data markers represent the actual score values, and the estimated values are represented by solid lines: (a) first eigenvector, (b) second eigenvector, (c) third eigenvector, (d) fourth eigenvector, and (e) fifth eigenvector.

HF and KOH before mixing. Note that eqs 26 and 27 are only approximations, because the volume of mixing has been neglected. Using equilibrium constants from Baumann,22 the expected concentrations for the 45 titration samples were calculated; Figure 10 shows the results. According to this model, HF is the dominant species for about the first third of the titration. Also of interest is the low concentration of H+, with a maximum value of approximately 0.16 M. Because the H+ concentration drops off rapidly, several samples with less than 2 mL of added KOH were included in the sample set. Other features to note in Figure 10 are the rise and decline of the bifluoride ion and the significant concentration of free fluoride past the midpoint of the titration. Fluoride-selective electrode measurements made on the titration samples showed excellent agreement with the equilibrium model except for the samples near the endpoint, where the hydroxide ion interfered with the electrode’s response. Estimation of the Equilibrium Constants. Optimal values for Ka and K2 were found by systematically varying both parameters until the best agreement was obtained between the first five scores and their corresponding estimates, which were based on the model concentrations. During the fitting process, the scores were weighted by the eigenvalues to decrease the influence of the fifth factor, which appeared to contain a significant amount of noise. The best fit was obtained when Ka ) 6.4 × 10-4 and K2 ) 3.12 (Figure 11). These values are slightly lower than those determined by Baumann (i.e., 6.85 × 10-4 for Ka and 5.0 for K2),22 but they fall well within the range of other published estimates.41 As indicated in Figure 11, the general agreement between the scores and the model is quite good. However, large residuals are apparent in the fifth factor, which makes the subsequent rotation of the scores and loadings into the titration curves and spectra somewhat ambiguous. As a result, the precision of the equilibrium constants is degraded.

Quantitative estimates of the uncertainties in the model parameters were obtained by performing a series of 50 Monte Carlo calculations. In each calculation, optimum values of Ka and K2 were determined for an artificial data set which was constructed by adding the original, second derivative titration spectra to a matrix containing simulated measurement errors. Each row of this latter matrix was comprised of normally distributed, random numbers that were scaled by the standard deviation spectrum, which was calculated from a set of 40 water spectra. To make the simulation as realistic as possible, the water spectra were measured and preprocessed in the same manner as the titration spectra. The standard deviations of the equilibrium constants obtained using this procedure were 1 × 10-5 for Ka and 0.06 for K2. These values are slightly lower than the uncertainties obtained by Baumann22 using ion-selective electrodes. Quantitation Results. Estimates of the concentrations were obtained by rotating the scores into the concentration space. Figure 12 shows the results of this transformation. Based on the quality of the fit in Figure 11, it is not surprising to see good agreement between the scores and the equilibrium model. The standard errors of estimation for each species are presented in Table 2. The measurement errors were under 10 mM for all five species. Also notable is the value for HF, 0.007 M, which agrees fairly well with the estimated error from the dilution standards’ calibration (0.005 M). In general, when the spectra of two or more analytes overlap, it is impossible to unambiguously determine both their extinction coefficient spectra and concentrations from mixture data. This is a well-known problem in factor analysis which has been addressed for two- and three-component mixtures by imposing the restriction that the spectra must be nonnegative at all wavelengths.42-44 With chemical-modeling regression, the degree

(41) Saloman, M.; Stevenson, B. K. J. Chem. Eng. Data 1974, 1, 42-44.

(42) Lawton, W. H.; Sylvestre, E. A. Technometrics 1971, 13, 617-633.

32


Figure 12. Estimated concentrations from chemical-modeling regression. Data markers denote the rotated scores, and the solid lines represent the model values: (a) HF, (b) F-, (c) HF2-, (d) H+, and (e) OH-.

Table 2. Standard Errors of Estimation (SEE) and Prediction (SEP) Obtained Using Chemical-Modeling Regression species

concentration range (M)

SEE (M)

unconstrained SEP (M)

constrained SEP (M)

HF FHF2H+ OH-

0.00-2.65 0.01-1.77 0.00-0.70 0.00-0.14 0.00-0.45

0.007 0.005 0.005 0.004 0.002

0.006 0.007 0.006 0.005 0.001

0.006 0.007 0.006 0.003 0.001

of ambiguity is diminished by constraining the results to obey a specific model. Nevertheless, uncertainties in the estimated concentrations and spectra arise from the uncertainties in the model parameters. Regions of ambiguity in the calibration standards’ concentrations were generated by calculating the model profiles that resulted after adding and subtracting the CMR uncertainties in the model parameters from Ka and K2. Figure 13 shows the result of subtracting the average concentration profiles from the upper and lower boundaries. HF showed the greatest uncertainty, with maximum values of approximately (0.004 M. Also of interest are the shapes of the boundaries, which appear to share common features. For example, the plots for HF, F-, and HF2- all contain a broad region of ambiguity whose positive component resembles the concentration profile of the bifluoride ion. This feature apparently results from the high degree of overlap in the species’ concentrations and indicates their relative dependence on each other. Pure-Component Spectra. The reconstructed second derivative spectra of the pure components are plotted in Figure 14. Although the spectra are highly overlapped and similar in appearance, they contain subtle shifts in shape and marked (43) Ohta, N. Anal. Chem. 1973, 45, 553-556. (44) Borgen, O. S.; Kowalski, B. R. Anal. Chim. Acta 1985, 174, 1-26.

