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Equilibrium modeling of ion adsorption based on Poisson–Boltzmann equation
Accepted Manuscript Title: Equilibrium modeling of ion adsorption based on Poisson-Boltzmann Equation Author: V. Steffen L. Cardozo-Filho E.A. Silva L.R. Evangelista R. Guirardello M.R. Mafra PII: DOI: Reference:
S0927-7757(14)00941-8 http://dx.doi.org/doi:10.1016/j.colsurfa.2014.11.065 COLSUA 19602
To appear in:
Colloids and Surfaces A: Physicochem. Eng. Aspects
Received date: Revised date: Accepted date:
14-9-2014 27-11-2014 28-11-2014
Please cite this article as: V. Steffen, L. Cardozo-Filho, E.A. Silva, L.R. Evangelista, R. Guirardello, M.R. Mafra, Equilibrium modeling of ion adsorption based on PoissonBoltzmann Equation, Colloids and Surfaces A: Physicochemical and Engineering Aspects (2014), http://dx.doi.org/10.1016/j.colsurfa.2014.11.065 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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*Graphical Abstract (for review)
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*Highlights (for review)
Highlights A phenomenological model to represent the bulk adsorption of ions is proposed. The model proposed was based on a previous surface adsorption one.
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There was improvement and simplification in the previous model. The model was successful in accurately describe the investigated cases.
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It reduced the number of adjustable parameters with no lose in the fit quality.
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Equilibrium modeling of ion adsorption based on Poisson-Boltzmann
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Equation V. Steffena, L. Cardozo-Filhoa, E. A. Silvab, L.R. Evangelistac, R. Guirardellod and M.R.
Departamento de Engenharia Química, Universidade Estadual de Maringá, Avenida
Colombo, 5790, 87020-900 Maringá, Paraná, Brazil.
Centro de Engenharia e Ciências Exatas, Universidade Estadual do Oeste do Paraná, Rua
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Mafrae
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Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790,
School of Chemical Engineering, State University of Campinas, PO Box 6066, 13083-970
Campinas, SP, Brazil.
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87020-900 Maringá, Paraná, Brazil.
Departamento de Engenharia Química, Universidade Federal do Paraná, Centro
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da Faculdade, 645, 85903-000 Toledo, Paraná, Brazil.
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Politécnico, Setor de Tecnologia, Jardim das Américas, 81531-990 Curitiba, Paraná, Brazil.
Corresponding author. Tel.: +55 44 3011 4749; fax: +55 44 3011 4792. E-mail adress: [email protected] (V. Steffen)
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Abstract A phenomenological model representing ion adsorption equilibrium is proposed. It is an extension of the model used to represent surface adsorption, and it is intended to interpret
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bulk adsorption data. The fundamental equations of the model were established for fluid and solid phases mass balance and for the spatial distribution of ions across the sample. The
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electric field distribution in the system, modeled on Poisson’s Equation, and the chemical
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potential behavior were determined as a function of fluid phase equilibrium concentration. The concentration of ions as a function of electric potential is given by Boltzmann
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distribution. The proposed model was successfully tested to represent equilibrium adsorption of metallic ions in zeolite, showing results similar to conventional isotherms.
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Keywords: Adsorption, Poisson Equation, Boltzmann Distribution, Poisson-Boltzmann,
Introduction
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1.
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Adsorption Isotherm.
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The investigation of sorbent performance is basic for industrial application and for design of adsorption process equipment. The analysis is normally carried out through equilibrium studies on the system of interest [1]. There is a large number of available
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isotherms in the literature that may represent adsorption equilibrium data, among which may be mentioned the ones of Langmuir [2], Freundlich [3], Sips [4], Redlich-Peterson [5], Khan [6], Tóth [7], Radke-Prausnitz [8], Dubinin-Radushkevich [9], Frumkin [10], Flory-Huggins [11, 12], and Fritz-Schluender [13]. The first two are the equilibrium adsorption isotherm models most widely accepted for single solute systems [14]. Due to the complexity of the adsorption mechanism it is rather difficult for researchers to choose isotherm models according to known mechanisms. The criterion for choosing an
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isotherm is, so far, mainly based on the use of statistical parameters of the model which best fit experimental data. However, a good curve, fitting in the sense of statistical evaluation, may not necessarily imply that the curve has true physical meaning, i.e., if a set of sorption data is
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analyzed by different isotherm equations, the best fit equation may not be the one reflecting the sorption mechanisms [15]. The importance of a model foregrounded on the physical
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phenomena involved in the process is thus evidenced.
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Here, we propose a model for selective adsorption phenomenon in an isotropic liquid that takes into consideration localized adsorption energies. The model basically consists of a
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set of equations that determine the profile of the electric field and the chemical potential. It is intended to use Poisson-Boltzmann equation, a model well known in the representation of
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surface adsorption, for treating bulk adsorption data, but due to the complexity of the model, it is necessary to approximate the adsorbent (composed by little porous particles) as a plane
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surface.
The adsorption of ions in the liquid phase involves the interaction between a solvent
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(usually water) containing a dissolved species, the adsorbate (metallic ions), and an adsorbent solid which must have high affinity to the adsorbate. Sorption reactions inside the pores and on the surface of the adsorbent depend on the type of dissolved ions in the aqueous medium,
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on the chemical nature of the adsorbate, and the nature of the adsorbent surface. Thus, electrolyte ions in the aqueous medium migrate to the electrically charged sorbent surface. The cations in the sorbent solid surface are substituted by the metallic ions in the solution. Another approach for the representation of metallic ions adsorption equilibrium data has been forwarded by Mafra et al. [16], whose mathematical model is based on the PoissonBoltzmann equation [17-19]. The model, used to represent surface adsorption, describes the electric field distribution of a sample limited by two adsorbent surfaces. However, the
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equations of the model [16] are valid only for the representation of batch equilibrium data since there are variables, such as adsorbent mass, system volume and initial concentration of metallic ions, in the formulation. Moreover, the model has four adjustable parameters.
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The model we are proposing here applies the Poisson-Boltzmann equation to ion adsorption and presents a mathematical formulation in terms only of adsorbate equilibrium
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concentrations in the fluid phase and on the adsorbent. It also evaluates strategies to reduce
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the number of parameters in the model when compared with the formulation developed by Mafra et al. [16]. The proposed model consists of a set of algebraic/differential equations that
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determine the profiles of electric and chemical potentials (differential equations) and the concentration on the adsorbent (algebraic equation).
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The model will be applied to the analysis of adsorption experimental data of Fe (III)
Basic theory
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2
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proposed model.
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and Zn (II) on zeolite NaY, provided by Ostroski et al. [20] to illustrate the usefulness of the
The Poisson equation is a partial differential equation of fundamental importance and wide usefulness in several research areas. It connects the electric potential (V) with the charge
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density (ρ) in the form: 2V
(1)
where is the dielectric permittivity of the medium. The liquid charge density is given by
q C C
(2)
where q is the charge of the ion and C± is the concentration of ± ions.
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By inserting Equation (2) into Equation (1), we obtain 2V
q
C
C
(3)
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One may observe that in Equation (3) the ions densities have to be related with the electric potential. One of the most used tools to establish this relation is the Boltzmann distribution,
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which holds true for low concentrations and high temperatures. It may be written here as
(4)
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N EE / k T e 0 B eE / kB T N0
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where N is the number of systems in the energy state E, N0 is the number of systems in the reference energy state E0, kB is the Boltzmann constant, and T is the absolute temperature.
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Since the modeled system involves ion adsorption, the more appropriate reference state for Boltzmann distribution is the ion equilibrium concentration in the bulk fluid phase (CE): q V / kB T
(5)
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C CE e
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where µ is the chemical potential. For convenience, the energies entering Equation (5) will be
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measured in kBT, i. e., µ/kBT → µ, and qV/kBT → ψ. Substitution of Equation (5) into Equation (3) yields
q 2 CE e e e kB T
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2
(6)
When the Poisson-Boltzmann equation is written in one dimension, we have d 2 e sinh , dz 2 L2
with L2
kB T 2 q 2 CE
(7)
defining an intrinsic length of the problem. Let us start by considering a system that consists of a fluid of dielectric permittivity containing ions. As already mentioned, the adsorbent is roughly approximated as a plane surface, so the system is limited by a flat adsorbent surface located at z=0. It is possible to
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obtain the spatial distribution of the electric field by means of the Poisson-Boltzmann equation and, consequently, the spatial distribution of the electric potential across the sample. It may be assumed that the surface selectively adsorbs only positive charges (cations), which
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will give rise to a surface electric field. The modeling of this system will result in an electric potential profile with a maximum value in the adsorbent surface (ψS), that decreases from z=0
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or rather, (dψ/dz)z=D=0 and ψ(zD)=ψD, as shown in Figure 1.
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to z=D. For an adsorbent surface distance greater than z=D, the electric potential is constant,
Since the ions positioned at z=0 are not necessarily adsorbed, in order to obtain a
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mathematical expression that gives the density of adsorbed ions (nS), the Boltzmann distribution has to be used again:
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nS N e A S
(8)
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where N is the number of sites per unit surface, ψS is the surface electrical potential, and A is
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the adsorption energy (A=Ead/kB T).
