A Generalized-Rate Model for Describing and Scaling Redox Kinetics

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Environmental Modeling

A Generalized Rate Model for Describing and Scaling Redox Kinetics in Sediments Containing Variable Redox Reactive Materials Fen Xu, Yuanyuan Liu, and Chongxuan Liu Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.7b06354 • Publication Date (Web): 30 Mar 2018 Downloaded from http://pubs.acs.org on March 31, 2018

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Environmental Science & Technology

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A Generalized Rate Model for Describing and Scaling Redox Kinetics in Sediments

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Containing Variable Redox Reactive Materials

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Fen Xu1, 2†, Yuanyuan Liu2, 3†, and Chongxuan Liu2,4*

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1

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University of Technology, Chengdu, 610059, China

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2

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3 Key

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Engineering, Nanjing University, Nanjing, 210023, China

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4School

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State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu

Pacific Northwest National Laboratory, Richland, WA 99354 Laboratory of Surficial Geochemistry (Ministry of Education), School of Earth Sciences and

of Environmental Science and Engineering, Southern University of Science and

Technology, Shenzhen, 518055, China

Submit to Environmental Science & Technology

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†These

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contributed to the modeling part.

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*

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of Science and Technology, Shenzhen, China. Email: [email protected]

authors contributed equally. Xu, F. contributed to the experimental part and Liu, Y.

Corresponding author: School of Environmental Science and Engineering, Southern University

1

ACS Paragon Plus Environment


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ABSTRACT: This study developed a generalized modeling approach to describe and scale redox

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reactions from reactive components to the sediments and their assemblages using Cr(VI) reduction

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as an example. Batch experiments were performed to characterize the rates of Cr(VI) reduction in

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four Fe(II)-containing sediments and their assemblages. The experimental data were first used to

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calibrate a generalized rate model of Cr(VI) reduction with generic rate parameters. The

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generalized rate model was then used to describe the kinetics of Cr(VI) reduction in the sediment

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assemblages by linearly scaling the rate parameters from the individual sediments. By comparing

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with the experimental results, this study found that the generalized rate model with generic rate

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parameters can describe Cr(VI) reduction in individual sediments and their assemblages with

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different redox reactivity toward Cr(VI) reduction. The sediment-associated Fe(II) and its

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reactivity were found to be the key variables in the generalized model for describing the Cr(VI)

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reduction in the studied sediments. A three-step extraction method was subsequently developed to

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estimate the rate-specific Fe(II) pools that can facilitate the application of the scaling approach in

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field systems.

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INTRODUCTION:

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A sediment is a composite of minerals or reactive components that have different properties

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of reactivity with respect to mineral dissolution and precipitation, redox transformation, and

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interaction with metals and contaminants.1-6 The heterogeneity of geochemical reactivity in the

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sediment poses a significant challenge to scale geochemical reaction parameters from laboratory

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to field-scale applications when a reactive transport model is used to predict geochemical processes

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in natural system.2, 4, 6-14 Multi-rate model has been widely used in reactive transport models to

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address the scaling challenge.2, 3, 5, 7, 15-20 Generalized composite (GC) and component additivity

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(CA) are two such concepts for scaling sorption processes in heterogeneous sediments.1, 21-28 The

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GC approach assumes that the sorptive properties of a sediment can be described using generic

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sorption reactions with parameters determined by fitting experimental data. However, the model

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parameters obtained from one field site cannot be applied to the others.29-31 The CA approach

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assumed that sorption to a sediment can be described by linearly adding the sorption to the

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individual components in the sediment.1, 26, 27, 32, 33 However, the applicability of the CA approach

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is limited because of the complexity of minerals and their interactions in a sediment that are often

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difficult to identify in the field. An effective approach to scale sorption process and parameters has

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yet to be developed.

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Relative to sorption process, the scaling behavior of redox reaction is understudied. In

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natural sediments, various reductants (Fe(II), S(II), organic materials, etc.) can coexist that can

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reduce redox sensitive metals and contaminants such as Cr(VI), Tc(VII), U(VI), and As(V) etc.3,

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34-40

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various reactive components,2, 16, 18, 20 or single component with species of different reactivity such

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as Fe(II).3, 34, 35 These models are mathematically analogue to the GC approach for scaling sorption

Multi-rate models have been used to describe reaction properties in a sediment containing

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processes. Because of the difficulty to identify all the redox species and their redox reactivity in

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natural sediments, the rate constants fitted from one sediment are usually not applicable to other

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sediments, even at the same field site. Development of effective approaches to scale redox

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processes is therefore critically needed to extrapolate laboratory redox reactions for field-scale

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applications.

