EQUILIBRIUM ISOTHERM STUDIES FOR THE SORPTION OF DIVALENT METAL IONS ONTO PEAT: COPPER, NICKEL AND LEAD SINGLE COMPONENT SYSTEMS Y. S. HO, J. F. PORTER and G. MCKAY∗ Department of Chemical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, SAR China (∗ author for correspondence, e-mail: kemckayg@ust.hk) (Received 5 July 2000; accepted 6 September 2001) Abstract. The sorption of three divalent metal ions – copper, nickel and lead – from aqueous solution onto peat in single component systems has been studied and the equilibrium isotherms determined. The experimental data have been analysed using the Langmuir, Freundlich, Redlich-Peterson, Toth, Temkin, Dubinin-Radushkevich and Sips isotherm models. In order to determine the best fit isotherm for each system, six error analysis methods were used to evaluate the data: the coefficient of determination, the sum of the errors squared, a hybrid error function, Marquardt’s percent standard deviation, the average relative error and the sum of absolute errors. The error values demonstrated that the Sips equation provided the best model for the three sets of experimental data overall. Keywords: copper, isotherm, lead, nickel, peat, sorption Nomenclature aF At aL aLF aR aS at BD bT bR Ce E KL KLF KR Freundlich isotherm constant; Temkin isotherm constant, dm3 mmol−1 ; Langmuir isotherm constant, dm3 mmol−1 ; Langmuir-Freundlich isotherm constant ; Redlich-Peterson isotherm constant, (dm3 mmol−1 )bR ; Sips isotherm constant; Toth isotherm constant; Dubinin-Radushkevich isotherm constant; Temkin isotherm constant; Redlich-Peterson isotherm exponent; Solution phase metal ion concentration at equilibrium, mmol dm− 3; Mean free energy of sorption, kJ mol−1 ; Langmuir isotherm constant, dm3 g−1 ; Langmuir-Freundlich isotherm constant; Redlich-Peterson isotherm constant, dm3 g−1 ; Water, Air, and Soil Pollution 141: 1–33, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands. 2 Y. S. HO ET AL. Kt ka kd nLF nS qd qe qm R R2 T t Toth isotherm constant; Rate constant of adsorption ; Rate constant of desorption; Langmuir-Freundlich isotherm exponent (–); Sips isotherm exponent (–); Dubinin-Radushkevich isotherm constant, mmol g−1 ; Solid phase metal ion concentration at equilibrium, mmol g−1 ; Langmuir monolayer saturation capacity, mmol g−1 ; Universal gas constant, 8.314 J/mol ◦ K; Statistical linear coefficient of determination (the square of the correlation coefficient, R); (–); Absolute temperature, ◦ K; Toth isotherm exponent (–). Greek letters θ Fractional surface coverage (–). 1. Introduction It is now widely recognised that sorption processes provide a feasible technique for the removal of pollutants from wastewaters (McKay, 1995). The problems associated with metal ions discharged in industrial effluents are considerable, due to both their toxic and carcinogenic properties (Luckey and Venugopal, 1977; Goldstein, 1990; Freedman et al., 1990). Furthermore, metal ion containing effluents are discharged from a wide range of industries, including microelectronics, electroplating, battery manufacture, dyestuffs, chemical, pharmaceutical, metallurgical and many others (Tchobanoglous and Burton, 1991; Volesky and Holan, 1995). There is a continuing search for cheap, high capacity sorbents for metal ions because the relatively high cost of commercial sorbents such as activated carbon (Macias-Garcia et al., 1993; Sasaki et al., 1995; Meyer and Lieser, 1995) and ionexchange resins (Bolto and Pawlowski, 1987) remains a major obstacle to the use of sorption processes. Many agricultural waste products and byproducts of cellulosic origin have been tested for metal ion adsorption (Randall et al., 1975; Kumar and Dara, 1981; Shukla and Sakhardande, 1991). Several workers have studied the removal of lead by sorption processes, including the sorption of lead onto waste tea leaves (Tan and Abd. Rahman, 1988), the sorption of lead onto china clay and wollastonite (Yadava et al., 1991), onto EQUILIBRIUM ISOTHERM STUDIES 3 modified groundnut (Arachi hypogea) husks (Okieimen et al., 1991), and bagasse pith from the sugar cane industry (Aly and Daifullah, 1998). However, there is very limited accurate data available in the literature for the design of sorption systems for the removal of lead ions. The removal of copper from effluents has been studied using many sorbent materials such as biomass (Roa et al., 1993; Volesky and MayPhillips, 1995), kaolinite (Spark et al., 1995), dry hyacinth roots (Low et al., 1994) and seafood shell waste (Findon et al., 1993). In a recent review paper over one hundred copper-sorbent systems were reported (McKay et al., 1999). There is very little detailed information in the literature based on nickel removal by sorption. Coconut coir has a limited capacity (Baes et al., 1996), sewage sludge has been utilised (Gao et al., 1997) and various peats have been tested (Bunzl et al., 1976; Zhipei et al., 1984; Gosset et al., 1986; Ho et al., 1995; McKay et al., 1998). Sorption equilibria provide fundamental physicochemical data for evaluating the applicability of sorption processes as a unit operation. Sorption equilibrium is usually described by an isotherm equation whose parameters express the surface properties and affinity of the sorbent, at a fixed temperature and pH. Thus an accurate mathematical description of the equilibrium isotherm, preferably based on a correct sorption mechanism, is essential to the effective design of sorption systems. Until relatively recently, for most two parameter isotherms, the accuracy of the fit of an isotherm model to experimental equilibrium data was typically assessed based on the magnitude of the coefficient of determination for the linear regression. I.e., the isotherm giving an R 2 value closest to unity was deemed to provide the best fit. Such transformations of non-linear isotherm equations to linear forms implicitly alter their error structure and may also violate the error variance and normality assumptions of standard least squares (Myers, 1990; Ratkowski, 1990). Previous observations of bias resulting from deriving isotherm parameters from linear forms of isotherm equations – namely Freundlich parameters producing isotherms which tend to fit experimental data