Using the Poisson-Boltzmann equation, represented by Equation (7), in the system
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under analysis, the values of the electric potential at any point of the z coordinate may be calculated. On the adsorbent surface (z=0), the solution of this equation obeys the established
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boundary conditions for the electric field (E) on surface, namely:
E z 0
q nS k B T d q dz z 0
(9)
Another important relation is given by the mass balance equation: N N nS AST C0 V 2 2
(10)
where AST is the total surface area, C0 is the initial concentration of ions in the bulk fluid phase, V is the volume of fluid phase, and N is the number of ions ± , given by:
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N AST
D
0
C z dz CE V CE D AST
(11)
Therefore, the fundamental equations of the model are Equations (7), (9) and (10), or
2.1
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rather, the equations connecting chemical and electric potentials. A model for biosorption
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In Ref. [16], a model a model to represent the biosorption equilibrium of heavy metals
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was recently proposed. The manner of obtaining the equations of this mathematical modeling will be summarized as follows. By integrating twice equation (7) and using the boundary
D
cosh cosh D
2 D e 2 L
(12)
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S
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conditions (dψ/dz)z=D=0 and ψ(zD)=ψD, we obtain
2
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By combining Equations (7), (8), and (9) , one obtains
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C L 2 A e 8 0 cosh S cosh D e s N
(13)
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Finally, by inserting Equations (8) and (11) into Equation (10), after a mathematical rearrangement, it is possible to show that e CE AST
D
0
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N AST e A S T cosh dz C0 CE V CE D AS 2
(14)
Equations (12), (13) and (14) connect ψD, ψS and μ. They are the basic equations of the
model proposed to describe biosorption and may be solved numerically. Consequently, the initial and equilibrium concentrations of adsorbate (respectively, C0 and CE) can be determined. In the model, ψD, ψS and μ are dependent variables whereas the adsorption energy, A, the number of sites per unit surface, N, the total surface area, AST, and the thickness of the interfacial region, D, are adjustable parameters.
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In addition to the equations connecting the electric and chemical potential, an equation connecting qE, CE and C0 is required. This relation is obtained from Equation (8) rewritten in specific terms:
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ms qE N e A s T AS
(15)
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where is the number of the ion elementary charges. The isotherm is represented by Equation
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(15), in which ψS and μ may be obtained from the solutions of Equations (12)-(14) governing the model. Extended model
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2.2
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Since isotherms are functions of the equilibrium concentrations of fluid and adsorbent phases only, it is convenient to modify the mathematical formulation developed by Mafra et
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al. [16], written in terms of initial concentration (C0), adsorbent mass (mS) and solution
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volume (V). These modifications will allow the model to describe the equilibrium adsorption
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of ions on any system regardless of initial conditions and absolute amounts. Another important aspect of the model developed by Mafra et al. [16] is the boundary conditions used to solve the Poisson-Boltzmann differential equation. These conditions are
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(dψ/dz)z=D=0 and ψ(z=D)=ψD. The values ψD so obtained are small (order of magnitude 10-5). This suggests that they may be evaluated by stating ψ(z=D)=0 as boundary condition in the solution of the Poisson-Boltzmann equation. Using this new boundary condition, the model may be simplified and becomes more attractive. Consequently, this boundary condition will be hereafter considered. In this way, two distinct models for the Poisson-Boltzmann isotherm will be proposed. They differ by the boundary conditions used for solving the differential equation representing the electric field of the system. In the first model, the boundary conditions are represented by (dψ/dz)z=D=0 and Page 10 of 30
ψ(z=D)=ψD , while in the second model by (dψ/dz)z=D=0 and ψ(z=D)=0. For convenience, the models will be denoted, respectively, by the short names PB1 and PB2. In this scenario, in order to eliminate the term mS (gads) from the equations, the surface
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area per mass of substrate AS (dm2/gads) will be used to substitute total surface area (AST) used in the Mafra et al. [16] model. Thus, the isotherm may be obtained from Equation (8), where
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nS is the density of ions in the surface (meq/dm2) and N is the number of sites per surface unit
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(meq/dm2). The Poisson-Boltzmann isotherm proposed here will be represented by qE AS N e A S
(16)
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where qE is the equilibrium concentration of adsorbed ions (meq/ gads).
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To eliminate the initial concentration (Co) and the solution volume (V), the mass balance for ions in a closed system will be employed, represented by the following equation: (17)
d
V C0 CE mS
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qE
With regard to the manner the value of qE (given the value of CE) is obtained, it should
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be underlined that isotherms are usually classified as explicit and implicit. When the isotherm is explicit, the calculation of qE is given by a simple and direct way. When the isotherm is implicit, the calculation of qE is given by numerical methods, in which the value of qE is
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corrected at each iteration until the error is very small. However, the Poisson-Boltzmann isotherm does not fit in any of these two groups. When CE value is established, a set of nonlinear equations is also solved, with such unknown factors as chemical (μ) and electrical potentials (ψS e ψD), whose values are used in Equation (16) to calculate the qE value. This set of nonlinear equations is different for the two models (PB1 and PB2), mainly due to the fact that the number of equations which form each set of equations is also different. In the case of PB1 model there are three equations whereas for PB2 model there are only two equations. By
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definition we have ψD=0 for this latter model. These equations are obtained as above for the model of Ref. [16], by taking into account the changes mentioned to eliminate the terms accounting for the initial concentration, adsorbent mass and volume of the solution.
D
cosh cosh D
2 CE L e 2 2
S
D
2 D e 2 L
cosh
N e A S d CE D 2 cosh cosh D
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d
S
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In the case of PB1 model, the three equations are:
2
C L 2 A e 8 E cosh S cosh D e S N
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(18)
(19)
(20)
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The solution of Equations (18), (19) and (20) is not simple due to the two integations that have to be performed, involving elliptic functions. Moreover, they have singularities at
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the lower integration limits. These singularities are contoured with a suitable change of
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variables. The algebraic expressions for these integrals are obtained by considering the
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approximation ψD≈0. The set of non-linear equations was solved using the method by Broyden [21].
sinh 2 v 1 sinh D 2 2
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(21)
In the case of PB2 model, the two equations are given by: D N e A S e cosh z dz D 2 CE 0
S 2 sinh 1 2 ln 1 2 ,
(22)
(23)
where
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2 1 e2 D 4 cosh z dz D 1 2 1 2 e2 D
(25)
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0
(24)
e 2 L
cr
D
N e 2 A S 4 CE L
(27)
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tanh S 4
(26)
In the second model (PB2), the set of equations is smaller than the one of the first
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model (PB1). As mentioned above, the PB2 model consists of only two equations, Equations (22) and (23), since it has one unknown less, which are simpler than the ones forming the
analytically done. Thus,
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PB1 model. Furthermore, the unique integration to be performed in this model may be PB2 model consists of algebraic equations only, and, in this
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surface may be given by
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framework, the profile of the electric potential as a function of distance from the adsorbent
2
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1 e z z ln z 1 e
(28)
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2.2.1 Parameters analysis
In the equations of the model, the terms N and A always appear together in the form of
a product (Ne-A), and thus a new variable NS=Ne-A may be defined as only one adjustable parameter. The isotherm represented by Equation (16) is rewritten as qE AS N S e S
(29)
and the fundamental equations representing the models may also be rewritten. In the mathematical formulation of PB1 model, Equation (18) does not change, but Equations (19)
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and (20) shall be represented, respectively, by
2 CE L e 2 2
S
D
cosh cosh cosh D
d
N S e S CE D 2
(30)
2
C L e 8 E cosh S cosh D e2 S NS
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(31)
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In the case of PB2 model, Equation (22) is modified, whereas Equation (23) and the
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definitions given by equations (25), (26) and (27) remain unchanged. Only the definition
D N e S e cosh z dz S D 2 CE 0
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N S e 2 S 4 CE L
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given by Equation (24) slightly changes and takes a new form. The new equations are: (32)
(33)
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In the Poisson-Boltzmann model, D is a parameter that represents the distance from
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the surface where the electric potential has constant value ψD. Simulations were performed to
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evaluate the effect of this parameter on the isotherms. The values used were D=5L, D=10L , and D=500L. Figure 2 shows the simulations results that using the following values for the other parameters: A=1, N=3 meq/dm2 and AS=2 dm2/g.
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The graphs in Figure 2 show that parameter D has only a slight influence on the Poisson-Boltzmann isotherm. In this manner, the number of adjustable parameters of the model could be reduced. We decided to fix the value of D in 10L, a large enough value to ensure that the electrical potential is very small at this distance from adsorbent surface. Thus, beyond the surface area (AS), the proposed model has only two adjustable parameters, NS and L, which can be interpreted, respectively, as the maximum number of active sites that may be occupied by ions and the distance related to the electric field decay near the adsorbent surface.
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Simulations for three sets of values of the parameters were performed to evaluate the relationship among NS, AS , and L. The first set is formed by the values NS=1.0 meq/dm2, L=1.0 dm and AS=1.0 dm2/g, the second is formed by NS =0.1 meq/dm2, L=0.1 dm and
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AS=10 dm2/g, and the third is formed by NS =10 meq/dm2, L=10 dm and AS=0.1 dm2/g. The results of these simulations are shown in Figure 3.
