Kinetics of Ion-Pair Formation on Variable-Charge Minerals Using the Frequency Domain Method

Soil Chemistry Kinetics of Ion-Pair Formation on Variable-Charge Minerals Using the Frequency Domain Method

Predicting nutrient behavior is ever more critical to understanding and management of the environment, Xiufu Shuai* particularly in highly weathered and tropical environments. Th e fate of nutrients in the environment seems heavily Water Resources Research Center infl uenced by the kinetics of ion-pair formation on the surfaces of variable-charge minerals coupled with transport Univ. of Hawaii at Manoa processes. Th e objective of this study was to generate the coupled processes in column experiments and estimate the Honolulu, HI 96822 reaction rates using the frequency domain method. Columns were packed with ground natural minerals (gibbsite, goethite, and hematite). Th e input signals were designed as a sinusoidal change in NaNO3 concentration within Russell S. Yost −1 −1 Dep. of Tropical Plant and Soil Sciences 0.1 and 0.2 mmol L and constant pH 4.0, and the highest frequency of the input signals was 0.714 min . − + Univ. of Hawaii at Manoa Th e input and output signals of NO3 and H concentrations were monitored by ultraviolet–visible light and Honolulu, HI 96822 pH detectors, respectively. A mathematical model was derived to describe the diff usion process of a counterion from aqueous solution to β plane, the recombination–dissociation reaction between a charged surface site and the counterion, and the coupled transport process described by the convection–dispersion equation. Results showed that the output signals were dominated by the designed frequency and thus the coupled processes were linear. Th e + − aqueous H concentration changed linearly with that of the aqueous NO3 concentration. Th e mathematical model fi t the measured transfer function of the processes. Th e estimated rates of recombination of ion pairs were 52.0, 30.5, and 8.0 L mol−1 min−1, and the estimated rates of dissociation of the ion pair were 0.189, 0.256, and 0.285 min−1 for the natural gibbsite, goethite, and hematite, respectively.

Abbreviations: CDE, convection–dispersion equation; RA, relative amplitude; TLM, triple-layer model; XRD, x-ray diff raction.

inerals with variable surface charge such as gibbsite, goethite, and hematite Mare abundant in soils and important in environmental chemistry, especially in highly weathered soils of the tropics (Uehara and Gillman, 1981). Surface complexation models, such as the constant capacitance model (Schindler and Kamber, 1968; Hohl and Stumm, 1976), the diff use layer model (DLM) (Stumm et al., 1970; Huang and Stumm, 1973; Dzombak and Morel, 1990), the triple-layer model (TLM) (Yates et al., 1974; Davis et al., 1978; Hayes et al., 1991), and the charge-distribution multisite complexation (CD-MUSIC) model (Hiemstra et al., 1989a,b, 1996; Hiemstra and Van Riemsdijk, 1996) are being used to describe the surface chemistry of variable-charge colloids. In both the DLM and the TLM, + + − − ions such as Na , K , NO3 , Cl can be nonspecifi cally adsorbed at the electric diff use layer by electrostatic attraction or repulsion. In the TLM, the counterions can be adsorbed at the β plane by ion-pair formation with the charged surface sites. A kinetic study has shown that ion-pair formation on the β plane in the TLM is a physical diff usion process through the electric double layer with a subsequent recombination–dissociation reaction between charged sites on the surface of goethite and the counterion on the β plane (Sasaki et al., 1983). Surface reactions are usually coupled with transport processes in chemically and physically heterogeneous soil systems, and thus modeling and parameter esti-

Soil Sci. Soc. Am. J. 74:1568–1576 Published online 3 Aug. 2010 doi:10.2136/sssaj2009.0161 Mention of a specifi c brand of equipment does not imply an endorsement by the University of Hawaii or the authors. Received 28 Apr. 2009. *Corresponding author ([email protected]). © Soil Science Society of America, 5585 Guilford Rd., Madison WI 53711 USA All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.

