Adsorption kinetics data explain the rate of the adsorption of metal ions on the solid phase. For a batch contact process, the rate of sorption of strontium onto the CF bead surface is proportional to the amount of strontium uptake from the solution phase. A pseudo-first-order equation can be employed to describe reactions, whereas the pseudo-second-order equation affords the sorption capacity on the solid phase. Therefore, a pseudo-first-order or a pseudo-second-order model equation can be used to explain the kinetics of strontium sorption onto the CF beads. Linear forms of the first-order and second-order models, based on the adsorption equilibrium capacity, are as follows:
where
is the amount of strontium uptake on CF bead at equilibrium and
is the uptake at any time,
. The pseudo-first-order rate constant
(min
−1) is obtained by plotting
vs.
whereas the pseudo-second-order rate constant
(g·mg
−1·min) can be obtained by plotting
versus
. The slope of the straight line of first-order and second-order kinetic plots provides the value of
and
, respectively. The kinetic parameters obtained from Equations 10 and 11 are given in
Table 6. It may be noted from
Table 6 that at solution pH 6.5, the value of the first-order rate constant,
, remained almost constant, whereas the value of second-order rate constant,
, increased with the increase of solution concentration. It is also observed that the second-order model has higher correlation coefficient values for all the solution concentrations that were used in these studies compared with the data obtained by the first-order model. In addition, the second-order rate constant model provides a close match between the theoretical and experimental strontium uptake (
) by the CF beads. This suggests that strontium adsorption onto the CF beads follows the second-order model and that the overall rate of strontium adsorption on the CF bead may be controlled by a chemical process [
41].
Elovich Equation
The Elovich equation was fitted to experimental data in order to investigate Sr
2+ uptake on the CF bead. This equation was first proposed by Roginsky and Zeldovich; but it is commonly known as the Elovich equation and is extensively applied to chemisorption data [
42]. The general expression of the Elovich equation is as follows:
Integrating Equation 12 for the boundary conditions
to
and
to
, gives:
where
is the sorption capacity at any time,
(mg/g),
is the initial sorption rate (mg/g·min) and
is related to the extent of surface coverage and the activation energy for chemisorption (g/mg). The constants can be obtained from the slope and intercepts of the plot of
vs.
.
Table 6 shows the kinetic constants along with higher correlation coefficient (R
2 > 96) value suggesting the applicability of Elovich equation.
Sorption kinetics of strontium on the CF beads using 220 mg/L (2.5 mM/L) strontium solution was evaluated at pH ~6.5 at different temperatures (293–308 K). The exposure time of 24 h was used during batch studies to ensure that equilibrium was attained. The data were fitted to the pseudo-first-order, pseudo-second-order, and Elovich equation. It was observed that the kinetic constants,
and
for both the first- and second-order processes increased with the increase in solution temperature (
Table 5). With a higher correlation coefficient, the experimental data showed better agreement for the pseudo-second–order, rather than the first–order, model for the entire adsorption time at different temperatures. These observations also suggest that the strontium removal kinetics followed the pseudo-second-order model involving specific interaction between strontium and the CF bead surface [
43,
44].
In addition, the Weber–Morris model was also used to investigate the rate constant of the intra-particle transport of Sr
2+ into the CF bead [
45].
In Equation (14),
is the amount of strontium uptake (mg/g) at any time,
,
is the rate constants of intra-particle transport (mg/g·min
1/2) and
(mg/g) is a constant related to the thickness of the boundary layer diffusion. If the intra-particle diffusion is the rate controlling step in the adsorption process, then the linear plot of
vs. amount of strontium uptake, according to Weber–Morris model, should pass through the origin.
Figure 9 shows a multilinear plot for the entire time range; therefore, it was assumed that there are various issues that influence the adsorption process. Based on the Weber–Morris plot, it is assumed that, in the first stage, the immobilization of strontium onto the CF bead surface occurs over time. As the surface sites become exhausted, the rate of intraparticle diffusion may control the adsorption process in the second stage. It is suggested that the rate of adsorption becomes relatively slow in the final removal stage due to the intra-particle diffusion of metal ions from the exterior to the interior sites of the adsorbent [
43]. A similar observation was made by Min et al., 2016, for the removal of trace arsenate from water using functionalized chitosan nanofiber [
46]. They reported that the first steep slope corresponds to the rate controlling stage as pore diffusion takes hold and the second moderate sorption stage is intra-particle diffusion control, influenced by the pore structure; finally, the slow sorption stage is related to strereo-hindrance effect derived from the adsorbed species. It is important to note that the plots do not pass through the origin as a slight deviation from the origin was observed at all temperatures (
Figure 9). The intra-particle rate constant,
, was obtained from the slope of the straight-line portion of the plot of
versus
for various solution temperatures. It was observed that the straight line did not pass through the origin and the
value decreased with the increase of solution temperatures (
Table 5). This indicates that pore diffusion is not the sole rate controlling step; other mechanisms may control the rate of adsorption [
47,
48]. It is observed that the correlation coefficient,
and
, for the Weber–Morris and Elovich equation models is 98, whereas the correlation coefficient value for the pseudo-second-order equation is
> 99 (
Table 5). Therefore, it is hypothesized that either chemisorption or the pore diffusion process may dominate at any stage of the adsorption process [
49].
The strontium sorption kinetics was further analyzed for the CF bead with radius,
, at any time,
, that can be expressed by the following the equation [
50]:
where
is the fractional attainment of the strontium ion. The ability of Equation (15) to describe the sorption kinetics is limited due to the complex pore size, pore size distribution, particle size, and physicochemical changes such as swelling that may occur in the CF beads [
51]. To investigate the function of diffusion in controlling the rate factor for the sorption of strontium onto the CF bead from aqueous solution, Equation (15) can be approximated to:
where
is the fraction of strontium uptake from the solution by the CF bead at any time (
). Therefore, in terms of the fraction of strontium remaining in the CF bead, Equation 16 can be rewritten as follows:
Taking
as a constant
so that it includes any anomalies arising out of factors such as swelling, and replacing
by a constant
, we can rewrite Equation (17) for a particle of radius
as:
where
= 0.5 represents Fickian diffusion and 1 >
> 0.5 represents anomalous diffusion.
The diffusion co-efficient was estimated from the slope (
Figure 10) and mean bead radius of 0.11 cm. For each temperature, the data were treated for n values 0.5 and 0.65 and it was observed that strontium uptake from the solution onto the CF bead showed a straight line when the n value was 0.65 (
Figure 10). The function yield values range from 1.56 × 10
−7 to 1.94 × 10
−7 cm
2/s for the particle diffusion coefficient for the temperature range of 293 K to 308 K.