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The isotherm curves in Figure 3 are identical to the three sets of values of the
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parameter NS, AS and L, with correlation among these parameters. Therefore, it is possible to
Results and discussion
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3
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consider only two adjustable parameters in the form NS.AS and L.AS.
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Conventional adsorption isotherms models tested to evaluate the proposed model
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performance are shown in Table 1.
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Equilibrium data for Fe(III) and Zn(II) adsorption on zeolite NaY, obtained by (Ostroski, 2009), were employed to evaluate the models based on Poisson-Boltzmann equation proposed here.
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All calculations were made for CE in meq/L units and qE in meq/gads units. FORTRAN
77 codes were written for calculations. An optimization algorithm based on the version of the conjugate gradient algorithm described by Powell [22] was used to minimize the objective function value, which may be expressed as N
Fo qiexp qicalc
2
(34)
i 1
Further, the coefficient of determination (R2) was used as comparison parameter. Thus, an effective comparison may be done among the models with different numbers of adjustable Page 15 of 30
parameters, an alternative measure of the coefficient of determination may be employed which penalizes the inclusion of adjustable parameters with little explanatory power. This is the adjusted coefficient of determination (R2a):
n 1 1 R 2 n p
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Ra2 1
(35)
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where n is the number of experimental observations and p is the number of adjustable
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parameters in the isotherm.
Another statistical parameter used to evaluate the model performance is the average
1 n exp qi qicalc n i 1
(36)
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AAD
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absolute deviation (AAD):
The parameters estimated for PB1 and PB2 models, the coefficients of determination,
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adjusted coefficients of determination and objective function are given in Table 2.
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Results of the statistical analysis in Table 2 show that there was no significant
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difference between PB1 and PB2 models. Consequently, the PB2 model should be employed due to its lower complexity and to the fact that the set of equations to be solved is composed by fewer equations (only two) and only non-linear algebraic equations.
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Figures 4, 5 and 6 show, respectively, the isotherms graphs, chemical potential and surface electric potential for iron and zinc ions adsorption by zeolite experimental data best fitted parameters by PB2 model. Results in Figure 4 and statistical analysis in Table 2 show that PB1 and PB2 models accurately describe the equilibrium adsorption data. Figures 5 and 6 show that the profiles of the chemical potential and surface electrical potential are similar, although it must be highlighted that the surface electrical potential (ψS) values are approximately ten times higher
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than the corresponding values of chemical potential (μ). Results in Figures 4, 5 and 6 show that higher ions sorption capacity (Zn+2>Fe+3) are related to lower values of the chemical and electrical potential of the respective ions, which may be easily proved by evaluating Equation
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(29). Profiles of electrical potential of adsorbed ions on the zeolite surface were similar to the ones that obtained by Mafra et al. [16] who used this approach to represent the biosorption
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equilibrium data of chromium and copper ions by the marine alga Sargassum sp.
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There are four adjustable parameters (A, AST, N, D) in the modeling presented by Mafra et al. [16], whereas in the present study there are only two parameters that could
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accurately represent the equilibrium data of the two systems evaluated.
To evaluate the performance of the proposed Poisson-Boltzmann isotherm, results in
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Table 2 will be compared with conventional adsorption isotherms in Table 1. Tables 3 and 4 show the best fit parameters value as well as coefficients of determination, adjusted
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coefficients of determination and objective function of the isotherms shown in Table 1 applied, respectively, to the iron and zinc experimental data on ion adsorption.
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Table 3 shows that in the case of iron adsorption the Flory-Huggins and FritzSchluender isotherms present the highest value of R2, even though Flory-Huggins shows slightly lower values of AAD and Fo . The highest R2a is provided by Tóth, Sips and Frumkin
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isotherms, with three parameters. In general, all models show good fits to the experimental data of iron adsorption, given that R2 values are higher than 0.98. The Freundlich and Dubinin-Radushkevich isotherms are the exception. Table 4 shows that, in the case of zinc adsorption, the Fritz-Schluender isotherm presents the highest R2 and R2a and the lowest AAD and Fo values. In general, all models show good fits to the experimental data of zinc adsorption, given that, with the exception of Langmuir and Dubinin-Radushkevich isotherms, R2 values are higher than 0.96.
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The comparison of results in Table 2 and those in Tables 3 and 4 shows that the isotherm proposed in this paper, in addition to having a strong theoretical basis, has a good
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fitting quality and may be compared to the isotherms with the same number of parameters. 5. Conclusions
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In this paper, we proposed a modification of the model by Mafra et al. [16] to
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represent the bulk adsorption of ions. It is a model based on the Poisson-Boltzmann equation that uses mathematical modeling of potential and adsorption energy to describe the
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distribution of ions and adsorbed amount. The proposed model was applied to iron and zinc experimental data on ion adsorption and the results were in good agreement. The model is
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phenomenological and takes into account the interaction between the adsorbed particles. In fact, it may be extended to multicomponent adsorption systems. It has only two adjustable
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parameters and the fitting quality is comparable to conventional adsorption isotherms with the
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Acknowledgment
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same number of adjustable parameters.
This work was partially funded by the Brazilian Agency, CAPES.
A
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Nomenclature
Adsorption energy
AS
Surface area, dm2 g-1
AST
Total surface area, dm2
C0
Initial concentration in the bulk fluid phase, meq dm-3
CE
Equilibrium concentration in the bulk fluid phase, meq dm-3
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Concentration of ± ions in the bulk fluid phase, meq dm-3
D
Thickness of the interfacial region, dm
Fo
Objective function
kB
Boltzmann constant, J K-1
L
Intrinsic length, dm
mS
Adsorbent mass, g
nS
Density of ions in the surface, meq dm-2
N
Number of sites per unit surface, meq dm-2
N
Number of ± ions, meq
q
Electric charge of the ion, C mol-1
qE
Concentration of ions on the adsorbent, meq g-1
T
Temperature, K
V
Volume, dm3
Dielectric permittivity, C2 N-1 m-2
Number of elementary charges of the ion
Chemical potential
Eletrical potential
S
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d te
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D
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C
Electrical potential at z=D Surface electrical potential
References
[1] E.A. Silva, E.S. Cossich, C.R.G. Tavares, L. Cardozo-Filho, R. Guirardello, Modeling of copper(II) biosorption by marine alga Sargassum sp. in fixed-bed column, Process Biochemistry, 38 (2002) 791-799.
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M
an
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cr
ip t
[2] I. Langmuir, The Constitution and Fundamental Properties of Solids and Liquids. Part I. Solids, J. Am. Chem. Soc., 38 (1916) 2221-2295. [3] H.M.F. Freundlich, Über die adsorption in lösungen, J. Phys. Chem., 57(A) (1906) 385–470. [4] R. Sips, On the Structure of a Catalyst Surface, Journal of Chemical Physics, 16 (1948) 490-495. [5] O. Redlich, D.L. Peterson, A Useful Adsorption Isotherm, Journal of Physical Chemistry, 63 (1959) 1024-1024. [6] A.R. Khan, R. Ataullah, A. AlHaddad, Equilibrium adsorption studies of some aromatic pollutants from dilute aqueous solutions on activated carbon at different temperatures, Journal of Colloid and Interface Science, 194 (1997) 154-165. [7] J. Tóth, State Equations of the Soilid-gas Ibterface Layers, Acta Chim Acad Sci Hung, 69 (1971) 311-317. [8] C.J. Radke, J.M. Prausnitz, Adsorption of Organic Solutes from Dilute Aqueous-Solution on Activated Carbon, Industrial & Engineering Chemistry Fundamentals, 11 (1972) 445-451. [9] M.M. Dubinin, The Potential Theory of Adsorption of Gases and Vapors for Adsorbents with Energetically Nonuniform Surfaces, Chem. Rev., 60 (1960) 235-241. [10] A.N. Frumkin, Electrocapillary Curve of Higher Aliphatic Acids and the State Equation of the Surface Layer, J. Phys. Chem., 116 (1925) 466-470. [11] P.J. Flory, Thermodynamics of High Polymer Solutions, J Chem Phys, 10 (1942) 51-61. [12] M.L. Huggins, Some Properties of Solutions of Long-chain Compounds, The Journal of Physical Chemistry, 46 (1942) 151-158. [13] W. Fritz, E.-U. Schluender, Simultaneous Adsorption Equilibria of Organic Solutes in Dilute Aqueous-Solutions on Activated Carbon, Chemical Engineering Science, 29 (1974) 1279-1282. [14] B. Volesky, Z.R. Holan, Biosorption of heavy metals, Biotechnology Progress, 11 (1995) 235-250. [15] Y. Liu, Y.-J. Liu, Biosorption isotherms, kinetics and thermodynamics, Separation and Purification Technology, 61 (2008) 229-242. [16] M.R. Mafra, L. Cardozo-Filho, F.A.P. Voll, L.R. Evangelista, R. Guirardello, G. Barbero, A model for selective adsorption with a localized adsorption energy, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 358 (2010) 149-152. [17] G. Barbero, A.K. Zvezdin, L.R. Evangelista, Ionic adsorption and equilibrium distribution of charges in a nematic cell, Physical Review E, 59 (1999) 1846-1849. [18] L.R. Evangelista, G. Barbero, Adsorption phenomenon and external field effect on an isotropic liquid containing impurities, Physical Review E, 64 (2001) 021101. [19] G. Barbero, L.R. Evangelista, Adsorption Phenomena and Anchoring Energy in Nematic Liquid Crystals, CRC Press2006. [20] I.C. Ostroski, M.A.S.D. Barros, E.A. Silva, J.H. Dantas, P.A. Arroyo, O.C.M. Lima, A comparative study for the ion exchange of Fe(III) and Zn(II) on zeolite NaY, J Hazard Mater, 161 (2009) 1404-1412. [21] C.G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Math Comput, 19 (1965) 557-&. [22] M.J.D. Powell, Restart Procedures for Conjugate Gradient Method, Math Program, 12 (1977) 241-254.
