Biosorption of As(III) and As(V) on the surface of TW/MnFe2O4 composite from wastewater: kinetics, mechanistic and thermodynamics
M. S. Podder, C. B. Majumder
Department of Chemical Engineering, Indian Institute of Technology, Roorkee.
Roorkee – 247667, India.
Email: ,
1. Theoretical background
1.1. Determining adsorption kinetic parameters by non–linear regression
1.1.1 Residual Sum of Squares Error (SSE)
(1)
A value near to 0 specifies that the model has a lesser random error component and the fit will be more suitable for forecasting (Tsai and Juang, 2000; Dixit et al, 2011; Soares et al., 2014).
1.1.2. Reduced Chi square (Reduced χ2)
(2)
Reduced χ2 will be a smaller number, if the model predicted data are almost analogous to the experimental data, but if they differ; reduced χ2 will be a bigger number (Ho et al., 2005; Galindo et al., 2013).
1.1.3. Coefficient of determination (R2)
(3)
R2 can be on any value in the range of 0 and 1. With a value closer to 1 signifies that a larger percentage of variance is accounted for by the model (Dixit et al, 2011; Hossain et al., 2012; Soares et al., 2014).
1.1.4. Adjusted R-square ()
(4)
can also vary between 0 and 1, with a value near to 1 indicating the more suitable fit for model prediction (Soares et al., 2014).
1.1.5. Correlation coefficient (R value)
(5)
R value lies between 0 and 1. A value is nearer to 1 suggests the better curve fitting (Bates and Watts, 1988; Ratkowsky, 1990; Seber and Wild, 2003).
1.1.6. Root mean square of the error or theStandard Deviation (Root–MSE)
(6)
The smaller Root–MSE value specifies the better curve fitting (Tsai and Juang, 2000; Padmesh et al., 2006).
1.2. Adsorption kinetic modelling
1.2.1. Fractional power model or Modified Freundlich model (FP)
The power function equation which is an empirical model that defines the relation between the amount of the adsorbate per unit adsorbent weight and time t is as follows (Aharoni, 1991; Ho and Mckay, 2003):
(7)
When v = 0.5, the equation converts to the identical to the Webber–Morris intraparticle diffusion model (Weber and Morris, 1963a).
1.2.2. Pseudo first order model (PFO)
The pseudo first order kinetic model, one of the utmost common and empirical models for adsorption kinetics that has been suggested by Lagergren (Lagergren, 1898) for the solid/liquid adsorption systems and its differential form of this model is expressed as follows:
(8)
Actually the rate of mass transfer of this model is denoted by a rate expression of simple linear driving force (Yang, 1987).
Integrating Eq. (8) and by applying the boundary conditions F (t = 0) = 0, it becomes (Ho and McKay, 1998a; Ho, 2004):
(9)
or (10)
As stated by the literature (Azizian, 2004; Liu and Shen, 2008), pseudo first order model may exhibit characteristics of Langmuir rate equation at the preliminary adsorption stage otherwise near to equilibrium (Azizian, 2004). In this situation the variation in the concentration of adsorbate in solution or alteration in the relative surface coverage throughout the process of adsorption is quite unimportant (Azizian, 2004; Marczewski, 2010a). Generally in most considered adsorption systems, the pseudo first order equation does not fit well over the entire adsorption period.
1.2.3. Pseudo second order model (PSO)
Pseudo second order model is another simple and renowned kinetic model, is employed widely in current time. This model has been offered empirically by Ho and Mckay (Ho and McKay, 1998b; Ho and McKay, 1999) and theoretically by Azizian (Azizian, 2004), has presented a well–fitting to the experimental data. This model is based on the assumption that the rate of adsorption follows pseudo second order chemisorption mechanism. It can be expressed as follows:
(11)
Integrating Eq. (11) and by applying the boundary conditions F (t = 0) = 0, it becomes:
(12)
or where (13)
If the initial sorption rate, h = qt/t while t tends to 0, h (mg/g min) is expressed as follows (Ho and McKay, 2000):
(14)
The rate coefficient kPSO, initial adsorption rate h and predicted qe can be calculated. As stated by a theoretical study confirmed by Azizian (Azizian, 2004), pseudo second order rate coefficient (kPSO) is a complex function of initial adsorbate concentration. Previously this result has also been verified experimentally (Ho and McKay, 2000). In addition it has been revealed that this model may define the adsorption kinetics while the change in concentration of adsorbate is visible (Azizian, 2004).
