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Modeling Non-linear Adsorption to Carbon with a Single Chemical Parameter: A Lognormal Langmuir Isotherm Craig Warren Davis, and Dominic M. Di Toro Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/es5061963 • Publication Date (Web): 02 Jun 2015 Downloaded from http://pubs.acs.org on June 15, 2015
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Environmental Science & Technology
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Modeling Non-linear Adsorption to Carbon with a Single Chemical Parameter: A
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Lognormal Langmuir Isotherm
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Craig Warren Davis1, Dominic M. Di Toro2
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1
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Department of Civil & Environmental Engineering
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University of Delaware
7
Newark, DE 19716
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2
9
Department of Civil & Environmental Engineering
Craig Warren Davis
Dominic M. Di Toro
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University of Delaware
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Newark, DE 19716
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Key Words: sorption, model, organic chemicals, activated carbon, graphite
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Abstract
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Predictive models for linear sorption of solutes onto various media, e.g., soil organic carbon, are
17
well-established. However, methods for predicting parameters for non-linear isotherm models,
18
e.g.; Freundlich and Langmuir models, are not. Predicting nonlinear partition coefficients is
19
complicated by the number of model parameters to fit n isotherms (e.g., Freundlich (2n) or
20
Polanyi-Manes (3n)). The purpose of this paper is to present a non-linear adsorption model with
21
only one chemical specific parameter. To accomplish this, several simplifications to a log-normal
22
Langmuir (LNL) isotherm model with 3n parameters were explored. A single sorbate-specific
23
binding constant, the median Langmuir binding constant, and two global sorbent parameters; the
24
total site density and the standard deviation of the Langmuir binding constant were employed.
25
This single-solute specific (ss-LNL) model (2 + n parameters) was demonstrated to fit adsorption
26
data as well as the 2n parameter Freundlich model. The LNL isotherm model is fit to four data
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sets comprised of various chemicals sorbed to graphite, charcoal, and activated carbon. The RMS
28
errors for the 3-, 2-, and 1-chemical specific parameter models were 0.066, 0.068, 0.069, and
29
0.113, respectively. The median logarithmic parameter standard errors for the four models were
30
1.070, 0.4537, 0.382, and 0.201 respectively. Further, the single-parameter model was the only
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model for which there were no standard errors of estimated parameters greater than a factor of 3
32
(0.50 log units). The surprising result is that very little decrease in RMSE occurs when two of the
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three parameters, 𝜎𝜅 and 𝑞𝑚𝑎𝑥 , are sorbate independent. However, the large standard errors
34
present in the other models is significantly reduced. This remarkable simplification yields the
35
single sorbate-specific parameter (ss-LNL) model.
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Introduction Many models have been developed for adsorption of organic chemicals onto soils and
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carbonaceous materials, including black carbon 1,2. These models can be separated into two
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major classes: empirical and mechanistic. Although empirical models (e.g. the Freundlich
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isotherm) typically offer better fits to experimental data, they are not based on a mechanistic
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representation of the sorption process. For example, the Freundlich isotherm lacks a total site
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density parameter. Conversely, mechanistic models (e.g. the Langmuir isotherm) employ a
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simplified model of the adsorption processes, but generally offer poor fits to aqueous adsorption
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onto black carbon1.
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Statistically derived isotherm models, while less popular than Langmuir and Freundlich
46
models, offer a compromise between empirical models and mechanistic models. The Langmuir-
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Freundlich (LF) isotherm, proposed in 1948 by Sips, is one such model3,4, which utilizes a
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probability distribution of the Langmuir binding constants. The LF isotherm offers the flexibility
49
of the empirical models with the mechanistic basis of a Langmuir isotherm. It has been utilized
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extensively in the modeling of metal-ligand interactions with humic ligands5-9 and to model the
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adsorption energy distributions for organic compounds onto various types of carbon10-14.
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The purpose of this paper is to develop and apply a statistical isotherm model that
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accurately reproduces nonlinear adsorption with only one chemical-specific parameter. The
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model employs the basic principles of the LF model. However, the distribution of Langmuir
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binding constants is assumed to be lognormally distributed13-15.
