Environ. Sci. Technol. 2004, 38, 3950-3957
Estimation of Diffusion Coefficient of Chromium in Colloidal Silica Using Digital Photography NETNAPID TANTEMSAPYA† AND J A Y N . M E E G O D A * ,‡ Environmental Engineering Department, Khon Kaen University, Khon Kaen, Thailand, and Department of Civil and Environmental Engineering, New Jersey Institute of Technology, Newark, New Jersey 07102
In order to study the effectiveness of using colloidal silica, NYACOL DP5110, to stabilize chromium-contaminated soil, the diffusion of chromium in colloidal silica gel was estimated from laboratory experiments. To measure diffusion coefficients of chromium in the colloidal silica gel, a new measurement method based on digital photography was introduced. A series of experiments were designed and conducted to validate this new method and to estimate the diffusion coefficients of chromium in the colloidal silica gel. Accuracy of the proposed method was evaluated by several different ways. It was found that the apparent diffusion coefficient of chromium in colloidal silica gel ranged from 1.76 to 8.48 × 10-10 m2/s depending mainly on the concentration of silica in the gel with chromium concentration less than 10-2 M. Higher silica concentrations yielded lower diffusion coefficients due to the obstruction to the free movement of chromium. The adsorption isotherm of chromate to colloidal silica gel was found to be linear at pH 7; the partition coefficient was calculated to be 0.549 L/g. Mass balance calculations were performed to evaluate the accuracy of the proposed method and found that the measuring error was less than 6.5%. Based on the test data, the estimation of diffusion coefficients for chromium in colloidal silica gel using digital photography seems to be accurate and precise. This method is suitable for analyzing colored chemicals inside clear/white gels. From the results, it can be concluded that the gel behaves as a porous material with silica network forming continuous solid phase and its pore space saturated with water. The chromium ions diffuse in porous silica gel on a tortuous path. Therefore, the bulk diffusion dominates. Thus, the silica can be represented as a fix and impenetrable immersion in the solution. The presence of these motionless silica chains leads to an increase in the mean path of the diffusing molecules between two points in the system. On the basis of the test results, it can also be concluded that colloidal silica, NYACOL DP5110, for in-situ treatment of chromium-contaminated soils seems to be ineffective. Further research of more realistic simulation of diffusion and refined gel formulation with the capacity to convert the chromium to an immobile form is recommended.
* Corresponding author phone: (973)596-2464; fax: (973)596-5790; e-mail:
[email protected]. † Khon Kaen University. ‡ New Jersey Institute of Technology. 3950
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Introduction Colloidal silica is a silicate-based material that is used to stabilize and treat contaminated soil. Due to its low viscosity, contaminated soil can be in-situ treated by direct injection or grouting. In addition, colloidal silica acts as a contaminant adsorbent due to its large surface area. Application of colloidal silica grouting to remedy contaminated soil was first introduced by Du Pont (1). The possibility of using colloidal silica to control migration of waste and the impact of soil properties on this technology were further investigated by Lawrence Berkeley National Laboratory (2-6). This research demonstrated that colloidal silica could be injected into soil creating an impermeable wall, preventing the further migration of contaminants. Persoff et al. (7) performed additional physical and numerical model studies to evaluate the possibility of using colloidal silica to control the movement of nonaqueous phase liquid (NAPLs). Although the contaminated soil can be stabilized by colloidal silica, contaminants may still diffuse through the gel. The diffusion of chemical through gel occurs through interior pores (8). There are two possible diffusion processes through colloidal silica pores: bulk diffusion and surface diffusion (9-11). The bulk diffusion occurs when the pore sizes and pressures are large, and the diffusion rate depends on the collision between molecules. Surface diffusion occurs when the molecules adsorb onto the surface of the pore and jump from one adsorption site to another. The transport of chromium by diffusion as a molar flux (the rate of transfer per unit area of section) can be numerically modeled using mass balance and Fick’s law. The diffusion coefficient of chromium in colloidal silica gel can be estimated by laboratory experiments. There are many laboratory techniques to estimate the diffusion coefficient of chemicals in a gel (8, 12, 13). However, these techniques are complicated and time-consuming. Recently, an optical method was proposed to measure the diffusion coefficient of metals in polymer (14-16). It involves taking digital photographs at different time intervals during diffusion and using prior calibrations. The color pixel values from digital photographs are converted to chemical concentrations to compute diffusion coefficient. The optical technique utilizes the recent advances of computer technology with graphic applications and eliminates the need for classical methods that are tedious and time-consuming for concentration measurements. Polymers have a similar structure to colloidal silica gel. Hence, in this study, the optical method was adopted to estimate the diffusion coefficient of chromium in colloidal silica gel. On the basis of the above discussion, in-situ solidification and stabilization of contaminated soils using colloidal silica is a well-established technology. However, the theoretical understanding of how colloidal silica functions as an effective stabilization medium or diffusion of chromium ions in gel mass after stabilization is not well understood. This paper presents a concise summary of the optical method, measurements, and results. This paper also presents a theoretical explanation of the role of diffusion in the colloidal silicatreated chromium-contaminated soil and a concise summary of the optical method, measurements, and results.