1568 SSSAJ: Volume 74: Number 5 • September–October 2010 mation of the coupled processes are important for more realistic sociation reaction. Th e recombination–dissociation reaction of descriptions of the fate of nutrients and contaminants. Th e ki- counterions with charged surface sites to form an ion pair in the netics of ion-pair formation of ions as counterions at the surface β plane is described as of variable-charge soils is very helpful in predicting their mobil- + ⎯⎯kr→ CIββSS←⎯⎯ -CI [2] ity and transport in the environment. Th e kinetic methods used kd to study surface reactions, however, such as the electric fi eld pulse technique (Sasaki et al., 1983) and the pressure-jump method where S and S–CIβ are the concentrations of charged surface (Astumin et al., 1981), cannot be used to study the coupled pro- sites and the ion pair in the β plane, respectively, and kr and kd cesses because the time range of the pressure jump is on a scale are the rates of the recombination and dissociation reactions, re- −5 of 10 s (Sparks, 1989), which is much faster than transport spectively. Typically, all species, CIβ and S–CIβ, in the β plane processes on a scale from minutes to days. Recently, a frequency are considered adsorbed at the surface of minerals. domain method was developed by Shuai and Yost (2007) to Th e intrinsic equilibrium constant of the diff usion process study the transport of inert tracers in columns using the sinu- through the electric double layer is soidal change in tracer concentrations with time as input. Th e ≡ kf K diffusion [3] advantage of this method is that the spectra of the input were of k the widest passband (the portion of a spectrum between limit- b ing frequencies that is transmitted with minimum relative loss Th e intrinsic equilibrium constant of the recombination–disso- or maximum relative gain in a dynamic system) and the high- ciation process of the counterion is est amplitude, which permitted stimulation of the transport k K int ≡ r process and measurements with a maximum signal/noise ratio counterion [4] kd (Shuai and Yost, 2007). Th is method can also be used to measure Transport Process of the coupled surface reactions and transport processes by applying an Convection–Dispersion Equation active tracer of interest to columns in the infl uent solution. Th e frequency domain approach has not been reported in column Th e transport process of the convection–dispersion equa- experiments for the study of the coupled processes of ion-pair tion (CDE) has been widely used to describe transport in soils formation and transport. (Nielsen and Biggar, 1961; Biggar and Nielsen, 1967): Th e objectives of this study were (i) to measure the kinet- ∂∂ccc2 ∂ ics of ion-pair formation at the surfaces of three variable-charge RD=− V [5] ∂∂tzz2 ∂ minerals (natural gibbsite, goethite, and hematite) coupled with a transport process using the frequency domain method, and (ii) where c is the resident concentration of solute, t and z are the to estimate the reaction rates of ion-pair formation at the sur- time and space coordinates, respectively, D is the dispersion coef- faces of the variable-charge minerals. fi cient, V is the pore-water velocity, and R is the retardation fac- tor, which was 1 in this study. MATHEMATICAL MODELS AND For a fi nite system for which the exit boundary condition ALGORITHM FOR PARAMETER ESTIMATION does not aff ect the solute concentration inside the column, the Process of Ion-pair Formation at the Surfaces initial and boundary conditions are of Minerals cz(),0= 0 Th ere are two processes in the mechanism of ion-pair for- ∞= mation at the surface of goethite (Sasaki et al., 1983). Th e fi rst ct(),0 [6] process is diff usion of a counterion from an aqueous phase to the ctut()()0, = β plane in the TLM; the counterion is bound electrostatically to cLt()(), = yt the particle surface but not to any specifi c site. In this study, the physical diff usion process through electrical layers is described as and the transfer function of the CDE to describe the relationship between the input u(t) and the output y(t) is ⎯⎯kf→ CIaq ←⎯⎯ CIβ [1] kb Ys() Gs()≡ Us() where CIaq and CIβ are the concentrations of counterions in [7] aqueous solution and in the β plane and k and k are the for- ⎡⎤LV⎛⎞4 Ds f b =−+exp⎢⎥⎜⎟ 1 1 ward and backward rate constants, respectively, for adsorption ⎣⎦⎢⎥2DV⎝⎠2 and desorption of the counterion through the electrical double layer (Sasaki et al., 1983). where L is the length of the column, Y(s) and U(s) are the Laplace Th e second process is the movement of the counterion in transform of y(t) and u(t), respectively, and s is the Laplace op- the β plane until it fi nds a charged surface site and becomes a erator (s = jω, where j is an imaginary unit and ω is the angular site-bound counterion, which is termed the recombination–dis- frequency) (Jury and Roth, 1990; Shuai and Yost, 2007).