intensity changes. The relative peak heights might be related to the number of water molecules associated with each species. For example, the potassium fluoride spectrum is approximately fourthirds as intense as the spectrum of HF and KOH, which suggests a 4:3 ratio in these species’ hydration numbers. Also of interest in Figure 14 is the positive intensity of the hydronium ion’s spectrum in the region from approximately 950 to 1115 nm. This unusual behavior has also been reported by Biermann and Gilmour,45 who studied the near-IR spectra of nitric and perchloric acids in the longer-wavelength region of 1300-1900 nm. They attributed the effect to a reorientation of water dipoles around the hydronium ions, resulting in a solution that is more transparent than pure water. Thus, the apparent negative (positive in the second derivative domain) absorbance is caused by a refractive index effect which is superimposed on the positive spectrum of the hydronium ion. This explanation appears to be consistent with our data, because in the titration of HF, water is not an independent species in the experimental design, and the water’s contribution to the spectra has not been removed. Ambiguities in the pure-component spectra were estimated from the uncertainties in Ka and K2 using the following procedure. First, the uncertainties in the equilibrium constants, δKa and δK2, were added to Ka and K2. The equilibrium model was then used to estimate the species’ concentrations, and the associated purecomponent spectra were calculated using eqs 9 and 13. This process was repeated to generate the analogous set of purecomponent spectra that result after subtracting δKa and δK2 from the model parameters. For each species, the resulting pair of spectra showed nearly imperceptible changes. Therefore, the average spectrum, calculated from the mean values of Ka and K2, was subtracted from each bounding spectrum (Figure 15). The spectrum of the hydronium ion is the most ambiguous, as expected from the low concentration of H+ in solution. (45) Biermann, W.; Gilmour, J. B. Can. J. Chem. 1959, 37, 1249-1253.


33

Figure 13. Regions of ambiguity in the species’ concentration profiles: (a) HF, (b) F-, (c) HF2-, and (d) H+.

Figure 14. Pure-component spectra obtained from chemicalmodeling regression.

Prediction Results. Validation of the CMR calibration was performed by estimating the concentrations in an independent set of 25 prediction samples. Each prediction sample was made to represent a single, randomly selected point in a titration. After the samples’ spectra were acquired, both unconstrained and constrained methods were used for the prediction, as described earlier. In the latter procedure, the constraints were of the form

(Ka + δKa) g [H+][F-]/[HF] g (Ka - δKa)

(28)

(K2 + δK2) g [HF2-]/[HF-][F-] g (K2 - δK2) (29) where δKa and δK2 are the uncertainties in Ka and K2 that were 34


determined from the Monte Carlo simulation. The standard errors of prediction (SEP) from both approaches are listed in Table 2. The prediction errors agree well with the errors in the calibration. Furthermore, except for H+, both the constrained and the unconstrained methods performed equally well. This latter result was somewhat surprising, since the constrained minimization increased the admissible range of concentrations allowed by considering the uncertainties in the equilibrium constants. However, we found that the minimization required a reasonable first guess in order for convergence to occur, and we based the initial guess on the unconstrained least-squares estimate. While this approach does not require prior knowledge about each sample, the initial guess typically fell within the constraints, and the minimization was usually completed without changing the concentration estimates far from the initial values. Thus, in practice, both the constrained and unconstrained methods yielded comparable errors. CONCLUSIONS This research has shown that SW-NIRS and chemical-modeling regression can be successfully applied to the simultaneous measurement of HF, F-, HF2-, H+, and OH- over broad ranges of concentration and pH. By reducing the drift in the spectrophotometer and the samples’ temperatures, we were able to achieve quantitation errors below 10 mM for each of the five species. However, because of the high degree of spectral overlap and the low concentration of H+, ambiguities in the equilibrium constants and concentration estimates were observed. These ambiguities, which are expressed as inaccuracies, require con-

Figure 15. Ambiguities in the pure-component spectra that result from the uncertainties in Ka and K2: (a) HF, (b) F-, (c) HF2-, (d) H+, and (e) OH-.

sideration in the interpretation of prediction errors which are based solely on random errors. An intriguing problem that has been raised in this study is the determination of the true identities of the species that contribute to the pure-component spectra. Our results appear consistent with the commonly accepted model for HF dissociation. However, because the hydration numbers of the species are unknown, the intensities of the reconstructed spectra depend on the concentration range studied. This is yet another manifestation of the ill-posed nature of this analysis, which is only partially alleviated by model-based regression. The measurement scheme presented in this paper should be adaptable for process measurements. Use of the SW near-IR permits the use of long path lengths (up to 10 cm), and fiber optics are available for noninvasive measurements. Of course, temperature fluctuations and the presence of additional acids or solutes

will interfere with the measurement. However, in principle, these effects can be measured and implicitly modeled with indirect calibration techniques such as MLR, PCR, and PLS. ACKNOWLEDGMENT This work was partially supported by the Idaho National Engineering Laboratory, under Contract C85-110744. We also thank Battelle for assisting with graphics and production costs.

Received for review May 8, 1996. Accepted October 12, 1996.X AC9604550 X

Abstract published in Advance ACS Abstracts, November 15, 1996.


35

Quantification of Hydrofluoric Acid Species by Chemical-Modeling

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