Kinetics of adsorption

Before reaching the equilibrium state, several steps affect the adsorption process [64,65]. First of all, the component to be adsorbed has to move from the bulk phase to the vicinity of the adsorbent surface. Secondly, the properties of the solution close to the adsorbing particles differ from the bulk phase (diffusion and Stern layer, Figure 1) and the adsorbates have to travel through this region to reach the surface. This is called a film diffusion or boundary diffusion.

Furthermore, most of the adsorbents have porous structure and therefore, diffusion of the

adsorbates in macro/meso/micropores can have a major impact on the rate of the adsorption.

Finally, after reaching the active site, adsorbate is bound to it via a chemical reaction or physical interactions. This step can be generally called as a surface reaction.

As was depicted above, adsorption is not a simple one step process. Usually, the effect of travel in the solution is eliminated by vigorous mixing, but the contribution of all the other stages should be investigated. Therefore, different kinetic models have been developed to predict the rate determining step of the adsorption. Similarly, as in the case of adsorption isotherms (Langmuir, Freundlich), adsorption kinetics has most often been fitted by the two simplest models: pseudo-first- and pseudo-second-order equations. These models were originally proposed empirically and have only recently been examined on their theoretical basis [66].

Besides of the pseudo kinetic models, which assume that the adsorption is governed by the surface reaction, a variety of diffusion based models have been derived [64]. Especially in the case of porous adsorbents, diffusion can greatly affect the adsorption rate, which is discussed in the end of this section.

Pseudo-first-order model

Legergren proposed the simplest empirical kinetic model at the end of 19^th century [67]. The model was also called the pseudo-first-order (PS1) model due to the fact that it was associated with the kinetics of one-site adsorption governed by the rate of the surface reaction. The equation is given as:

) ( _e

1 q q

dt k

dq = − (39)

By integrating above equation at the boundary conditions q=0 at t=0 and q=qt at t=t, one will obtain a linear form of the PS1 model:

t k q q

q_{e t}) ln _{e 1}

ln( − = − (40)

where q_t and q_e are the adsorption capacity (mmol/g) at time t and at equilibrium respectively, while k1 represents the PS1 rate constant (1/min). For fitting of experimental data, however, it is

better to use the non-linear equation rearranged from eq. (40). This is because for the linear fitting the left hand side q_e (term ln(q_e-q_t)) has to be taken from the experimental data when there are actually two values for q_e in Eq. (40).

Earlier the PS1 model was derived for special cases such as ion-exchange by natural zeolites [68,69] and biosorption by aerobic granular sludge [70]. General analytical derivation was presented by Azizian [71] for high solution concentrations. Based on his derivation, k₁ was a combination of adsorption and desorption rate constants and a linear function of the initial concentration of adsorbate. Liu and Liu [15], however, denoted that Azizian’s assumptions about the first order adsorption and desorption reactions with respect to the available and occupied adsorption sites, were not justified. They also stated that this derivation was only valid for pure solutions, which is rarely the case in the real systems. Moreover, the dependence of k₁ on the initial concentration of adsorbate has not always been observed even if PS1 model has been well fitted [64].

Finally, Rudzinski and Plazinski [72] derived a general kinetic equation based on the Statistical Rate Theory and showed that the PS1 model was a special case of this new equation.

Similarly to Azizian’s approach, this was achieved by assuming high solution concentration, which essentially remained constant during the kinetic experiment. The PS1 rate constant was determined as a function of initial concentration of the adsorbate and the energy of the adsorption.

It should be noted that even if most of the derivations of PS1 model were achieved from the assumption that the surface reaction governs the adsorption process, mathematically equivalent equations have also been obtained for film or pore diffusion [68,73]. Based on this, Plazinski et al. [65] suggested that the PS1 model is a general formula describing the experimental data measured not too far from the equilibrium. Authors also proved this point by investigating the properties of mechanism independent general rate equation.

