4.5 Modeling adsorption isotherms
4.5.1 One-component systems
In Papers I-V the studied isotherm models were Langmuir, Freundlich, Sips, Redlich-Peterson, and BiLangmuir. To further compare different models we also tested other isotherm equations
presented in section 1.1.5. Furthermore, to study the effect of linearization, four different linear forms of Langmuir model (Table 3) were applied. Modeling results for Co(II) adsorption by EDTA- and DTPA-silica gels are shown in Figures 14 and 15 and Tables 13 and 14.
0 1 2 3 4 5
0.0 0.1 0.2 0.3 0.4
qe (mmol/g)
Experimental Langmuir Freundlich Sips
Redlich-Peterson Toth
qe (mmol/g)
Ce (mmol/L)
Ce (mmol/L) qe (mmol/g)
Ce (mmol/L) (a)
0 1 2 3 4 5
0.0 0.1 0.2 0.3
(b) Experimental
Temkin Biosorption Dubinin -Radushkevich
Fritz-Schlunder BiLangmuir qe (mmol/g)
Ce (mmol/L) 0.0 0.2 0.4 0.6 0.8 0.10
0.15 0.20 0.25 0.30
0.0 0.2 0.4 0.6 0.8 0.10
0.15 0.20 0.25 0.30
Figure 14. Fitting results of different isotherms for Co(II) adsorption by EDTA-silica gel 40-63 µm. Dose: 2 g/L, pH: 3, Co(II) concentration: 0.017-5.1 mM. Most of the isotherms simulated for comparison (see Table 13).
Figure 14 shows that many of the isotherm models appeared to fit to the experimental data well.
Clear deviations were seen in the Langmuir, Freundlich, Temkin, and Dubinin-Radushkevich models. The poor fittings of the Langmuir and Freundlich models were also seen in Papers I and II. By comparing the predicted qm values and magnitudes of selected error functions, the best fitting model appeared to be the BiLangmuir model. It was also observed that the Sips and general adsorption models gave overlapping curves. This is because the general adsorption
isotherm can be rearranged to the Sips equation [29]. Furthermore, it was noticed that Fritz-Schlunder model was poorly fitted at high concentrations (Figure 14b) even though the equations with a higher amount of parameters are generally better fitted. In this case a higher amount of adjustable parameters seemed to increase weighting towards low concentration range (small ME, see Tables 13 and 14). From linearized Langmuir models, the fourth form seemed to offer the closest approximation to the non-linear regression. The best linear plot was obtained using the first form, however, and the curve drawn based on the estimated parameters gave the best apparent fit to the experimental data amongst all of the other forms of Langmuir model (Figure 15). El-Khaiary also presented similar observations [61].
The differences between linearized Langmuir models are due to the different weighting of the data points [61]. For example, the second linear model did not give any reasonable results, because the lowest qe values (ie. high 1/qe) had extreme weights resulting in highly biased estimation. In the first and fourth form the x- and y-axis were no longer independent, strengthening the correlation. The fourth model with qe/Ce as the y-axis gave higher values and thus higher weighting at low concentrations, however, while the first model with Ce/qe as the y-axis gave higher values and weighting at high concentrations. The latter led to a better apparent fit of the model. Moreover, it seems that the non-linear regression of the Langmuir model was also weighted towards low qe values (Figure 15).
Table 13. Fitting results of different isotherms for Co(II) adsorption by EDTA-silica gel 40-63 µm. qm,exp = 0.30 mmol/g. Uncertainty of the parameters obtained by Origin. Total differential used for linear equations.
Equilibrium model qm (mmol/g) K (L/mmol) n χ2 R2 ME (%)
Langmuir 0.29 ± 0.01 41.98 ± 8.19 0.0035 0.949 18.93
Langmuir-lin1 0.31 ± 0.01 14.90 ± 2.25 0.0219 0.999a/
0.900b 16.02
Langmuir-lin2 1.00 ± 0.29 5.42 ± 102.72 0.2894 0.720a/
- b 156.17
Langmuir-lin3 0.27 ± 0.01 66.18 ± 11.30 0.0034 0.637a/
0.929b 24.42
Langmuir-lin4 0.29 ± 0.09 43.44 ± 7.41 0.0032 0.637a/
0.947b 19.73
Freundlich 0.0007 ±
0.0005d 5.43 ± 0.55 0.0066 0.910 62.37
Sips 0.33 ± 0.01 5.46 ± 1.63 0.57 ± 0.06 0.0026 0.981 29.59
Redlich-Petersonc 0.28 ± 0.01 82.22 ± 13.87 0.91 ± 0.01 0.0016 0.994 21.80 General adsorption 0.33 ± 0.01 0.18 ± 0.05e 0.57 ± 0.06 0.0026 0.981 29.60
qm (mmol/g) aT (mmol/L) nT
Toth 0.35 ± 0.02 9.95 ± 1.51 0.44 ± 0.06 0.0022 0.984 26.58
qm (mmol/g) BDR (mmol2/J2) Dubinin–
Radushkevich 0.29 ± 0.01 0.0089 ± 0.0007 0.0025 0.967 24.53
AT (L/mmol) BT (J/mmol)
Temkin 1439 ± 306 0.038 ± 0.001 0.0021 0.980 25.56
qm
(mmol/g)
KFS
(L/mmol) nFS mFS
Fritz-Schlunder 0.27 ± 0.01 436.51 ± 501.21 1.30 ± 0.21 1.19 ± 0.22 0.0014 0.987 17.22 qm1
(mmol/g)
KBiL1
(L/mmol)
qm2
(mmol/g)
KBiL2
(L/mmol)
BiLangmuirc 0.19 ± 0.02 110.67 ± 22.92 0.13 ± 0.01 1.88 ± 0.75 0.0015 0.996 20.83 aR2 value of linear fitting. bR2 value calculated after non-linear fitting using parameters obtained from linear fitting.
