Pseudo-Isotherms Using a Second Order Kinetic Expression Constant
YUH-SHAN HO
School of Public Health, Taipei Medical University, 250 Wu-Hsing Street, Taipei 11014, Taiwan ysho@tmu.edu.tw
Received March 12, 2003; Revised January 16, 2004; Accepted March 29, 2004
Abstract. The kinetics of four sorption systems, Cu/tree fern, Pb/tree fern, AB9/activated clay and BR18/activated clay have been studied based on the assumption of a pseudo-second order rate law. Pseudo-isotherms using the pseudo-second order kinetic expression constant have been developed to describe the four liquid-solid sorption systems. The experimental results have been analyzed using a pseudo-Langmuir and a pseudo-Redlich-Peterson isotherm. Both isotherms were found to represent the measured sorption data well. According to the evaluation using the pseudo-Langmuir equation, the monolayer sorption capacities were obtained to be 13.9, 46.6, 124 and 105 mg g−1for copper, lead, AB9 and BR18 respectively.
Keywords: pseudo-isotherm, pseudo-second order, kinetics, equilibrium, adsorption
Introduction
The equilibrium sorption isotherm is fundamentally important in the design of sorption systems. Equilib-rium relationships between sorbent and sorbate are de-scribed by sorption isotherms, usually the ratio between the quantity sorbed and the remaining of the solution at a fixed temperature at equilibrium. In many cases, the equilibrium sorption capacity is unknown and when chemisorption tends to become unmeasurably slow, the amount sorbed is still significantly smaller than the equilibrium amount (Ungarish and Aharoni, 1981). On the other hand, equilibrium takes a long time to ob-tain in some sorption systems. For example, sorption of naphthalene on solid coal tar particles takes more than 60 h to reach equilibrium (Bayard et al., 2000). In 1995, Ho developed a pseudo-second order expression for solid-liquid sorption systems. The pseudo-second order equation has the following advantages: it does not have the problem of assigning an effective sorp-tion capacity; the sorpsorp-tion capacity, rate constant of pseudo-second order and the initial sorption rate all can be determined from the equation without knowing any parameter beforehand. An earlier and yet important
discussion of the use of this model in bi-metal/solutions has been given by Ho et al. (1996). Several applications to metal (Manohar et al., 2002; Reddad et al., 2002); dye (Aksu and Tezer, 2000; Fu and Viraraghavan, 2001) and organic (Kelleher et al., 2001) sorption systems have also been reported. More recently, Ho and Chiang (2001) have published extensively on the use of the model in mixed sorbents.
Tree fern is a complex material containing lignin and cellulose as major constituents (Newman, 1997). Chemical sorption can occur by the polar functional groups of lignin, which include alcohols, aldehydes, ketones, acids phenolic hydroxides and ethers as chem-ical bonding agents (Adler and Lundquist, 1963). It is, therefore, concluded that tree fern can be a suitable sorbate for sorption of metal cations because of its po-lar and acid characters. The removal of divalent metal ions, such as Zn(II), Cu(II), Pb(II) by using tree fern was first reported by Ho et al. (2002, 2004) and Cd(II) (Ho and Wang, 2004).
This paper studies four sorption systems: Cu/tree fern, Pb/tree fern, Acid Blue 9/activated clay, and Basic Red 18/activated clay, where a batch sorption pseudo-isotherm was also developed. Pseudo-pseudo-isotherms were
derived from a pseudo-second order kinetic expres-sion constant and were used to predict the monolayer sorption capacities and sorption constants. A com-parison of the reaction order rate parameter is also discussed.
Materials and Methods
Metal/Tree Fern Sorption Systems
A range of copper(II) concentrations from 52.5 to 328 mg dm−3 and lead(II) concentrations from 74.1 to 350 mg dm−3 were used and agitation was carried out for 2 h. All contact investigations were executed in a baffled agitated 2 dm3sorber vessel. Samples (3 mL) were withdrawn at suitable time intervals and filtered through a 0.45µm membrane filter and then analysed. A 6.8 g sample of tree fern (74–88µm) was added to each 1.7 dm3 of metal ion solution and an agitation speed of 300 rpm was used for all experiments. The temperature was controlled with a water bath of 20◦C for all studies.