better at low concentrations and Langmuir isotherms tending to fit the data better at higher concentrations (Richter et al., 1989) – appear to support this hypothesis. Non-linear optimisation provides a more complex, yet mathematically rigorous, method for determining isotherm parameter values (Seidel and Gelbin, 1988; Seidel-Morgenstern and Guiochon, 1993; Malek and Farooq, 1996; Khan et al., 1997), but still requires an error function assessment, in order to evaluate the fit of the isotherm to the experimental results. In this study, five different non-linear error functions were examined and in each case the isotherm parameters were determined by minimising the respective error function across the concentration range studied using the solver add-in with Microsoft’s spreadsheet, Excel (Microsoft, 1995). Four two-parameter equations – the Freundlich, Langmuir, Temkin and Dubinin-Radushkevich isotherms – and three three-parameter equations – the Redlich-Peterson, Toth and Sips isotherms – were examined for their ability to model the equilibrium sorption data. 4 Y. S. HO ET AL. 2. Equilibrium Isotherms Equilibrium isotherm equations are used to describe experimental sorption data. The equation parameters and the underlying thermodynamic assumptions of these equilibrium models often provide some insight into both the sorption mechanism and the surface properties and affinity of the sorbent. The symbols and coefficients used in the equations are defined in the Nomenclature section. 2.1. T HE F REUNDLICH ISOTHERM In 1906, Freundlich presented the earliest known sorption isotherm equation (Freundlich, 1906). This empirical model can be applied to nonideal sorption on heterogeneous surfaces as well as multilayer sorption and is expressed by the following equation: qe = aF CebF . (1) The Freundlich isotherm has been derived by assuming an exponentially decaying sorption site energy distribution (Zeldowitsch, 1934). It is often criticised for lacking a fundamental thermodynamic basis since it does not reduce to Henry’s law at low concentrations. 2.2. T HE L ANGMUIR ISOTHERM In 1916 Langmuir developed a theoretical equilibrium isotherm relating the amount of gas sorbed on a surface to the pressure of the gas (Langmuir, 1916). The Langmuir model is probably the best known and most widely applied sorption isotherm. It has produced good agreement with a wide variety of experimental data and may be represented as follows: qe = qm aL Ce 1 + aL Ce or, alternatively qe = KL Ce . 1 + aL Ce (2) One can readily deduce that at low sorbate concentrations it effectively reduces a linear isotherm and thus follows Henry’s law. Alternatively, at high sorbate concentrations, it predicts a constant – monolayer – sorption capacity. 2.3. T HE R EDLICH -P ETERSON ISOTHERM Jossens and co-workers modified the three parameter isotherm first proposed by Redlich and Peterson (1959) to incorporate features of both the Langmuir and Freundlich equations (Jossens et al., 1978). It can be described as follows: qe = KR Ce 1 + aR CebR . (3) EQUILIBRIUM ISOTHERM STUDIES 5 At low concentrations the Redlich-Peterson isotherm approximates to Henry’s law and at high concentrations its behaviour approaches that of the Freundlich isotherm. 2.4. T HE T EMKIN ISOTHERM The derivation of the Temkin isotherm assumes that the fall in the heat of sorption is linear rather than logarithmic, as implied in the Freundlich equation (Aharoni and Ungarish, 1977). The Temkin isotherm has generally been applied in the following form (Aharoni and Sparks, 1991): qe = RT ln(AT Ce ) . bT (4) 2.5. T HE D UBININ -R ADUSHKEVICH ISOTHERM This isotherm is generally expressed as follows (Dubinin, 1960):
2    1 qe = qD exp −BD RT ln 1 + . Ce (5) Radushkevich (1949) and Dubinin (1965) have reported that the characteristic sorption curve is related to the porous structure of the sorbent. The constant, BD , is related to the mean free energy of sorption per mole of the sorbate as it is transferred to the surface of the solid from infinite distance in the solution and this energy can be computed using the following relationship (Hasany and Chaudhary, 1996): 1 E=√ 2BD (6) 2.6. T HE T OTH ISOTHERM Derived from potential theory, this isotherm (Toth, 1962) has proven useful in describing sorption in heterogeneous systems such as phenolic compounds on carbon. It can be represented by the following equation: qe = Kt Ce (at + Ce )1/t (7) It assumes an asymmetrical quasi-Gaussian energy distribution with a widened left-hand side, i.e. most sites have a sorption energy less than the mean value. 6 Y. S. HO ET AL. 2.7. T HE S IPS , OR L ANGMUIR -F REUNDLICH ISOTHERM (Sips, 1948) Langmuir also considered the case of a molecule occupying two sites, e.g. for the dissociative sorption of hydrogen on platinum. In such cases a derivative of the original Langmuir isotherm can be determined assuming that the rates of adsorption and desorption are proportional to (1 − θ)2 and θ 2 , respectively, where θ is the fractional surface coverage. Therefore, the rate equation becomes: dθ = ka C(1 − θ)2 − kd θ 2 . dt (8) Generalising to m(= 1/n) sites and assuming equilibrium produces the Sips sorption isotherm: qe = qm (as Ce )ns . 1 + (aS Ce )ns (9) Or, in the Langmuir-Freundlich form: qe = KLF CenLF . 1 + (aLF Ce )nLF (10) The Langmuir-Freundlich name derives from the limiting behaviour of the equation. At low sorbate concentrations it effectively reduces to a Freundlich isotherm and thus does not obey Henry’s law. At high sorbate concentrations, it predicts a monolayer sorption capacity characteristic of the Langmuir isotherm. 