In contrast to the pseudo first order, the pseudo second order model forecasts the adsorption behaviour over the whole adsorption time (Ho, 2006). Though there are many parameters influencing the adsorption capacity including the solution pH, the particle size and dose of adsorbent and the nature of the adsorbents, the initial adsorbate concentration and the adsorption temperature, an adsorption kinetic model is simply in relation to the influence of noticeable factors onto the overall rate.
1.2.4. Elovich model
Elovich's equation developed by Zeldowitsch (Zeldowitsch, 1934) has been extensively employed to define the adsorption of gases onto solid materials (Rudzinski and Panczyk, 2000; Heimberg et al., 2001). This semi–empirical equation is also utilized effectively for defining kinetic of second order supposing that the actual surfaces of solid are energetically heterogeneous; however the equation does not recommend any certain adsorbate–adsorbent mechanism (Aharoni and Ungarish, 1976; Sparks, 1989). The model defines the kinetics of the chemisorption process (Low, 1960; Chien and Clayton, 1980; Cheung et al., 2001). The Elovich equation has hardly been practised to liquid state adsorption. The Elovich equation is another rate equation based on the adsorption capacity which is written as follows (Low, 1960):
(15)
Its integrated form of Eq. (15) is:
(16)
However because of the complexity of the original Elovich equation, Chien and Clayton (Chien and Clayton,1980) has tried for making it simpler by supposing that aEbEt > 1 and by applying the boundary conditions of qt = 0 at t = 0 and qt = qt at t = t (Ho, 2006), formerly its integrated form of Eq. (16) is:
(17)
with (18)
aE signifies the chemisorption rate at zero coverage, bE represents the desorption constant corresponding to the extent of surface coverage and activation energy for chemisorption (g/mg) (Ho and McKay, 1998a; Zhang and Stanforth, 2005; Gupta and Babu, 2009).
The Elovich kinetic equation defines the fractionary kinetic orders and some kinetic parameters replicating probable variations in the rate of adsorption as function of the time of adsorption and initial concentration. The value of 1/bE replicates the number of sites accessible for adsorption process while the value of 1/bE ln (aEbE) specifies the measure of adsorption while ln t = 0 (Ahmad et al., 2014).
1.2.5. Avrami kinetic model
The Avrami equation has been employed for confirming exact variations of kinetic parameters as functions of the time and temperature of adsorption. It is furthermore revision of kinetic thermal decomposition modelling (Avrami, 1940). The Avrami kinetic model equation expresses the Avrami exponential which is a fractionary number associated with the probable variations of the mechanism of adsorption that occur throughout the process of adsorption. Thus the adsorption mechanism could follow multiple kinetic orders that are altered throughout the contact of the adsorbate with the adsorbent. The Avrami kinetic model is expressed as follows:
(19)
Its integrated form of Eq. (19) is:
(20)
Value of nAV can be utilized to prove probable interactions of the mechanisms of adsorption in relative to the contact time and the temperature.
1.2.6. Modified second order model (MSO)
This rate equation has been well explained by Ritchie (Ritchie, 1997) while a number of surface sites, nR, are employed by each adsorbate. It is given as follows:
(21)
So the Ritchie equation can be acquired by integrating Eq. (21) and taking nR = 1
(22)
(23)
where θ = qt/qe and β1R = (1– θ0). The adsorbent surface coverage has been typically presumed to be zero (θ0 = 0).
The Ritchie equation becomes for nR = 2, 3, . . . .k
(24)
(25)
where θ = qt/qe, and . If t0 = 0 and βnR ≠ 0, the nth order equation Eq. (25) can be simplified as follows:
(26)
So the modified second order equation can be acquired for nR = 2 as follows (Cheung et al., 2001):
(27)
1.2.7. Ritchie second order model
When the preadsorbed step has not occurred (i.e. θ0 = 0, then β2R = 1), the modified second order equation (Eq. (27)) has been converted to the Ritchie second order equation (Cheung et al., 2001) as follows:
(28)
where k′′2R = k′2R
1.2.8. Exponential kinetic model (EXP)
The shape of rate of adsorption with time proposes an exponential form of kinetic equation which may define this drift for some adsorption systems. The rate expression of exponential kinetic model has been recommended as follows (Haerifar and Azizian, 2012a; Haerifar and Azizian, 2013; Haerifar and Azizian, 2014):
(29)
or (30)
where kEXP = k′EXPqe. Its integrated form of Eq. (8) at boundary conditions (qt = 0 at t = 0 and qt = qt at t = t) is:
(31)
or (32)
where kEXP is constant but qe (or initial concentrations) are different.