Dye/Activated Clay Sorption Systems
In this study, the experimental data of Acid Blue 9 and Basic Red 18 are analyzed through the use of a pseudo-second order kinetic mechanism had been re-ported previously in the literature by Ho et al. (2001). Kinetics Model
Adsorption kinetics deals with the rate at which sorp-tion occurs, where the rate is defined as the change of a given quantity over specific period of time. Never-theless, after a period of time the sorption is reversible when enough sorbate had been sorbed and desorption had become more active. At this point, the concen-tration of the sorbate depends on the rate differences between the sorption and desorption reactions. If one neglects the desorption reaction, then the reaction rate will depend only on the concentrations of the sorbate. Reaction rate of a chemical reaction is defined as the change in concentration of a reactant or product per unit time. The concentrations of the products do not appear in the rate law because the reaction rate is being studied under the conditions where the reverse reactions do not contribute to the overall rate. The re-action order and rate constant must be determined by
experiments. In 1995, Ho presented a pseudo-second order rate law expression, which demonstrated how the rate depended on the sorption equilibrium capacity but not the concentration of the sorbate.
Tree fern contains polar functional groups such as aldehydes, ketones, acids, and phenolics. These groups can be involved in chemical bonding and are respon-sible for the cation exchange capacity of the tree fern (Ho, 2003). Thus, the tree fern-copper reaction may be represented in two ways:
2(TF)−+ Cu2+↔ Cu(TF)2 (1)
and
2H(TF)+ Cu2+↔ Cu(TF)2+ 2H+ (2) where (TF)−and (TF) are polar sites on the tree fern surface.
Rate expression for the sorption described by Eqs. (1) and (2) is d(TF)t dt = k2[(TF)0− (TF)t] 2 (3) or d[H(TF)]t dt = k2[(H(TF))0− (H(TF))t] 2 (4)
where (TF)t and [H(TF)]t are the number of active
sites occupied on the tree fern at time t, and (TF)0and [H(TF)]0are the number of equilibrium sites available on the tree fern.
The kinetic rate equations can be rewritten as fol-lows:
dqt
dt = k(qe− qt)
2 (5)
An integrated pseudo-second order rate law can be ob-tained from Eq. (5) for the boundary conditions t= 0 to t= t and qt= 0 to qt = qt, and is given by:
1 (qe− qt)
= 1
qe
+ kt (6)
Equation (6) can be rearranged to obtain a linear form:
t qt = 1 kq2 e + 1 qe t (7)
where qeis the amount of sorbate sorbed at equilibrium
of sorbate sorbed at time t (mg g−1); k is the equilibrium rate constant of pseudo-second order sorption (g mg−1 min−1).
The pseudo-second order rate law equation shows how the sorption capacity of sorbate depends on time. If the equilibrium sorption capacity of sorbate and the rate constant k are known, then the sorption capacity of sorbate at any time can be calculated. Using the linear form equation, Eq. (7), and plotting t/qt versus t, will
give a straight line with slope 1/qeand intercept 1/kq2e.
Isotherm Models
Langmuir Isotherm
The Langmuir sorption isotherm has been successfully applied to many pollutant sorption processes and it is the most commonly used sorption isotherm for the sorption of a solute from a liquid solution (Langmuir, 1916). The saturated monolayer isotherm can be rep-resented as:
qe=
qmKaCe
1+ KaCe
(8) The above equation can be rearranged to the follow-ing linear form:
Ce qe = 1 Kaqm + 1 qm Ce (9)
where Ceis the equilibrium concentration (mg dm−3);
qeis the amount of metal ion sorbed (mg g−1); qmis
qefor a complete monolayer (mg g−1); Kais sorption
equilibrium constant (dm3mg−1). A plot of C
e/qe
ver-sus Ceshould give a straight line with slope of 1/qm
and an intercept of 1/Kaqm.