2.8. D ETERMINING ISOTHERM PARAMETERS BY LINEARISATION The simplest method to determine isotherm constants for two parameter isotherms is to transform the isotherm variables so that the equation is converted to a linear form and then to apply linear regression. The experimental results are shown in Figure 1 with the Langmuir and Freundlich linear regression curves. Although a linear analysis is not possible for a three-parameter isotherm, a trial and error procedure has previously been applied to a pseudo-linear form of the Redlich-Peterson isotherm to obtain values for the isotherm constants (McKay et al., 1984). The method is based on the following equation:   Ce − 1 = bR ln(Ce ) + ln(aR ) , ln KR qe (11) and involves varying the isotherm parameter, KR , to obtain the maximum value of the correlation coefficient for the regression of against. Figure 2 shows the coefficient of determination, R 2 , as a function of KR for lead. EQUILIBRIUM ISOTHERM STUDIES 7 Figure 1. Linear Freundlich and Langmuir isotherms for lead, nickel and copper on peat. 8 Y. S. HO ET AL. TABLE I Isotherm constants for two-parameter models by linear regression Isotherm Langmuir Freundlich Temkin Dubinin-Radushkevich Transformed X-values Y-values Ce Ln(Ce ) Ln(Ce ) (Ln(1 + 1/Ce ))2 Ce /qe Ln(qe ) qe Ln(qe ) Slope Intercept aL /kL bF R.T ./bT −BD R 2 T 2 1/KL Ln(aF ) R.T .Ln(AT )/bT Ln(qD ) Figure 2. Redlich-Peterson equation isotherms of metal ions sorbed on peat. 9 EQUILIBRIUM ISOTHERM STUDIES 2.9. D ETERMINING ISOTHERM PARAMETERS BY NON - LINEAR REGRESSION Due to the inherent bias resulting from linearisation, alternative isotherm parameter sets were determined by non-linear regression. This provides a mathematically rigorous method for determining isotherm parameters using the original form of the isotherm equation (Seidel and Gelbin, 1988; Seidel-Morgenstern and Guiochon, 1993; Malek and Farooq, 1996; Khan et al., 1996). Most commonly, algorithms based on the Levenberg-Marquardt or Gauss-Newton methods (Edgar and Himmelblau, 1989; Hanna and Sandall, 1995) are used. The optimisation procedure requires the selection of an error function in order to evaluate the fit of the isotherm to the experimental equilibrium data. The choice of error function can affect the parameters derived – error functions based primarily on absolute deviation bias the fit towards high concentration data and this weighting increases when the square of the deviation is used to penalise extreme errors. This bias can be offset partly by dividing the deviation by the measured value in order to emphasise the significance of fractional deviations. In this study, five non-linear error functions were examined and in each case a set of isotherm parameters were determined by minimising the respective error function across the concentration range studied. The error functions employed were as follows: 1. The Sum of the Squares of the Errors (ERRSQ): p   i=1 qe,meas − qe,calc 2 i . (12) 2. A Composite Fractional Error Function (HYBRD):  p   (qe,meas − qe,calc )2 . qe,meas i i=1 (13) 3. A Derivative of Marquardt’s Percent Standard Deviation (MPSD) (Marquardt, 1963):  p   qe,meas − qe,calc 2 . q e,meas i i=1 (14) 4. The Average Relative Error (ARE) (Kapoor and Yang, 1989): p  qe,meas − qe,calc qe,meas i=1 . i (15) 10 Y. S. HO ET AL. 5. The Sum of the Absolute Errors (EABS): p  i=1 qe,meas − qe,calc i . (16) As each of the error criteria is likely to produce a different set of isotherm parameters, an overall optimum parameter set is difficult to identify directly. Hence, in order to try to make a meaningful comparison between the parameter sets, a procedure of normalising and combining the error results was adopted producing a so-called ‘sum of the normalised errors’ for each parameter set for each isotherm. The calculation method for the ‘sum of the normalised errors’ was as follows: (a) select one isotherm and one error function and determine the isotherm parameters that minimise that error function for that isotherm to produce the isotherm parameter set for that error function; (b) determine the values for all the other error functions for that isotherm parameter set; (c) calculate all other parameter sets and all their associated error function values for that isotherm; (d) select each error measure in turn and ratio the value of that error measure for a given parameter set to the largest value of that error from all the parameter sets for that isotherm; and (e) sum all these normalised errors for each parameter set. The parameter set thus providing the smallest normalised error sum can be considered to be optimal for that isotherm provided: • There is no bias in the data sampling – i.e. the experimental data are evenly distributed, providing an approximately equal number of points in each concentration range; and • There is no bias in the type of error methods selected. 3. Materials and Methods 3.1. A DSORBENT The experiments were conducted with sphagnum moss peat sourced from New Zealand. The peat was washed and then dried at a temperature of 105±5 ◦ C for 8 hr and finally screened to obtain a particle size range of 500–710 µm before use. 11 EQUILIBRIUM ISOTHERM STUDIES TABLE II Metal concentrations in sphagnum peat moss Metal Concentration (µg g−1 of peat) Metal Concentration (µg g−1 of peat) Iron Aluminium Sodium Manganese Zinc Lead Nickel 890 660 285 24 19 13.5 5.5 Cadmium Potassium Chromium Copper Mercury Silver Zirconium 1.1 <1 <1 <1 <1 <1 <1 3.2. A DSORBENT CHARACTERISATION An elemental analysis was carried out on the peat and the results, on a dry mass basis, were: carbon, 57.2%; hydrogen 5.7%; oxygen 36.0%; nitrogen 0.7%; and sulphur, 0.4%. The BET (Brunnauer, Emmelt and Teller, nitrogen isotherm) surface area was determined to be 26.5 m2 g−1 and the pore volume 0.73 × 10−6 m3 g−1 . The absolute density, measured in paraffin oil, was found to be 1220 kg m−3 . The metal content of the peat was determined by digesting it by heating in nitric acid and filtering. The metal ion concentrations were determined by inductively coupled plasma atomic emission spectroscopy (ICP-AES). The results are shown in Table II. 3.3. R EAGENTS Analytical grade reagents supplied by Aldrich Chemicals were used in all experiments. Stock solutions of copper (II) sulphate (CuSO4 ·5H2 O), nickel (II) sulphate (NiSO4 ·5H2 O) and lead (II) nitrate (Pb(NO3 )2 ) were prepared in deionised water. All solutions used in this study were diluted with deionised water as required. 3.4. E XPERIMENTAL SYSTEM The sorption experiments for the three systems were carried out in a thermostatically controlled shaking water bath using capped 0.125 dm3 Erlenmeyer conical flasks maintained at 25±1 ◦ C. In the sorption isotherm tests, 0.055 g of peat was thoroughly mixed with 0.1 dm3 of the metal ion solution. The concentration ranges studied for each system are shown in Table III. As previous contact time studies indicated equilibrium was reached in two hours under similar experimental conditions (Allen, 1987; Ho et al., 1994, 1995), the mixtures were agitated for three hours to ensure that equilibrium was achieved. 