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Mathematical Model for Ion-Pair Formation at ⎡⎤⎛⎞kS⎣⎦⎡⎤ the Surfaces of Minerals Coupled with Transport Hs()=+ s⎢⎥11 K ⎜⎟ +r diffusion ⎜⎟+ [18] ⎢⎥⎝⎠skd Ion-pair formation at the surface of a variable-charge min- ⎣⎦ eral was coupled with transport (the CDE) along the columns. Th e relationship between the input U(s), the output Y(s), For the coupled diff usion process Eq. [1] and the reaction Eq. and the transfer function G(s) described by Eq. [16] or [17] is [2], the following kinetic equations can be given: illustrated in Fig. 1. When there is no noise in the measurement

2 of output, the general relationship between input and output can ∂∂ccc ∂ aq=−−+DVkckc aq aq [8] be expressed as ∂∂tzz2 ∂faq bβ ∂ ()=−t ( )() cβ yt∫ gtτττ u d [19] =−+kc() k k⎣⎦⎡⎤ S c + kc [9] 0 ∂t faq b rβ dip ∂ where g(t) is the impulse response function of the system and the cip =−kSc⎣⎦⎡⎤ kc [10] inverse Laplace transform of G(s). ∂t rdipβ

where caq and cβ are the concentrations of the counterion in Spectral Analysis of Signals, Linearity Test, and aqueous solution and the β plane, respectively, cip is the con- Measured Transfer Function of the Processes centration of the ion pair, and [S ] is the concentration of the For an input signal that is a sine function, sin(2πf1t), with charged surface site at equilibrium. Th e initial and boundary fundamental frequency f1, its spectral components (subharmon- conditions of the partial diff erential equations (PDEs) above are ics) with frequencies of the output signal can be calculated with identical to Eq. [6]. the algorithm for fast Fourier transform (FFT) in MATLAB Th e PDEs are then Laplace transformed with respect to t: (Cooley and Tukey, 1965; Th e MathWorks, Natick, MA). Th eir ∂∂2 amplitudes for the subharmonics f , 2f , 3f , …, Nf are noted as CCaq aq 1 1 1 1 ()skC+−=− kC D V [11] A , A , …, A , where N is an integer. Th e linearity of the system faqbβ ∂∂2 zz 1 2 N was tested on the basis of the relative amplitude (RA) of the nth ++⎡⎤ − = subharmonic with frequency nf1 to its fundamental frequency f1: ()skbr k⎣⎦ S Cβ kC dipfaq kC [12] = An RAn [20] A ⎡⎤ −+() = 1 kSCrdip⎣⎦β skC 0 [13] where n is an integer 2, 3, …, N (Shuai and Yost, 2007). If the where Caq, Cβ, and Cip are the Laplace transforms of caq, cβ, and maximum values of the RAs of both the input and output signals cip, respectively. Th ese equations can be simplifi ed to are <<1, the system can be regarded as linear (Schoukens and ∂∂2 Pintelon, 1991; Shuai and Yost, 2007) and the transfer function CCaq aq DVHsC−=() [14] can be used. ∂∂2 zz aq For a linear system, the following equation is used to calcu- and late the measured process transfer function Gm: ⎡⎤kskS()++⎣⎦⎡⎤ k Yj()ω =+⎢⎥fr d ()= m k Hs() s1 [15] Gjmkkω [21] ⎢⎥2 ++()⎡⎤ ++ Uj()ω ⎣⎦sskkSkbr⎣⎦ d kk bd m k

where H(s) is determined by the ion-pair formation process. where Ym( jωk) and Um( jωk) are the spectral components of the Similarly to Shuai and Yost (2007), the solution to the ordinary kth fundamental frequency ωk obtained from the FFT for the diff erential equation is measured output signals ymk(t) and input signal umk(t), respectively. ⎪⎪⎧⎫LV⎡⎤4 D ()=−+ () ⎨⎬() [16] Algorithm for Estimation of Parameters in the Ys Usexp⎢⎥ 1 1 2 Hs ⎩⎭⎪⎪2DV⎣⎦ Transfer Function or When there was noise in the measurement of output, the Ys() transfer functions Eq. [17] and [18] are the model used to Gs()≡ Us() [17] ⎪⎪⎧⎫LV⎡⎤4 D =−+exp⎨⎬⎢⎥ 1 1 Hs() ⎩⎭⎪⎪2DV⎣⎦2 where G(s) is the transfer function to describe the transport process of the CDE coupled with the ion-pair formation process H(s). When both kf and kb are much greater than ω, kr[S ], and Fig. 1. Model structure for system identifi cation of the transport kd, Eq. [15] can be approximated as process through a column coupled with the ion-pair formation process.

