Kinetic modeling of the adsorption process of Pd(II) complex ions onto activated carbon
Abstract
In this work, the results of kinetic studies of Pd(II) chloride complex ion adsorption process on activated carbon are presented. The experiments were conducted for different temperatures, initial concentrations of Pd(II) complex ions as well as for different amount of activated carbon. These results confirmed that the mechanism of the adsorption process is complex and can be described by two step reaction model. A new form of the adsorption isotherm based on kinetic mechanism is suggested and can describe the observed process. Moreover, it was shown that for specific conditions, the proposed isotherm can be transformed into the Freundlich’s isotherm. The activation energies of the subsequent stages of the studied process were determined and are equal to E_{1} = − 9 ± 9, E_{2} = − 6 ± 1.4 and E_{3} = − 85 ± 8.9 J mol^{−1}, respectively. It is also suggested that the observed positive Gibbs energy change during adsorption can be related only to the first step of this process which does not lead to the final product.
Keywords
Adsorption Palladium(II) chloride ions Recovery Adsorption isotherm Equilibrium Kinetic studiesIntroduction
At present, activated carbon is often applied to the heavy metals ions removal form the aqueous solutions. There are two main application areas. The first one, related to recycling of metals [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and the second one, related to the environmental protection [12, 13, 14, 15, 16, 17]. Our work refers to the first area mentioned above.
The process of Pd(II) chloride complex ions adsorption on activated carbon was described in our previous paper [18]. In the case of acidic solutions containing chloride ions, palladium forms chloride complexes. There are several reports which indicate that during the adsorption process on activated carbon, palladium(II) is reduced to the metallic form [5, 6, 9, 11].
Simonov et al. [11] have investigated the process of palladium(II) chloride complex ions sorption on graphitelike carbon materials. They have shown that in such a system, two processes occur simultaneously. The first one is related to the formation of πcomplexes of PdCl_{2} with fragments of the carbon matrix, and the second one is related to the reduction reaction of Pd(II) to Pd(0).
Consequently, we studied the conditions under which such a recovery process is possible. In our previous work [18], we investigated and described the conditions of Pd(II) ion recovery through adsorption on activated carbon used as the sorbent.
More than a century ago, Herbert Freundlich published the paper [19] in which the isotherm equation describing adsorption process was presented. Since then, the Freundlich isotherm has been one of the most often used equations to describe this process.
We found that Freundlich adsorption isotherm describes the adsorption of Pd(II) complex ions much better then Langmuir equation.
The determined parameters of Freundlich equation are equal to β = 0.035 and 1/p = 0.49, at 294 K, and β = 0.062 and 1/p = 0.53 at 323 K. However, these investigations are time consuming. For example, experiments with desorption at 295 K, took about 10 months.
Moreover, using this description of the experimental data, it has been difficult to clarify the thermodynamics of the adsorption process. The analysis of the adsorption product suggested that PdCl_{4}^{2−} adsorbed on the carbon surfaces undergoes irreversible transformation into solid product. However, in the model of the adsorption process suggested by us, thermodynamic considerations were limited only to the first step of the suggested mechanism of adsorption. Thus, there is a question if more information about the process can be drawn from the kinetic data. If this approach may lead to an adsorption isotherm, it would be much faster. Consequently, in this paper, we demonstrate that adsorption isotherm can be derived from the kinetics considerations.
This newly proposed approach gives a direct relationship between the initial concentration of absorbed substance, the amount of the used absorber and equilibrium constant of the adsorption process.
Materials and methods
In all our experiments, commercially available activated carbon, Norit GF40 (AC) in nonmodified form was used. Palladium(II) chloride complex was obtained according to the methodology described in our previse paper [18].
The measurements of the rate of Pd(II) chloride complex ions adsorption onto activated carbon were carried out in the cyclic glass reactor kept in the thermostat at constant temperature (± 0.2 °C). After the constant temperature in the system was reached, suitable amount of activated carbon was introduced into the aqueous solution containing fixed concentration of Pd(II) chloride complex ions. The total volume of the solution was equal to 300 mL. The sample of 3 mL of the solution was taken periodically and analyzed spectrophotometrically (Shimadzu, model PC 2501, Japan) to detect changes of Pd(II) chloride ions concentration. The absorbance level was monitored and read out at the wavelength 279 nm. Next, the absorbance level was used to calculate the concentration of Pd(II), assuming molar absorption coefficient equal to 5980 dm^{3} cm^{−1} mol^{−1} [20]. After each UV–Vis analysis, the sample was returned to the cyclic reactor to maintain constant volume of the reagents. The solution was mixed by the glass stirrer dipped into the reactor at ca. 2 cm distance from its bottom. The rotor speed was adjusted and controlled using CAT R50D stirrer.
Conditions applied to kinetic measurements of the adsorption process
Initial concentration of reagents  T (K)  Mr (rpm)  