cData from Paper IV, other results simulated for comparison.
dUnit: (mmol/g)/(L/mmol)nF eUnit: mmol/L
Table 14. Fitting results of different isotherms for Co(II) adsorption by DTPA-silica gel 40-63 µm. qm,exp = 0.28 mmol/g. Uncertainty of the parameters obtained by Origin. Total differential used for linear equations.
Equilibrium model qm (mmol/g) K (L/mmol) n χ2 R2 ME (%)
Langmuir 0.28 ± 0.01 57.93 ± 8.84 0.0021 0.963 12.53
Langmuir-1a 0.30 ± 0.01 17.31 ± 3.22 0.0223 0.999a/
0.906b 15.23
Langmuir-2a 0.57 ± 0.76 15.99 ± 22.77 0.1007 0.868a/
-b 78.29
Langmuir-3a 0.27 ± 0.01 77.70 ± 10.18 0.0019 0.744a/
0.955b 15.04
Langmuir-4a 0.28 ± 0.07 57.48 ± 8.04 0.0021 0.725a/
0.959b 13.17
Freundlich 0.00039 ±
0.00033d 5.83 ± 0.62 0.0067
0.903 57.47
Sips 0.31 ± 0.01 7.71 ± 1.85 0.60 ± 0.05 0.0016 0.989 20.39
Redlich-Petersonc 0.27 ± 0.01 97.93 ± 11.68 0.92 ± 0.01 0.0009 0.997 12.01 General adsorption 0.31 ± 0.01 0.13 ± 0.03e 0.60 ± 0.05 0.0016 0.989 20.39
qm (mmol/g) aT (mmol/L) nT
Toth 0.32 ± 0.01 12.58 ± 1.55 0.47 ± 0.05 0.0012 0.992 17.16
qm (mmol/g) BDR (mmol2/J2) Dubinin–
Radushkevich 0.28 ± 0.01 0.049 ± 0.003 0.0014 0.982 17.34
AT (L/mmol) BT (J/mmol)
Temkin 2324 ± 522 0.035 ± 0.001 0.0015 0.981 18.68
qm
(mmol/g)
KFS
(L/mmol) nFS mFS
Fritz-Schlunder 0.27 93.29 0.99 0.92 0.0002 0.993 12.20
qm1 (mmol/g)
KBiL1 (L/mmol)
qm2
(mmol/g)
KBiL2 (L/mmol)
BiLangmuirc 0.19 ± 0.01 137.37 ± 23.39 0.12 ± 0.01 2.78 ± 0.91 0.0008 0.998 11.69 aR2 value of linear fitting. bR2 value calculated after non-linear fitting using parameters obtained from linear fitting.
cData from Paper IV, other results simulated for comparison.
dUnit: (mmol/g)/(L/mmol)nF eUnit: mmol/L
0 1 2 3 4 5 0.0
0.1 0.2 0.3
qe (mmol/g)
Ce (mmol/L)
Experimental Langmuir non-linear
Langmuir-lin1 Langmuir-lin3 Langmuir-lin4 qe (mmol/g)
Ce (mmol/L)
0.0 0.2 0.4 0.6 0.8 0.05
0.10 0.15 0.20 0.25 0.30
Figure 15. Fitting results of the linear and non-linear Langmuir models for Co(II) adsorption by EDTA-silica gel 40-63 µm. Dose: 2 g/L, pH: 3, Co(II) concentration: 0.017-5.1 mM (Unpublished data simulated for comparison).
