Quantification of minerals from ATR-FTIR spectra with

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 181 (2017) 7–12

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Quantification of minerals from ATR-FTIR spectra with spectral interferences using the MRC method Francisco Bosch-Reig ⁎, José Vicente Gimeno-Adelantado, Francisco Bosch-Mossi, Antonio Doménech-Carbó ⁎ Departament de Química Analítica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Valencia, Spain

a r t i c l e

i n f o

Article history: Received 16 September 2016 Received in revised form 2 February 2017 Accepted 5 February 2017 Available online 07 February 2017 Keywords: ATR-FTIR Quantification Sodium feldspar Potassium feldspar Quartz Kaolin

a b s t r a c t A method for quantifying the individual components of mineral samples based on attenuated total reflectance – Fourier transform infrared spectroscopy (ATR-FTIR) is described, extending the constant ratio (CR) method to analytes absorbing in a common range of wavenumbers. Absorbance values in the spectral region where the analytes absorb relative to the absorbance of an internal standard absorbing at a wavenumber where the analytes do not absorb, permits the quantification of N analytes using measurements at N fixed wavenumbers. The method was tested for mixtures of albite, orthoclase, kaolin and quartz. © 2017 Published by Elsevier B.V.

1. Introduction The identification and quantification of individual components in mixtures is an obvious analytical demand in mineral analysis. Infrared spectroscopy is a well-established technique for the identification of organic and inorganic species [1,2] of extended application in mineral analysis [3–7]. Fourier- Transform infrared spectroscopy (FTIR) is of particular interest in the case of complex systems containing organic and inorganic components [8,9] where there is an increasingly growth use of measurements in the attenuated total reflectance (ATR) mode [10,11]. The use of infrared spectroscopy for quantification purposes is, however, relatively scarce in the context of mineral analysis, with few studies specifically devoted to quantification [12–15]. In the simplest case of a binary mixture of two components displaying overlapping bands, quantification can be in principle obtained using the Vierordt's method from absorbance measurements at two selected wavenumbers. The same scheme is of application to a N-component system using absorbance measurements at N wavenumbers. The application of this method in conventional Vis-UV spectrophotometry is straightforward because the optical path is constant in this technique. In ATR-FITR, however, the situation is different because the absorbance/transmittance response depends, among other factors, on anomalous dispersion and the shape and size

⁎ Corresponding authors. E-mail addresses: [email protected] (F. Bosch-Reig), [email protected] (A. Doménech-Carbó).

http://dx.doi.org/10.1016/j.saa.2017.02.012 1386-1425/© 2017 Published by Elsevier B.V.

distribution of the sample [16,17] so that the optical path depends in general of the thickness, granulometry and composition of the entire sample, including any possible IR-transparent matrix (a frequent situation in mineral samples). In these circumstances, the Vierordt's method cannot strictly be applied. In order to provide ‘absolute’ concentrations for a system containing N absorbing substances, we described in a previous report [18] a quantification method based on the addition of an internal absorbing standard in known and constant concentration. In this method, the use of the ratio between the absorbances of the problem and the internal standard permits to avoid the problem of the variable optical path. The socalled constant ratio method (CR), was applied to Fourier-transform infrared spectroscopy (FTIR) for quantifying individual components in mineral mixtures, being applied to calcite/quartz binary mixtures. The previously reported method [18] assumed that all the compounds of interest (the A, B, C, … analytes and the added standard) displayed non-overlapping bands. In the current report, we describe a generalized CR method for mineral quantification, based on ATR-FTIR measurements, to be applied for mixtures of components whose absorption bands overlap. This modified constant ratio method, MCR, ideally permits the quantification of N analytes in a mixture providing that their separate spectra are known, using an internal standard which provides an absorption band separated from the spectra of the analytes from the absorbance measurement at N wavenumbers where all analytes absorb. As a result, in contrast with the Vierordt's method, the absolute concentrations of the A, B, C, … analytes can be determined in the presence of a non-absorbing matrix with no influence of the matrix-dependent optical path.