[PdCl_{4}^{2−}]_{0} (mmol dm^{−3})  [C]_{0} (g/L)  
0.383  1.67  323  600 
0.229  
0.036  
~ 0.036  0.33  323  1200 
1.67  
3.33  
6.67  
~ 0.036  1.67  313  1200 
323  
348  
~ 0.036  1.67  323  600 
900  
1200 
After the absorbance measurement, the concentration of Pd(II) complex ions at each instant of time was calculated using earlier determined PdCl_{4}^{2−} absorption coefficient [20].
Results
Having concentration vs time data points registered experimentally, the kinetic curve was determined. Then, the proposed integral form resulting from the assumed model was fitted to the obtained kinetic curve, and respective parameters related to the process were derived.
Taking into account the results obtained earlier [18] it was assumed that the studied process consists of two steps. This assumption is supported by the fact, that calculated Gibbs free energy for assumed single step process is positive. This clearly suggests that the studied process has to consist of at least two steps. Moreover, structural investigation of final product indicates substrates transformation into solid product. The first one is related to the adsorption of Pd(II) chloride complex ions on the surface of the activated carbon. The second step is related to the reaction on the surface.
It was also shown previously [18], that the formation of PdCl_{2} on the surface was observed, and this chemical compound is the final product of the reaction.
Here n[C]_{org} corresponds to the concentration of functional groups on the surface of applied activated carbon, and k_{x} is the rate constant (where x = 1, 2, 3), and m, n correspond to stoichiometry indexes. The speciation as well as concentration of functional groups at the surface of AC was analyzed in our previous paper [4]. It was shown that the total concentration of acidic forms of functional groups is equal to 17.96 mM/g, where the concentration of alkaline type functional groups are equal to 4.1 mM/g. The main fractions of the acidic groups are carbonylic (13.99 mM/g) and phenolic (2.63 mM/g) [4].
Parameters such as A, B, γ_{1}, γ_{2} were determined using the TableCurve software by fitting Eq. (7) to all experimental data shown in Fig. 2. Next, parameters corresponding to the observed rate coefficients k_{1,obs}, k_{2,obs} and k_{3,obs}, were calculated using Mathcad software by solving the system of Eqs. (8)–(11). The initial concentration \([PdCl_{4}^{2  } ]_{0}\) was determined spectrophotometrically, and this value was used during calculations. Having rate coefficients determined from this model, their dependence on various experimental parameters such as AC initial concentration, Pd(II) initial concentration and temperature was analyzed.
Effect of AC initial concentration on the reaction rate
Moreover, it can be seen that fitted equations pass exactly through the origin of the coordinates system. This in turn confirms, that the decrease of absorbance [concentration of Pd(II)] is directly related to the amount of activated carbon present in the solution. Furthermore, it can also be suggested, that the order of the reaction with respect to the concentration of AC = 1. It can also be seen that the k_{3,obs} is in fact independent of AC concentration, which seems to confirm, that the second step (Eq. 4) is related to the transformation step, which is very slow.
Influence of Pd(II) initial concentration on the reaction rate
It should be pointed out that at t = 0, initial concentration of \([PdCl_{4}^{2  } ]_{ads}\) and \([PdCl_{2} ]_{ads}^{{}}\) is equal to 0. Therefore, in Eq. (12), this concentration is not taken into account.
Such linearization methods are outdated now in chemical kinetics, and nonlinear least squares fitting to the untransformed original equation should be rather used [22]. Therefore, we determine these parameters by using those two approaches. In the case of fitting the equation in its untransformed form, the order of the reaction was found to be equal to 1.7 ± 0.22.
Therefore, the observed rate coefficients obtained from the solution of equations system (Eq. 8) were also analyzed as a function of initial Pd(II) concentration. The initial rate method yields rate coefficient only for one way irreversible reaction, while the analytical solution corresponds to the steady state which can be different for different initial Pd(II) concentration. Comparing k_{1obs} one can find that taking into account the error bar, the rate coefficients obtained from both methods are of the same order [k_{1,obs} = 0.01299, obtained from initial rate, and k_{1,obs} = 0.01282 from the solution of Eq. (8)].
It can also observed that the increase of Pd(II) initial concentration results in a decrease of Pd(II) observed adsorption rate constant k_{1,obs}, as well as it increases the observed rate constant k_{2,obs}. In case of k_{3,obs} the effect of initial concentration of Pd(II) can be neglected. This is an explanation of fractional value of determined order of the reaction.
Influence of the temperature on the reaction rate
Finally, the influence of the temperature on the rate constants was also investigated. Using Arrhenius dependence, the activation parameters were determined.
Here R is the gas constant, T is the temperature in kelvin, k_{x,obs} is the observed rate constant of xth reaction (x = 1, 2, 3), E_{x,a} is the activation energy of xth reaction, A is the Arrhenius constant.
Influence of temperature on the observed rate constants
T (K)  k_{1,obs} (min^{−1})  k_{2,obs} (min^{−1})  k_{3,obs} (min^{−1}) 