Even if the BiLangmuir model gave the best approximation for the experimental isotherm curves, the applicability of the equation should be verified. As presented in section 1.1.5 the BiLangmuir model assumes that the adsorbent surface has two different active sites with different affinities. In Paper I, the high affinity sites were assigned to the EDTA- or DTPA-groups and low affinity sites to primary amino groups of APTES-functionalities. This was justified by comparing the amount of determined surface coverages of chelating agents and the total amount of metals adsorbed. In Paper V, however, the model fits to the equilibrium curves measured at a higher pH (pH 4) indicated that the amount of high energy active sites was much lower than the amount of low energy active sites. Based on this, the interpretations presented in Paper I were reconsidered and the low and high energy active sites assigned to the different speciations of EDTA and DTPA surface groups (see Papers IV and V). The latter was supported by speciation calculations, which gave similar relations for the two most abundant speciations compared to the relations of qm1 and qm2 values obtained from the BiLangmuir model (Papers IV and V). Furthermore, the surface coverages calculated based on the elemental analysis did not necessarily give the most reliable results, which was seen in the case of the hybrid materials (see section 4.1). Therefore, the experimental verification presented in Paper I can be questioned. Lastly, it should be noted that speciation calculations that are considered for aqueous species are not directly applicable for
immobilized chelates. Therefore, as with the acidity of the carboxyl groups [122], stability constants of surface bound chelates can also differ from those of aqueous chelates causing different speciation distributions.
For modified chitosans, the Sips model gave the best fitting results for both Co(II) and Ni(II) (Papers II and V). This suggested a heterogeneous adsorption and can be assigned to the crosslinking effect as well as some amount of functionalities (other than chelating groups) such as -NH2 and -OH on the adsorbent surface (see FTIR-spectra in Figure 8). The isotherm curves were slightly S-shaped, which can be attributed to the lower affinity of surface groups towards metal ions at low concentrations. A crosslinking effect (see sections 4.2.1, 4.2.2), which was enhanced at low concentrations and inhibited due to the fast metal chelation at high concentrations, might have played an important role in the adsorption phenomena on modified chitosans.
We observed that fitting results for hybrid materials were more related to the type of metal than the type of material (Paper III). Adsorption isotherms of Co(II) and Cd(II) were better fitted by the BiLangmuir model and those of Ni(II) and Pb(II) by the Sips model (except Ni(II) adsorption on Chi:TEOS 2:60). The latter suggested a more heterogeneous adsorption for Ni(II) and Pb(II) compared to the two other metals. This was supported by the Ni(II) adsorption on modified chitosans observations, which showed higher heterogeneity for Ni(II) adsorption compared to that of Co(II) (Table 4 in Paper II). At pH 2, Ni(II)EDTA prefers the speciation with one negatively charged carboxyl group (Appendix IA). Paper II suggested that this group participated in the binding of another Ni(II) ion thus increasing a system heterogeneity. At pH 3, however, all the studied metals form EDTA-chelates with negatively charged carboxyl groups (Appendix IA). In this case, a lower heterogeneity for Co(II) and Cd(II) adsorption can be assigned to a lower affinity of carboxylates and amino-bearing groups towards these cations than towards Ni(II) and Pb(II) [92]. Lastly, it should be noted that a similar heterogeneity as above was not seen for EDTA- and DTPA-silica gels most likely due to their rigid structure and lower surface coverage of functional groups.
Effect of error function
The effect of error functions was discussed in section 1.1.7. Most of the modeling in this work was conducted by minimizing ERRSQ error function, because it is programmed in the Origin software. For the comparison, however, simple simulations with the Excel Solver were used to test all error functions presented in Table 2.
Figure 16 shows that the BiLangmuir fits using error functions other than MPSD and ARE were rather similar. ERRSQ and R2 gave exactly the same results followed by EABS and χ2 (Table 15). The poor fittings obtained with MPSD and ARE arises from their weighting towards low concentrations. Therefore, ME values shown in Table 15 are low for these functions. Similar results are presented in Table 4 in Paper II and Tables 3 and 4 in Paper IV. Tables 3 and 4 in Paper IV also show that the MPSD error function predicted that both of the actives sites on functionalized silica gels were similar thus turning the BiLangmuir model into a simple one-component Langmuir isotherm. In conclusion, the use of ERRSQ error function throughout this study seems to be a proper choice.
0 1 2 3 4 5
0.0 0.1 0.2 0.3
qe (mmol/g)
Ce (mmol/L)
Experimental ERRSQ HYBRID MPSD ARE EABS R2 χ2 qe (mmol/g)
Ce (mmol/L) 0.0 0.2 0.4 0.6 0.8 0.10
0.15 0.20 0.25 0.30
Figure 16. Effect of the error function on the BiLangmuir fits for Co(II) adsorption by EDTA-silica gel 40-63 µm. Dose: 2 g/L, pH: 3, Co(II) concentration: 0.017-5.1 mM (Unpublished data simulated for comparison).