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In this study, we evaluated a generalized multirate model with linear parameter additivity

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concept to describe and scale redox kinetics determined from individual sediments to their

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assemblages using Cr(VI) reduction as an example. Four individual sediments with different redox

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reactivity collected from the Columbia River hyporheic zone (HZ) at the US DOE’s Hanford Site

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were first used to experimentally study Cr(VI) reduction, which were then used to calibrate the

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generalized multirate model with a set of generic rate constants. The individual sediments were

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then composited with different mass ratios to experimentally study Cr(VI) reduction kinetics in

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the assemblages. The experimental results were used to evaluate the applicability of the

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generalized multirate model with the set of generic rate constants from the individual sediments.

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The results demonstrated the applicability of the generalized rate model. The key variables in the

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generalized multirate model for the studied sediments are the Fe(II) contents in different rate

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pools.34 A three-step extraction method was subsequently developed to experimentally quantify

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these variables so that the generalized model with the generic rate constants can be readily

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validated or directly used for field-scale applications of Cr(VI) reduction by sediment-associated

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Fe(II).

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MATERIALS AND METHODS

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Sediment and Sample Preparation. Sediments were collected from the Columbia River

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HZ at the US Department of Energy’s Hanford site 300 Area (Figure 1). The detailed information

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about the study site and the redox properties of the sediments from our previous study are provided

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in supporting information (SI).

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The first sediment, which mainly consists of silt and clay, was frozen-collected in the HZ

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(Figure 1B, white circle) and was stored at –80 °C until use. The frozen collection method was

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described elsewhere.36 This sediment sample was named as Sediment A, hereafter.

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The second sediment was a composite of 20 sediments frozen-collected from 20 different

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locations in the HZ (Figure 1, red circles). The collected sediments were sieved and C>B>A). The reduction

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rate in Sediment D was significantly higher than that in Sediment C (Figure 2c and d). Within 20

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hours, 400 μM Cr(VI) was reduced by Sediment D; while only 40 μM Cr(VI) was reduced by

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Sediment C. Although HCl-extractable Fe(II) in Sediment D was only 4 times higher than in

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Sediment C, Cr(VI) reduction rate in Sediment D was 10 times faster than in Sediment C. The

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result suggested that the freshly generated Fe(II) (biogenic Fe(II)) was more reactive than that in

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the original sediment. The importance of biogenic Fe(II) on Cr(VI) reduction has been

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demonstrated in previous studies.34, 35, 42, 48

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The experimental results of Cr(VI) reduction in the sediments were described using the

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generalized three-rate model developed in our previous studies (Eqs 1 and 2).34, 35 In the previous

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study,35 we showed that at least three rates were needed to describe Cr(VI) reduction in the

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sediments from the Hanford HZ. The fraction of rate component i ( f i ) and its corresponding rate

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constant (ki and KCr(VI)) were fitted from the experimental data (Figure 2a−d). First, the parameters

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were estimated by the initial and late time reduction kinetics. Specifically, the fraction of the fastest

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rate component and its rate constant (k1) were estimated from the initial rate of the measured Cr(VI)

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reduction profiles (data within the first ~5 h). Then, the fraction of Fe(II) in the slowest rate

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component and its rate constant (k3) were estimated from the measured Cr(VI) reduction profiles

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at the late times (after ～20 h). The rest of the Fe(II) in the sediment was assigned to the second

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rate component with a rate constant k2, which was estimated by fitting the overall Cr(VI) reduction

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profile. In the generalized rate model, all the sediments share the same set of rate constants, but

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with different fractions of Fe(II) in different rate components. Consequently, all the parameters in

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the generalized model were further tuned to minimize the overall least error between the measured

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and calculated Cr(VI) concentrations for all the individual sediments in Figure 2a-d.