Pseudo-second-order model

The PS1 model was generalized to two-site-occupancy adsorption to form a pseudo-second-order (PS2) equation:

(

)

2 q q

dt k

dq = − (41)

where k2 is the PS2 rate constant (g/mmol min) and k2qe2

represents the initial sorption rate. An integration of Eq. (41) at the boundary conditions, q = 0 at t=0 and q=qt at t=t, gives a linear form:

e 2 e t 2

1 q

t q q k

t = + (42)

Azizian [71] derived the PS2 model using classical theory of adsorption/desorption by assuming a low initial concentration of adsorbate. Based on this derivation, k₂ was a complicated function of the initial concentration of adsorbate, its equilibrium surface capacity as well as the rates of the adsorption and desorption. Later on Azizian’s assumptions were debated by Liu and Liu [15], who stated that his derivation was applicable in the initial stage of the adsorption (small value of t) and did not necessarily require the assumption of a low initial concentration.

The direct mathematical derivation of the PS2 model could not be done using the Statistical Rate Theory. Convergence of the general model derived by Rudzinski and Plazinski [72] and the PS2 model, however, was nearly perfect in certain conditions indicating that the PS2 model was a special case of this general equation.

The PS2 model has also been assigned to a special case of the more general rate law [74]

or the Langmuir kinetic model [75]. In addition, it has been noted that this model is able to estimate experimental qe values quite well and is not very sensitive for the influence of the random errors [65]. The latter is also one of the reasons the second-order model is usually better fitted on the experimental data than the first-order model.

Elovich model

The Elovich model was proposed by Roginsky and Zeldovich in the early 1930’s:

(

)

q B _{E E}

E

t= 1 ln1+ (43)

where AE (mmol/min g) and BE (g/mmol) are the Elovich constants. These constants were related to the rate of the chemisorption and surface coverage, respectively [76,77].

Interpretations of the Elovich equation are usually connected to the heterogeneous surfaces. This assumption was also used by Rudzinski and Plazinski when they derived an Elovich-like equation using the Statistical Rate Theory [72,73,78]. One interesting observations the authors made [78] was that the PS2 model and Elovich-type model were essentially identical when the surface coverage of the adsorbate was lower than 0.7 times the equilibrium coverage.

Intraparticle diffusion model

Weber and Morris [64] proposed that a plot of the adsorption capacity vs. the square root of contact time should give a straight line if the pore-diffusion was the rate limiting step of the adsorption process. The most commonly applied pore-diffusion model is the intraparticle diffusion model:

( )

t k

q= _dif ^1/2 + (44)

where k_dif (mmol/gmin^1/2) is the rate constant of intraparticle diffusion and C (mmol/g) represents the thickness of the boundary layer.

In many studies, the plot of the intraparticle diffusion model has shown multi-linearity, which has been assigned the following diffusion steps during the adsorption process [64,79,80]:

(i) an external surface adsorption or film diffusion, (ii) a gradual adsorption, where intraparticle diffusion is controlled, and (iii) a slow diffusion of the adsorbates from larger pores to micropores. Different linear portions can also be assigned to diffusion into macropores, then mesopores and finally into micropores [80].

Ho et al. [64] presented some more criteria to confirm the diffusion mechanism. The constant k_dif should vary linearly with reciprocal particle diameter and the product of k_dif and the adsorbent mass should vary linearly with the adsorbent mass. Moreover, the rate of the reaction governed by surface reaction depends more on the temperature than the diffusion governed process.

Other kinetic models

Plenty of different kinetic models have been derived for adsorption processes. Besides those presented above, these include for example: Langmuir kinetics [65], Freundlich kinetics [81], the modified PS1 model [82], different forms of the PS2 model [83], the Bangham diffusion model [20,84], the particle diffusion model [64,68], and the Ritchie equation [83,85]. Previous models, however, are rarely applied and therefore not discussed further.

Linear or non-linear fittings

Kinetic modeling is usually done by the linear fitting of PS1 and PS2 models (Eq. 40 and 42).

After comparing the linear and non-linear methods several authors concluded that the non-linear method is a better way to estimate the model parameters [86-88]. When comparing the linear forms of pseudo-equations Plazinski et al. [65] noted that the linear PS2 model tends to “smooth”

the experimental data due to the contribution of inherent errors in t/qt not dependent on the system’s closeness to equilibrium. In the case of the PS1 model, however, errors in ln(qe-qt) are inversely proportional to the (q_e-q_t) when most of the scattering in experimental data (ln(q_e-q_t) vs. t) ought to occur close to equilibrium. In addition, to conduct linear fitting, q_e in this term is taken from the experimental data and can differ from the estimated value (right hand side qe in Eq. 40).