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The application of this methodology is illustrated here by the quantification of individual minerals in mixtures of albite and orthoclase using ATR-FTIR measurements. The method can be also applied to other FTIR measurement modes. Albite, NaAlSi3O8, and orthoclase, KAlSi3O8, are the extreme members of the plagioclase minerals forming, respectively, sodium and potassium feldspars. The composition of sodium and potassium feldspars is an important parameter in regard to the densification behavior on thermal treatment of pastes for porcelain bodies [19], but is of importance in environmental [20], mineralogical [21,22] and archaeological [23–25] studies. In this work, potassium hexacyanoferrate(II) was chosen as the internal standard. This compound displays an absorption band at 2115 cm−1, due to the vibration of the triple carbon-nitrogen bond, which does not interfere with the absorptions of quartz and silicates, which are concentrated in the wavenumber region between 450 and 1200 cm−1 [26]. In the current report, the MCR method was applied to the quantification of albite, orthoclase, quartz and/or kaolin in synthetic mixtures of these minerals. 2. Experimental Potassium hexacyanoferrate(II) (Carlo Erba) was used as a standard for ATR-FTIR measurements. Mixtures of analyte/standard were prepared in different proportions, weighing each component on an analytical balance to obtain different binary, ternary, etc. mixtures. The standards were: sodium feldspar (LGC Standards, BCS-CRM N° 375/1), potassium feldspar (NCS DC 61102, China National Analysis Center for Iron and Steel), quartz (high purity silica, BCS-CRM 313/2) and kaolin (NCS DC 60123a, China National Analysis Center for Iron and Steel). The certified composition of such standards is summarized in Table 1. KBr (Merck) was used as a non-absorbing diluting species for relative uncertainty determination after preparing binary mixtures with the reference minerals. For obtaining ATR-FTIR spectra of reference minerals, an amount of 0.200 g of the mineral was finely powdered with an agatha mortar and pestle and mixed with 0.25 mL of nujol oil until forming a homogeneous paste; then a fraction was transferred onto the diamond window of the ATR device. Synthetic mixtures were prepared as before by mixing weighted amounts (± 0.0001 g) of the different minerals to a total mass of 0.200 g. Infrared spectra were obtained between 4000 and 400 cm−1, using an average of 32 scans and 4 cm−1 resolution. The spectra were recorded with respect to a background taken on the air or on a drop of nujol oil previously to each spectrum and under the same measurement conditions. In the registers obtained for each

Table 1 Certified composition of the standards used in this study. Nafeld: sodium feldspar (LGC Standards, BCS-CRM N° 375/1); Kfeld: potassium feldspar (NCS DC 61102, China National Analysis Center for Iron and Steel); Kao: kaolin (NCS DC 60123a, China National Analysis Center for Iron and Steel); Qua: quartz (BCS-CRM 313/2). nd non detected. wt%

Nafeld

Kfeld

Kao

Qua

SiO2 TiO2 Al2O3 Fe2O3

69.26 0.313 17.89 0.291

66.26 0.048 18.63 0.19

99.73 0.0243 0.068 0.0229

Fe2O3T Na2O K2O CaO MgO MnO BaO SrO SO3 H2O P2O5 CO2 LOI

nd 8.89 1.47 0.78 0.180 nd nd nd nd nd 0.226 nd 0.72

nd 3.69 9.60 0.76 0.054 nd nd nd nd nd nd nd 0.86

45.30 0.060 37.70 Fe2O3T 0.35 FeO (0.026) 0.35 0.045 0.042 0.064 0.021 0.0018 nd nd 0.76 15.26 0.16 (0.026) 14.81

nd 0.0057 0.0108 0.0160 0.0038 0.00032 0.00067 0.00024 nd nd nd nd nd

sample, the absorbances of the peaks chosen for the analyte or analytes and the standard were measured at the selected wavenumbers taking a horizontal base line defined in the region between 1400 and 1200 cm−1. In order to determine the uncertainty in absorbance measurements, five independent measurements were taken on different standard + KBr mixtures having mass fractions of KBr between 0 and 1. 3. Description of the method Let us consider the case of a sample containing a binary mixture of two components, R, S, which display a series of overlapping absorption bands. In principle, the ‘absolute’ concentrations of R and S in the mixture can be determined from absorbance measurements at two selected wavenumbers. Here, we consider the more realistic case in which the components R and S are accompanied by a non-absorbing matrix T of unknown composition. In this system, one can assume that there will be a linear dependence of the absorbance associated to each one of the absorbing components on its concentration, cJ. Then, the absorbance contribution of the J-component at a given wavenumber, AJ(υk) will be: A J ðυk Þ ¼ ε J ðυk Þbc J ; J ¼ R; S