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Modeling and Experimental Data
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Lognormal Langmuir (LNL) Isotherm 3 ACS Paragon Plus Environment
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The Langmuir isotherm 16 is conveniently expressed as:
𝑞 (𝑐 ) =
𝑞𝑚𝑎𝑥 𝐾𝐿 𝑐 1 + 𝐾𝐿 𝑐
(1)
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where q(c) is the sorbed concentration (mmol/kg sorbent), KL is the Langmuir binding constant
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(L/mmol sorbate), c is the aqueous concentration (mM) and qmax is the saturated monolayer
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sorption capacity (mmol/kg-sorbent). The Freundlich isotherm 16 is: 𝑞(𝑐 ) = 𝐾𝐹 𝑐 𝑣
(2)
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where KF is the empirical Freundlich constant (mmol/kg)(mM)1/ν, c is the aqueous concentration
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(mM), and v is the Freundlich exponent. The Langmuir-Freundlich isotherm 3 is a superposition
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of Langmuir isotherms ∞
𝑞 (𝑐) = 𝑞𝑚𝑎𝑥 ∫ −∞
𝑐𝐾𝐿 𝑞𝑚𝑎𝑥 (𝐾𝐿𝐹 𝑐)𝑣 ( ) 𝑓 𝐾𝐿 𝑑𝐾𝐿 = 1 + 𝑐𝐾𝐿 1 + (𝐾𝐿𝐹 𝑐)𝑣
(3)
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where 𝑞𝑚𝑎𝑥 is the total site density (mmol/kg-sorbent), 𝐾𝐿 is the local Langmuir binding constant
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(L/mmol-sorbate), and f(KL) is the probability density function (pdf) of the Langmuir binding
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constant. The approach that Sips employed was to derive a f(KL) which produces the LF
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isotherm, the right-hand side of Eq. (3). 𝐾𝐿𝐹 is the LF binding constant (L/mmol sorbate), and v
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is an empirical exponent. When v = 1 the isotherm reduces to the Langmuir equation, and for
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𝐾𝐿𝐹 c 95%). RMS errors for the models are 0.0659,
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0.0680, 0.0690, and 0.113, respectively. (B) Boxplots of the logarithmic estimated standard
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errors of the estimated parameters for models 1-4 above. Dotted line represents 1 order of
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magnitude error in the estimated parameters, semi-dashed lines represent 2 order of magnitude
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error in the estimated parameters. (C) Bar plot of the total number of estimated parameters for
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models 1-4 above.
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Figure 3. Adsorption isotherms for the single sorbate-specific parameter Lognormal Langmuir
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(ss-LNL) model onto graphite, charcoal, and Darco GAC. The sorbate symbols are the same for
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figures 1 and 3. Solid lines represent the fit to the LNL isotherm model. Model parameters are
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presented in supplemental information (Tables A1-a,b).
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Figure 4. (A) Effect of systematically varying the standard deviation of the median binding
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constant, 𝜎𝜅 , on the shape of the Normalized ss-LNL isotherm. (B),(C) Normalized Isotherms for
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Graphite, Charcoal, and Darco GAC (B) and F400 GAC (C). Fitted sorbent isotherm parameters
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and associated standard errors (fitted median binding constants and standard errors can be found
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in tables A-1a,b and A2-d in supplemental information).
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Figure 5. Plot of logarithmic single-chemical parameter LNL median binding constants, sorted
337
from smallest to largest for graphite (A), charcoal (B), and Darco GAC (C). Boxplots of the
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LNL isotherm model residuals (𝑙𝑜𝑔(𝑞𝑝𝑟𝑒𝑑 ) − 𝑙𝑜𝑔(𝑞𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 ) for graphite (D), charcoal (E), and
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Darco GAC (F). The solid line represents 1:1 agreement between qpred and qobserved adsorbed
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concentrations, dashed lines represent +/- 0.3 log-units difference between observed and
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modeled adsorbed. IQR contains 50% of the data, whiskers represent the 5th and 95th percentile
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of the residuals, and points represent outliers (< 5% and >95%).
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Figure 6. (A) Plot of logarithmic single-chemical parameter LNL median binding constants,
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sorted from smallest to largest for F400 GAC. (B) Boxplots of the LNL isotherm residuals
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(𝑙𝑜𝑔(𝑞𝑝𝑟𝑒𝑑 ) − 𝑙𝑜𝑔(𝑞𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 ) for F400 GAC. The solid line represents 1:1 agreement between
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qpred and qobserved adsorbed concentrations, dashed lines represent +/- 0.3 log-units difference
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between observed and modeled adsorbed. IQR contains 50% of the data, whiskers represent the
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5th and 95th percentile of the residuals, and filled points represent outliers (< 5% and >95%).