12 Y. S. HO ET AL. TABLE III Sorption systems studied Metal Mass of peat (g) Test solution volume (dm3 ) Metal concentration range (mg dm−3 ) Pb Cu Ni 0.055 0.055 0.055 0.100 0.100 0.100 35–210 5–150 10–150 TABLE IV Final pH values at various initial metal ion concentrations Initial concentrations (Co mg dm−3 ) 25 50 75 100 150 200 209 250 300 309 410 504 Copper solutions 3.60 3.50 3.40 3.30 3.20 3.15 3.10 3.05 Nickel solutions 3.60 3.45 3.40 3.35 3.30 3.25 3.22 3.22 Lead solutions 5.38 5.19 4.80 4.63 4.42 At the end of the experiment, the samples were filtered (Whatman No. 1) and the filtrates were analysed for the respective metal ion concentrations using ICP-AES. The initial and final solution pH values were measured. Initial pH values were set at 4.50±0.10 and final pH values appeared to show some variation with initial solution concentration. The values are given in Table IV. Although significant pH changes were observed for several of the systems, none resulted in precipitation (Ho, 1995). 4. Results and Discussion 4.1. T HE F REUNDLICH ISOTHERM The linear Freundlich isotherm plots for the sorption of the three metals onto peat are presented in Figure 3. Examination of the plot suggests that the linear Freundlich isotherm is a good model for the sorption of lead but not for nickel or copper. Table V shows the linear Freundlich sorption isotherm constants, coefficients of determination (R 2 ) and error values. Based on the R 2 values, the linear form of the Freundlich isotherm appears to produce a reasonable model for sorption in all three systems, with the lead and 13 EQUILIBRIUM ISOTHERM STUDIES Figure 3. Freundlich equation isotherms of metal ions sorbed on peat. TABLE V Linear Freundlich isotherm parameters Freundlich: aF Freundlich: bF Correlation coefficient R 2 Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Lead Nickel Copper 0.639589 0.125618 0.972028 0.002143 0.004477 0.009927 0.260984 0.122768 0.135100 0.215808 0.961755 0.000233 0.001755 0.013889 0.321260 0.041996 0.185859 0.150492 0.915671 0.000871 0.005069 0.030982 0.450704 0.076003 14 Y. S. HO ET AL. nickel isotherms seeming to fit the experimental data better than copper. The error function values do not support this assertion, however. The Freundlich isotherm constants determined by non-linear regression are shown in Table VI. The results demonstrate that the values of the constants – bF and especially aF – obtained by non-linear regression are remarkably consistent and quite similar to the linear transform values from Table V. In each case the MPSD parameters are closest to those obtained by linearisation, typically within 2%, and these provide either the second or third best fit to the experimental data, based on the sum of the normalised errors. On this basis it would thus seem that the linear Freundlich model does give a reasonable approximation to the optimum parameter set found by non-linear regression. The minimum error occurs for the constants determined using the HYBRD error measure in the cases of copper and nickel and those determined using the ERRSQ measure for lead. The values for most error measures suggest that the Freundlich isotherm produces a better fit to the nickel data than it does for lead or copper. 4.2. T HE L ANGMUIR ISOTHERM The sorption data were analysed according to the linear form of the Langmuir isotherm. The plots of specific sorption (Ce /qe ) against the equilibrium concentration (Ce ) for copper, nickel and lead are shown in Figure 4 and the linear isotherm constants – aL , qm , KL – and the error values are presented in Table VII. These isotherms were found to be linear over the whole concentration range studied with extremely high coefficients of determination (Table VII). The R 2 values suggest that the Langmuir isotherm provides a good model of the sorption system. The sorption constant, KL , and sorption capacity, qm , for lead are higher than those for copper and nickel. The same sorption capacity order has been reported for metal cation sorption on a modified coconut coir as Pb(II) > Cu(II) > Ni(II) although the order of KL was given as Cu(II) > Pb(II) > Ni(II) and this was interpreted as meaning that not all inorganic sites may be available for copper(II) binding (Baes et al., 1996). The Langmuir data from the other five error analysis methods are presented in Table VIII. The individual constants, KL and aL , are significantly different, although the ratios give monolayer saturation capacities very close to those obtained by the linear error analysis approach. In all cases, all the error values for any parameter set are lower than the same errors determined for the linear form of the isotherm. The only parameter values that are even close to those obtained by linearisation are the EABS set for nickel. Considering the comparative magnitudes of the error values together with the range of variation in the isotherm parameters suggests that the Langmuir isotherm does not provide a particularly good model for the sorption of lead on peat. Lastly, the HYBRD parameter set produces the best overall fit for both nickel and lead whereas 15 EQUILIBRIUM ISOTHERM STUDIES TABLE VI Non-linear Freundlich isotherm parameters Method/error function (parameter set) ERRSQ HYBRD MPSD ARE EABS 0.185265 0.131816 0.000736 0.005108 0.038883 0.430381 0.066189 3.224603 0.185031 0.143146 0.000776 0.004780 0.031566 0.427924 0.069557 3.199412 0.184987 0.152419 0.000864 0.004981 0.029834 0.427502 0.072452 3.307742 0.179765 0.163444 0.001272 0.007168 0.040634 0.416427 0.075044 3.930186 0.185279 0.094257 0.001215 0.011342 0.109226 0.449630 0.056134 4.703244 0.135381 0.198179 0.000204 0.001792 0.017113 0.308700 0.037656 3.756599 0.134985 0.208052 0.000214 0.001700 0.014508 0.31496 0.039998 3.751730 0.13476 0.216674 0.000236 0.001765 0.013829 0.321143 0.042113 3.903084 0.135255 0.172131 0.000274 0.002874 0.032076 0.292270 0.031409 4.654606 0.135255 0.172112 0.000275 0.002875 0.032092 0.292274 0.031406 4.655863 0.630586 0.118623 0.001936 0.004517 0.011321 0.242862 0.110567 3.951712 0.634548 0.122366 0.001994 0.004367 0.010217 0.254818 0.118135 3.986851 0.638871 0.125919 0.002156 0.004493 0.009897 0.264773 0.124669 4.136891 0.647390 0.124467 0.002558 0.005369 0.012227 0.232601 0.109142 4.325719 0.642898 0.118531 0.002684 0.006278 0.016016 0.246093 0.108842 4.802496 Copper Freundlich: aF Freundlich: bF Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Nickel Freundlich: aF Freundlich: bF Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Lead Freundlich: aF Freundlich: bF Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors the EABS produces the best fit for copper based on the sum of the normalised errors. 