ð1Þ

where εJ(υk) represents the molar or mass absorptivity of the Jcomponent at the wavenumber υk, and b the unknown optical path. In principle, the concentrations of two analytes R and S, cR, cS, can be calculated, using the Vierordt's method, by solving the following twoequation system from absorbance measurements at two wavenumbers υ1, υ2, in the problem R + S + T mixture: Aðυ1 Þ ¼ εR ðυ1 ÞbcR þ εS ðυ1 ÞbcS

ð2Þ

Aðυ2 Þ ¼ εR ðυ2 ÞbcR þ εS ðυ2 ÞbcS

ð3Þ

To determine the concentrations cR, cS, the absorptivities εR(υ1), εR(υ2), εS(υ1), εS(υ2), have to be independently determined from absorbance measurements in R + T and S + T mixtures of known composition. The application of the above scheme, however, is only possible if the optical path b is the same for all the above experiments. In ATR-FTIR spectroscopy, however, several effects, in particular anomalous dispersion and those associated to the shape and size distribution of solid particles [15,16], make variable the optical path which will be in general dependent on the granulometry and composition of the analytes and the matrix; i.e., the value of b will be in principle different for each measurement, being function of the thickness of the sample and the granulometry all the components of the sample. Accordingly, the absorptivity coefficients calculated in calibration experiments at R + T and S + T mixtures do not equal those operating in the R + S + T problem mixture (the “optical path problem”) so that, strictly, the Vierordt's method cannot be applied to ATR-FTIR. In these circumstances, one can write b = b(∀ cJ, cT), ∀ cJ representing the concentration of all Jabsorbing components and cT the (unknown) concentration of the matrix. Accordingly, the Vierordt's method cannot be applied and Eq. (1) should be rewritten as:
A J ðυk Þ ¼ ε J ðυk Þb ∀c J ; cT c J ;

J ¼ R; S

ð4Þ

The problem of the variable optical path can be avoided by means of the use of an internal standard, as previously described for the case of mixtures of components displaying non-overlapping bands [18]. For this purpose, a known amount of the R + S + T sample will be mixed with a known amount of a new substance, the internal reference material, P, which absorbs at a wavenumber υP with no spectral interference with R and S. For several applications, the above system will be accompanied by a dispersing substance, typically nujol oil, with no absorption at the wavenumbers υ1, υ2, υP; this system will be labeled as the

F. Bosch-Reig et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 181 (2017) 7–12

Fig. 1. Absorbance ATR-FTIR spectra of: a) sodium feldspar (Nafeld) and b) potassium feldspar (Kfeld) reference materials.

‘modified problem’. The ratio between the absorbance of this system at a given wavenumber υk and the absorbance of the internal standard at its characteristic wavenumber υP will be: Aðυk Þ εR ðυk ÞbðcR ; cS ; cT ÞcR þ εS ðυk ÞbðcR ; cS ; cT ÞcS ¼ εP ðυP ÞbðcA ; cB ; cT ÞcP AP ðυP Þ

ð5Þ

This ratio is independent on b(∀ cJ, cT) so that, taking absorbance measurements at two wavenumbers,υ1, υ2: Aðυ1 Þ εR ðυ1 ÞcR þ εS ðυ1 ÞcS cR cS ¼ ¼ K RP ðυ1 υP Þ þ K SP ðυ1 υP Þ εP ðυP ÞcP cP cP AP ðυP Þ

ð6Þ

Aðυ2 Þ εR ðυ2 ÞcR þ εS ðυ2 ÞS cR cS ¼ ¼ K RP ðυ2 υP Þ þ K SP ðυ2 υP Þ cP cP AP ðυP Þ εP ðυP ÞcP

ð7Þ

In these equations, KRP(υ1υP) (= εR(υ1)/εP(υP)) and KSP(υ1υP) (= εS(υ1)/εP(υP)) are the ‘specific absorbance parameters’ which replace the absorptivity coefficients. Accordingly, the ‘absolute’ concentrations cR, cS can be calculated by solving the above system of two equations providing that the K-coefficients are determined from calibration experiments in R + P and S + P mixtures optionally incorporating a given matrix T. In these experiments: ε J ðυk ÞbðcR ; cS ; cT Þc J cJ Aðυk Þ ¼ ¼ K JP ðυk υP Þ …; J ¼ R; S cP AP ðυP Þ εP ðυP ÞbðcR ; cS ; cT ÞcP

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Fig. 2. Absorbance ATR-FTIR spectrum of a) quartz and b) kaolin reference materials. Two replicate spectra are superimposed.