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As shown in Figure 2a-d, the generalized three-rate model can describe Cr(VI) reduction

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kinetics in Sediment A-D using the same set of rate constants (ki and KCr(VI), i = 1, 2, and 3 in Eqs

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1 and 2). However, the fraction of Fe(II) in each rate component ( f i m , i = 1, 2, and 3 and m = A,

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B, C, and D) was different between different sediments, reflecting their overall difference in

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sediment reactivity toward Cr(VI) reduction. The estimated rate constants (ki =1, 2, 3) for different

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rate components were different, indicating that each rate component had different reactivity

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towards Cr(VI) reduction. The rate constant of the fastest component (k1 = 0.4 ± 0.03 M-1∙h-1) was

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100 times larger than the moderate rate component (k2 = 0.003 ± 0.001 M-1∙h-1), which was 10

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times larger than the slowest component (k3 = 0.0003 ± 0.0001 M-1∙h-1). The fitted value of KCr(VI)

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was 1.5 ± 0.2×10-4 M.

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Cr(VI) reduction rate is proportional to both the initial Fe(II) and its corresponding rate

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constant for each rate component (Eqs 1 and 2). Since the rate constant for the fastest component

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was much larger than other components, the overall rate of Cr(VI) reduction quickly decreased

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when the Fe(II) in the fastest component was exhausted (Figure 2a-d). The average rate constant

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(< k >) for a sediment was calculated from the following equation: N

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k   fi k i

(5)

i 1

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Table 2 shows that the average rate increased with increasing amount of Fe(II) in the fastest

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component. The overall rate of Cr(VI) reduction in Sediment C was much faster than in Sediment

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A and B (Figure 2a−c) because Sediment C contained a higher amount of Fe(II) (Table 1) and a

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larger fraction of Fe(II) was associated with the fastest component than that in Sediment A and B

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(Table 2). Fresh biogenic Fe(II) in Sediment D is a strong reductant for Cr(VI) reduction.34, 42, 48 11



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Model fitting found that 40% of the fresh biogenic Fe(II) in Sediment D was associated with the

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fastest rate component (Table 2) that was much higher than that in Sediment C (15%).

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Scaling of Cr(VI) Reduction Kinetics to the Sediment Assemblages. The generalized

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multirate model was evaluated for scaling Cr(VI) reduction rates with parameters determined from

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the individual sediments to the sediment assemblages. In the generalized rate model, the rate

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constants were the same for all the individual sediments so that the rate additivity model (Eq 3)

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becomes the Fe(II) additivity model for each rate component (Eq 4). With the calculated Fe(II)

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concentration for each rate component from individual sediments (Tables 1 and 2), equations 1

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and 2 was used to calculate the rate of Cr(VI) reduction in the assemblages.

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As a validation of the additivity model for Fe(II), the calculated total Fe(II) in Sediment

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Assemblage 1, 2, 3, and 4 was compared with HCl-extracted Fe(II) concentrations. The results

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indicated that the calculated HCl-extractable Fe(II) concentrations generally matched with the

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measured HCl-extractable Fe(II) concentrations (Table 1).

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The generalized rate model with linearly scaled Fe(II) concentrations was used to predict

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Cr(VI) reduction in Sediment Assemblage 1-4 (Figure 2e-h). The predicted Cr(VI) reduction

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matched well with the experimental results for all the sediment assemblages, indicating that the

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proposed approach can be applied to scale Cr(VI) reduction with a generic set of rate constants

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and Fe(II) concentrations scaled from individual sediments. As expected, the rate of Cr(VI)

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reduction follows an order of Assemblage 4>3>2>1, consistent with their trends in total Fe(II)

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content and Fe(II) in the fastest rate component. It is also consistent with the average rate constants

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(k) in the sediment assemblages ranged from 0.114 to 0.152 M-1h-1 in Sediment Assemblage 1 to

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4 (Table 2).

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Calculation of Cr(VI) Reduction Rate Using Experimentally Determined Fe(II) in the

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Sediments. The advantage of the generalized model is that the rate formulation and rate constants

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can be used to describe the individual sediments and their assemblage. This provides a way to

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extrapolate the generalized rate for field scale application. The remaining key variables for scaling

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the model are the total reactive Fe(II) and its fractions in different rate components. These

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parameters may be difficult to determine in field sediments without extensive characterization of

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sediment properties. The extraction approach described in the method section is to avoid this

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problem by directly determining Fe(II) contents in different rate components so that the

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generalized rate model with rate constants determined from laboratory experiments can be directly

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applied in field.