16 Y. S. HO ET AL. Figure 4. Langmuir equation isotherms of metal ions sorbed on peat. TABLE VII Linear Langmuir isotherm parameters Langmuir: KL (dm3 g−1 ) Langmuir: aL (dm3 mmol−1 ) Saturation capacity: qm Coefficient of determination, R 2 Sum of errors2 Hybrid error function YMarquardt’s PSD Average relative error Sum of absolute errors Lead Nickel Copper 45.75918 77.53212 0.590197 0.999392 0.026728 0.080102 0.242729 0.784015 0.286497 1.035939 6.260953 0.165460 0.998830 0.000160 0.001823 0.021504 0.232811 0.023962 4.652902 23.34904 0.199276 0.999640 0.000550 0.005296 0.051826 0.309219 0.037545 17 EQUILIBRIUM ISOTHERM STUDIES TABLE VIII Non-linear Langmuir isotherm parameters Method/error function (parameter set) ERRSQ HYBRD MPSD ARE EABS Copper Langmuir: KL Langmuir: aL Saturation capacity: qm Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors 6.84352 35.0648 0.195168 0.000194 0.001250 0.008370 0.170779 0.026442 4.808365 7.10994 36.5389 0.194585 0.000198 0.001217 0.007648 0.163665 0.026625 4.679145 7.28845 37.5613 0.194041 0.000205 0.001233 0.007511 0.161987 0.027298 4.723174 7.49222 38.3845 0.195189 0.000217 0.001323 0.008101 0.130987 0.02309 4.580714 7.29922 37.3742 0.195301 0.000207 0.001266 0.007805 0.139412 0.02254 4.48369 1.22088 7.55792 0.161537 0.000109 0.000967 0.009183 0.210910 0.024842 4.260104 1.27775 7.96428 0.160435 0.000113 0.000925 0.008006 0.206782 0.025414 4.158057 1.32445 8.30854 0.159408 0.000122 0.000953 0.007720 0.20498 0.026105 4.239416 1.40764 8.80566 0.159856 0.000148 0.001167 0.009351 0.181438 0.023992 4.537514 1.15011 7.02848 0.163636 0.000119 0.001171 0.012281 0.210351 0.023133 4.687898 100.5017 181.1768 0.554716 0.006893 0.015684 0.037471 0.465177 0.208822 4.327903 107.7296 196.1349 0.549263 0.007061 0.015278 0.034543 0.465526 0.213802 4.294782 114.5102 210.7899 0.543243 0.007561 0.015648 0.033672 0.466953 0.219476 4.387715 91.47266 165.1662 0.553822 0.007247 0.017421 0.044517 0.456389 0.201928 4.546698 88.29985 154.6331 0.571028 0.008019 0.019851 0.051284 0.463050 0.196249 4.885809 Nickel Langmuir: KL Langmuir: aL Saturation capacity: qm Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Lead Langmuir: KL Langmuir: aL Saturation capacity: qm Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors 18 Y. S. HO ET AL. Figure 5. The coefficient of determination as function of KR for sorption of lead onto peat. 4.3. T HE R EDLICH -P ETERSON ISOTHERM The Redlich-Peterson isotherm plots for sorption of the three metals onto peat are presented in Figure 2. Examination of the plot shows that the Redlich-Peterson isotherm accurately describes the sorption behaviour of copper, nickel and lead on peat over the concentration ranges studied. The Redlich-Peterson isotherm constants – KR , aR and bR – and the coefficients of determination are presented in Table IX. Since the method used to derive the isotherm parameters maximises the linear correlation coefficient, it is unsurprising that in all cases the Redlich-Peterson isotherms exhibit extremely high R 2 values – indicating, superficially at least, that it produces a considerably better fit compared to the preceding two parameter isotherms. Figure 5 shows the coefficient of determination, R 2 , as a function of KR for lead. 19 EQUILIBRIUM ISOTHERM STUDIES TABLE IX ‘Linear’ Redlich-Peterson isotherm parameters R–P: KR R–P: aR R–P: bR Coefficient of determination, R 2 Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Lead Nickel Copper 288.6086 470.4830 0.913060 0.999939 0.00029589 0.00055725 0.00106101 0.07899636 0.04095156 2.027348 13.55103 0.906189 0.999716 3.02328E-05 0.00021183 0.00149554 0.08610689 0.01209342 9.475487 49.22798 0.952612 0.999821 7.17191E-05 0.00039820 0.00222289 0.10518300 0.01903796 The non-linear error analyses are shown in Table X. Similar to the non-linear Freundlich analysis, the Redlich-Peterson isotherm constants are very consistent across the range of error methods and the actual parameter values derived by non-linear regression are very close to those obtained using the linear analysis – particularly those determined using the MPSD error measure. In several cases the error values determined by non-linear regression are greater than those for the linear isotherm parameters. This indicates that the linear equation analysis method may be a suitable approach to use for this three parameter isotherm. The lowest sums of normalised errors are obtained using the EABS parameter set for copper and nickel and the HYBRD parameter set for lead. 4.4. T HE T EMKIN ISOTHERM The sorption data were analysed according to the linear form of the Temkin isotherm and the linear plots are shown in Figure 6. Examination of the data shows that the Temkin isotherm provides a close fit to the lead and nickel sorption data, but the copper experimental data are not modelled as well across the concentration range studied. The linear isotherm constants and coefficients of determination are presented in Table XI and the non-linear error analyses are shown in Table XII. Excepting the EABS parameter set for copper, the non-linear analysis methods show the AT parameter to be quite consistent and comparable with the values derived by linearisation. The non-linear bT parameters are less consistent, particularly for copper. For all three metals the ERRSQ parameter sets are within 1% of the linear values for this isotherm and it is quite interesting to notice that several nonlinear parameter sets result in worse errors than the linear set – possibly indicating that, in this case, the linearisation method