1570 SSSAJ: Volume 74: Number 5 • September–October 2010 estimate the reaction rates of the ion-pair formation process MATERIALS AND METHODS based on inputs and outputs. Because G( jω) is complex, its am- Chemicals plitude and phase are separated to form a vector in real numbers, Analytical reagent grade NaNO3 and volumetric stan- −1 and a cost function is defi ned: dard 0.100 mol L HNO3 and NaOH were used. Nanopure water was degassed before use. ()PVfP=−⎡⎤⎡⎤ ( )T VfP − ( ) K ⎣⎦⎣⎦mmωω,, [22] Th e natural gibbsite, goethite, and hematite (Ward’s Natural Science, Rochester, NY) were selected for more realistic description with of the fate of nutrients and contaminants in chemically heterogeneous V =⎡ soils. Th ey were ground and wet sieved with deionized water, and the m1m11m⎣γγGG,,… F , [23] fraction between 25 and 45 μm was collected and freeze-dried. Th e spe- ∠∠]T γγ2m12mGG,,… F cifi c surface area measurements of the natural minerals were made by the and Brunauer–Emmett–Teller method of adsorption of N2. Th e amorphous material contents were measured by extracting the natural minerals with fP()=⎡ () P() P −1 ωγω,,,,,,⎣ 11Gj… γω 1 GjF 0.2 mol L ammonium oxalate at pH 3.0 (Hodges and Zelazny, 1980). [24] T X-ray diff raction (XRD) analysis of the natural minerals indicated γω∠∠⎤Gj(),,,PP… γω Gj ( , )⎦ 21 2F that they were not pure and contained small amount of other minerals. Th e natural gibbsite shown in Fig. 2 contained kaolinite. Th e natural V P where m is the measured transfer function, is the parameter goethite shown in Fig. 3 contained hematite, quartz, pyrolusite, augelite P vector in the transfer function Eq. [17] and [18], K( ) is the sum [Al2(PO4)(OH)3], and other unidentifi ed minerals. Th e natural hema- of squared residuals, |x| is the amplitude, ∠x is the phase of a tite shown in Fig. 4 contained goethite and quartz. complex function, and γ1 and γ2 are weights. Because the am- To qualitatively identify phosphate in the natural gibbsite and he- plitude and phase have diff erent units, the weights are defi ned as matite, the three natural minerals were extracted with 0.1 mol L−1 NaOH at a solid/water ratio of 1:20 for 24 h at room temperature. Th e concen- = 1 γ1 tration of NaOH in this study was much lower than the concentration max()GG ,… , [25] m1 mF in an alkaline wet process for the solubilization of aluminum phosphate and ore (Horita, 1993), and the extractable phosphate in this study was probably from the desorption of adsorbed phosphate and the dissolu- 1 = tion of augelite at the solid surfaces. Phosphate in the solutions was ana- γ2 [26] ()∠∠… maxGGm1 , , mF lyzed by inductively coupled plasma mass spectroscopy. Phosphate was found in all three extracted solutions at concentrations of 0.038, 0.093, Th e reaction rates of the ion-pair formation process are esti- and 0.049 mmol L−1 for the natural gibbsite, goethite, and hematite, re- mated by minimizing the cost function using the Levenberg– spectively. Even though phosphate was found in the NaOH-extractable Marquardt method (Seber and Wild, 1989).

Fig. 2. X-ray diffraction pattern of natural gibbsite (Gib is gibbsite and Fig. 3. X-ray diffraction pattern of natural goethite (Aug is augelite, Kao is kaolinite). Goe is goethite, Hem is hematite, Pyr is pyrolusite, and Qua is quartz).