1.2.9. Mixed 1,2 order model (MOE)
The actual difficulty with rate equations of first and second order is that they are usually utilized since mostly they own simple analytic forms and nevertheless they appropriately signify the boundary conditions of usual kinetic studies. Other more common equations are typically hard to analyze and apply, as they do not suggest simple analytic formulas otherwise it is complex as well as problematic for solving, if analytic solution are present (such as solution of the Langmuir rate equation) (Azizian, 2004; Liu and Shen, 2008).
Marczewski has exhibited that the rate equation may be assessed around the equilibrium point by a Taylor series and so he has presented a combined form of pseudo first and second order equations i.e. called mixed 1,2 order rate equation as follows (Marczewski, 2010b):
(33)
where and and f2 (f2 < 1) is defining the part of second order term in the rate equation.
The integrated form of Eq. (33) is as follows (Marczewski, 2007; Marczewski, 2010b):
(34)
or (35)
The MOE model is the combined form of pseudo first and pseudo second order equations. In recent times the Langmuir kinetic equation has been solved analytically by Marczewski and he has derived a modest equation for homogeneous surfaces (Marczewski, 2010a). The acquired equation is analogous to Eq. (35) from mathematical point of view nonetheless f2 is substituted by feq, where feq is θe(1 – Ce/C0), where θe is equilibrium surface coverage. So if the acquired f2 = feq, at that time the kinetic model of system is Langmurian in addition the surface is homogeneous; however if f2 ≠ feq then the kinetics is MOE as well as the surface is heterogeneous. Conversely MOE may be preserved as purely empirical equation for energetically heterogeneous surfaces otherwise as Langmuir equation for energetically homogeneous surfaces (Marczewski, 2010a).
1.2.10. Fractal−like mixed 1,2 order model (FMOE)
The Langmuir kinetic equation (Langmuir, 1918) is one of the renowned kinetic models and various techniques has been offered for obtaining its rate constants of both adsorption and desorption (Azizian et al., 2007; Azizian et al., 2009). In the Langmuir equation, feq = ueqθe, where ueq = (1 − Ceq/C0) and θe = qe/qm. Nevertheless f2 in Eq. (36) is the contribution of second order kinetic model. Newly Marczewski (Marczewski, 2010a) has exhibited that the Langmurian rate equation can be reorganised to an equation integrated kinetic Langmuir equation (IKL) which is mathematically same as Eq. (37). Nevertheless the key dissimilarity between them is substitution of feq by f2.
Haerifar and Azizian (Haerifar and Azizian, 2012a; Haerifar and Azizian, 2012b) have extended the IKL equation by bearing in mind the fractal−like adsorption kinetics in the kinetic Langmuir model (Langmuir, 1918).
(36)
where . dθ/dt, is the overall adsorption rate. By description of equilibrium constant, KL = ka,obs/kd,obs.
Haerifar and Azizian (Haerifar and Azizian, 2012a; Haerifar and Azizian, 2012b) have introduced the next equation for the acquired adsorption rate coefficient as follows:
(37)
where (0 ≤ h ≤ 1) (t ≥ 1).
After rearrangement and substitution of Eq. (36) by Eq. (37), it becomes:
(38)
by rearrangement:
(39)
where α = 1–h, 0 ≤ h <1 and k′a,0 = ka,0/α.
At this moment as stated by the derivation method of IKL equation from Langmuir kinetic model (Marczewski, 2010a), Eq. (38) can be rearranged as follows:
(40)
or
(41)
or (42)
and (43)
Its integrated form of Eq. (40) is as follows:
(44)
or (45)
Then Eq. (44) is called fractal−like IKL equation.
IKL and mixed 1,2 order model have the same mathematical forms. Thus the fractal−like form of mixed 1,2 order equation is also identical to Eq. (48) and (49), nevertheless feq should be substituted by f2 for fractal−like mixed 1,2 order model. Therefore the fractal−like mixed 1,2 order model is expressed as follows:
(46)
1.2.11. Fractal−like pseudo first order model (FPFO)
Moreover as said by Azizian’s theoretical analysis of the Langmuir rate equation (Azizian, 2004) and derivation of mixed 1,2 order model by Marczewski (Marczewski, 2010b), the pseudo first order model can be derived from Langmuir kinetic model (Azizian, 2004) or MOE (Marczewski, 2010b) under the distinct considerations. Thus its fractal−like forms can also be derived from fractal−like Langmuir kinetic model or fractal−like MOE as follows (Haerifar and Azizian, 2012a; Haerifar and Azizian, 2012b):
(47)
where (0 ≤ h < 1) (t ≥ 1) (48) Its integrated form of Eq. (47) is as follows:
(49)