has little effect on the error assumptions 20 Y. S. HO ET AL. TABLE X Non-linear Redlich-Peterson isotherm parameters Method/error function (parameter set) ERRSQ HYBRD MPSD ARE EABS Copper on peat Redlich-Peterson: KR Redlich-Peterson: aR Redlich-Peterson: bR Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors 9.12546 47.3294 0.95609 5.87048E-05 0.000337 0.001957 0.087120 0.014915 4.912457 9.32887 48.4361 0.953872 5.90988E-05 0.000334 0.001906 0.085894 0.015042 4.879921 9.49796 49.3592 0.951802 6.01127E-05 0.000336 0.001893 0.085851 0.015273 4.910657 9.22545 47.8311 0.957427 5.96449E-05 0.000344 0.001998 0.079263 0.013999 4.818668 9.22574 47.8328 0.957424 5.96443E-05 0.000344 0.001997 0.079269 0.013999 4.818586 1.89294 12.5524 0.915323 2.92544E-05 0.000210914 0.001549 0.097286 0.013145 4.919283 1.95122 12.9865 0.911438 2.94452E-05 0.000209141 0.001504 0.092130 0.012672 4.798940 2.00198 13.3663 0.907896 2.99284E-05 0.000210419 0.00149255 0.087992 0.012272 4.740070 2.06997 13.8559 0.903325 3.10E-05 2.16229E-04 0.001519 0.083206 0.011821 4.734614 2.06995 13.8558 0.903326 3.10E-05 2.1601E-04 0.001519 0.083208 0.011820 4.734610 Nickel on peat Redlich-Peterson: KR Redlich-Peterson: aR Redlich-Peterson: bR Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Lead on peat Redlich-Peterson: KR 265.5835 276.3955 285.7632 302.5026 271.9999 Redlich-Peterson: aR 434.5717 451.5460 466.1403 498.4159 448.7846 Redlich-Peterson: bR 0.916566 0.914918 0.913461 0.913740 0.918845 Sum of errors2 0.000283423 0.000286278 0.000293306 0.000438201 0.00033956 Hybrid error function 0.000556362 0.000549559 0.000554329 0.000818418 0.000662247 Marquardt’s PSD 0.001115482 0.001071412 0.001060007 0.001533878 0.001308956 Average relative error 0.081038781 0.079921612 0.079218564 0.074622077 0.074886167 Sum of absolute errors 0.040161059 0.040480136 0.040839923 0.038825702 0.037307825 Sum of normalised errors 4.037198 4.000696 4.015261 4.871499 4.275031 of least squares. Overall, the HYBRD error analysis method gives the minimum sum of the normalised errors for all three metals. 21 EQUILIBRIUM ISOTHERM STUDIES Figure 6. Temkin equation isotherms of metal ions sorbed on peat. TABLE XI Linear Temkin isotherm parameters Temkin: AT (dm3 mmol−1 ) Temkin: RT /bT Temkin: bT Coefficient of determination, R 2 Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Lead Nickel Copper 0.056137 59997.08 0.041318 0.990944 0.000618 0.001317 0.003005 0.138397 0.065069 0.025044 241.9233 10.24692 0.983955 7.67066E-05 0.000581 0.004682 0.176790 0.023040 0.021799 5285.085 0.469050 0.943494 0.000430 0.002669 0.017900 0.328875 0.053543 22 TABLE XII Non-linear Temkin isotherm parameters Method/error function (parameter set) ERRSQ HYBRD MPSD ARE EABS Copper on peat 0.0218325 5219.68 2.105586 0.000408 0.002545 0.017166 0.307380 0.049539 2.834586 0.022752 3716.99 1.49941 0.000422 0.002425 0.014554 0.297195 0.050282 2.786959 0.0233999 2991.34 1.206688 0.000446 0.002480 0.014055 0.292891 0.051267 2.824536 0.024103 2302.69 0.928891 0.000502 0.002761 0.015283 0.275246 0.050444 2.880696 0.016604 72484.2 29.23967 0.000839 0.007709 0.073524 0.375712 0.047456 4.925654 Nickel on peat Temkin: AT Temkin: RT /bT Temkin: bT Sum of errors2 0.025044 241.918 0.097588 7.67048E-05 0.025509 218.659 0.088206 7.83658E-05 0.025867 203.352 0.082031 8.18064E-05 0.026223 185.301 0.074749 9.03253E-05 0.024994 225.4382 0.090940 0.00010974 Y. S. HO ET AL. Temkin: AT Temkin: RT /bT Temkin: bT Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors TABLE XII (continued) Method/error function (parameter set) ERRSQ HYBRD MPSD ARE EABS Nickel on peat (continued) 0.000581 0.004682 0.176788 0.023040 4.195716 0.000563 0.004193 0.174351 0.023452 4.107515 0.000573 0.004089 0.172633 0.023770 4.136505 0.000634 0.004477 0.169602 0.024086 4.353279 0.000793 0.005801 0.172528 0.022996 4.930682 0.056136 60004.69 0.041313 0.000618 0.001317 0.003004 0.138380 0.065063 4.150878 0.056895 53354.24 0.046462 0.000631 0.001284 0.002769 0.140198 0.066898 4.123511 0.057600 48180.58 0.051452 0.000664 0.001310 0.002704 0.140898 0.068112 4.179681 0.058683 44731.00 0.055419 0.000930 0.001746 0.003442 0.123538 0.060851 4.680385 0.055292 70475.00 0.035175 0.000655 0.001512 0.003781 0.130443 0.060093 4.378426 Lead on peat Temkin: AT Temkin: RT /bT Temkin: bT Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors EQUILIBRIUM ISOTHERM STUDIES Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors 23 24 Y. S. HO ET AL. Figure 7. Dubinin-Radushkevich equation isotherms of metal ions sorbed on peat. 4.5. T HE D UBININ -R ADUSHKEVICH ISOTHERM The linear Dubinin-Radushkevich isotherm plots for the sorption of the three metals onto peat are presented in Figure 7 and examination of the data shows that the Dubinin-Radushkevich isotherm provides a very accurate description of the data for copper ions over the concentration range studied. However, for the sorption of lead and nickel the experimental data do not correlate as well with the DubininRadushkevich equation and this is confirmed by the coefficients of determination shown in Table XIII. The values of E calculated using Equation (6) are 7.7, 4.9 and 11.5 kJ mol−1 for copper, nickel and lead, respectively. The typical range of bonding energy for ion-exchange mechanisms is 8–16 kJ mol−1 , indicating that chemisorption may play a significant role in the adsorption process. The qD values are consistent with the non-linear qm values previously determined for the Langmuir isotherm. 