Redlich-Peterson Isotherm
The Redlich-Peterson isotherm contains three parame-ters and incorporates the features of the Langmuir and the Freundlich isotherms (Redlich and Peterson, 1959). The Redlich-Peterson isotherm has a linear dependence on concentration in the numerator and an exponen-tial function in the denominator. It can be described as follows:
qe=
ACe
1+ BCeg
(10)
It has three isotherm constants, namely, A, B and g (0<
g< 1), which characterize the isotherm. Equation (10)
can be converted to a linear form by taking natural logarithms: ln ACe qe − 1 = g ln (Ce)+ ln (B) (11)
Three isotherm constants, A, B and g can be evaluated from the linear plot represented by Eq. (11) using a trial and error optimization method. A general trial and error procedure which is applicable to computer op-eration was developed to determine the coefficients of determination, r2, for a series of values of A for the linear regression of ln(Ce) on ln[A(Ce/qe)− 1] and to
obtain the best value of A which yields a maximum ‘optimized’ value of r2.
Results and Discussions
The experimental data were analysed according to the pseudo-second order equation. Figure 1 shows linear variation of t/qtwith time, t, for the sorption system of
Pb/tree fern at various initial concentrations. Good lin-earization of the experimental data was observed from the graph, thereby implying that a pseudo-second or-der reaction is involved. Figure 2 shows the sorption of copper(II) onto tree fern with various initial copper(II) concentration. The solid curves in the figure were the theoretical plots of the pseudo-second order equation. Table 1 listed all four sorption systems included the pseudo-second order rate constant, k, the equilibrium sorption capacity, qe, and the coefficient of
determina-tion, r2, at various initial concentration of sorbate, C 0. These data were calculated from the intercept and the slope of a straight line according to Eq. (7). The data illustrated good compliance with the pseudo-second order rate law based on sorption capacity because the coefficients of determination were higher than 0.995 for all the systems in this study.
Generally true equilibrium conditions are not re-flected in many sorption systems so the equilibrium sorption capacity, qe, is unknown (Bayard et al., 2000;
Ungarish and Aharoni, 1981). Equilibrium takes a long time to obtain in some sorption systems. For instance, sorption of acid dyes onto activated carbon such as Acid Red 114, Polar Yellow and Polar Blue RAWL, will take 21 days to achieve equilibrium (Choy et al., 1999). In the proposed pseudo-second order equation, by fitting Eq. (7) to the experimental data, qe can be obtained
Figure 1. Pseudo-second order sorption kinetics of lead onto tree fern at various initial concentrations.
Table 1. Kinetic parameters for four sorption systems. System C0(mg dm−3) Ce(mg dm−3) qe(mg g) k (g mg−1min−1) r2 AB9/Activated clay 59.7 0.583 14.8 0.660 1.000 114 1.78 28.0 0.272 1.000 356 22.8 83.3 0.0329 1.000 475 45.7 107 0.0138 1.000 552 86.9 116 0.0110 1.000 BR18/Activated clay 111 1.91 27.2 3.48 1.000 213 2.37 52.7 0.238 1.000 326 13.1 78.3 0.0439 1.000 443 72.5 92.7 0.0300 1.000 520 104 104 0.0338 1.000 Cu/Tree fern 52.5 26.1 6.60 0.0180 0.995 79.5 43.8 8.92 0.0238 1.000 123 78.9 11.0 0.0124 0.999 183 138 11.1 0.0253 0.998 328 278 12.7 0.0178 0.999 Pb/Tree fern 74.1 1.9 18.0 0.0681 1.000 111 7.3 26.0 0.0219 1.000 196 57.6 34.7 0.0070 0.999 350 188.3 40.5 0.0044 1.000 537 351.3 46.3 0.0039 1.000
from the slope of the plot of t/qt against t in a shorter
time than the traditional method. For this reason, it is reasonable to use pseudo-second order parameters to obtain the equilibrium sorbate concentration, Ce, to
analyse the pseudo-isotherms.
Four sorption systems, Cu/tree fern, Pb/tree fern, AB9/activated clay and BR18/activated clay were tested by applying the Langmuir and pseudo-Redlich-Peterson isotherms using a pseudo-second order kinetic expression constant. The amount of sor-bate sorbed at equilibrium can be obtained from the pseudo-second order equation and the equilibrium liq-uid phase sorbate concentration can be calculated from the equation below:
Ce= C0−
qems
V (12)
where Ceis the equilibrium liquid phase sorbate
con-centration, mg dm−3; C0is the initial liquid phase sor-bate concentration, mg dm−3; qe is the equilibrium
sorption capacity, mg g−1; V is solution volume, dm3, and ms is the mass of mixed sorbent, g.