Materials and Methods Physical Diffusion Model. The chemical gradient and flux across a gel medium can be modeled using Fick’s law. For 10.1021/es0342704 CCC: $27.50
2004 American Chemical Society Published on Web 06/05/2004
Cwater ) C0
t ) 0, z f -∞
Equation 3 can be solved by Laplace transformation to obtain the following solutions (19):
Cgel )
Cwater )
FIGURE 1. Experimental condition for diffusion study. a one-dimensional diffusion with linear adsorption:
∂C ) ∂t
D ∂ 2C 2 F 1 + K ∂z θ
(1)
where C is a chemical concentration in water (mg/L); D is a diffusion coefficient (m2/s); K is a partition coefficient (L/ g); t is time (s); z is a distance in the diffused direction (m); and F is a mass density (g/L). A new diffusion coefficient Dtotal can be defined where
Dtotal )
D F 1+ K θ
(2)
and θ is porosity (volume of void/ total volume). Thus, eq 1 becomes
∂C ∂2 C ) Dtotal 2 ∂t ∂z
(3)
The diffusion coefficient can be estimated by making the following assumptions and using the appropriate boundary condition to analyze the data obtained from digital photography: (i) Transport occurs in two different media, which are water (-∞ < z < 0) and gel (0 < z < ∞). (ii) Transport is controlled only by diffusion and adsorption (no advection). To avoid the convection transport, the interface between water and gel is as large as possible to make sure that the concentration at the center is not affected by possible advection at the cell walls. Note that the charge of the ions might contribute to the diffusion, making the diffusive transport non-Fickian especially for high pH gels. (iii) The interface resistance is where the initial concentration of chemical entering the gel media is equal to a fraction of the chemical concentration in water and gel. (iv) Chemical concentration at the two ends of the cell are zero for z ) ∞ and C0 for z ) -∞ throughout the experiment (17). Thus, the cell can be assumed to be semi-infinite (18). The experimental condition is shown as Figure 1 where the concentration of the chemical is a solid line. By applying the following boundary conditions:
Cgel )K Cwater Dgel
t g 0, z ) 0
∂Cgel ∂Cwater ) Dwater t g 0, z ) 0 ∂x ∂x
where Cgel is the concentration of chemical in a gel phase (mg/g) and Cwater is the concentration of chemical in a water phase (mg/L):
Cgel ) 0
t ) 0, z f ∞
KC0 Dgel 1+K Dwater
( )
1/2
z
erfc
( ) ( ) {
C0 Dgel 1+K Dwater
1/2
1-K
(4)
x4Dgelt
Dgel Dwater
1/2
erf
}
z
x4Dwatert
(5)
The diffusivity of chromium in water (Dwater) is available in the literature (20-22). The partition coefficient (K) can be obtained from the experiment. Thus, the concentration profile of the two phases can be determined using eqs 4 and 5 to estimate the diffusion coefficient Dgel from experimental results. Materials. Colloidal Silica. Colloidal silica used in this research is NYACOL DP5110 from Eka Chemical Inc. It was specially made for the demonstration site (Viscous Barrier) at Brookhaven National Laboratory. NYACOL DP5110 is a mixture of colloidal silica, water, antifreeze, and alkaline material. The alkaline material was added to increase pH and to make the gel stable. However, the colloidal silica works in stabilizing the soil with little contribution from soil pH. The physical properties of this colloidal silica are as follows: density silica content pH viscosity
1.205 g/mL 30% by wt (300 g/L) 9 9.5 cps
Calcium Chloride (CaCl2). Reagent grade dihydrated calcium chloride from Fisher Scientific was used as a destabilizer. One part of 1.0 M calcium chloride to four parts of 300 g/L by weight colloidal silica was required to destabilize colloidal silica to form a gel within 3 h (23). One molar calcium chloride solution was prepared and used throughout the research. Chromium Solution (Cr6+). Reagent grade potassium dichromate (K2Cr2O7) crystals from Mallinckrodt Inc. were mixed to form the chromium solution. The crystals were oven-dried at 103 °C and transferred to a desiccator prior to use. A 1000 mg/L, pH 7, chromium stock solution was prepared and used throughout the research.