Page 20 of 30
Figure captions
ip t
Figure 1 – Profile of electrical potential ψ(z) in the sample.
cr
Figure 2 – Influence of Parameter D in the Poisson-Boltzmann isotherm.
us
Figure 3 – Poisson-Boltzmann isotherm for three sets of values of parameters A S, NS and L.
M
an
Figure 4 – Iron and zinc ions adsorption isotherms (PB2 model).
d
Figure 5 – Chemical potential for iron and zinc ions adsorption isotherms (PB2 model).
Ac
ce p
te
Figure 6 – Surface electrical potential for iron and zinc ions adsorption isotherms (PB2 model).
Page 21 of 30
Tables
Table 1 – Adsorption isotherms.
qE
Freundlich
qE
n
a and n
K S CE S
KS ,
1 aS C E S
RedlichPeterson
qE
K RP CE 1 aRP CE RP
Khan
qE
qMax bK CE
RadkePausnitz
qE
qE
1
Ac
Frumkin
Flory-Huggins
M 1 nT
aR
qDR e
rR CE R 1
1 K DR RT ln 1 C E
e f K F CE , where
nFH 1
nFH
qE
q0 a1 CE1 1
[4]
RP
[5]
qMax , bK and nK
[6]
qMax , bT and nT
[7]
a R , rR and R
[8]
2
qDR and
K DR R T
qE qMax
e2 nFH FH K FH CE
[3]
b
FritzSchluender
nT
aR rR CE R
ce p
DubininRadushkevich
1 bT CE
aS and S
K RP , aRP and
nK
qMax bT CE
te
Tóth
1 bK CE
d
qE
[2]
1 b CE
a CE
qMax and b
qMax b CE
us
qE
Sips
Reference
cr
Langmuir
Parameters
ip t
Equation
an
Isotherm
b a2 C E 2
2
qMax , K F and f qMax , K FH , nFH and FH
q0 , a1 , a2 , b1 and b2
[9]
[10]
[11, 12]
[13]
Page 22 of 30
Table 2 – Calculated parameters for PB1 and PB2 models for iron and zinc ions adsorptions data. Zinc(PB1)
Iron(PB2)
Zinc(PB2)
0.771 3.58 0.0413 0.987 0.985 0.0626
0.238 3.07 0.0857 0.968 0.963 0.0680
0.761 3.60 0.0422 0.987 0.985 0.0628
0.237 3.07 0.0844 0.968 0.963 0.0678
ip t
Iron(PB1)
cr
Parameters L.AS(dm3 g-1) NS.AS(meq g-1) Fo R2 R2a AAD
Fo
R2
R2a
AAD
0.0520
0.984
0.981
0.0622
0.0894
0.972
0.968
0.0838
0.0395
0.988
0.983
0.0628
0.0407
0.987
0.982
0.0636
te
us
Table 3 – Calculated parameters for iron adsorption data.
0.0390
0.988
0.983
0.0625
aRP (dm3 meq 1 ) 2.03
0.0407
0.987
0.982
0.0636
0.0416
0.987
0.982
0.0641
0.252
0.922
0.909
0.152
0.0400
0.988
0.983
0.0629
0.0322
0.990
0.982
0.0514
0.0337
0.990
0.976
0.0525
Isotherm
Parameters
b(dm3 meq 1 ) 1.19
an
qMax (meq g 1 ) 2.92
Langmuir
a(dm3 g 1 ) 1.46 n 0.421
Freundlich
M
qMax (meq g 1 ) 3.89
Tóth
bT (dm3 meq 1 ) 1.46 nT 1.58 rR (meq g 1 ) 2.32 R 0.154
d
Radke-Pausnitz
aR (meq g 1 ) 4.71
RedlichPeterson Khan
K RP (dm3 g 1 ) 4.71 RP 0.845 qMax (meq g 1 ) 1.64
bK (dm3 meq 1 ) 2.62 nK 0.791
Ac
DubininRadushkevich
aS (dm3 meq 1 ) 0.786
ce p
Sips
K S (dm3 g 1 ) 2.77 S 0.800
Frumkin
qDR (meq g 1 ) 2.27
K DR R T 0.509 2
qMax (meq g 1 ) 3.61
K F (dm3 meq 1 ) 1.28 f -1.23 qMax (meq g 1 ) 2.35
Flory-Huggins
K FH (dm3 meq 1 ) 209.0
nFH 0,0106 FH -176 q0 (meq g 1 ) 2.46 FritzSchluender
a1 (dm3 meq 1 ) 0.630 b1 0.538 1
a2 (dm meq ) 0.0204 b2 2.12 3
Page 23 of 30
Table 4 – Calculated parameters for zinc ion adsorption data. R2
R2a
AAD
b(dm3 meq 1 ) 3.07
0.170
0.935
0.926
0.0934
a(dm3 g 1 ) 1.98 n 0.289
0.103
0.961
0.955
0.0929
0.0746
0.972
0.962
0.0732
Freundlich
qMax (meq g 1 ) 5.60
Tóth
bT (dm3 meq 1 ) 20.1 nT 3.12
Radke-Pausnitz
Khan
K S (dm3 g 1 ) 3.96 S 0.542 aS (dm3 meq 1 ) 0.939 K RP (dm3 g 1 ) 26.7 RP 0.788 aRP (dm3 meq 1 ) 12.2 qMax (meq g 1 ) 1.04
bK (dm3 meq 1 ) 20.2 nK 0.766 qDR (meq g 1 ) 2.45
DubininRadushkevich
d
qMax (meq g 1 ) 5.15
K F (dm3 meq 1 ) 4.96 f -5.10
te
Frumkin
K DR R T 0.242 2
0.0838
0.958
0.0798
0.973
0.964
0.0702
0.0838
0.968
0.958
0.0798
0.0875
0.967
0.956
0.0822
0.367
0.861
0.841
0.173
0.0785
0.970
0.960
0.0744
0.0782
0.970
0.952
0.0735
0.0466
0.982
0.965
0.0616
0.0705
an
RedlichPeterson
rR (meq g 1 ) 2.20 R 0.211
M
Sips
aR (meq g 1 ) 26.7
cr
qMax (meq g 1 ) 2.82
Langmuir
0.968
us
Parameters
ip t
Fo
Isotherm
qMax (meq g 1 ) 4.15 K FH (dm3 meq 1 ) 8.77
ce p
Flory-Huggins
nFH 0,674 FH -6.11 q0 (meq g 1 ) 2.88
a1 (dm3 meq 1 ) 0.727 b1 0.363 1
a2 (dm meq ) 0.0169 b2 2.30
Ac
FritzSchluender
3
Page 24 of 30
M
an
us
cr
ip t
Figure 1
Ac ce p
te
d
S
(z)
D
0
D Page 25 of 30
z
an
us
cr
ip t
Figure 2
d
M
2,0
te Ac ce p
1,0
E
q (meq/g)
1,5
0,5
0,0 0
1
2
C
E
D=5L
3
4
(meq/L) Page 26 of 30
D=10L
D=500L
us
cr
ip t
Figure 3
M
an
1,0
te
0,6
Ac ce p
E
q (meq/g)
d
0,8
0,4
0,2
0,0 0
1
2
C
E
3
4
(meq/L) Page 27 of 30
Set 1
Set 2
Set 3
us
cr
ip t
Figure 4
an
3,0
M
2,5
d te Ac ce p
1,5
E
q (meq/g)
2,0
1,0
0,5
0,0 0
1
2
C
E
3
4
(meq/L)
Iron - PB2 model
Iron - Exp. (Ostroski, Page 2009) 28 of 30
Zinc - PB2 model
Zinc - Exp. (Ostroski,
2009)
us
cr
ip t
Figure 5
M
an
0,30
d
0,25
Ac ce p
te
0,20
0,15
0,10
0,05
0,00 0
1
2
3
4
C (meq/L) E
Page 29 of 30
Iron
Zinc
us
cr
ip t
Figure 6
an
3,0
d
M
2,5
Ac ce p
te
2,0
1,5 S
1,0
0,5
0,0 0
1
2
3
4
C (meq/L) E
Page 30 of 30
Iron
Zinc
S0927-7757(14)00941-8 http://dx.doi.org/doi:10.1016/j.colsurfa.2014.11.065 COLSUA 19602
To appear in:
Colloids and Surfaces A: Physicochem. Eng. Aspects
Received date: Revised date: Accepted date:
14-9-2014 27-11-2014 28-11-2014
Please cite this article as: V. Steffen, L. Cardozo-Filho, E.A. Silva, L.R. Evangelista, R. Guirardello, M.R. Mafra, Equilibrium modeling of ion adsorption based on PoissonBoltzmann Equation, Colloids and Surfaces A: Physicochemical and Engineering Aspects (2014), http://dx.doi.org/10.1016/j.colsurfa.2014.11.065 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Ac
ce
pt
ed
M
an
us
cr
i
*Graphical Abstract (for review)
Page 1 of 30
*Highlights (for review)
Highlights A phenomenological model to represent the bulk adsorption of ions is proposed. The model proposed was based on a previous surface adsorption one.
ip t
There was improvement and simplification in the previous model. The model was successful in accurately describe the investigated cases.