The MCR method can be extended to an N-component system based on the above assumptions. Thus, the absorbance of the sample at a given wavenumber, υk, relative to the absorbance of the internal standard at its characteristic wavenumber, υP, will be: Aðυk Þ A1 ðυk Þ þ    þ AN ðυk Þ ¼ AP ðυP Þ AP ðυP Þ c1 cN ¼ K 1P ðυ1 υP Þ þ    þ K NP ðυN υP Þ cP cP

ð9Þ

If we dispose of absorbance measurements at N fixed wavenumbers, we obtain a system of N equations (one for each k wavenumber) with N unknowns (the c1, c2, … , cN concentrations of the analytes) which can be solved if the K-coefficients are determined from the spectra of each one of the individual J-components enriched with the internal standard from absorbance measurements at J + P mixtures of known composition. In these experiments:
ε J ðυk Þb ∀c J ; cT c J cJ Aðυk Þ ¼ ¼ K JP ðυk υP Þ cP AP ðυP Þ εP ðυP ÞbðcA ; cB ; cT ÞcP

ð10Þ

ð8Þ

The relevant point to emphasize is that, as the optical path terms cancel for all series of measurements, such K-parameters are independent of the optical path so that they are independent on sample thickness, granulometry, etc. an essential property for analyzing solid materials by ATR-FTIR. Notice that A(υk) and AP(υP) are directly measured in the spectrum of the P-enriched sample and KRP(υ1υP), KRP(υ2υP) and KSP(υ1υP), KSP(υ2υP) can be determined from calibration experiments in R + P and S + P binary mixtures, respectively. Table 2 Absorption bands in the ATR-FTIR spectra of the mineral standards used in this study. st strong band; sh shoulder. Mineral

Wavenumber at the absorption maximum/cm−1

Nafeld Kfeld Kao Qua K4Fe(CN)6

1145sh, 1095sh, 1040st, 994st, 786, 761, 743, 724, 692, 648 1140, 1090sh, 1045sh, 992st, 767, 725, 647 1114, 1024st, 998st, 935, 910, 788, 750, 678, 645 1160sh, 1090sh, 796, 777, 690 2115st

Fig. 3. Absorbance ATR-FTIR spectra of samples a) Nafeld and b) Kfeld spiked with 17 wt% K4Fe(CN)6. The arrows mark the band associated to the vibration of the triple carbonnitrogen bond of the K4Fe(CN)6 internal standard.

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Fig. 4. Absorbance ATR-FTIR spectrum of a synthetic mixture of sodium (50.58 wt%) and potassium feldspar (36.28 wt%) plus K4Fe(CN)6 (13.14 wt%). Three replicate spectra are superimposed. The arrow marks the band associated to the vibration of the triple carbon-nitrogen bond of the K4Fe(CN)6 internal standard.

4. Results and discussion Fig. 1 shows the ATR-FTIR spectra of Nafeld and Kfeld reference materials in the region between 600 and 1400 cm−1 where all absorption bands are located. The infrared spectra of quartz, feldspars and kaolin is dominated by a series of overlapping bands between 800 and 1200 cm− 1, accompanied by less intense bands between 600 and 800 cm−1. The bands in the range between 1200 and 900 cm− 1 can be assigned to the asymmetric stretching vibration of the Si\\O groups (maximum at 1080 cm−1), the symmetric stretch at 800 and 780 cm−1, and the symmetric and asymmetric Si\\O bending mode at 695, 520, and 450 cm−1, respectively [26]. Table 2 summarizes the wavenumbers at which the absorption maxima appear in the spectra of the studied reference minerals. The spectrum of K4Fe(CN)6 consisted, as previously noted [10], on a unique band at 2115 cm−1. Then, no spectral interference with the clays under study is associated to this compound so that it can be used as an internal standard for absorbance measurements. This can be seen in Fig. 3 where the spectra of samples Nafeld and Kfeld enriched with 17 wt% K4Fe(CN)6 are shown. The repeatability of the measurements is a critical aspect to be considered. Under our experimental conditions, the ATR-FTIR spectra of both reference materials and their mixtures exhibited an excellent repeatability, as can be seen in Fig. 2, where four independently recorded spectra of quartz and kaolin reference materials are superimposed. As expected, the spectra of mixtures of the above components displayed a series of overlapped bands corresponding to the

Fig. 5. Absorbance ratio vs. concentration ratio plots for Nafeld reference material spiked with K4Fe(CN)6 forming nujol oil-embedded pellets using absorbance measurements at υ = 1090 (squares) and 745 (triangles) cm−1 and υFe = 2115 cm−1.