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Tables
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Table 1. Summary of experimental data used in construction of the Log-Normal Langmuir
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isotherm model (Graphite 21, Charcoal 21, Darco GAC 19, and F400 GAC 20). # of Sorbent
Sorbate Chemical Classes
Reference
Sorbates Graphite
13
PAHs, Nitroaromatics, Chlorinated Aromatics
(21)
Wood Char
11
PAHs, Nitroaromatics, Chlorinated Aromatics
(21)
Darco GAC
14
Hydrocarbons, Ketones, Ethers, Mostly Non(19)
aromatic PAHs, Nitroaromatics, Chlorinated Aromatics, F400 GAC
44
(20)
Pesticides 352 353
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̃𝐿 ) standard errors, individual, and combined RMS errors (Eq. (15)) Table 2. Summary of 𝑙𝑜𝑔(𝐾
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̃𝐿 are for the ss-LNL model fits onto graphite, charcoal, Darco GAC, and F400 GAC. Units of 𝐾
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[𝑚𝑀]−1 Sorbent
# of Parameters Total RMSE1
RMSE Range2
̃ L))3 Range of SE(log(𝑲
Graphite
15
0.0530
0.00912 - 0.0937
0.384 – 0.404
Charcoal
13
0.0515
0.0293 – 0.0699
0.121 – 0.132
Darco GAC
16
0.121
0.0336 – 0.214
0.423 – 0.477
F400 GAC
46
0.113
0.0258 – 0.441
0.112 – 0.416
𝑅𝑀𝑆𝐸𝑡𝑜𝑡𝑎𝑙 = √
357
1
358
2
∑(log(𝑞𝑝𝑟𝑒𝑑 )−log(𝑞𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 ))2 𝑛
Range of RMS Errors for each sorbate isotherm, i:
∑𝑖 (log(𝑞𝑝𝑟𝑒𝑑 ) − log(𝑞𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 ))2 𝑅𝑀𝑆𝐸𝑠𝑜𝑟𝑏𝑎𝑡𝑒=𝑖 = √ 𝑛𝑖 359
̃ L) for each isotherm Range of standard error of log(𝐾
3
360
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Table 3. Comparison of conventional Freundlich and single-chemical parameter LNL isotherm
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models RMS errors (Eq. (15)) for Graphite, Charcoal, Darco GAC, and F400 GAC.
Sorbent Graphite Charcoal Darco GAC F400 GAC
N 13 11
RMSE Freundlich LNL 0.142 0.053 0.118 0.0515
14 44
0.0941 0.0674
0.121 0.113
Combined 82
0.0895
0.0985
363 364
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Supporting Information
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The following information is included as supplemental information to accompany this text:
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Appendix A
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A1. Log-normal Langmuir Fitted Isotherm Parameters
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a. Graphite/Charcoal
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b. Darco GAC
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A2. Log-normal Langmuir Fitted Isotherm Parameters – F400 GAC
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̃𝐿 ), 𝜎𝜅 , and 𝑞𝑚𝑎𝑥 – Model (1) a. Chemical-specific log(𝐾
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̃𝐿 ) and 𝑞𝑚𝑎𝑥 – Model (2) b. Chemical-specific log(𝐾
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̃𝐿 ) and 𝜎𝜅 – Model (3) c. Chemical-specific log(𝐾
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̃𝐿 ) – Model (4) d. Chemical-specific log(𝐾
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A3. Discussion of Standard Errors of Fitted Parameters – Model (4)
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A4. Fitted ss-LNL Isotherms for F400 GAC
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Appendix B
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B1. Log-normal Langmuir Visual Basic Code
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B2. Comparison of Freundlich and LNL Isotherm Residuals
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B3. Tables of Fitted Freundlich Isotherm Parameters and Associated Standard Errors
382
This material is available free of charge via the Internet at http://pubs.acs.org.
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Corresponding Author
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Dominic M. Di Toro
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[email protected] 388
Author Contributions
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The manuscript was written through contributions of all authors. All authors have given approval
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to the final version of the manuscript.