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Three-step extraction method was developed based on the rate of Fe(II) release in the

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presence of 1 M sodium acetate (Figure S1), which, together with 1 M HCl extraction result, was

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used to determine Fe(II) concentration for each rate component. The Fe(II) extracted with 1 M

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sodium acetate includes soluble Fe(II), exchangeable Fe(II), and carbonate-associated Fe(II) that

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are all reactive toward Cr(VI) reduction.49-52 DI water is often used to extract water soluble fraction

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of Fe(II), and MgCl2/CaCl2 to extract ion exchangeable Fe(II).45-47 Using these extraction methods,

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it was found that there was no detectable dissolved Fe(II) in the studied sediments. The

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exchangeable fractions of Fe(II) were seldom detected in Sediment A, B, and C. The extracted

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Fe(II) using 1 M sodium acetate (Figure S1) increased with increasing time. The majority of Fe(II)

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was extracted within the first hour. After 1 h, the extracted Fe(II) concentration only increased

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slowly until 24 h, and then stabilized after that. Intuitively, Fe(II) extracted by 1 M sodium acetate

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within the first hour should be readily available for Cr(VI) reduction. The rate-limited release of

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Fe(II) after the first hour may be due to the effect of chemical heterogeneity (mineral phases that

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have different reactivity in sodium acetate solution) or physical heterogeneity (mass transfer

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processes such as diffusion in internal pores of mineral grains). Based on this extraction procedures,

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Fe(II) extracted within the first hour was used as the Fe(II) in the fastest rate component, and Fe(II)

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extracted between 1 to 24 h was considered as that in the moderate rate component, and the

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difference between HCl (1M HCl for 2 h) extractable and sodium acetate (1 M sodium acetate for

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24 h) extractable Fe(II) was used as the Fe(II) for the slowest rate component. The Fe(II) assigned

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to three rate components using this method (Table 3) matched well with the additivity model-fitted

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value (Table 2). Other extraction procedures were tried, but the fractions of Fe(II) determined

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using the approach as described here matched the best with the model.

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Using the experimentally determined Fe(II) for each rate component, the simulated results

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from the generalized rate model with the same rate constants as described before generally matched

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with the experimental data of Cr(VI) reduction for Sediment A, B, and D (Figure 3a-c). However,

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the predicted Cr(VI) concentration for Sediment B was lower than experimental results because

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the experimentally determined fraction of Fe(II) in the fastest component in Sediment B was

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significantly over-estimated. In general, the average rate constants, , of Sediments A, B, and

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D predicted using experimentally determined Fe(II) in the rate components (Table 3) was greater

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than those fitted from Cr(VI) reduction kinetics (Table 2).

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Cr(VI) reduction in the assemblage sediments was also calculated using the experimentally

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determined Fe(II) fractions for Sediment A, B, and D and the reactive Fe(II) components fitted

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from Cr(VI) reduction kinetics for Sediment C. Some of Sediment C was preserved in a stainless

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steel container and stored in a 4 °C refrigerator for 9 months before extraction method was

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developed. After 9 month storage in a 4 °C refrigerator, HCl-extractable Fe(II) concentration in

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Sediment C decreased from 15.42 ± 2.61 to 9.51 ± 0.88 μmol∙g-1 as a result of oxidation by O2.35,

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40, 53 Since the redox reactivity of the sediment

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components in Sediment C was fitted from Cr(VI) reduction kinetics. The predicted Cr(VI)

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reduction in the sediment assemblages matched the major trend of the experimental results of

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Cr(VI) reduction (Figure 3d-g). However, the prediction over-estimated initial Cr(VI) reduction

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rate due to over-estimation of Fe(II) in the fastest rate component by the three-step extraction

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method. The discrepancy increased with the mass ratio of Sediment D because Sediment D

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contained higher fraction of Fe(II) in the fastest component.

changed during long time storage, the reactive Fe(II)

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Environmental Implications. Natural sediments usually contain multiple redox reactive

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species with different reactivity, which requires significant effort to quantify redox kinetics, rate

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models and rate properties, especially at field site with heterogeneous sediment reaction properties.

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This study developed a method that used a generalized multirate model concept with a generic set

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of rate constants to quantitatively describe redox kinetics in sediments with different redox

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reactivity, and to scale the redox kinetics with parameters determined from individual sediments

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to sediment assemblages, providing a framework for extrapolating redox properties derived from

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laboratory studies for field scale applications.