Ac
ce pt
ed
M
an
us
cr
It reduced the number of adjustable parameters with no lose in the fit quality.
Page 2 of 30
Equilibrium modeling of ion adsorption based on Poisson-Boltzmann
ip t
Equation V. Steffena, L. Cardozo-Filhoa, E. A. Silvab, L.R. Evangelistac, R. Guirardellod and M.R.
Departamento de Engenharia Química, Universidade Estadual de Maringá, Avenida
Colombo, 5790, 87020-900 Maringá, Paraná, Brazil.
Centro de Engenharia e Ciências Exatas, Universidade Estadual do Oeste do Paraná, Rua
an
b
us
a
cr
Mafrae
c
Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790,
School of Chemical Engineering, State University of Campinas, PO Box 6066, 13083-970
Campinas, SP, Brazil.
te
d
d
87020-900 Maringá, Paraná, Brazil.
Departamento de Engenharia Química, Universidade Federal do Paraná, Centro
ce p
e
M
da Faculdade, 645, 85903-000 Toledo, Paraná, Brazil.
Ac
Politécnico, Setor de Tecnologia, Jardim das Américas, 81531-990 Curitiba, Paraná, Brazil.
Corresponding author. Tel.: +55 44 3011 4749; fax: +55 44 3011 4792. E-mail adress: [email protected] (V. Steffen)
Page 3 of 30
Abstract A phenomenological model representing ion adsorption equilibrium is proposed. It is an extension of the model used to represent surface adsorption, and it is intended to interpret
ip t
bulk adsorption data. The fundamental equations of the model were established for fluid and solid phases mass balance and for the spatial distribution of ions across the sample. The
cr
electric field distribution in the system, modeled on Poisson’s Equation, and the chemical
us
potential behavior were determined as a function of fluid phase equilibrium concentration. The concentration of ions as a function of electric potential is given by Boltzmann
an
distribution. The proposed model was successfully tested to represent equilibrium adsorption of metallic ions in zeolite, showing results similar to conventional isotherms.
M
Keywords: Adsorption, Poisson Equation, Boltzmann Distribution, Poisson-Boltzmann,
Introduction
te
1.
d
Adsorption Isotherm.
ce p
The investigation of sorbent performance is basic for industrial application and for design of adsorption process equipment. The analysis is normally carried out through equilibrium studies on the system of interest [1]. There is a large number of available
Ac
isotherms in the literature that may represent adsorption equilibrium data, among which may be mentioned the ones of Langmuir [2], Freundlich [3], Sips [4], Redlich-Peterson [5], Khan [6], Tóth [7], Radke-Prausnitz [8], Dubinin-Radushkevich [9], Frumkin [10], Flory-Huggins [11, 12], and Fritz-Schluender [13]. The first two are the equilibrium adsorption isotherm models most widely accepted for single solute systems [14]. Due to the complexity of the adsorption mechanism it is rather difficult for researchers to choose isotherm models according to known mechanisms. The criterion for choosing an
Page 4 of 30
isotherm is, so far, mainly based on the use of statistical parameters of the model which best fit experimental data. However, a good curve, fitting in the sense of statistical evaluation, may not necessarily imply that the curve has true physical meaning, i.e., if a set of sorption data is
ip t
analyzed by different isotherm equations, the best fit equation may not be the one reflecting the sorption mechanisms [15]. The importance of a model foregrounded on the physical
cr
phenomena involved in the process is thus evidenced.
us
Here, we propose a model for selective adsorption phenomenon in an isotropic liquid that takes into consideration localized adsorption energies. The model basically consists of a
an
set of equations that determine the profile of the electric field and the chemical potential. It is intended to use Poisson-Boltzmann equation, a model well known in the representation of
M
surface adsorption, for treating bulk adsorption data, but due to the complexity of the model, it is necessary to approximate the adsorbent (composed by little porous particles) as a plane
te
d
surface.
The adsorption of ions in the liquid phase involves the interaction between a solvent
ce p
(usually water) containing a dissolved species, the adsorbate (metallic ions), and an adsorbent solid which must have high affinity to the adsorbate. Sorption reactions inside the pores and on the surface of the adsorbent depend on the type of dissolved ions in the aqueous medium,
Ac
on the chemical nature of the adsorbate, and the nature of the adsorbent surface. Thus, electrolyte ions in the aqueous medium migrate to the electrically charged sorbent surface. The cations in the sorbent solid surface are substituted by the metallic ions in the solution. Another approach for the representation of metallic ions adsorption equilibrium data has been forwarded by Mafra et al. [16], whose mathematical model is based on the PoissonBoltzmann equation [17-19]. The model, used to represent surface adsorption, describes the electric field distribution of a sample limited by two adsorbent surfaces. However, the
Page 5 of 30
equations of the model [16] are valid only for the representation of batch equilibrium data since there are variables, such as adsorbent mass, system volume and initial concentration of metallic ions, in the formulation. Moreover, the model has four adjustable parameters.
ip t
The model we are proposing here applies the Poisson-Boltzmann equation to ion adsorption and presents a mathematical formulation in terms only of adsorbate equilibrium
cr
concentrations in the fluid phase and on the adsorbent. It also evaluates strategies to reduce
us
the number of parameters in the model when compared with the formulation developed by Mafra et al. [16]. The proposed model consists of a set of algebraic/differential equations that
an
determine the profiles of electric and chemical potentials (differential equations) and the concentration on the adsorbent (algebraic equation).
M
The model will be applied to the analysis of adsorption experimental data of Fe (III)
Basic theory
ce p
2
te
proposed model.
d
and Zn (II) on zeolite NaY, provided by Ostroski et al. [20] to illustrate the usefulness of the
The Poisson equation is a partial differential equation of fundamental importance and wide usefulness in several research areas. It connects the electric potential (V) with the charge
Ac
density (ρ) in the form: 2V
(1)
where is the dielectric permittivity of the medium. The liquid charge density is given by
q C C
(2)
where q is the charge of the ion and C± is the concentration of ± ions.
Page 6 of 30
By inserting Equation (2) into Equation (1), we obtain 2V
q
C
C
(3)
ip t
One may observe that in Equation (3) the ions densities have to be related with the electric potential. One of the most used tools to establish this relation is the Boltzmann distribution,
cr
which holds true for low concentrations and high temperatures. It may be written here as
(4)
us
N EE / k T e 0 B eE / kB T N0
an
where N is the number of systems in the energy state E, N0 is the number of systems in the reference energy state E0, kB is the Boltzmann constant, and T is the absolute temperature.
M
Since the modeled system involves ion adsorption, the more appropriate reference state for Boltzmann distribution is the ion equilibrium concentration in the bulk fluid phase (CE): q V / kB T
(5)
d
C CE e
te
where µ is the chemical potential. For convenience, the energies entering Equation (5) will be
ce p
measured in kBT, i. e., µ/kBT → µ, and qV/kBT → ψ. Substitution of Equation (5) into Equation (3) yields
q 2 CE e e e kB T
Ac
2
(6)
When the Poisson-Boltzmann equation is written in one dimension, we have d 2 e sinh , dz 2 L2
with L2
kB T 2 q 2 CE
(7)
defining an intrinsic length of the problem. Let us start by considering a system that consists of a fluid of dielectric permittivity containing ions. As already mentioned, the adsorbent is roughly approximated as a plane surface, so the system is limited by a flat adsorbent surface located at z=0. It is possible to
Page 7 of 30
obtain the spatial distribution of the electric field by means of the Poisson-Boltzmann equation and, consequently, the spatial distribution of the electric potential across the sample. It may be assumed that the surface selectively adsorbs only positive charges (cations), which
ip t
will give rise to a surface electric field. The modeling of this system will result in an electric potential profile with a maximum value in the adsorbent surface (ψS), that decreases from z=0
us
or rather, (dψ/dz)z=D=0 and ψ(zD)=ψD, as shown in Figure 1.
cr
to z=D. For an adsorbent surface distance greater than z=D, the electric potential is constant,
Since the ions positioned at z=0 are not necessarily adsorbed, in order to obtain a
an
mathematical expression that gives the density of adsorbed ions (nS), the Boltzmann distribution has to be used again:
M
nS N e A S
(8)
d
where N is the number of sites per unit surface, ψS is the surface electrical potential, and A is
te
the adsorption energy (A=Ead/kB T).