SSSAJ: Volume 74: Number 5 • September–October 2010 1571 solutions of all three natural minerals, crystalline Table 1. Properties of the natural minerals and the columns. phosphate minerals were not identifi ed in the Natural Solvent Estimated dispersion Solid Surface Point of Amorphous samples of gibbsite or hematite by XRD, possibly mineral velocity coeffi cient concentration area zero charge materials due to the low crystallinity of augelite. cm min−1 cm2 min−1 g L−1 m2 g kg−1 Gibbsite 9.42 1.497 1364 321.8 5.00 36.4 Th e points of zero charge (pH0) of the three natural minerals were measured using the proce- Goethite 9.77 1.438 2137 379.1 4.70 198.9 Hematite 8.50 1.561 1614 650.2 3.58 10.6 dure in Uehara and Gillman (1981). Th e procedure is briefl y described as follows. Each natural mineral of 1.0 g was mixed with diff erent volumes of stock solutions of NaNO3, ics of the effl uent solution. Th e pH detector (Sensorex Corp., Garden HNO3, and NaOH to form diff erent pH and ionic strength treatments; Grove, CA) was composed of a pH electrode and a fl ow cell. Th e fl at- the total volume in each tube was 20.0 mL. Dinitrogen gas was used to surface electrode (Sensorex Model 450C) was a combination pH–refer- replace the air above the solution in the tubes, and the tubes were closed ence electrode with a double reference junction design. Th e reference with air-tight caps. Aft er 24 h of shaking, the pH values of the suspensions electrode was a sealed, gel-fi lled design and included a peripheral, semi- were measured. Th e adsorbed H+ or OH− was plotted against the pH porous, polyethylene junction. Th e fl ow cell had 50-μL internal volume, for each electrolyte concentration to fi nd the pH0, the common point at which was achieved by locating the fl at-surface pH electrode at the top which the curves intersected (Uehara and Gillman, 1981). of the fl ow cavity. Th e resulting rectangular cross-section fl ow path had no protruding parts that could interfere with a clean, sweeping fl ow of Experiment Setup liquid. Th e pH detector was supplied with inlet and outlet fi ttings sized Th e columns in this study were previously used in the identifi ca- for standard 1.6 mm (1/16 inch) o.d. tubing, which was connected to tion of convection–dispersion as the transport process of an inert tracer the outlet of the UV/Vis detector. Th e electrode was connected to an using the frequency domain method (Shuai and Yost, 2007). Th e col- Accumet Research pH meter (Fisher Scientifi c, Hampton, NH), which umns were washed with pH 4.0 HNO3 solution until the effl uent pH was programmed to collect pH data automatically and continuously at was 4.0 ± 0.1 before use in this study. Th e amounts of natural minerals equal time intervals. Th e pH electrode was calibrated with buff er solu- packed in the columns, the water contents, and the solvent velocity are tions before use. Th e measured pH values and time data from the pH listed in Table 1. detector were automatically stored in a computer. A schematic representation of the experimental setup is shown in Fig. 5. High-performance liquid chromatography (HPLC) was used as a Design of Input and the Measurement of Input solvent delivery system and an ultraviolet–visible light (UV/Vis) detec- and Output Signals − tor was used to measure the concentration of NO3 at 210-nm wave- Eleven input sinusoidal signals with periods of 12.8, 7.2, 4.8, 3.6, length. Th e measured absorbance and time data at the UV/Vis detector 3.0, 2.4, 2.2, 2.0, 1.8, 1.6, and 1.4 min were obtained by changing the −1 were automatically stored in a computer. An additional pH detector was relative composition of Solutions A (0.100 mmol L HNO3) and B −1 −1 connected to the outlet of UV/Vis detector to monitor the pH dynam- (0.200 mmol L NaNO3 and 0.100 mmol L HNO3) as a sinusoidal function with period T (min):

cos⎣⎦⎡⎤() 2π Tt+ 1 A%= 100 [27] 2

B%=− 100 A [28]

Th e input signal for the H+ concentration at the inlet of the column was constant at 0.1 mmol L−1. Th e pH 4.0 of the infl uent solutions was

selected to be close to the pH0 values of the three natural minerals to ensure low surface charge densities at their surfaces. Th e input signals were determined with the column disconnected from the setup, whereas the output signals were obtained with the column connected between

Fig. 5. Schematic representation of the experimental setup for Fig. 4. X-ray diffraction pattern of natural hematite (Goe is goethite investigating the transport process g(t) coupled with the ion-pair and Qua is quartz). formation process h(t).

1572 SSSAJ: Volume 74: Number 5 • September–October 2010 RESULTS AND DISCUSSION Estimating the Point of Zero Charge of the Selected Natural Minerals Th e results of titrations of the natural minerals using the method of Uehara and Gillman (1981) are shown in Fig. 6. Th e surface charge of the three natural minerals was positive at low pH and became negative as the pH of aqueous solution increased. Th e surface charge of the three minerals increased as the ionic strength increased. Th e pH0 values of the natural minerals shown in Table 1 were determined as the common point at which the curves at diff erent ionic strengths intersected. Th e pH0 values of the natural gibbsite and goethite were >4.0 and the pH0 value of the natural hematite was <4.0. Th e pH0 values of the three natural minerals obtained in this study were much lower than the values of synthetic minerals prepared in the lab Fig. 6. The surface charge of the three natural minerals at different pH and ionic strengths (IS). with reagent-grade chemicals (Sverjensky the outlet of the HPLC and the inlet of the UV/Vis detector. Th e data and Sahai, 1996). Th is diff erence might collection frequency was one recording per second for the UV/Vis de- be due to impurities that accompany naturally derived materi- tector and one recording per 3 s for the pH detector. als compared with synthetic minerals of relatively higher purity (Parks, 1965; Schwertmann and Fechter, 1982; Rabung et al., 1998). Th e pH0 is a surface phenomenon and refl ects not so much the mineral as it does its surface charge characteristics; it is not a constant for a mineral but is a highly variable parameter that is subject to change by surface contamination (G. Uehara, personal communication, 2009). Phosphate was identifi ed in the three natural minerals used in this study by extraction with −1 0.1 mol L NaOH or XRD, and the highly reduced pH0 values of the three natural minerals used in this study were probably due to the adsorption of phosphate at the surfaces and the addition of negative surface charge (Mekaru and Uehara, 1972; Wann and Uehara, 1978; Shuai and Zinati, 2009).