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Reactive transport models are typically used to simulate and predict solute and contaminant

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transport in field. In applying a reactive transport model, a simulation domain is typically divided

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into many numerical grids. The sediment properties within each numerical grid may be

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heterogeneous, but are often assumed to be homogeneous. Under this condition, the generalized

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rate model approach described here can be used to calculate the sediment properties within a

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numerical grid based on the redox properties of a few generic reactive components that can be

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determined at laboratory. If the mass fractions of such generic reactive components can be

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measured across the model domain, the redox properties in all the numerical grids can be calculated 15



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using the developed approach. In this study, we also developed an extraction approach to

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determine the mass fractions of the reactive components (i.e., sediment-associated Fe(II) with

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different reactivity). This would facilitate the application of the generalized approach in reactive

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transport simulations.

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The method developed in this study was focused on Cr(VI) reduction by sediment-

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associated Fe(II) with three generic Fe(II) components. Cr(VI) can be reduced by other

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reductants such as S(-II),40, 54-56 and organic matters.57, 58 Consequently, the number of the

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generic reactive components and associated rate constants derived from this study are likely site-

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specific. However, the concept of the generalized multirate model with a set of generic rate

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constants to extrapolate laboratory results for field scale applications is applicable for other sites

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and to other redox sensitive contaminants such as Tc, U, As, etc. For these systems, the

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generalized rate expression (Eqs 1 and 2) may need to be modified, for example, by increasing

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the number of rate components to incorporate other redox reaction mechanisms. Further study,

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especially at the field scale, is needed to further evaluate the applicability of the generalized

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model framework as proposed here.

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Supporting Information. The details of experimental method, the chemical composition of

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SRW, the grain size fraction, and the sodium acetate extraction kinetics are provided in the

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supporting information. This material is available free of charge via the Internet at

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http://pubs.acs.org

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ACKNOWLEDGMENTS

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This research is supported by the U.S. DOE, Office of Science, Biological and Environmental

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Research (BER) as part of the Subsurface Biogeochemical Research (SBR) Program through

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Pacific Northwest National Laboratory (PNNL) SBR Science Focus Area (SFA) Research 16


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Project. F Xu, Y Liu, and C Liu also acknowledge the supports from Research Funds from

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National Natural Science Foundation of China (No.41502233, No. 41572228, NO. 41521001,

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and No. 41773111) and China Postdoctoral Science Foundation (No. 2015M582305). C Liu also

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acknowledges the support from Southern University of Science Technology (G61296001) and

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from Guangdong Provincial Key Laboratory of Soil and Groundwater Pollution Control.

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25



B

A

522 523

Figure 1. Sediment sampling locations (the right panel) at the HZ of the U.S. DOE’s Hanford

524

Site in Washington State. The top left panel was the study area. The bottom left panel was the

525

schematic diagram of the geologic formation of the HZ at the site. Sediment A was frozen

526

collected from an impermeable zone (white circle in the right panel), Sediment B was

527

homogenized from the sediments frozen collected from 20 different locations (red circles), and

528

Sediment C was fresh collected from the shore of the Columbia River (white cross).

26


Page 26 of 32

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529

Table 1. The mass ratios of Sediment A−D in the sediment assemblages and HCl-extractable

530

Fe(II) content in the sediment samples.

Sediment A Sediment B Sediment C Sediment D Sediment Assemblage 1 Sediment Assemblage 2 Sediment Assemblage 3 Sediment Assemblage 4

A:B:C:D 1:0:0:0 0:1:0:0 0:0:1:0 0:0:0:1 1:2:2:1 1:1:1:1 0:1:1:2 0:1:1:4

Measured Fe(II) (μmol∙g-1) 1.52 ± 0.08 4.58 ± 0.35 15.42 ± 2.61 62.28 ± 3.22 15.18 ± 2.02 20.24 ± 1.37 33.25 ± 2.11 38.30 ± 1.42

Predicted Fe(II) (μmol∙g-1) NA NA NA NA 17.15 ± 1.54 20.90 ± 1.57 35.80 ± 2.34 44.45 ± 2.63

531

Measured Fe(II) in the sediments were experimentally determined; predicted Fe(II) in sediment

532

assemblages were calculated from mass ratios and Fe(II) concentrations of the individual

533

sediments.