Using the Poisson-Boltzmann equation, represented by Equation (7), in the system
ce p
under analysis, the values of the electric potential at any point of the z coordinate may be calculated. On the adsorbent surface (z=0), the solution of this equation obeys the established
Ac
boundary conditions for the electric field (E) on surface, namely:
E z 0
q nS k B T d q dz z 0
(9)
Another important relation is given by the mass balance equation: N N nS AST C0 V 2 2
(10)
where AST is the total surface area, C0 is the initial concentration of ions in the bulk fluid phase, V is the volume of fluid phase, and N is the number of ions ± , given by:
Page 8 of 30
N AST
D
0
C z dz CE V CE D AST
(11)
Therefore, the fundamental equations of the model are Equations (7), (9) and (10), or
2.1
ip t
rather, the equations connecting chemical and electric potentials. A model for biosorption
cr
In Ref. [16], a model a model to represent the biosorption equilibrium of heavy metals
us
was recently proposed. The manner of obtaining the equations of this mathematical modeling will be summarized as follows. By integrating twice equation (7) and using the boundary
D
cosh cosh D
2 D e 2 L
(12)
M
d
S
an
conditions (dψ/dz)z=D=0 and ψ(zD)=ψD, we obtain
2
d
By combining Equations (7), (8), and (9) , one obtains
te
C L 2 A e 8 0 cosh S cosh D e s N
(13)
ce p
Finally, by inserting Equations (8) and (11) into Equation (10), after a mathematical rearrangement, it is possible to show that e CE AST
D
0
Ac
N AST e A S T cosh dz C0 CE V CE D AS 2
(14)
Equations (12), (13) and (14) connect ψD, ψS and μ. They are the basic equations of the
model proposed to describe biosorption and may be solved numerically. Consequently, the initial and equilibrium concentrations of adsorbate (respectively, C0 and CE) can be determined. In the model, ψD, ψS and μ are dependent variables whereas the adsorption energy, A, the number of sites per unit surface, N, the total surface area, AST, and the thickness of the interfacial region, D, are adjustable parameters.
Page 9 of 30
In addition to the equations connecting the electric and chemical potential, an equation connecting qE, CE and C0 is required. This relation is obtained from Equation (8) rewritten in specific terms:
ip t
ms qE N e A s T AS
(15)
cr
where is the number of the ion elementary charges. The isotherm is represented by Equation
us
(15), in which ψS and μ may be obtained from the solutions of Equations (12)-(14) governing the model. Extended model
an
2.2
M
Since isotherms are functions of the equilibrium concentrations of fluid and adsorbent phases only, it is convenient to modify the mathematical formulation developed by Mafra et
d
al. [16], written in terms of initial concentration (C0), adsorbent mass (mS) and solution
te
volume (V). These modifications will allow the model to describe the equilibrium adsorption
ce p
of ions on any system regardless of initial conditions and absolute amounts. Another important aspect of the model developed by Mafra et al. [16] is the boundary conditions used to solve the Poisson-Boltzmann differential equation. These conditions are
Ac
(dψ/dz)z=D=0 and ψ(z=D)=ψD. The values ψD so obtained are small (order of magnitude 10-5). This suggests that they may be evaluated by stating ψ(z=D)=0 as boundary condition in the solution of the Poisson-Boltzmann equation. Using this new boundary condition, the model may be simplified and becomes more attractive. Consequently, this boundary condition will be hereafter considered. In this way, two distinct models for the Poisson-Boltzmann isotherm will be proposed. They differ by the boundary conditions used for solving the differential equation representing the electric field of the system. In the first model, the boundary conditions are represented by (dψ/dz)z=D=0 and Page 10 of 30
ψ(z=D)=ψD , while in the second model by (dψ/dz)z=D=0 and ψ(z=D)=0. For convenience, the models will be denoted, respectively, by the short names PB1 and PB2. In this scenario, in order to eliminate the term mS (gads) from the equations, the surface
ip t
area per mass of substrate AS (dm2/gads) will be used to substitute total surface area (AST) used in the Mafra et al. [16] model. Thus, the isotherm may be obtained from Equation (8), where
cr
nS is the density of ions in the surface (meq/dm2) and N is the number of sites per surface unit
us
(meq/dm2). The Poisson-Boltzmann isotherm proposed here will be represented by qE AS N e A S
(16)
an
where qE is the equilibrium concentration of adsorbed ions (meq/ gads).
M
To eliminate the initial concentration (Co) and the solution volume (V), the mass balance for ions in a closed system will be employed, represented by the following equation: (17)
d
V C0 CE mS
te
qE
With regard to the manner the value of qE (given the value of CE) is obtained, it should
ce p
be underlined that isotherms are usually classified as explicit and implicit. When the isotherm is explicit, the calculation of qE is given by a simple and direct way. When the isotherm is implicit, the calculation of qE is given by numerical methods, in which the value of qE is
Ac
corrected at each iteration until the error is very small. However, the Poisson-Boltzmann isotherm does not fit in any of these two groups. When CE value is established, a set of nonlinear equations is also solved, with such unknown factors as chemical (μ) and electrical potentials (ψS e ψD), whose values are used in Equation (16) to calculate the qE value. This set of nonlinear equations is different for the two models (PB1 and PB2), mainly due to the fact that the number of equations which form each set of equations is also different. In the case of PB1 model there are three equations whereas for PB2 model there are only two equations. By
Page 11 of 30
definition we have ψD=0 for this latter model. These equations are obtained as above for the model of Ref. [16], by taking into account the changes mentioned to eliminate the terms accounting for the initial concentration, adsorbent mass and volume of the solution.
D
cosh cosh D
2 CE L e 2 2
S
D
2 D e 2 L
cosh
N e A S d CE D 2 cosh cosh D
us
cr
d
S
ip t
In the case of PB1 model, the three equations are:
2
C L 2 A e 8 E cosh S cosh D e S N
an
(18)
(19)
(20)
M
The solution of Equations (18), (19) and (20) is not simple due to the two integations that have to be performed, involving elliptic functions. Moreover, they have singularities at
d
the lower integration limits. These singularities are contoured with a suitable change of
te
variables. The algebraic expressions for these integrals are obtained by considering the
ce p
approximation ψD≈0. The set of non-linear equations was solved using the method by Broyden [21].
sinh 2 v 1 sinh D 2 2
Ac
(21)
In the case of PB2 model, the two equations are given by: D N e A S e cosh z dz D 2 CE 0
S 2 sinh 1 2 ln 1 2 ,
(22)
(23)
where
Page 12 of 30
2 1 e2 D 4 cosh z dz D 1 2 1 2 e2 D
(25)
ip t
0
(24)
e 2 L
cr
D
N e 2 A S 4 CE L
(27)
us
tanh S 4
(26)
In the second model (PB2), the set of equations is smaller than the one of the first
an
model (PB1). As mentioned above, the PB2 model consists of only two equations, Equations (22) and (23), since it has one unknown less, which are simpler than the ones forming the
analytically done. Thus,
M
PB1 model. Furthermore, the unique integration to be performed in this model may be PB2 model consists of algebraic equations only, and, in this
te
surface may be given by
d
framework, the profile of the electric potential as a function of distance from the adsorbent
2
ce p
1 e z z ln z 1 e
(28)
Ac
2.2.1 Parameters analysis
In the equations of the model, the terms N and A always appear together in the form of
a product (Ne-A), and thus a new variable NS=Ne-A may be defined as only one adjustable parameter. The isotherm represented by Equation (16) is rewritten as qE AS N S e S
(29)
and the fundamental equations representing the models may also be rewritten. In the mathematical formulation of PB1 model, Equation (18) does not change, but Equations (19)
Page 13 of 30
and (20) shall be represented, respectively, by
2 CE L e 2 2
S
D
cosh cosh cosh D
d
N S e S CE D 2
(30)
2
C L e 8 E cosh S cosh D e2 S NS
ip t
(31)
cr
In the case of PB2 model, Equation (22) is modified, whereas Equation (23) and the
us
definitions given by equations (25), (26) and (27) remain unchanged. Only the definition
D N e S e cosh z dz S D 2 CE 0
M
N S e 2 S 4 CE L
an
given by Equation (24) slightly changes and takes a new form. The new equations are: (32)
(33)
d
In the Poisson-Boltzmann model, D is a parameter that represents the distance from
te
the surface where the electric potential has constant value ψD. Simulations were performed to
ce p
evaluate the effect of this parameter on the isotherms. The values used were D=5L, D=10L , and D=500L. Figure 2 shows the simulations results that using the following values for the other parameters: A=1, N=3 meq/dm2 and AS=2 dm2/g.
Ac
The graphs in Figure 2 show that parameter D has only a slight influence on the Poisson-Boltzmann isotherm. In this manner, the number of adjustable parameters of the model could be reduced. We decided to fix the value of D in 10L, a large enough value to ensure that the electrical potential is very small at this distance from adsorbent surface. Thus, beyond the surface area (AS), the proposed model has only two adjustable parameters, NS and L, which can be interpreted, respectively, as the maximum number of active sites that may be occupied by ions and the distance related to the electric field decay near the adsorbent surface.