Input and Output Signals with Time Th e input and output signals when the period of the signals was the designed 4.80 min are shown in Fig. 7 as an example. − Th e input signal of the NO3 concentration was a sine wave of period 4.80 min and that of H+ was a constant 0.100 mmol L−1. − + Th e output signals of the NO3 and H concentrations were sinusoidal with the same period of 4.80 min and amplitude reduction and phase shift compared with the input signal of the − NO3 concentration. Th ere was a delay of π between the out- + − put signals of the H and NO3 concentrations for the natural gibbsite and goethite, whereas no delay was observed for the nat- + − ural hematite. Th ere were linear relationships of amplitudes and Fig. 7. The measured dynamics of H and NO3 concentrations in the infl uent solutions (input signals) and effl uent solutions (output signals) phases between the H+ and NO − concentrations in the effl uent − 3 when the period of the sine function of the input NO3 concentration solutions, as shown in Fig. 8. Th e phase diff erence between the was 4.8 min in the column experiments using the three natural minerals.

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+ − output signals of the H and NO3 concentrations was a constant close to −π for the natural gibbsite and goethite and close to 0 for the natural hematite. Th ese linear relationships of amplitudes and phases be- + − tween the H and NO3 concentrations in the effl uent solutions were interpreted as the eff ects of ionic strength on the surface charge. At pH 4.0, the surface charges of the natural gibbsite and goethite were positive because their pH0 values were >4.0 − and NO3 was the counterion; the surface charge of the natural + hematite was negative because its pH0 value was <4.0 and Na was the counterion. As the aqueous NaNO3 concentration increased, the positive surface charge on the natural gibbsite and goethite increased due to increased adsorption of H+ onto the surfaces and, thus, the H+ concentration in aqueous solution decreased (Uehara and Gillman, 1981; Dzombak and Morel, 1990); similarly, the negative surface charge on the natural hematite increased due to the desorption of H+ from its surface and, thus, the H+ concentration in aqueous solution increased. Th ese linear relationships of amplitudes and phases between + − H and NO3 concentrations in the effl uent solutions were interpreted as ion-pair formation. Th e ion-pair formation in the β plane of the natural minerals also resulted in a reduction in the concentration of the available charged surface sites according to Fig. 8. Linear relationship of the amplitudes and phases between + + − Eq. [2], and then the following reaction of H adsorption on the the output signals of the concentrations of H and NO3 in effl uent natural gibbsite and goethite was enhanced: solutions of the column experiments using the three natural minerals.

+ ++ SOH HR SOH2 [29] the natural minerals were virtually linear and thus justifying their treatment as linear systems. or the desorption of H+ on the natural hematite was enhanced:

SOHR SO−++ H [30]

For the natural hematite at pH 4.0, the aqueous concentrations of the counterion Na+ was not measured; its concentration was calculated based on the charge bal- + − + ance of H , NO3 , and Na in the aqueous solution.

Spectral Components of the Input and Output Signals

Th e RAs of the nth subharmonics of the input and − output signals of the NO3 concentration are shown in Fig. 9. Th e subharmonics with a frequency of greater than fi ve times the fundamental frequency were negligible in this study. Th e RAs of the input signal of the − NO3 concentration were <2%, which indicated that the input signal was of the designed frequency. Th e − RAs of the output signals of the NO3 concentration were <11, 6, and 12% for the natural gibbsite, goethite, and hematite, respectively. Even though the values of the RAs of the output signals were greater than that of the input signal of the NO − concentrations, they were 3 − Fig. 9. Relative amplitude of subharmonics (RAS) of the input signals (NO3 <<1, indicating that the processes inside the columns of − concentrations in the infl uent solutions) and output signals (NO3 concentrations in the effl uent solutions) in the column experiments using the three natural minerals.