391
Funding Sources
392
Graduate Assistance in Areas of National Need (GAANN) Fellowship Grant #P200A090174.
393
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Sorbed Concentration [mmol/kg]
104
Darco GAC ●● ● ●● ●● ● ● ●● ● ● ● ● ●● ●● ●●
101
10−1 10−7
●● ● ●
Charcoal
● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●●
●●
● ●● ●● ● ● ● ● ● ● ●
Graphite
●●
10−5
10−3
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10−1
~ Normalized Conentration − cKL
101
Page 29 of 34 101
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(A) ● ●● ● ●
●● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●
10−1 4
●
●
10−7
10
10−5
10−3
0
●
10−1
(B)
103
●
●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●
102 101
●
Phenanthrene 2,4,6−TNT Naphthalene 1,2,4−Trichlorobezene 2,4−DNT 1,2,4−Trimethylbenzene p−Nitrotoluene 1,2−Dichlorobenzene Xylene Chlorobenzene Toluene Benzonitrile Benzene
100 10−1
●
10−6
104
(C) 103
10−4
10−2
100
● ● ● ●● ● ● ● ●●● ● ●● ● ●●● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●
0
Sorbed Concentration [mmol/kg−C] Sorbed Concentration [mmol/kg−C] Sorbed Concentration [mmol/kg C]
100
● ● ● ●●
102
●
●
●
1
10
●
100 10−6
Hexane Heptane Benzene Toluene 0 Nitrobenzene Tetrachloroethylene 1,1,2−Trichloroethylene Methyl tert−butyl Ether Diethyl Ether Diisopropyl Ether 2−Heptanone 3−Hexanone 1−Heptanol 1−Hexanol
−2 0 Paragon Plus Environment 10−4 ACS10 10 102
Aqueous Concentration [mM] Aqueous Concentration [mM]
0
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−1 Sorbate−specific Parameters: (A) ~ (1) KL,σκ,qmax (2)
~ KL,qmax
(3)
~ KL,σκ
(4)
~ KL●
Log(Predicted) − Log(Observed)Page 30 of 34 Sorbed Concentration [mmol/kg] −0.5 0 0.5 1
●
● ●● ● ●●
● ● ● ●● ● ● ●
● ● ● ● ● ● ●●
●
● ●● ●●
● ●● ●●●● ●●
●●● ●●
● ● ● ● ●● ● ●● ●
● ● ●●
●●●● ●● ●
●
(B) ~ (1) KL,σκ,qmax (C)
~ KL,qmax
●●●
● 150
● 100
(3) (4)
~ KL,σκ ~ KL
●●●
● ●
● 50
●
(4) (3) (2) (1)
# of Parameters
(2)
●● ● ●
0
Plus 0 ACS Paragon 1 2 Environment 3 4 5 Standard Errors of Logarithmic Estimated Parameters
● ●
● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●
10−1 10−7
10−5
10−3
3
Charcoal
10
102
101
100 10−6
10−4
10−2
4
10
Darco GAC
3
10
102
Sorbed Concentration [mmol/kg−C] Sorbed Concentration [mmol/k
Graphite
100
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● ● ● ●●
● ●●
Sorbed Concentration [mmol/kg−C] Sorbed Concentration [mmol/k
Sorbed SorbedConcentration Concentration[mmol/kg−C] [mmol/kg Sorbed C] Concentration [mmol/k
1 Page1031 of 34
10−1
100
● ● ● ● ●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●● ● ● ●
10−7
10−5
10−3
10−1
10−7
10−5
●
●
● ● ● ● ● ● ●
10−6
10−4
10−2
● ● ● ●
100
10−3
10−1
●●
●
●● ● ● ●● ● ● ● ●
10−6
10−4
10−2
100
● ●● ●●●
●
101 100 10−6 10−4 10−2
100
2 −6 Paragon 10ACS 10 10−4Plus 10−2Environment 100 102
10−6 10−4 10−2
100
102
Aqueous Concentration [mM] Aqueous Concentration [mM] Aqueous Concentration [mM] Aqueous Concentration [mM]
(A)
10−0