25 EQUILIBRIUM ISOTHERM STUDIES TABLE XIII Linear Dubinin-Radushkevich isotherm parameters D–R: qD D–R: BD × R 2 × T 2 D–R: BD D–R: E (J mol−1 ) Coefficient of determination, R 2 Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Lead Nickel Copper 0.581122 0.023172 3.77071E-09 11515.25 0.991634 0.000709 0.001428 0.002945 0.131666 0.063628 0.152386 0.128840 2.09657E-08 4883.493 0.967911 0.000201 0.001532 0.012011 0.241464 0.031340 0.194962 0.051478 8.37683E-09 7725.830 0.993349 7.27602E-05 0.000428 0.002550 0.107694 0.018612 Table XIV shows the best fit qD and BD parameters for the five error methods – the qD values are remarkably consistent for each metal, with the exception of the EABS set for lead. The values are also generally close to those for the linear parameter set, particularly those determined using the MPSD error method which provides the second or third best fit of the non-linear parameter sets. Based on the sum of the normalised errors, the HYBRD parameter set produces the best fit for nickel, whereas the ARE set produces the best fit for copper and lead. 4.6. T HE T OTH ISOTHERM This three parameter isotherm has been analysed by the five error methods and the values of the constants are presented in Table XV. Once again, similar to several of the preceding isotherm results, the parameter values do not vary significantly across the range of error methods. The sums of the normalised errors indicate that the closest fits to the copper, nickel and lead experimental data are produced by the ARE/EABS, MPSD and ERRSQ parameter sets respectively. 4.7. T HE L ANGMUIR -F REUNDLICH FORM OF THE S IPS EQUATION Table XVI shows the values of the Langmuir-Freundlich constants using the five analysis methods. While the three constants KLF , aLF and nLF are quite similar across the range of error functions, there appears to be two distinct groupings according to the parameter values obtained. This appears to be related to the form of the error function since the first three error methods (ERRSQ, HYBRD, MPSD) employ the square 26 Y. S. HO ET AL. TABLE XIV Non-linear Dubinin-Raduskevich isotherm parameters Method/error function (parameter set) ERRSQ HYBRD MPSD ARE EABS 0.195388 0.052755 6.53952E-05 0.000408 0.002638 0.104990 0.016797 4.856741 0.195133 0.052014 6.63078E-05 0.000400 0.002459 0.099644 0.016518 4.715189 0.194880 0.051490 6.82482E-05 0.000404 0.002422 0.096535 0.016434 4.703748 0.195526 0.051068 7.18628E-05 0.000431 0.002597 0.085699 0.014994 4.693398 0.195526 0.051068 7.18628E-05 0.000431 0.002597 0.085699 0.014994 4.693398 D–R: qD D–R: BD × R 2 × T 2 Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Lead 0.153637 0.138301 0.000187 0.001552 0.013622 0.256926 0.031074 4.737768 0.152857 0.132932 0.000191 0.0015033 0.012280 0.246658 0.030991 4.589190 0.152067 0.128966 0.000202 0.001534 0.011972 0.249805 0.032537 4.695822 0.152546 0.124010 0.000218 0.001674 0.013065 0.220184 0.029555 4.724451 0.152546 0.124010 0.000218 0.001674 0.013065 0.220184 0.029555 4.724451| D–R: qD D–R: BD × R 2 × T 2 Sum of errors2 Hybrid errorfunction Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors 0.583958 0.023824 0.000668 0.001437 0.003213 0.140608 0.064213 4.629516 0.582399 0.023469 0.000680 0.001407 0.003000 0.135813 0.063927 4.527346 0.580763 0.023164 0.000713 0.001430 0.002943 0.132621 0.064125 4.546646 0.585493 0.023152 0.000794 0.001656 0.003489 0.117059 0.056780 4.716763 0.585494 0.023152 0.000794 0.001656 0.003489 0.117059 0.056780 4.716763 Copper D–R: qD D–R: BD × R 2 × T 2 Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Nickel of the difference between the calculated and measured q values whereas the last two error methods (ARE, EABS) use the absolute value of the difference. For all three metal ion systems the MPSD error method produces the lowest sum of the normalised errors. 27 EQUILIBRIUM ISOTHERM STUDIES TABLE XV Non-linear Toth isotherm parameters Method/error function (parameter set) ERRSQ HYBRD MPSD ARE EABS 0.209239 0.063027 0.619460 5.03443E-05 0.000281 0.001583 0.089526 0.015854 4.920121 0.210009 0.064388 0.607436 5.05624E-05 0.000280 0.001556 0.087284 0.015674 4.865674 0.210752 0.065696 0.596812 5.11254E-05 0.000281 0.001548 0.085874 0.015568 4.853426 0.210941 0.065055 0.602320 5.21982E-05 0.000289 0.001610 0.081725 0.014833 4.848424 0.210941 0.065055 0.602320 5.21982E-05 0.000289 0.001610 0.081725 0.014833 4.848424 0.188969 0.184819 0.565458 2.44437E-05 0.000171471 0.001218 0.087139 0.012056 4.871167 0.190549 0.186288 0.552727 2.45E-05 0.000170654 0.001197 0.083659 0.011735 4.786564 0.192083 0.187806 0.541474 2.47604E-05 0.000171258 0.001192 0.080836 0.011461 4.739218 0.191520 0.186323 0.538451 2.61037E-05 0.000180 0.001242 0.077666 0.011219 4.821852 0.191520 0.186323 0.538451 2.61037E-05 0.000180 0.001242 0.077666 0.011219 4.821852 0.739442 0.062242 0.289165 0.000132747 0.000270147 0.000563315 0.058308 0.028105 4.650009 0.742158 0.062698 0.286650 0.00013288 0.000269824 0.000561862 0.058791 0.028298 4.662527 0.743244 0.062885 0.285637 0.000133035 0.000269893 0.000562170 0.058879 0.028329 4.666122 0.751869 0.064666 0.278049 0.000150 0.000303 0.000633 0.054741 0.026666 4.871044 0.751869 0.064666 0.278049 0.000150 0.000303 0.000633 0.054741 0.026666 4.871044 Copper Toth Kt Toth: at Toth: t Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Nickel Toth: Kt Toth: at Toth: t Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Lead Toth: Kt Toth: at Toth: t Sum of errors2 Hybrid error function Marquardt’s PSD Average Relativeerror Sum of absolute errors Sum of normalised errors 28 Y. S. HO ET AL. TABLE XVI Non-linear Langmuir-Freundlich isotherm parameters Method/error function (parameter set) ERRSQ HYBRD MPSD ARE EABS 2.14991 10.3345 0.673314 3.82427E-05 2.14742E-04 0.0012159 0.0712637 0.0124994 4.1194485 2.08274 9.98363 0.664207 3.8412E-05 2.13607E-04 0.0011946 0.0690375 0.0122898 4.0577340 2.02256 9.66912 0.656161 3.8852E-05 2.14466E-04 0.0011891 0.0675713 0.0121539 4.0334520 1.59041 7.42852 0.59441 5.69E-05 3.05038E-04 0.0016369 0.0604497 0.0113882 4.7593535 1.58715 7.40996 0.593951 5.67691E-05 3.04548E-04 0.0016340 0.0605878 0.0113613 4.7543318 0.592046 3.22598 0.662162 2.3424E-05 1.63176E-04 0.0011490 0.0845587 0.0117719 4.6767189 0.580038 3.14329 0.653107 2.34948E-05 1.62524E-04 0.0011323 0.0814457 0.0114842 4.6011570 0.569123 3.06809 0.645185 2.36756E-05 1.63004E-04 0.0011280 0.0789199 0.0112394 4.5563464 0.516854 2.69653 0.611076 2.70E-05 1.83503E-04 0.0012473 0.0715295 0.0103810 4.7277609 0.516854 2.69653 0.611076 2.70E-05 1.83503E-04 0.0012473 0.0715295 0.0103810 4.7277609 4.170931 5.966065 0.380987 1.25349E-04 2.6168E-04 5.63066E-04 0.059261 0.027873 4.506014 4.149833 5.930056 0.380112 1.25375E-04 2.61623E-04 5.62975E-04 0.059255 0.027851 4.504986 4.155374 5.940039 0.380370 1.25389E-04 2.61644E-04 5.62913E-04 0.0592190 0.0278629 4.5048663 3.867534 5.451214 0.370238 1.49E-04 3.13E-04 6.82E-04 0.054393 0.026122 4.855024 3.867534 5.451214 0.370238 1.49E-04 3.13E-04 6.82E-04 