313  0.01245  0.01332  0.00332 
323  0.01243  0.01343  0.00381 
348  0.01282  0.01355  0.00418 
E_{a} (J/mol)  − 9 ± 9  − 6 ± 1.4  − 85 ± 8.7 
A (min^{−1})  1 × 10^{−2} ± 2 × 10^{−3}  1 × 10^{−2} ± 4 × 10^{−4}  4 × 10^{−4} ± 1 × 10^{−4} 
Effect of mixing rate
As it can be seen, there is no visible influence of the mixing rate on the observed rate constants. This suggest, that the process rate is not controlled by the mass transfer phenomenon. Therefore, the influence of this parameter can be neglected in further considerations.
Adsorption isotherm
Since the slowest process in our reaction scheme 2 is described by step 2, i.e., the formation of PdCl_{2}, one can assume that the adsorption stage (i.e., step 1) is at equilibrium, and the ratio of the observed rate constant may in fact correspond to the equilibrium constant.
Considering the above relation, one can further assume that adsorption isotherm should also contain the equilibrium constant. A similar approach was used by Langumir. However, his isotherm was not related to the process in the solvent. Generally speaking, in most cases, the derived isotherms describe only a relation between concentration of absorbent and adsorbed substances, but they do not contain the information about the equilibrium constant.
Then, the relation (18) can be further rearranged in the following way:
Her:\(\left[ {PdCl_{4}^{2  } } \right]_{\text{soln}}\) is the equilibrium concentration of Pd(II) ions in the solution, \(\left[ {PdCl_{4}^{2  } } \right]_{0}\) is the initial concentration of Pd(II).
Taking into account that the concentration of acidic functional groups is about \([C]_{org,0}\) = 18 mM, and the initial concentration of Pd(II) as well as the amount of formed product \(\left[ {PdCl_{{4_{{\frac{m}{n},ads}} }} \cdot C_{org} } \right]\) on the surface of AC is of the order 5 × 10^{−4} M, the changes of [C] _{org,0} concentration can be neglected.
The parameters m and n can’t be determined directly, however m/n ratio can be determined from the slope of fitted line, while from the intercept of fitted line the equilibrium constant can also be calculated.
From the slope of the plot \(\log \left( {\frac{{[PdCl_{4}^{2  } ]_{0}  [PdCl_{4}^{2  } ]_{\text{soln}} }}{{[C]^{{}}_{org,0} }}} \right)\) versus \(\log \left( {[PdCl_{4}^{2  } ]_{sol} } \right)\), the \(K = \frac{{k_{1} }}{{k_{2} }}\) value can be determined. The value of \(K = \frac{{k_{1} }}{{k_{2} }}\) at the temperature 323 K is equal to 1.04 ± 0.23 and for temperature 298 K is equal to 0.96 ± 0.4. It is obvious that this ratio \(\frac{{k_{1} }}{{k_{2} }}\) is practically constant. These k ratios can be also derived from kinetic rate constant obtained by solving the system of Eqs. (8)–(11).
Calculation of equilibrium constant
T (K)  k_{1,obs} (min^{−1})  k_{2,obs} (min^{−1})  \(K = \frac{{k_{1,obs} }}{{k_{2,obs} }}\) 