Table 15. Effect of the error function on the BiLangmuir model parameters for Co(II) adsorption by EDTA/DTPA-silica gel 40-63 µm (Unpublished data simulated for comparison).
Equilibrium model qm1 (mmol/g) KBiL1
(L/mmol) qm2 (mmol/g) KBiL2
(L/mmol) χ2 R2 ME (%)
EDTA-silica gel
ERRSQ 0.19 110.55 0.13 1.89 0.0015 0.992 20.88
HYBRID 0.23 52.21 0.10 0.90 0.0029 0.980 16.17
MPSD 0.28 20.62 0.07 0.15 0.0155 0.930 15.02
ARE 0.01 55.67 0.30 10.62 0.0349 0.883 15.02
EABS 0.20 100.00 0.13 1.56 0.0015 0.991 19.71
R2 0.19 110.55 0.13 1.89 0.0015 0.992 20.88
χ2 0.21 93.82 0.12 1.43 0.0015 0.991 19.43
DTPA-silica gel
ERRSQ 0.19 136.85 0.12 2.78 0.0008 0.996 11.76
HYBRID 0.22 83.27 0.09 1.36 0.0010 0.992 10.36
MPSD 0.28 39.55 0.00 21.21 0.0049 0.959 12.10
ARE 0.24 68.27 0.08 0.78 0.0014 0.988 10.18
EABS 0.22 100.00 0.09 1.54 0.0008 0.994 10.62
R2 0.19 136.85 0.12 2.78 0.0008 0.996 11.76
χ2 0.20 117.13 0.11 2.15 0.0007 0.995 11.12
Effect of data range
In Paper IV, it was observed that the error between simulated and experimental data was highest at low concentrations. Therefore, simulations for the BiLangmuir model were conducted by omitting the first equilibrium point (Table 16), which considerably improved the MPSD and ARE fittings. Other error functions were less affected by the change of the data range, although ME values in these cases also decreased significantly.
Table 16. Effect of the error function on the BiLangmuir model parameters for Co(II) adsorption by EDTA/DTPA-silica gel 40-63 µm. First equilibrium point omitted from simulations (Unpublished data simulated for comparison).
Equilibrium model qm1 (mmol/g) KBiL1
(L/mmol) qm2 (mmol/g) KBiL2
(L/mmol) χ2 R2 ME (%)
EDTA-silica gel
ERRSQ 0.18 138.58 0.14 2.16 0.0002 0.996 2.98
HYBRID 0.19 117.86 0.13 1.78 0.0002 0.995 2.90
MPSD 0.21 101.10 0.12 1.35 0.0003 0.994 3.03
ARE 0.19 122.23 0.13 1.88 0.0002 0.995 2.80
EABS 0.20 100.00 0.13 1.56 0.0004 0.993 2.98
R2 0.18 138.58 0.14 2.16 0.0002 0.996 2.98
χ2 0.19 122.93 0.13 1.84 0.0002 0.995 2.88
DTPA-silica gel
ERRSQ 0.18 159.10 0.12 3.20 0.0001 0.996 1.92
HYBRID 0.18 152.49 0.12 3.01 0.0001 0.996 1.86
MPSD 0.19 146.04 0.12 2.75 0.0001 0.996 1.81
ARE 0.20 131.64 0.11 2.45 0.0001 0.995 1.67
EABS 0.22 100.00 0.09 1.54 0.0003 0.993 2.77
R2 0.18 159.10 0.12 3.20 0.0001 0.996 1.92
χ2 0.19 152.13 0.12 3.02 0.0001 0.996 1.85
It is also interesting to compare the BiLangmuir model fittings of Papers I and IV, due to the higher amount of experimental points measured in the latter. Around a 10 to 15% differences was seen in the predicted qm1/2 values, while the differences between KBiL1/2 values ranged from 25 to 50%. Therefore, as Cernik et al. proposed earlier [53], small changes in the experimental points may noticeably influence the fitting results. It should be noted, however, that the used data range did not change the type of the best fitted model or the relations of simulated qm1/2 and KBiL1/2
values and therefore fitting results in Papers I and IV rather support each other.
Effect of initial guess values
Effect of initial guess values was investigated in Paper IV. The initial guesses for the three-parameter Redlich-Peterson model did not affect the simulation results (see section 4.1 in Paper IV). The four-parameter BiLangmuir model, however, gave a worse fit when the initial guesses of qm1/2 were not set close to those obtained experimentally (Figure 3 and Tables 3 and 4 in Paper IV). Therefore, it can be concluded that the higher amount of estimated parameters makes the isotherm model more sensitive to the initial guesses and use of experimentally obtained qm values gives a proper initialization for the simulations.