Experimental Procedures Partition Coefficient and Adsorption Isotherms. Colloidal silica, 300 g/L, was mixed with CaCl2 to form a gel (4:1 by volume). After mixing, 0.5 g (approximate 0.4 mL) of solution was transferred to a 50-mL round clear plastic tube with 3 cm diameter. From the setting, the large interface area is obtained to ensure the greatest adsorption. The gel formation time was expected to be 3 h (24). The gel samples were left for 24 h after the mixing to ensure complete curing before the measurements were taken. Once the colloidal silica gel was set, 20 mL of chromium solution (at different concentrations) was poured on top of the gel. Chromium concentrations ranged from 10-8 to 10-2 M/L. NaNO3 was added to maintain ionic strength at 10-3 M/L. The pH of the solution was kept at 7. The samples were shaken at 150 rpm and 20 °C for up to 72 h before taking chromium measurements. This was to ensure that the water was well mixed or there was no concentration gradient at the interface. Chromium concentrations were measured using an atomic absorption spectrophotometer (AAS PerkinElmer 370) for samples with chromium concentration higher VOL. 38, NO. 14, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 3. Isotherm of chromium sorption to colloidal silica gel.
FIGURE 2. Digital image from the diffusion study. than 10-6 g/L, and an atomic absorption graphite furnace (AAGF Perkin-Elmer 4110ZL) was used for the remaining samples. Measurements were taken after a range of contact times (1, 3, 6, 12, 24, 48, and 72 h). The concentration of chromium in gel was calculated by the difference. To study the optimum pH value for adsorption of chromium to colloidal silica gel, the pH of the chromium solution was varied from 3 to10. The pH of the solution was adjusted prior to adding to the gel. The sample preparation and the measurement were performed the same as above. Diffusion with Different Silica Contents. Different silica concentrations (240, 210, 180, 150, and 120 g/L) were prepared by mixing colloidal silica solution with different amounts of deionized water. Then, colloidal silica was mixed with CaCl2 to form a gel (4:1 by volume). After preparing the gel solutions, 2 mL of the colloidal silica solutions with different silica concentration was transferred to 5-mL clear plastic vials (1 × 1 × 5 cm3). The gel samples were left for 24 h for curing before the measurements. Once the colloidal silica gel was set, 0.20 mg of chromium (in 1 cm3 water) was poured on top of the set gel in a plastic vial and placed in place with constant light source. To determine the contribution to the measured diffusion coefficient due to the convective flow of chromium, the above experiment was repeated with 240 g/L silica content but with double the volume of chromium solution on top of the gel. The light used in the experiment is from Fluorescent Lamps. Then, the diffusion from aqueous phase to gel phase was observed and captured using a digital camera (Olympus Digital Camera D-460 zoom). Digital pictures were taken every 10 min for 360 min to record the color change with time. The experimental cell is shown in Figure 2. To convert digital colors to concentration values, calibration curves were prepared for each gel concentration by mixing the gel with various amounts of chromium to obtain the desired concentrations and then transferring them to vials. The partial concentrations of chromium used were 200, 180, 160, 140, 120, 100, 80, 60, 40, 20, and 0 mg of chromium/g of gel, respectively. The resolution of digital pictures was 1600 × 1200 pixels. Note that for the experimental setup used, 1 cm of digital picture contained approximately 300 pixels (or 1 pixel = 3.3 × 10-5 m). Pictures were then downloaded and analyzed for the color intensity changes corresponding to the chromium travel distance. A sufficient travel distance (1.50 cm) was 3952
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used to ensure the semi-infinite slab. Measurements were taken at seven adjacent points for every 0.01 cm of travel distance to ensure the precision and to minimize wall effects. The color intensities from digital image were converted to numerical values using computer software SigmaScan Pro from Sigma Inc. Then, the color intensity values were converted to concentration values using calibration curves. The diffusion coefficient (D) was then estimated using time/ distance/concentration data by least-squares optimization for the best fit for eq 4. The smoother technique using S-Plus 2000 from MathSoft Engineering and Education Inc. was applied to data sets. Diffusion with Different Gelation Times. The preparation of colloidal silica gel and diffusion measurements was the same as described before. Silica concentration in the gel was 240 g/L for this test. The gelation time was allowed to vary for 3, 6, 12, 18, 24, and 48 h. After the required gelation time was reached, 0.20 mg of chromium solution was added to colloidal silica gel in plastic vials. Then, the observation and data collection proceeded as described before. Confirmation Test for the Reliability of Measurements. Since the accuracy of digital photography was not known, a confirmation test was performed. The preparation of colloidal silica gel and the instruments used were similar to previous experiments. Silica concentration in the gel was 240 g/L for this test. However, the allowed diffusion time was different for each sample. It varied from 30, 60, 90, 120, 150, to 180 min. After the gel was completely set, 0.20 mg of chromium was added to colloidal silica gel in a plastic vial. The data were collected as described before. At the end of each desired diffusion time and after a picture was taken, a sample of chromium solution was taken for chemical analysis. Chromium samples were analyzed by AAGF to compare to the concentration value obtained from digital image.