0.054393 0.026122 4.855024 Copper on peat Langmuir-Freundlich: KLF Langmuir-Freundlich: aLF Langmuir-Freundlich: nLF Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Nickel on peat Langmuir-Freundlich: KLF Langmuir-Freundlich: aLF Langmuir-Freundlich: nLF Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors Lead on peat Langmuir-Freundlich: KLF Langmuir-Freundlich: aLF Langmuir-Freundlich: nLF Sum of errors2 Hybrid error function Marquardt’s PSD Average relative error Sum of absolute errors Sum of normalised errors 29 EQUILIBRIUM ISOTHERM STUDIES TABLE XVII Linear isotherm comparison – Sum of normalised errorsa Linear isotherm Lead Nickel Copper Freundlich Langmuir Temkin Dubinin-Radushkevich Redlich-Peterson 0.938344 5.000000 0.455592 0.446509 0.266096 4.608361 3.983077 1.964325 3.757032 0.871335 4.554900 3.811958 2.777875 0.697433 0.684328 a Denotes normalised across the range of linear isotherms (instead of error functions). TABLE XVIII Linear isotherm comparison – Regression coefficients of determination (R 2 ) Linear isotherm Lead Nickel Copper Freundlich Langmuir Temkin Dubinin-Radushkevich Redlich-Peterson 0.972028 0.999392 0.990944 0.991634 0.999939 0.961755 0.998830 0.983955 0.967911 0.999716 0.915671 0.999640 0.943494 0.993349 0.999821 4.8. E RROR ANALYSIS – L INEAR SYSTEMS Unsurprisingly, possibly because of the method used to determine its parameters, the data in Table XVII show that the three parameter Redlich-Peterson isotherm produced the best fit for all three metals based on the measured errors. For both the lead and nickel equilibrium data, the linear parameter set for the Temkin isotherm provides the second closest fit over all the error methods. For copper, the linear Dubinin-Radushkevich parameters produce the second closest fit. The linear parameter sets providing the worst fit over all the error methods were the Langmuir, Freundlich and Freundlich isotherm parameters for the lead, nickel and copper data respectively. As Table XVIII shows, the highest coefficients of determination (R 2 ) were obtained for all three systems using the Redlich-Peterson isotherm, with the Langmuir isotherm generally producing a close second. Excepting the Langmuir and Redlich-Peterson isotherms, the R 2 values appear to give a reasonable indication of the relative quality of fit of the linear isotherm based on the sums of normalised errors. A possible explanation lies in the fact that linearisation of these three remaining isotherms requires taking logarithms, introducing similar effects to the error structure. Taking logarithms may also lead 30 Y. S. HO ET AL. to better fits at the extremes of concentrations since linear regression implicitly minimises the sum of the squares of the errors to determine the equation parameters and thus the error in a value of 103 will have a similar weighting to the error in a value of 10−3 . 4.9. E RROR ANALYSIS – N ON - LINEAR SYSTEMS Based on the non-linear regression results, selection of the linear isotherm transformation with the highest linear regression coefficient of determination does not appear to be the most appropriate method to choose a model for sorption equilibria. Based on any of the error measures, better fits can be obtained for most two parameter isotherms by using non-linear regression – although, except for the Langmuir and Temkin isotherms, the linear parameters do provide reasonably close estimates to the optimised non-linear solutions for all three metal ion systems. Using the normalised error criteria presented here to combine non-linear errors demonstrates that the parameters obtained by linearisation are not bad and in certain cases better than the values derived by non-linear regression. For the four two-parameter isotherms examined, the HYBRD error measure produced the parameter set providing the lowest sum of normalised errors in nine out of the twelve systems (examining four isotherms for each of three metals) and the MPSD error measure was second best for six of the twelve systems. For the three parameter isotherms, the MPSD error measure produced the parameter set providing the lowest sum of normalised errors in four out of the nine systems with the HYBRD error measure giving the second lowest value in each case an therefore providing the next best fit isotherm constants. The three parameter isotherms were always found to provide a better match to the experimental data than the two parameter isotherms and the incorporation of the degrees of freedom of the system – the number of data points minus the number of isotherm parameters – as a divisor in the error function did not alter this. The Sips or Langmuir-Freundlich isotherm equation employing the parameter set derived using the MPSD error provided the best model overall for all three experimental systems. 5. Conclusions A detailed error analysis was carried out to determine the best isotherm models for sets of equilibrium sorption data for three metal ions on peat. In all cases the three parameter Sips – or Langmuir-Freundlich – isotherm equation was found to provide the closest fit to the equilibrium data and the optimum parameter values were produced by non-linear regression using the MPSD error function. This optimum isotherm-error function combination was identified using the sum of normalised errors. EQUILIBRIUM ISOTHERM STUDIES 31 Regarding the results presented for the two parameter isotherms examined, several comments can be made: • excepting the Langmuir isotherm, the order of linear coefficients of determination was observed to provide a good indication of the relative ranking of the linear isotherm fits; • excepting the Langmuir and Temkin isotherms, the linear derived parameters produced reasonable estimates of the values derived by non-linear regression and were generally closest to those determined using the MPSD error function; and • the HYBRD error function generally appears to produce the best fit isotherm parameter values for two parameter isotherms. For the three parameter isotherms examined the MPSD error function generally produces isotherm parameter values that give the lowest sum of normalised errors, most often followed by the HYBRD error function. 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