1574 SSSAJ: Volume 74: Number 5 • September–October 2010 faces of the three natural minerals coupled with transport along the column profi le were able to be stimulated by the input signals and identifi ed si- multaneously. In Sasaki et al. (1983), the rates of ion-pair recombination and dissociation for synthesized goethite were at a scale of 105 L mol−1 s−1 and 104 s−1, respectively, which were much higher than the rates at scales of 10 L mol−1 min−1 and 10−1 min−1 obtained in this study. Th is diff erence in reaction rates might be due to the surfaces of the natural minerals used in this study. Th e natural minerals used in this study were obtained by grinding the natural products, and amorphous material probably developed during grinding, whereas the amount of amorphous material might be negligible at the crystal surface of synthetic goethite (Ryden et al., 1977). Th e amorphous material Fig. 10. Measured transfer function (symbols) and the modeling of ion-pair formation at the surfaces of contents of the three natural minerals the three natural minerals coupled with a transport process. are shown in Table 1. Th e structural Estimation of the Diffusion Equilibrium Constant porosity characteristic of amorphous and Recombination–Dissociation Reaction Rates material and its coatings on crystalline mineral particles (Jones Th e transfer function of ion-pair formation was measured and Uehara, 1973) might result in a reduction of the movement based on inputs and outputs, and the model of Eq. [17] and [18] of a counterion in the β plane around the surface of minerals with was fi tted by minimizing the cost function Eq. [22] as shown in two degrees of translational freedom to fi nd a charged surface site Fig. 10. Th e estimates of the dispersion coeffi cients in the CDE before it becomes a site-bound counterion. − are shown in Table 1. Th e equilibrium constant of the diff usion When NO3 was the counterion, the intrinsic equilibrium process, Kdiff usion, the recombination–dissociation reaction rates constant of the recombination–dissociation process in ion-pair int and their equilibrium constant Kcounterion are shown in Table 2. formation, log Kcounterion , at the surface of the natural goethite Th e highest angular frequency of the input was 2π/12.8 in this study was 2.08—within the range of the experimental val- = 4.49 rad min−1. Th e passbands of the CDE (the frequency ues, ?1.7–2.4, in Sahai and Sverjensky (1997). For the natural int at which the amplitude of the CDE transfer function is 0.707) gibbsite, log Kcounterion was 2.44 in this study, higher than the were 2.8, 3.0, and 2.4 rad min−1 for the natural gibbsite, goe- predicted value of 1.91 in Sahai and Sverjensky (1997). When + int thite, and hematite, respectively. Th e recombination rate kr[S ] Na was the counterion, log Kcounterion at the surface of the −1 0 −1 and dissociation rate kd were at a scale of 10 ? 10 rad min . natural hematite was 1.45 in this study, lower than the range Th e reaction rates were at the same scale as the passbands of the of the experimental values, ?1.6–2.9, in Sahai and Sverjensky int CDE, and they were less than the highest angular frequency of (1997). Th e value of Kcounterion at the surface of the natural the input. Th us, the processes of ion-pair formation on the sur- gibbsite was 2.3 times that of the natural goethite, which is con-

Table 2. Estimated rates of diffusion processes and recombination–dissociation reactions of counterions with charged surface sites to form an ion pair at the β plane.

Recombination rate‡ Natural mineral Counterion K † Dissociation rate, k log K int = log(k /k ) diffusion k d counterion r d kr[S ] r min−1 L mol−1 min−1 min−1 − Gibbsite NO3 0.196 1.179 51.952 0.189 2.44 − Goethite NO3 0.202 1.186 30.491 0.256 2.08 Hematite Na+ 1.130 0.141 8.028 0.285 1.45 † Diffusion process equilibrium constant.

‡ kr, recombination reaction rate; [S ], concentration of the charged surface site at equilibrium.