Environmental Science & Technology
Page 32 of 34
q / qmax
10−2
σ
10−4
5.0 4.0 3.0 2.0 1.0 0.0
10−6 10−8 10−10 10−9
d$q Concentration [mmol/kg C] Sorbed
103
10−7
10−5
10−3
10−1
101
Darco GAC
(B)
● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●
101 ●●
● ●● ● ● ● ●● ●● ●● ● ●● ● ●● ● ●● Charcoal
Graphite
10−1 10−6 103
10−4
10−2
●
10−0
●● ● ●● ● ●● ●●● ●● ●●●●● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●●● ●● ● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● κ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●●● ●● ● ● ●● ● ●● ●● ● ● ● ● ●
102
(C)
σ
101
Graphite Charcoal Darco GAC F400 GAC
10−1 10−10
10−8
qmax
3.48 (0.292) 2.61 (0.107) 3.97 (0.294) 4.96 (0.001)
16.0 (3.93) 978 (79.7) 5065 (1255.2) 10199(120.4)
ACS Paragon Plus Environment 10−6 10−4 10−2 100
~ cKL
●
● ●
1.0
1:length(graphchem)
0.0
(E)
0.5
●
●
● ● ●
−0.5
−1.0
ACS Paragon Plus Environment
●
● ●
● ●
1:length(charchem)
unique(d4s$logK)
● ●
●
0
●
1.0
−0.5
Methyl−tert−butyl Ether Diethyl Ether 3−Hexanone 1−Hexanol Diisopropyl Ether 1−1−2−Trichloroethane 1−Heptanol Tetrachloroethane Benzene Nitrobenzene Toluene 2−Heptanone Hexane Heptane
●
0
Log(Model) − Log(Observed)
● ●
Naphthalene
−1.0 ●
(B)
1−2−4−Trichlorobenzene
0 1
2−4−Dinitrotoluene
0.5 ●
m−Nitrotoluene
−0.5
(D) ● ● ●
1−2−Dichlorobenzene
0.0 ● ●
TNT
●
1−2−4−Trimethylbenzene
1.0 ● ● ●
Xylene
● ●
2
Benzonitrile
2
unique(d3s$logK)/2.303
3
●
Toluene
−1
(A)
Benzene
1
Log(Model) − Log(Observed)
~ unique(d2s$logK)/2.303 Log( KL ) 4
Benzene Benzonitrile Toluene Chlorobenzene Xylene 1−2−Dichlorobenzene m−Nitrotoluene 1−2−4−Trimethylbenzene 2−4−Dinitrotoluene Naphthalene 1−2−4−Trichlorobenzene TNT Phenanthrene
log(qPred) − log(qObserved)
LNL Model Residuals Log(Model) − Log(Observed)
Page 33 of 34 Environmental Science & Technology
(C)
●
(F)
● ●
−1
●
●
● ●
● ● ●
● ● ● ●
−2 ●
1:length(gacchem)
0.5
0.0 ●
● ●
−1.0
LNL Model Residuals log(qPred) − log(qObserved) ~ Log( KL ) 1
−5
0.0
methylene chloride 1−2−dichloroethane dibromomethane 1−1−dichloroethane chloroform tert butyl methyl ether cis 1−2−dichloroethylene 1−1−1−trichloroethane bromodichloromethane 1−2−dichloropropane carbon tetrachloride trans 1−2 dichloroethene 1−1−dichloroethene dibromochloromethane 1−2−dibromoethane 1−3−dichloropropane bromoform 1−1−1−2−tetrachloroethane 1−1−dichloropropene 1−2−3−trichloropropane benzene trichloroethene tetrachloroethene isophorone toluene dibromochloropropane aldicarb chlorobenzene metribuzin 2−4−5 trichlorophenoxy acetic acid 1−3−dichlorobenzene ethyl benzene carbofuran oxamyl metolachlor cyanazine bromobenzene 1−2−dichlorobenzene atrazine styrene simazine hexachlorocyclopentadiene alachlor 1−3−5−trichlorobenzene
Log(Model) − Log(Observed)
Environmental Science & Technology
0
−1 ● ● ● ● ● ●
−2
−3
−4 ● ● ● ● ● ● ● ● ● ● ● ●
●
● ● ● ● ●
● ● ●
ACS Paragon Plus Environment
Page 34 of 34
(A) ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ●
● ● ● ● ● ● ● ● ●
●
1.0
(B)
0.5
●
●
−0.5
−1.0