From kinetic measurements  
323  0.01243  0.01343  0.92554 
Derived from new adsorption isotherm  
323  –  –  1.04 ± 0.23 
In the studied case, m/n ratio is equal to 0.52 ± 0.014 and 0.48 ± 0.027 for T = 323 K and T = 298 K.
Discussion
Comparison of two different method of k_{1,obs} determination
[C]_{0} (g/L)  T (K)  \(\left[ {PdCl_{4}^{2  } } \right]_{0}^{{}}\) (M)  V_{0}  k_{1,obs} calculated form V_{0}  k_{1,obs} from the fit of Eqs. (8)–(11) to the experimental data 

1.67  348  3.55 × 10^{−5}  3.34 × 10^{−7}  9.41 × 10^{−3}  7.96 × 10^{−3} 
1.67  348  2.29 × 10^{−4}  2.97 × 10^{−6}  1.30 × 10^{−2}  1.23 × 10^{−2} 
1.67  348  3.3 × 10^{−4}  5.84 × 10^{−6}  1.74 × 10^{−2}  1.25 × 10^{−2} 
6.67  323  3.99 × 10^{−5}  1.29 × 10^{−6}  3.24 × 10^{−2}  5.21 × 10^{−2} 
1  323  3.61 × 10^{−5}  6.78 × 10^{−7}  1.88 × 10^{−2}  2.55 × 10^{−2} 
1.67  323  3.87 × 10^{−5}  7.39 × 10^{−7}  1.91 × 10^{−2}  1.54 × 10^{−2} 
0.33  323  3.53 × 10^{−5}  9.86 × 10^{−8}  2.80 × 10^{−3}  2.44 × 10^{−3} 
It can be seen that there is an insignificant difference between the observed rate coefficients determined from initial rate method and calculated using the proposed model. It should be pointed out, that expected error is higher in case of initial rate method since, this method works well when Δt approaches 0, which is not fulfilled in our case. In the studied case Δt ~ 2 min.
The proposed isotherm contains the equilibrium constant which is defined by the ratio of k_{xobs}. Thanks to that it can be determined from kinetic measurements. The obtained form of Eq. (20) can be easily reduced to Freundlich isotherm, if K ~ β, and \(\frac{m}{n}\sim \frac{1}{n}\). The ratio \(\frac{m}{n}\) as defined by us, corresponds to the number of Pd(II) ions reacted with n number of functional groups (active sites). It is clear (Fig. 7) that the \(\frac{m}{n}\) ratio is not equal to 1 but is ca. 0.5. It may mean that two functional groups are required to immobilize one Pd(II) complex ion.
It can be seen that Gibbs free energy of formation of the final product may have a negative value. This, in turn, confirms that the overall process may be spontaneous and formation of a stable final product is compatible with thermodynamic requirements.
Studies shown by Kalmar et al. [27, 28] were related to the kinetic studies of methylene blue adsorption on quartz substrate. They have shown that kinetic approaches might to be good way to shed light to the equilibrium state.
Conclusions
It was demonstrated that using kinetic studies adsorption isotherm can be derived from the obtained results. The derived Eq. (21) is in fact a new isotherm equation. What is the most important, this equation can be easily transformed to Freundlich isotherm equation. In fact, we have found a kinetic scheme explaining the form of Freundlich equation. This approach seems to be attractive because kinetic studies are faster than typical equilibrium adsorption experiments.

The determined value of the equilibrium constant using proposed isotherm is identical with the value of equilibrium constant calculated from kinetic data i.e., rate constants.

m and n parameters can be determined only by using a numerical method or by assuming the detailed mechanism of the studied process.

Using the assumed kinetic model for two step adsorption reaction it is possible now to explain its positive contribution to \(\Delta G^{0,*}\) obtained in our previous paper [18].

The obtained data and proposed isotherm model may find application in designing new technology of precious metals recovery, water purification etc.

The proposed isotherm model can be extended into complex system in which several parallel reactions take place.
Notes
Acknowledgements
Authors thank Mrs. Ewa Zalecka (Brenntag Polska) for kind supply of activated carbon samples. This work was supported by the National Science Center of Poland under Grand Number 2016/23/D/ST8/00668 Sontata 12.
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