534

27


40 35 30 25 20 15 10 5 0

a) Sediment A

Page 28 of 32

b) Sediment B 60 Cr(VI) (M)

Cr(VI) (M)


0

100

200 Time(h)

20

0

300

0

50 Time(h)

1500

c) Sediment C

300

40

100

d) Sediment D

Cr(VI) (M)

Cr(VI) (M)

250 200 150 100

1000

500

50 0

0

100

200

0

300

0

200

Time(h) 800

e) SA1_1:2:2:1

600

Cr(VI) (M)

Cr(VI) (M)

800

400 200 0

600 400 200

0

100

200

300

0

400

0

100

200

300

Time(h) 800

g) SA3_0:1:1:2

600

Cr(VI) (M)

Cr(VI) (M)

800

400 200

535

600

f) SA2_1:1:1:1

Time(h)

0

400 Time(h)

h) SA4_ 0:1:1:4

600 400 200

0

50

100

150

200

250

0

0

100

Time(h)

200

300

Time(h)

536

Figure 2. Measured (symbols) and simulated (lines) Cr(VI) reduction in Sediment A−D and

537

Sediment Assemblage 1−4.

28


Page 29 of 32


538

Table 2. Fractions of the three redox reactive components with different rate constants in the

539

sediment samples.

Sediment A Sediment B Sediment C Sediment D Sediment Assemblage 1 Sediment Assemblage 2 Sediment Assemblage 3 Sediment Assemblage 4 540 541 542

f1 0.01±0.005 0.01±0.002 0.15±0.005 0.40±0.010 0.28 0.32 0.36 0.38

f2 0.20±0.10 0.20±0.10 0.55±0.17 0.15±0.04 0.28 0.23 0.20 0.17

f3 0.79±0.10 0.79±0.10 0.30±0.17 0.45±0.05 0.44 0.45 0.44 0.45

0.005 0.005 0.062 0.161 0.114 0.131 0.145 0.152

f1, f2, and f3 are the fractions of the fastest, moderate, and slowest rate components. Fraction values in Sediment A-D were model-fitted, and in Sediment Assemblage1 to 4 were calculated from Eq 4.

543

29



Page 30 of 32

544

Table 3. Fractions of the three reactive Fe(II) components determined by the three-step

545

extraction method and the average rate constants calculated from the experimental determined

546

fractions.

Sediment A Sediment B Sediment C* Sediment D 547 548 549

f1 0.03±0.01 0.07±0.06 0.15±0.005 0.44±0.11

f2 0.16±0.01 0.30±0.09 0.55±0.17 0.27±0.11

f3 0.81±0.01 0.63±0.04 0.30±0.17 0.29±0.03

0.013 0.029 0.062 0.177

f1, f2, and f3 are the fractions of the fastest, moderate, and slowest rate components. *Fractions of Sediment C were fitted from Cr(VI) reduction kinetics because the redox reactivity of the sediment changed during long time storage.

30


Page 31 of 32


10 0

0

100

200

1000

40

20

0

300

Cr(VI) (M)

20

0

20

Time(h)

60

600 400 200

0

100

300

551 552 553 554

0

200

400 Time(h)

600

e) SA2_1:1:1:1

600 400

0

400

0

100

800

f) SA3_0:1:1:2

600

Cr(VI) (M)

Cr(VI) (M)

200 Time(h)

400 200

550

0

100

200

800

0

80

800

d) SA1_1:2:2:1 Cr(VI) (M)

Cr(VI) (M)

40

500

Time(h)

800

0

c) Sediment D

60 Cr(VI) (M)

Cr(VI) (M)

30

1500

b) Sediment B

a) Sediment A

40

200 Time(h)

300

g) SA4_0:1:1:4

600 400 200

0

50

100

150

200

250

0

Time(h)

0

100

200

300

Time(h)

Figure 3. Experimental and predicted Cr(VI) reduction in Sediment A, B, and D and Sediment Assemblage 1−4. The fractions of the Fe(II) components in Sediment A, B, and D were determined by the three-steps extraction method, and in Sediment Assemblage1 to 4 were calculated from Eq 4.

555

31



t

3

i  3 ki CFe(II) i 1

800

C

A,i Fe(II)

K Cr(VI)  CCr(VI)

  C m

m ,i Fe(II)

40

Sed. A

20 0

m

0

100 200 300

Time(h)

400 0

0

200

Time(h)

400

Cr(VI) (M)

Cr(VI) (M)

Sed. A:B:C:D=1:2:2:1

CCr(VI)

Cr(VI) (M)

CCr(VI)

Page 32 of 32

1500 1000

Sed. D

500 0

0 200 400 600

Time(h)


A Generalized-Rate Model for Describing and Scaling Redox Kinetics

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