Page 14 of 30
Simulations for three sets of values of the parameters were performed to evaluate the relationship among NS, AS , and L. The first set is formed by the values NS=1.0 meq/dm2, L=1.0 dm and AS=1.0 dm2/g, the second is formed by NS =0.1 meq/dm2, L=0.1 dm and
ip t
AS=10 dm2/g, and the third is formed by NS =10 meq/dm2, L=10 dm and AS=0.1 dm2/g. The results of these simulations are shown in Figure 3.
cr
The isotherm curves in Figure 3 are identical to the three sets of values of the
us
parameter NS, AS and L, with correlation among these parameters. Therefore, it is possible to
Results and discussion
M
3
an
consider only two adjustable parameters in the form NS.AS and L.AS.
d
Conventional adsorption isotherms models tested to evaluate the proposed model
te
performance are shown in Table 1.
ce p
Equilibrium data for Fe(III) and Zn(II) adsorption on zeolite NaY, obtained by (Ostroski, 2009), were employed to evaluate the models based on Poisson-Boltzmann equation proposed here.
Ac
All calculations were made for CE in meq/L units and qE in meq/gads units. FORTRAN
77 codes were written for calculations. An optimization algorithm based on the version of the conjugate gradient algorithm described by Powell [22] was used to minimize the objective function value, which may be expressed as N
Fo qiexp qicalc
2
(34)
i 1
Further, the coefficient of determination (R2) was used as comparison parameter. Thus, an effective comparison may be done among the models with different numbers of adjustable Page 15 of 30
parameters, an alternative measure of the coefficient of determination may be employed which penalizes the inclusion of adjustable parameters with little explanatory power. This is the adjusted coefficient of determination (R2a):
n 1 1 R 2 n p
ip t
Ra2 1
(35)
cr
where n is the number of experimental observations and p is the number of adjustable
us
parameters in the isotherm.
Another statistical parameter used to evaluate the model performance is the average
1 n exp qi qicalc n i 1
(36)
M
AAD
an
absolute deviation (AAD):
The parameters estimated for PB1 and PB2 models, the coefficients of determination,
d
adjusted coefficients of determination and objective function are given in Table 2.
te
Results of the statistical analysis in Table 2 show that there was no significant
ce p
difference between PB1 and PB2 models. Consequently, the PB2 model should be employed due to its lower complexity and to the fact that the set of equations to be solved is composed by fewer equations (only two) and only non-linear algebraic equations.
Ac
Figures 4, 5 and 6 show, respectively, the isotherms graphs, chemical potential and surface electric potential for iron and zinc ions adsorption by zeolite experimental data best fitted parameters by PB2 model. Results in Figure 4 and statistical analysis in Table 2 show that PB1 and PB2 models accurately describe the equilibrium adsorption data. Figures 5 and 6 show that the profiles of the chemical potential and surface electrical potential are similar, although it must be highlighted that the surface electrical potential (ψS) values are approximately ten times higher
Page 16 of 30
than the corresponding values of chemical potential (μ). Results in Figures 4, 5 and 6 show that higher ions sorption capacity (Zn+2>Fe+3) are related to lower values of the chemical and electrical potential of the respective ions, which may be easily proved by evaluating Equation
ip t
(29). Profiles of electrical potential of adsorbed ions on the zeolite surface were similar to the ones that obtained by Mafra et al. [16] who used this approach to represent the biosorption
cr
equilibrium data of chromium and copper ions by the marine alga Sargassum sp.
us
There are four adjustable parameters (A, AST, N, D) in the modeling presented by Mafra et al. [16], whereas in the present study there are only two parameters that could
an
accurately represent the equilibrium data of the two systems evaluated.
To evaluate the performance of the proposed Poisson-Boltzmann isotherm, results in
M
Table 2 will be compared with conventional adsorption isotherms in Table 1. Tables 3 and 4 show the best fit parameters value as well as coefficients of determination, adjusted
te
d
coefficients of determination and objective function of the isotherms shown in Table 1 applied, respectively, to the iron and zinc experimental data on ion adsorption.
ce p
Table 3 shows that in the case of iron adsorption the Flory-Huggins and FritzSchluender isotherms present the highest value of R2, even though Flory-Huggins shows slightly lower values of AAD and Fo . The highest R2a is provided by Tóth, Sips and Frumkin
Ac
isotherms, with three parameters. In general, all models show good fits to the experimental data of iron adsorption, given that R2 values are higher than 0.98. The Freundlich and Dubinin-Radushkevich isotherms are the exception. Table 4 shows that, in the case of zinc adsorption, the Fritz-Schluender isotherm presents the highest R2 and R2a and the lowest AAD and Fo values. In general, all models show good fits to the experimental data of zinc adsorption, given that, with the exception of Langmuir and Dubinin-Radushkevich isotherms, R2 values are higher than 0.96.
Page 17 of 30
The comparison of results in Table 2 and those in Tables 3 and 4 shows that the isotherm proposed in this paper, in addition to having a strong theoretical basis, has a good
ip t
fitting quality and may be compared to the isotherms with the same number of parameters. 5. Conclusions
cr
In this paper, we proposed a modification of the model by Mafra et al. [16] to
us
represent the bulk adsorption of ions. It is a model based on the Poisson-Boltzmann equation that uses mathematical modeling of potential and adsorption energy to describe the
an
distribution of ions and adsorbed amount. The proposed model was applied to iron and zinc experimental data on ion adsorption and the results were in good agreement. The model is
M
phenomenological and takes into account the interaction between the adsorbed particles. In fact, it may be extended to multicomponent adsorption systems. It has only two adjustable
d
parameters and the fitting quality is comparable to conventional adsorption isotherms with the
ce p
Acknowledgment
te
same number of adjustable parameters.
This work was partially funded by the Brazilian Agency, CAPES.
A
Ac
Nomenclature
Adsorption energy
AS
Surface area, dm2 g-1
AST
Total surface area, dm2
C0
Initial concentration in the bulk fluid phase, meq dm-3
CE
Equilibrium concentration in the bulk fluid phase, meq dm-3
Page 18 of 30
Concentration of ± ions in the bulk fluid phase, meq dm-3
D
Thickness of the interfacial region, dm
Fo
Objective function
kB
Boltzmann constant, J K-1
L
Intrinsic length, dm
mS
Adsorbent mass, g
nS
Density of ions in the surface, meq dm-2
N
Number of sites per unit surface, meq dm-2
N
Number of ± ions, meq
q
Electric charge of the ion, C mol-1
qE
Concentration of ions on the adsorbent, meq g-1
T
Temperature, K
V
Volume, dm3
Dielectric permittivity, C2 N-1 m-2
Number of elementary charges of the ion
Chemical potential
Eletrical potential
S
cr us
an
M
d te
ce p
Ac
D
ip t
C
Electrical potential at z=D Surface electrical potential
References
[1] E.A. Silva, E.S. Cossich, C.R.G. Tavares, L. Cardozo-Filho, R. Guirardello, Modeling of copper(II) biosorption by marine alga Sargassum sp. in fixed-bed column, Process Biochemistry, 38 (2002) 791-799.
Page 19 of 30
Ac
ce p
te
d
M
an
us
cr
ip t
[2] I. Langmuir, The Constitution and Fundamental Properties of Solids and Liquids. Part I. Solids, J. Am. Chem. Soc., 38 (1916) 2221-2295. [3] H.M.F. Freundlich, Über die adsorption in lösungen, J. Phys. Chem., 57(A) (1906) 385–470. [4] R. Sips, On the Structure of a Catalyst Surface, Journal of Chemical Physics, 16 (1948) 490-495. [5] O. Redlich, D.L. Peterson, A Useful Adsorption Isotherm, Journal of Physical Chemistry, 63 (1959) 1024-1024. [6] A.R. Khan, R. Ataullah, A. AlHaddad, Equilibrium adsorption studies of some aromatic pollutants from dilute aqueous solutions on activated carbon at different temperatures, Journal of Colloid and Interface Science, 194 (1997) 154-165. [7] J. Tóth, State Equations of the Soilid-gas Ibterface Layers, Acta Chim Acad Sci Hung, 69 (1971) 311-317. [8] C.J. Radke, J.M. Prausnitz, Adsorption of Organic Solutes from Dilute Aqueous-Solution on Activated Carbon, Industrial & Engineering Chemistry Fundamentals, 11 (1972) 445-451. [9] M.M. Dubinin, The Potential Theory of Adsorption of Gases and Vapors for Adsorbents with Energetically Nonuniform Surfaces, Chem. Rev., 60 (1960) 235-241. [10] A.N. Frumkin, Electrocapillary Curve of Higher Aliphatic Acids and the State Equation of the Surface Layer, J. Phys. Chem., 116 (1925) 466-470. [11] P.J. Flory, Thermodynamics of High Polymer Solutions, J Chem Phys, 10 (1942) 51-61. [12] M.L. Huggins, Some Properties of Solutions of Long-chain Compounds, The Journal of Physical Chemistry, 46 (1942) 151-158. [13] W. Fritz, E.-U. Schluender, Simultaneous Adsorption Equilibria of Organic Solutes in Dilute Aqueous-Solutions on Activated Carbon, Chemical Engineering Science, 29 (1974) 1279-1282. [14] B. Volesky, Z.R. Holan, Biosorption of heavy metals, Biotechnology Progress, 11 (1995) 235-250. [15] Y. Liu, Y.-J. Liu, Biosorption isotherms, kinetics and thermodynamics, Separation and Purification Technology, 61 (2008) 229-242. [16] M.R. Mafra, L. Cardozo-Filho, F.A.P. Voll, L.R. Evangelista, R. Guirardello, G. Barbero, A model for selective adsorption with a localized adsorption energy, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 358 (2010) 149-152. [17] G. Barbero, A.K. Zvezdin, L.R. Evangelista, Ionic adsorption and equilibrium distribution of charges in a nematic cell, Physical Review E, 59 (1999) 1846-1849. [18] L.R. Evangelista, G. Barbero, Adsorption phenomenon and external field effect on an isotropic liquid containing impurities, Physical Review E, 64 (2001) 021101. [19] G. Barbero, L.R. Evangelista, Adsorption Phenomena and Anchoring Energy in Nematic Liquid Crystals, CRC Press2006. [20] I.C. Ostroski, M.A.S.D. Barros, E.A. Silva, J.H. Dantas, P.A. Arroyo, O.C.M. Lima, A comparative study for the ion exchange of Fe(III) and Zn(II) on zeolite NaY, J Hazard Mater, 161 (2009) 1404-1412. [21] C.G. Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations, Math Comput, 19 (1965) 557-&. [22] M.J.D. Powell, Restart Procedures for Conjugate Gradient Method, Math Program, 12 (1977) 241-254.