Fig. 6. Variation with the mass fraction of KBr of the relative uncertainty in absorbance measurements at the indicated wavenumbers in mixtures of KBr plus Nafeld (squares, 994 cm−1), Kfeld (solid squares, 992 cm−1) and Kao (triangles, 998 cm−1). Values of ΔA(υ)/A(υ) from five independent spectra for each KBr plus mineral mixture. Bars correspond to deviations of 0.01. Dotted line corresponds to an averaged uncertainty value.

superposition of the above signals. This can be seen in Fig. 4, where the spectrum a of mixture of sodium feldspar (50.58 wt%), potassium feldspar (36.28 wt%) and K4Fe(CN)6 (13.14 wt%) is depicted. It is pertinent to note, however, that unless quartz, the reference materials were no ‘pure’ minerals (see Table 1) so that the K(υkυP) values are standardsensitive. It is also pertinent to note that the absorbance ratios varied slightly with the granulometry of the materials, a phenomenon also observed in infrared spectra on KBr pellets [18]. To minimize this effect, the spectra were recorded on samples of finely powdered minerals and prepared by pasting the solid material with nujol oil. The optimum dosage, balancing repeatability and amount of sample (logically the use of a minimal amount of sample as possible is desirable) was obtained for a total mass of solid materials of 0.20000 ± 0.00010 g pasted with 0.25 mL of nujol oil, as described in the Experimental section. Under these conditions, the ATR-FTIR measurements on K4Fe(CN)6-enriched mineral specimens presented a high repeatability, as can be seen in Fig. 4, where three independent measurements of the aforementioned mixture of Nafeld plus Kfeld and K4Fe(CN)6 are depicted. Calibration plots were performed on binary mixtures of sodium and potassium feldspars with K4Fe(CN)6 using absorbance measurements at the absorption maxima previously indicated in the feldspar spectra and the absorbance at 2115 cm− 1, corresponding to the absorption of K4Fe(CN)6. Fig. 5 depicts the results for Nafeld reference material using two wavenumbers, 1090 (squares) and 745 (triangles) cm−1. The plots of the ratio between the absorbance of the feldspar at the

Fig. 7. Theoretical variation of the relative uncertainty in the concentration ratio, Δ(c/cP)/ (c/cP), on the concentration ratio, c/cP, from Eq. (15) assuming ΔA/Ao° = 1% and Δε/εo = 1%, for different values of the εP(υP)/ε(υ) ratio.

F. Bosch-Reig et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 181 (2017) 7–12

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Table 3 Peaks and K(υkυP)-parameters for quantification at selected wavenumbers, using potassium hexacyanoferrate(II) standard. Number between parenthesis represent the standard deviation in the last significant figure. Component

υ (cm−1)

υ (cm−1)

υ (cm−1)

υ (cm−1)

υ (cm−1)

υ (cm−1)

υ (cm−1)

K(υkυP) Nafeld Kfeld Kao Qua

1150 0.494 (5) 0.257 (3) 0.067 (1) 0.342 (3)

1135 0.523 (5) 0.308 (3) 0.160 (2) 0.417 (4)

1095 0.534 (5) 0.317 (3) 0.167 (2) 0.667 (7)

1035 0.861 (8) 0.566 (6) 0.693 (7) 0.874 (9)

990 0.941 (9) 0.857 (9) 0.707 (8) 0.842 (8)

935 0.540 (5) 0.360 (4) 0.413 (4) 0.595 (6)