Test Results A closer examination of digital photos with diffusion chromium did not show any volume change with time, indicating that with the movement of chromium through colloidal silica gel there was no swelling or shrinkage of the gel. Partition Coefficient. Figure 3 shows the experimental test data from the partition coefficient measurements. It shows the isotherm of chromium sorption to colloidal silica gel with ionic strength of 10-3 M/L, PCO2 of 10-3.5 atm, CaCl2 concentration of 0.25 M/L, SiO2 concentration of 4.8 g/L, pH 7, and 20 °C temperature. It shows that there are no differences in the amount Cr(VI) sorbed after 1 h. This suggests that the sorption reached equilibrium within 1 h. When the average measured data were plotted on a log-log scale and fitted using least-squares method, the isotherm was found to be linear over 6 orders of magnitude of chromium concentration. The assumption of the mass
FIGURE 4. Comparison of predicted and measured concentration for various silica gel concentrations.
FIGURE 5. Comparison of different color intensities used in calibration curve. balance is based on the linear adsorption isotherm. Thus, eqs 4 and 5 are valid for the reported experiment conditions. The partition coefficient of chromium between gel and water calculated from linear regression was found to be K ) 0.55 L/g with R 2 ) 0.93. This indicates a low adsorption of chromium to silica gel surface confirming the validity of one assumption. Calibration Curve. It was found that color intensity is linearly proportional to chromium concentration up to 300 mg/g of gel. Beyond this value, color intensity from the digital image remained constant. Thus, chromium concentration values up to 200 mg/g of gel were used. In addition, this range was based on the result from the partition coefficient obtained from the previous experiment, and the concentration of chromium leached from the chromium ore processing residue (COPR) is about 110 mg/L (25). Hence it was found that the chromium sorbed to colloidal silica treated COPR from New Jersey was approximately 60 mg/g. Calibration curves were developed for each silica concentration, which were 240, 210, 180, 150, and 120 g/L. The measured color intensity in a mixture of the three main colors (green, red, and blue) was evaluated with multiple regression to determine the calibration curves. Multiple regression was conducted using Microsoft Excel. The concentration value obtained from each calibration curve was compared to their actual value as shown in Figure 4. The calibration shows high correlation between chromium concentration and color intensity. From the regression for 240-120 g/L silica in the gel, the R 2 values were 0.997, 0.968, 0.988, and 0.973; percentage errors were 3.51%, 6.27%, 7.08%, and 6.88%, respectively. This indicates that the results from
the proposed method do not depend on the silica content, which is a major benefit of the proposed method. As can been seen from Figure 5, for different gel concentrations, similar color intensity values were obtained for each chromium concentration. Since the color of chromium is yellow, the calibration was based on blue color. Note that with higher chromium concentration blue color intensity was lower. This indicated that there is no variation between various gel concentrations and the color intensity calibration. Diffusion with Different Silica Concentrations. Figure 6 shows raw data and those predicted using eq 4 for 240 g/L silica content at 50 min. The raw data are averaged over the seven points collected from the digital photography and converted to concentration values using the calibration curve. The diffusion coefficient was determined by least-squares error method. Diffusion coefficients at six different time intervals (50, 100, 150, 200, 250, and 300 min) were estimated to calculate the average value. The diffusion coefficient values estimated from the experiments ranged from 4.91 to 8.48 × 10-10 m2/s, which are comparable to those reported in the literature. In literature, the diffusion coefficient ranged from 10-13 to 10-9 m2/s (8, 17, 20, 26, 27). The variation of the diffusion coefficient with silica content in the gel is shown in Figure 7. The diffusion coefficient of chromium in gel decreases linearly with increasing silica content. Lakatos et al. (20) reported similar findings. It is interesting to note that, upon extrapolation, the diffusion of chromium in water is obtained at zero silica content, which validates the experimental data obtained form VOL. 38, NO. 14, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 6. Comparison between predicted and measured chemical concentration.