Figure 3 shows the linear plot of Ce/qeversus Ce.
The result was a straight line with slope of 1/qm and
intercept of 1/Kaqmfor dye/activated clay sorption
sys-tems. The values of the pseudo-Langmuir constants
qmand Ka are presented in Table 2 for the four
sorp-tion systems. These values of the coefficients of de-termination, r2, indicated a strong positive relation-ship between the data and the linear equation where the solid-liquid sorption data followed the Langmuir sorption isotherm. Figure 4 shows pseudo-Langmuir and pseudo-Redlich-Peterson isotherms for
Table 2. Pseudo-Langmuir isotherm constants for four sorption systems. qm Ka System (mg g−1) (dm3mg−1) r2 AB9/Activated clay 124 0.147 0.995 BR18/Activated clay 105 0.215 0.995 Cu/Tree fern 13.9 0.0378 0.997 Pb/Tree fern 46.6 0.0857 0.994
Figure 3. Pseudo-Langmuir isotherm for sorption of dyes onto activated clay.
Table 3. Pseudo-Redlich-Peterson isotherm constants for four sorption systems. A B System (dm3g−1) (dm3mg−1)g g r2 AB9/Activated clay 45.6 1.20 0.729 0.999 BR18/Activated clay 1.23× 109 3.81× 107 0.739 0.969 Cu/Tree fern 0.510 0.0363 1.000 0.995 Pb/Tree fern 40.2 1.84 0.877 1.000
the metal/tree fern sorption systems. The monolayer saturation capacities, qm, and sorption constants, Ka,
were determined from the slope and intercept of the pseudo-Langmuir equation listed in Table 2. The monolayer saturation capacities, qm, for the four
sorption systems were determined to be 13.9, 46.6, 124 and 105 mg g−1 for copper/tree fern, lead/tree fern, AB9/activated clay and BR18/activated clay respectively.
Experimental results also showed that the pseudo-Redlich-Peterson isotherm was an appropriate de-scription of the data for both metal/tree fern and dye/activated clay sorption over the concentration ranges studied. Table 3 shows the pseudo-Redlich-Peterson isotherm constants A, B and g as well as the coefficients of determination, r2, for four sorption systems. In all cases, the pseudo-Redlich-Peterson isotherm exhibited a high coefficient of determination.
Conclusion
This investigation was based on the assumption of a pseudo-second order mechanism in relation to four sorption systems, Cu/tree fern, Pb/tree fern, AB9/activated clay and BR18/activated clay. The re-sults from the study showed high coefficients of determination. Pseudo-Langmuir and Redlich-Peterson isotherm were derived from the pseudo-second order kinetic expression constant and were used to describe the liquid-solid sorption systems. The re-sults also showed an excellent fit with the experimental data for both pseudo-isotherms. According to the eval-uation using the pseudo-Langmuir eqeval-uation, the mono-layer sorption capacities were obtained to be 13.9 mg g−1 for Cu/tree fern, 46.6 mg g−1 for Pb/tree fern, 124 mg g−1 for AB9/activated clay and 105 mg g−1 for BR18/activated clay.
Nomenclature
(TF)−and (TF) Polar sites on the tree fern surface
(TF)0and [H(TF)]0 The number of equilibrium sites available on the tree fern (TF)t and [H(TF)]t The number of active sites
occupied on the tree fern at time t
A Redlich-Peterson isotherm
constant (dm3g−1)
B Redlich-Peterson isotherm
constant (dm3mg−1)g
C0 Initial liquid phase sorbate
concentration (mg dm−3)
Ce Equilibrium concentration
(mg dm−3)
g Redlich-Peterson isotherm
constant (0< g < 1)
k Pseudo-second order rate
constant (g mg−1min−1)
Ka Sorption equilibrium constant
(dm3mg−1)
ms Mass of mixed sorbent (g)
qe Amount of sorbate sorbed at
equilibrium (mg g−1)
qm qefor a complete monolayer
(mg g−1)
qt Amount of sorbate sorbed at
time t (mg g−1)
r2 Coefficient of determination
t Reaction time (min)
V Solution volume (dm3)
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