905 0.366 (4) 0.223 (2) 0.693 (7) 0.401 (4)

selected wavenumber, ANafel(υ), and the absorbance of the internal standard, AFe(υFe) vs. the ratio of the respective concentrations, cNafel/ cFe, exhibited an excellent linearity in terms of linear correlation coefficients, but the slopes differed slightly but significantly, as expected, from the reference sodium feldspar to the sample. Table 3 summarizes the K(υkυP) values determined from ATR-FTIR spectra of the reference minerals at a set of selected wavenumbers. The uncertainty in the K(υkυP) values was estimated from the standard deviation in the absorbance values from three independent measurements in experiments such as in Fig. 2. Using such data, the composition of different binary mixtures prepared from weighted amounts of the standards was calculated. It is pertinent to note that the differences in such coefficients using replicate spectra were lower than the 5% in the average K(υkυP) value. The results obtained for different binary and ternary mixtures of the studied reference materials are summarized in Table 4 where standard deviations in concentrations were calculated from those in absorbance measurements combined with those estimated for K(υkυP) coefficients in Table 3. A satisfactory agreement was obtained in the studied systems between the composition from weight of the standards and that calculated from ATR-FTIR spectra. Apart from contamination, general error sources in FTIR spectroscopy are associated to atmospheric intrusion, interference fringes, stray light, anomalous dispersion, and detector non-linearity [1,2,16,17], among others, which should be separately considered for transmission, diffuse reflectance, and ATR. In this last mode, irregular reflection features [1,2,26,27], instability in the wavenumber scale [28] and particle size and particle size heterogeneity [16] are of importance. As a result, the uncertainty in absorbance measurements varies in general with the wavenumber defining an instrument/mode-characteristic error curve. In the context of mineral/inorganic analysis, analysis of errors has been focused on linear regression [13–15] and multivariate [8–11] methods. To evaluate errors associated to the proposed MRC method, it will be assumed here that the uncertainty (in the following represented by Δ) in the concentration of a given analyte, c, relative to the internal standard in concentration cP depends on the uncertainties of the respective absorbances A(υ) and AP(υP) at the indicated wavenumbers and those of the absorptivity coefficients, ε(υ) , ε(υP) (or, equivalently, the K-coefficients). Then:       Δðc=cP Þ Δ½AðυÞ=AP ðυP Þ Δ½εðυÞ=εP ðυP Þ ¼ þ c=cP AðυÞ=AP ðυP Þ ε ðυÞ=ε P ðυP Þ

ð11Þ

Experimental data for absorbance measurements carried out in mixtures of the studied mineral standards and K4Fe(CN)6 with a nonabsorbing compound, KBr, revealed that the relative uncertainty, ΔA(υ)/A(υ), was maintained within the 0.01 to 0.04 range, increasing slowly upon increasing the proportion of KBr. Fig. 6 shows the variation of this quantity, determined from five independent measurements for each KBr + mineral mixture, with the mass fraction of KBr (fKBr) in mixtures of KBr plus Nafeld (squares, 994 cm− 1), Kfeld (solid squares, 992 cm−1) and Kao (triangles, 998 cm−1). Since the values of the relative uncertainty remain confined into a narrow range, it is in principle plausible to assume, for simplicity, that the uncertainty in the individual measurements of the absorbances AP(υP) and A(υ) is the same (ΔA) so that:     Δ½AðυÞ=AP ðυP Þ 1 1 ¼ ΔA þ AðυÞ=AP ðυP Þ AðυÞ AP ðυP Þ

ð12Þ

Additionally, one can assume that the sum of the absorbances of the problem compound and the standard become approximately equal to an averaged value Ao. Then, designating by ξ the A(υ)/AP(υP) ratio, the individual absorbances would be A(υ) = ξAo/(1 + ξ) and AP(υP) = Ao/(1 + ξ). Accordingly, one can obtain:   2 Δ½AðυÞ=AP ðυP Þ ΔA ð1 þ ξÞ ¼ AðυÞ=AP ðυP Þ Ao ξ

ð13Þ

The main result is that, assuming that the ΔA/Ao ratio, representing the averaged relative uncertainty in absorbance measurements, is constant, the uncertainty in the absorbance ratio A(υ)/AP(υP) tends to a minimum value for ξ = 1. This reasoning can in principle be extended to the absorptivity ratio so that:   Δ½εP ðυP Þ=εðυÞ Δε ð1 þ ζ Þ2 ¼ ζ εP ðυP Þ=εðυÞ εo