FIGURE 7. Comparison of diffusion coefficient at different silica contents. the proposed method. The repeated experiment with 240 g/L silica content but with double the volume of chromium solution on top the gel gave identical results as those above, suggesting insignificant contribution to diffusion coefficient due to the convective flow of chromium. Colloidal silica gel has a nonuniform structure composed of silica chains and water. Silica chains form a porous network with water inside the pore spaces. The network of polymerized silicate forms a flexible structure of gel, and its “pore space” is filled with solventsin this case it is water (28). The pore size of silica gel has a wide distribution. The size of pore space depends on the concentration of silica, pH, and salt content. The average diameter of colloidal silica gel pore is approximately micropore (φ < 2.0 × 10-9 m) or mesopore (2.00 × 10-9 m < φ < 5.0 × 10-8 m) (29) while the size of hydrated chromium ion is φ ∼ 1.38 × 10-10 m for trivalent chromium and φ ∼ 1.04 × 10-10 m for hexavalent chromium (30), which is 10 times smaller than average pore diameter of gel. Therefore, the metal ions transport in gel medium can be seen as small particle movement in water that is in pores of gel network. It is reasonable to assume that, during the transport, the obstruction to free movement of chromium can occur inside the pore of the gel (31). The estimated diffusion coefficient ranged from 4.91 × 10-10 to 8.48 × 10-10 m2/s, which is 1 order of magnitude lower than diffusion of ions in water that ranged from 1 × 10-9 to 2 × 10-9 m2/s (32). From the discussion above, the gel behaves as a porous structure with the silica network forming continuous solid phase, and its pore space is saturated with water. Therefore, the diffusion dominates by bulk diffusion. In other words, it can be characterized as bulk diffusion of chromium ions in porous material with tortuosity. Thus, the silica can be represented as a fix and impenetrable immersion in the 3954
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FIGURE 8. Porosity, retardation factor, and tortuosity at different silica contents. solution. The presence of this motionless silica chains leads to an increase in the mean path of the diffusing molecules between two points in the system (8). The diffusion of chromium in gel media can also be defined as
Dgel )
Dwater F 1+ K θ
(6)
where the term 1 + (F/θ)K can be assumed as a retardation factor due to the transport sorption (33). Hence the term can be calculated from the experimental results. An expression of gel geometry as a function of diffusion coefficient was given by Lakatos and Lakatos-Szab (34) and can be expressed as
Dwater τ2 ) Dgel θ
(7)
The bulk diffusivity of chromium in aqueous phase is reported as D ) 1.12 × 10-9 m2/s (21). The density of commercial amorphous silica in colloidal silica ranged from 2.2 to 2.3 g/cm3 (35). Thus, the porosity, the retardation factor, and the tortuosity as a function of silica concentration can be determined from eqs 6 and 7. The calculated results are shown in Figure 8, which showed that porosity of colloidal silica gel ranged from 0.9 to 0.8. Hence the porosity of colloidal silica gel is inversely related to colloidal silica concentration in gel. Extrapolation produces a porosity value of 1.0 at zero silica concentration. The variation of retardation factor ranged from 4 to 14 and increased linearly with silica content in gel. Extrapolation produces a zero retardation factor at zero silica concentration. Tortuosity ranged from 1.25 to 2 and increased
TABLE 1. Amount of Chromium in Water and Gel time (min)
Cr in solution (mg)a
Cr in gel (mg)b
total Cr (mg)
error (%)
30 60 90 120 150 180
0.190 0.196 0.196 0.192 0.186 0.174
0.011 0.016 0.017 0.018 0.021 0.022
0.201 0.212 0.212 0.210 0.206 0.196
0.57 5.96 6.22 4.89 3.11 2.23
a Wet analysis. b Total area under curve of concentration vs distance obtained from digital image.
