Prediction of Equillibrium Sorption Isotherm: Comparison of Linear

Feb 10, 2010 - A comparison of the linear least-squares method and a trial-and-error nonlinear method of the four isotherms was done. The Langmuir iso...
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Ind. Eng. Chem. Res. 2010, 49, 2882–2888

Prediction of Equillibrium Sorption Isotherm: Comparison of Linear and Nonlinear Methods Siddharth Parimal,† Murari Prasad,*,‡ and Ujjwal Bhaskar† AdVanced Materials and Processes Research Institute (C.S.I.R.), Hoshangabad Road, Bhopal-462064 (India), and Department of Chemical Engineering, B.I.T.S., Pilani, India

Sorption of three divalent toxic metal ions, viz. copper, lead, and zinc, onto a low-cost aluminosilicate mineral, pyrophyllite, was studied in a single component system at four different temperatures. Four isothermssFreundlich, Langmuir, Redlich-Peterson, and Temkinswere examined in order to test their suitability for the given sets of experimental data. A comparison of the linear least-squares method and a trial-and-error nonlinear method of the four isotherms was done. The Langmuir isotherm had four different linear forms, and all of them were individually studied and compared with the others. The parameters of all the four forms of the Langmuir isotherm obtained by the linear method were different. Langmuir-I is the most popular form of the Langmuir isotherm having the highest coefficient of determination values (0.996886) compared with other linear forms of the Langmuir isotherm. The Redlich-Peterson isotherm produced a higher coefficient of determination values (0.999234) in comparison to all the isotherms studied. 1. Introduction The present study explores the avenues for the application of waste minerals generated, for sorption of toxic divalent ions. This study aims at finding a suitable isotherm that describes the adsorption process in the case of pyrophyllite. The adsorption process, in general, encompasses various mechanisms such as external mass transfer of solute, intraparticle diffusion, liquid film diffusion, and complexation. Unless extensive experimental data are available concerning the specific adsorption application, determining the rate-controlling step is impossible. Therefore, empirical design procedures based on adsorption equilibrium data are the most common method to predict adsorber size and performance. The different parameters and the underlying thermodynamic assumption of these equilibrium models often provide some insight into both the adsorption mechanism and the surface properties and affinity of adsorbent.1 The most appropriate method in designing the adsorption systems and assessing the performance of the adsorption systems is to have an idea about the adsorption isotherms. Sorption isotherms predict the equilibrium relationships between sorbates and sorbents at a fixed temperature and pH. Batch reactions at fixed temperatures can be studied to obtain data to calculate parameters for these isotherms. The most commonly used method to determine the best-fitting sorption isotherm is linear regression. The linear regression method with linearly transformed isotherm equations has been widely used to confirm experimental data and isotherms using coefficients of determination, R2. In general, for the cases involving most of the two parameter isotherms, the magnitude of the coefficient of linear regression is the most commonly used parameter for estimating the accuracy of the fit of an isotherm model to experimental equilibrium data. In such a case, an isotherm giving an R2 value closest to unity is accepted as the best fit isotherm in that particular case. * To whom correspondence should be addressed. E-mail: [email protected]; [email protected]. Telephone no.: 91-755-2457105. Fax no.: 91-755-2457042/2488985/2488323. † B.I.T.S. ‡ Advanced Materials and Processes Research Institute (C.S.I.R.).

However, such transformations of nonlinear isotherm equations to linear forms bring alterations in their error structure and may also violate the error variance and normality assumptions of standard least-squares.2,3 So, it will be an inappropriate technique to use the linearization method for estimating the equilibrium isotherm parameters. Instead of using linear methods, nonlinear optimization provides for a more complex, yet rigorous method for isotherm parameter determination. However, this requires an error function assessment, in order to evaluate the fit of the isotherm to the experimental results. In recent years, several error analysis methods, such as coefficients of determination, the sum of errors squared, a hybrid error function, Marquardt’s percent standard deviation, the average relative error, the sum of the absolute errors, and Chisquare have been used to determine the best-fitting isotherm equation by several investigators.4-6 In the present study, the linear least-squares method and a nonlinear method of four isotherms (Freundlich, Langmuir, Redlich-Peterson, and Temkin) were compared in an experiment examining adsorption of copper, lead, and zinc onto pyrophyllite. Data were also studied across four different temperatures. A trial-and-error procedure7-9 was used for the nonlinear method using the solVer add-in with Microsoft’s spreadsheet program, Microsoft Excel. 2. Materials and Methods 2.1. Adsorbent Materials. Representative samples of High alkali pyrophyllite were procured from M/S Eastern Mineral, Jhansi (U.P.), India. The particle size of the powder samples was in the range -45 to +75 µm. The samples (1 kg) were sized and ground separately for experimental work. The representative ground samples, taken after coning and quartering, were subjected to wet chemical analysis. An electric rotary shaking machine and Systronic digital pH meter were used for equilibration and pH measurement of the solutions. For all the three heavy metals, 0.5 g sample was used as adsorbent. 2.2. Methods. Stock lead, copper, and zinc ion solutions (1000 mg L-1 each) were prepared from analytical grade lead nitrate, copper nitrate, and zinc nitrate using double distilled water and serially diluted to prepare solutions of varying initial