Page 20 of 30
Figure captions
ip t
Figure 1 – Profile of electrical potential ψ(z) in the sample.
cr
Figure 2 – Influence of Parameter D in the Poisson-Boltzmann isotherm.
us
Figure 3 – Poisson-Boltzmann isotherm for three sets of values of parameters A S, NS and L.
M
an
Figure 4 – Iron and zinc ions adsorption isotherms (PB2 model).
d
Figure 5 – Chemical potential for iron and zinc ions adsorption isotherms (PB2 model).
Ac
ce p
te
Figure 6 – Surface electrical potential for iron and zinc ions adsorption isotherms (PB2 model).
Page 21 of 30
Tables
Table 1 – Adsorption isotherms.
qE
Freundlich
qE
n
a and n
K S CE S
KS ,
1 aS C E S
RedlichPeterson
qE
K RP CE 1 aRP CE RP
Khan
qE
qMax bK CE
RadkePausnitz
qE
qE
1
Ac
Frumkin
Flory-Huggins
M 1 nT
aR
qDR e
rR CE R 1
1 K DR RT ln 1 C E
e f K F CE , where
nFH 1
nFH
qE
q0 a1 CE1 1
[4]
RP
[5]
qMax , bK and nK
[6]
qMax , bT and nT
[7]
a R , rR and R
[8]
2
qDR and
K DR R T
qE qMax
e2 nFH FH K FH CE
[3]
b
FritzSchluender
nT
aR rR CE R
ce p
DubininRadushkevich
1 bT CE
aS and S
K RP , aRP and
nK
qMax bT CE
te
Tóth
1 bK CE
d
qE
[2]
1 b CE
a CE
qMax and b
qMax b CE
us
qE
Sips
Reference
cr
Langmuir
Parameters
ip t
Equation
an
Isotherm
b a2 C E 2
2
qMax , K F and f qMax , K FH , nFH and FH
q0 , a1 , a2 , b1 and b2
[9]
[10]
[11, 12]
[13]
Page 22 of 30
Table 2 – Calculated parameters for PB1 and PB2 models for iron and zinc ions adsorptions data. Zinc(PB1)
Iron(PB2)
Zinc(PB2)
0.771 3.58 0.0413 0.987 0.985 0.0626
0.238 3.07 0.0857 0.968 0.963 0.0680
0.761 3.60 0.0422 0.987 0.985 0.0628
0.237 3.07 0.0844 0.968 0.963 0.0678
ip t
Iron(PB1)
cr
Parameters L.AS(dm3 g-1) NS.AS(meq g-1) Fo R2 R2a AAD
Fo
R2
R2a
AAD
0.0520
0.984
0.981
0.0622
0.0894
0.972
0.968
0.0838
0.0395
0.988
0.983
0.0628
0.0407
0.987
0.982
0.0636
te
us
Table 3 – Calculated parameters for iron adsorption data.
0.0390
0.988
0.983
0.0625
aRP (dm3 meq 1 ) 2.03
0.0407
0.987
0.982
0.0636
0.0416
0.987
0.982
0.0641
0.252
0.922
0.909
0.152
0.0400
0.988
0.983
0.0629
0.0322
0.990
0.982
0.0514
0.0337
0.990
0.976
0.0525
Isotherm
Parameters
b(dm3 meq 1 ) 1.19
an
qMax (meq g 1 ) 2.92
Langmuir
a(dm3 g 1 ) 1.46 n 0.421
Freundlich
M
qMax (meq g 1 ) 3.89
Tóth
bT (dm3 meq 1 ) 1.46 nT 1.58 rR (meq g 1 ) 2.32 R 0.154
d
Radke-Pausnitz
aR (meq g 1 ) 4.71
RedlichPeterson Khan
K RP (dm3 g 1 ) 4.71 RP 0.845 qMax (meq g 1 ) 1.64
bK (dm3 meq 1 ) 2.62 nK 0.791
Ac
DubininRadushkevich
aS (dm3 meq 1 ) 0.786
ce p
Sips
K S (dm3 g 1 ) 2.77 S 0.800
Frumkin
qDR (meq g 1 ) 2.27
K DR R T 0.509 2
qMax (meq g 1 ) 3.61
K F (dm3 meq 1 ) 1.28 f -1.23 qMax (meq g 1 ) 2.35
Flory-Huggins
K FH (dm3 meq 1 ) 209.0
nFH 0,0106 FH -176 q0 (meq g 1 ) 2.46 FritzSchluender
a1 (dm3 meq 1 ) 0.630 b1 0.538 1
a2 (dm meq ) 0.0204 b2 2.12 3
Page 23 of 30
Table 4 – Calculated parameters for zinc ion adsorption data. R2
R2a
AAD
b(dm3 meq 1 ) 3.07
0.170
0.935
0.926
0.0934
a(dm3 g 1 ) 1.98 n 0.289
0.103
0.961
0.955
0.0929
0.0746
0.972
0.962
0.0732
Freundlich
qMax (meq g 1 ) 5.60
Tóth
bT (dm3 meq 1 ) 20.1 nT 3.12
Radke-Pausnitz
Khan
K S (dm3 g 1 ) 3.96 S 0.542 aS (dm3 meq 1 ) 0.939 K RP (dm3 g 1 ) 26.7 RP 0.788 aRP (dm3 meq 1 ) 12.2 qMax (meq g 1 ) 1.04
bK (dm3 meq 1 ) 20.2 nK 0.766 qDR (meq g 1 ) 2.45
DubininRadushkevich
d
qMax (meq g 1 ) 5.15
K F (dm3 meq 1 ) 4.96 f -5.10
te
Frumkin
K DR R T 0.242 2
0.0838
0.958
0.0798
0.973
0.964
0.0702
0.0838
0.968
0.958
0.0798
0.0875
0.967
0.956
0.0822
0.367
0.861
0.841
0.173
0.0785
0.970
0.960
0.0744
0.0782
0.970
0.952
0.0735
0.0466
0.982
0.965
0.0616
0.0705
an
RedlichPeterson
rR (meq g 1 ) 2.20 R 0.211
M
Sips
aR (meq g 1 ) 26.7
cr
qMax (meq g 1 ) 2.82
Langmuir
0.968
us
Parameters
ip t
Fo
Isotherm
qMax (meq g 1 ) 4.15 K FH (dm3 meq 1 ) 8.77
ce p
Flory-Huggins
nFH 0,674 FH -6.11 q0 (meq g 1 ) 2.88
a1 (dm3 meq 1 ) 0.727 b1 0.363 1
a2 (dm meq ) 0.0169 b2 2.30
Ac
FritzSchluender
3
Page 24 of 30
M
an
us
cr
ip t
Figure 1
Ac ce p
te
d
S
(z)
D
0
D Page 25 of 30
z
an
us
cr
ip t
Figure 2
d
M
2,0
te Ac ce p
1,0
E
q (meq/g)
1,5
0,5
0,0 0
1
2
C
E
D=5L
3
4
(meq/L) Page 26 of 30
D=10L
D=500L
us
cr
ip t
Figure 3
M
an
1,0
te
0,6
Ac ce p
E
q (meq/g)
d
0,8
0,4
0,2
0,0 0
1
2
C
E
3
4
(meq/L) Page 27 of 30
Set 1
Set 2
Set 3
us
cr
ip t
Figure 4
an
3,0
M
2,5
d te Ac ce p
1,5
E
q (meq/g)
2,0
1,0
0,5
0,0 0
1
2
C
E
3
4
(meq/L)
Iron - PB2 model
Iron - Exp. (Ostroski, Page 2009) 28 of 30
Zinc - PB2 model
Zinc - Exp. (Ostroski,
2009)
us
cr
ip t
Figure 5
M
an
0,30
d
0,25
Ac ce p
te
0,20
0,15
0,10
0,05
0,00 0
1
2
3
4
C (meq/L) E
Page 29 of 30
Iron
Zinc
us
cr
ip t
Figure 6
an
3,0
d
M
2,5
Ac ce p
te
2,0
1,5 S
1,0
0,5
0,0 0
1
2
3
4
C (meq/L) E
Page 30 of 30
Iron
Zinc