ð14Þ

where ζ (= εP(υP)/ε(υ)) represents the absorptivity ratio and Δε/εo an averaged relative uncertainty of the absorptivity values. Again, the lower uncertainty value would be reached for ε(υ) = εP(υp). This means that, ideally, the lower uncertainty in the c/cP ratio would be obtained when the above condition applies simultaneously to A(υ) = AP(υp). Both conditions lead to a minimum uncertainty for c/cP = 1,

Table 4 Statistical data for the analysis of different binary and ternary mixtures of the studied reference minerals from ATR-FTIR data using K4Fe(CN)6 as a internal standard. Components (wt%)

(1st)a (±0.02)

(2nd)a (±0.02)

Nafel + Kfel Nafel + Kfel Nafel + Kfel Nafel + Kao Kafel + Kao Nafel + Qua Nafel + Kao + Qua Kfel + Kao + Qua

42.57 26.33 77.81 51.22 55.37 46.77 33.16 34,08

57.43 73.67 22.19 48.78 44.63 53.23 38.44 35.23

a b

Composition for wighted amounts of the standards (all ±0.001 g). composition from ATR-FTIR data using the K(υkυP) values in Table 3.

(3rd)a (±0.02)

28.40 30.69

(1st)b 44.0 27.2 79.0 50.8 58.0 42.3 32.8 35.0

± ± ± ± ± ± ± ±

(2nd)b 0.8 0.5 1.6 1.1 1.2 0.8 0.6 0.7

57.0 72.8 21.0 49.2 42.0 57.7 38.9 35.6

± ± ± ± ± ± ± ±

(3rd)b 1.2 1.5 0.4 1.0 0.8 1.2 0.7 0.7

28.3 ± 0.5 29.4 ± 0.6

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but this condition, by practical reasons, is not operative.   Δðc=cP Þ ΔA ð1 þ ζ Þ2 Δε ð1 þ ζ Þ2 ¼ þ ζ ζ c=cP Ao εo

ð15Þ

Fig. 7 compares the variation of the relative uncertainty in the concentration ratio, Δ(c/cP)/(c/cP), on the concentration ratio, c/cP, assuming ΔA/Ao = 1% and Δε/εo = 1%, for different values of the εP(υP)/ε(υ) ratio. One can see that the uncertainty in the concentration increases rapidly on decreasing c/cP ratio below 0.2 whereas increases more slowly for c/cP ratio larger than 1. 5. Conclusions ATR-FTIR spectroscopy can be used for quantifying the individual components of mineral mixtures based on absorbance measurements at fixed wavenumbers where all components absorb avoiding the problem of the variable optical path. The application of the method requires the measurement of the absorbance of an internal standard, added in known proportion to the sample, at a wavenumber where the analytes do not absorb significantly. The method can ideally be applied to determine the content of N analytes in a multicomponent mineral sample from absorbance measurements at N wavenumbers. Application to mixtures of sodium feldspar, potassium feldspar, kaolin and/or quartz provide satisfactory results under the described experimental conditions thus denoting the suitability of ATR-FTIR as a technique usable for quantification in mineral anlysis. Acknowledgements Financial support from the MINECO Project CTQ2014-53736-C3-2-P, which is supported with ERDF funds is gratefully acknowledged. References [1] U.P. Fringeli, ATR and reflectance IR spectroscopy, applications, in: J. Lindon, G.E. Tranter, D. Koppenaal (Eds.), Encyclopedia of Spectroscopy and Spectrometry, Academic Press, San Diego 2000, pp. 58–75. [2] P.R. Griffiths, J.A. Haseth, Fourier Transform Infrared Spectroscopy, second ed. Wiley-Interscience, Hoboken, 2007. [3] M.E. Böttcher, P.-L. Gehlken, The vibrational spectra of BaMg(CO3)2 (norsethite), Appl. Spectrosc. 51 (1997) 130–131. [4] J. Madejova, FTIR techniques in clay mineral studies, Vib. Spectrosc. 31 (2003) 1–10. [5] R.L. Frost, M.L. Weier, M.E. Clissold, P.A. Williams, Infrared spectroscopic study of natural hydrotalcites carboydite and hydrohonessite, Spectrochim. Acta A 59 (2003) 3313–3319. [6] R.L. Frost, S. Bahfenne, J. Graham, Infrared and infrared emission spectroscopic study of selected magnesium carbonate minerals containing ferric iron–implications for the geosequestration of greenhouse gases, Spectrochim. Acta A 71 (2008) 1610–1616.

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