linearly with silica content in the gel. Extrapolation produces a tortuosity value of 1.0 at zero silica concentration. Diffusion with Different Gelation Times. The estimated diffusion coefficient values for gelation times of 3, 6, 12, 18, 24, and 48 h are 5.07, 5.26, 5.25, 5.09, 4.97, and 5.13 × 10-10 m2/s, respectively. No correlation between gelation time and diffusion coefficient was found. Thus, it can be concluded that gel formation time has no relation to chemical diffusion. Assuming that the gelation time has no contribution to diffusion, the average diffusivity value was calculated from different gelation times. The calculated average diffusivity value of 5.13 ( 0.11 × 10-10 m2/s is comparable to values reported in the literature. The reported values ranged from 10-9 to 10-13 m2/s (8, 17, 20, 26, 27, 36). Hence, it appears that there is no significant change in gel structure after 3 h. The average diffusion coefficient of chromium with silica concentration (240 g/L) and initial chromium concentration (200 mg/L) obtained from the two experiments, respectively, was 5.01 ( 0.15 × 10-10 m2/s with 95% confidence. The standard deviation was 1.50 × 10-11 m2/s, which yielded 2.90% error. Confirmation Test for the Reliability of Measurements. For the mass balance calculations, the total amount of chemical in the system at anytime is constant. Therefore, the area under curve of variation of chromium concentration with distance at anytime should be a constant. The total chromium mass in the system should be 0.200 mg. The analysis of chemical in water by wet analysis together with the analysis of chemicals using digital photography should yield the same amount of chemical at any time in the system. The different time intervals were used for confirmation. The chromium concentration in the solution was determined directly from the wet analysis using AAGF. The amount of chromium in gel was calculated from the area under curve multiplied by the cross section of the vials. The concentration profile in two media could be calculated based on the diffusion coefficient obtained from the previous experiments. The concentration profiles can be calculated using eqs 4 and 5. The diffusion coefficient of chromate ions in water and in the gel and the partition coefficient used in the calculation were respectively 1.12 × 10-9 m2/s (21), 5.01 × 10-10 m2/s, and 0.55 L/g, respectively. The results are shown in Table 1. From Table 1, the total amount of chromium in each sample was close to 0.200 mg. The error ranged from 0.57 to 6.22%. Hence, it can be concluded that determination of chromium concentration from the digital photography calculation is comparable to that from wet analysis. Consequently, the proposed digital photography method seems to be a reliable method to determine diffusion coefficients of chromium in gel. Method Application. Consider typical application of grout curtain as shown in Figure 9 to contain chromiumcontaminated soils. Due to the low permeability, there is virtually no advective movement; hence, contaminants move through the barrier by diffusion. One-dimensional diffusion was assumed in this simulation.
FIGURE 9. Simulation of a gel barrier method application. Here, the following assumptions were made: (i) At the interface between soil and gel, a chemical in solid state would dissolves until it reaches a saturated concentration. The chemical at the interface diffuses into the gel media. (ii) The concentration of chromium in gel at the groundwater interface is zero at all time. This is based on the assumption that the groundwater mass is large as compared to the gel mass. Hence the chemical concentration in gel at any distance can be calculated from this equation (18):
z C ) C0 erfc x4Dt erfc(x) ) 1 -
2 xπ
∫
x
0
(8)
exp(-u2) du
(9)
From eqs 8 and 9, the chromium concentration as a function of distance and time can be calculated. Hence, the leachate concentration, which is an amount of chemical diffusing, can be obtained. The input parameters required for eqs 8 and 9 are the concentration of chemical entering gel (C0) and diffusion coefficient (D). Concentration of chemical entering gel can be determined from the solubility of chromium compounds and the partition coefficient (K). Chromium compounds of interest in this simulation are COPR. Rock et al. (25) analyzed chemical properties of COPR from Kearney, NJ, and found that the waste has high levels of soluble hexavalent chromium, 940 ( 40 µM (109.04 mg/ L), and low levels of trivalent chromium,