SSSAJ: Volume 74: Number 5 • September–October 2010 1575 sistent with the favored outer sphere TLM on gibbsite for the Hiemstra, T., W.H. Van Riemsdjk, and G.H. Bolt. 1989b. Multisite proton adsorption modeling at the solid/solution interface of (hydr)oxides: A adsorption of B while an inner sphere TLM was formed on goe- new approach: I. Model description and evaluation of intrinsic reaction thite (Goldberg, 2005). constants. J. Colloid Interface Sci. 133:91–104. In Qafoku and Sumner (2001, 2002) and Qafoku et al. Hiemstra, T., P. Venema, and W.H. Van Riemsdijk. 1996. Intrinsic proton affi nity of reactive surface groups of metal (hydr)oxides: Th e bond valence (2004), salt adsorption was investigated using clay-fraction min- principle. J. Colloid Interface Sci. 184:680–692. eralogy dominated by kaolinite and Al and Fe oxides. In the sys- Hodges, S.C., and L.W. Zelazny. 1980. Determination of noncrystalline soil tem with oppositely charged surfaces, ion-pair formation may components by weight diff erence aft er selective dissolution. Clays Clay Miner. 28:35–42. be a mechanism for the counterion to be adsorbed in the Stern 2+ Hohl, H., and W. Stumm. 1976. Interaction of Pb with hydrous γ-Al2O3. J. layers of the surfaces. Th e diff erence in the equilibrium constants Colloid Interface Sci. 55:281–288. of recombination–dissociation reactions may result in diff erent Horita, G.I. 1993. Process for recovery of phosphorus and aluminum compounds counterion charge densities at diff erent mineral surfaces, and from Maranhao aluminum phosphate rock. Fert. Res. 34:79–84. Huang, C., and W. Stumm. 1973. Specifi c adsorption of cations of hydrous thus, interactions between oppositely charged surfaces may oc- γ-Al2O3. J. Colloid Interface Sci. 43:409–420. cur to balance the simultaneous adsorption of a cation and an Jones, R.C., and G. Uehara. 1973. Amorphous coatings on mineral surfaces. Soil anion (Qafoku and Sumner, 2002). Sci. Soc. Am. Proc. 37:792–798. Jury, W.A., and K. Roth. 1990. Transfer functions and solute movement through soil: Th eory and applications. Birkhauser Verlag, Basel, Switzerland. CONCLUSIONS Mekaru, T., and G. Uehara. 1972. Anion adsorption in ferruginous tropical soils. Th e processes of ion-pair formation at surfaces of natural Soil Sci. Soc. Am. J. 36:296–300. gibbsite, goethite. and hematite and a coupled transport pro- Nielsen, D.R., and J.W. Biggar. 1961. Miscible displacement in soils: I. Experimental information. Soil Sci. Soc. Am. Proc. 25:1–5. cess were identifi ed in column experiments using the frequency Parks, G.A. 1965. Th e isoelectric points of solid oxides, solid hydroxides, and domain method. Th is is a general technique applicable to other aqueous hydroxo complex systems. Chem. Rev. 65:177–198. single or multiple counterions, such as Cl−, K+, or SO 2−, to Qafoku, N.P., E. van Ranst, A. Noble, and G. Baert. 2004. Variable charge soils: 4 Th eir mineralogy, chemistry and management. Adv. Agron. 84:159–215. form an ion pair with a charged sites at the surface of minerals Qafoku, N.P., and M.E. Sumner. 2001. Retention and transport of calcium and soils. Th e factors determining the rates of ion-pair formation nitrate in variable charge subsoils. Soil Sci. 166:297–307. include the counterion concentration in aqueous solution, the Qafoku, N.P., and M.E. Sumner. 2002. Adsorption and desorption of indiff erent ions in variable charge subsoils. Soil Sci. Soc. Am. J. 66:1231–1239. ionic strength of the aqueous solution, the surface charge density, Rabung, T., H. Geckeis, J. Kim, and H.P. Beck. 1998. Sorption of Eu(III) on a and the amorphous material content. natural hematite: Application of a surface complexation model. J. Colloid Interface Sci. 208:153–161. ACKNOWLEDGMENTS Ryden, J.C., J.R. McLaughlin, and J.K. Syers. 1977. Time-dependent sorption of phosphate by soils and hydrous ferric oxides. Eur. J. Soil Sci. 28:585–595. We are grateful to Dr. Jaw-Kai Wang and Dr. Jingyu Chen for help in Sahai, N., and D.A. Sverjensky. 1997. Solvation and electrostatic model for specifi c the experiments, Dr. Youjun Deng for x-ray diff raction analysis, and electrolyte adsorption. Geochim. Cosmochim. Acta 61:2827–2848. Prof. Ning-shou Xu for reviewing earlier draft s of the manuscript. Th is Sasaki, M., M. Moriya, T. Yasunaga, and R.D. Astumain. 1983. A kinetic study research was made possible through support provided by the Offi ces of of ion-pair formation on the surface of α-FeOOH in aqueous suspensions Agriculture and Natural Resources Management, Bureau for Economic using the electric fi eld pulse technique. J. Phys. Chem. 87:1449–1453. Growth, Agriculture and Trade, and the U.S. Agency for International Schindler, P., and H.R. Kamber. 1968. Die Aciditat von Silanolgruppen. Helv. 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1576 SSSAJ: Volume 74: Number 5 • September–October 2010