10.1021/ie9013343  2010 American Chemical Society Published on Web 02/10/2010

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Table 1. Isotherms and Their Linear Forms isotherm Qe ) kFCe

1/n

Freundlich

Langmuir-1

Qe )

linear form

plot

log(Qe) ) log(kF) + 1/nlog(Ce)

log(Qe)vs log(Ce)

Ce Ce 1 ) + Qe Qs KaQs

Ce vs Ce Qc

QsKaCe 1 + KaCe

( )

1 1 1 1 ) + Qe KaQs Ce Qs

Langmuir-2

( )

1 Qe Ka Ce

Langmuir-3

Qe ) Qs -

Langmuir-4

Qe ) KaQs - KaQe Ce

Redlich-Peterson

Temkin

Qe )

Qe )

(

ACe

ln A

1 + BCe

s

ln(aTC ) ( RT bT ) e

concentration for experimental work. Pyrophyllite mineral samples were equilibrated separately with 100 mL solutions of different concentrations of copper, lead, and zinc ions as described elsewhere.10,11 The suspensions were shaken on a mechanical shaker at 25 °C for 40 min. Samples were collected after 5 min intervals from the suspensions and then filtered

)

Ce - 1 ) g ln(Ce) + ln(B) Qe

Qe )

1 1 vs Qe Ce

RT RT ln(aT) + ln(Ce) bT bT

Qe vs

Qe Ce

Qe vs Qe Ce

(

ln A

)

Qe - 1 vs ln(Ce) Ce Qe vs ln(Ce)

through Whatman filter paper (No. 42). The filtrates were analyzed for Cu2+, Pb2+, and Zn2+ concentrations using an atomic absorption spectrophotometer (GBC make model no. 902). Samples were collected at regular intervals of 5 min. For arriving at the adsorption equilibrium, the sampling time was extended up to 40 min. Experiments were repeated for different

Figure 1. Langmuir isotherms obtained for sorption of Cu(II) at different temperatures: (a) Langmuir-1, (b) Langmuir-2, (c) Langmuir-3, and (d) Langmuir-4.

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Figure 2. Langmuir isotherms obtained for sorption of Pb(II) at different temperatures: (a) Langmuir-1, (b) Langmuir-2, (c) Langmuir-3, and (d) Langmuir-4.

Figure 3. Langmuir isotherms obtained for sorption of Zn(II) at different temperatures: (a) Langmuir-1, (b) Langmuir-2, (c) Langmuir-3, and (d) Langmuir-4.

37.96643 0.104209 0.997486 40.20268 0.084256 0.975623 39.06337 0.092967 0.928145 39.92568 0.086287 0.928145 0.295258 10.02757 0.898888 1.372523 0.011458 2.260462 0.999429 1.356297 7.551094 0.932838 38.45303 0.055674 0.978607 44.84539 0.037863 0.934119 38.84184 0.054305 0.713773 44.49274 0.038761 0.713773 0.410974 5.645084 0.852022 2.768262 1.36 × 10-5 1.087 0.992284 0.464198 9.020012 0.898436 35.08897 0.043801 0.973269 39.0534 0.033961 0.930998 34.23189 0.047201 0.713541 39.31314 0.03368 0.713541 0.421396 4.555948 0.869863 3.187184 1.82 × 10-6 0.829234 0.98832 0.3736 8.203258 0.909619 83.39844 0.058558 0.874592 61.1227 0.128795 0.860694 64.59869 0.11802 0.63279 77.26306 0.074682 0.63279 0.440642 11.42018 0.927579 0.559358 70516098 8.05 × 108 0.953788 0.679434 17.29411 0.876052 73.34382 0.05035 0.944109 60.52497 0.078954 0.924265 63.14993 0.073386 0.765995 70.73375 0.056213 0.765995 0.465421 8.699325 0.96303 0.534579 61714090 5.37 × 108 0.971724 0.481678 16.37517 0.933514 65.40354 0.04439 0.939589 55.51969 0.064852 0.920312 57.33507 0.061793 0.750155 64.92382 0.046354 0.750155 0.467486 7.294901 0.95139 0.532514 4.27 × 108 3.11 × 109 0.962114 0.406385 14.89075 0.92433 37.88447 0.105943 0.999521 38.64112 0.0978 0.997969 38.42882 0.09972 0.99396 38.4928 0.099118 0.99396 0.285295 10.39176 0.952274 1.086788 0.057492 3.231541 0.999991 1.479553 7.378333 0.974507 52.11902 0.108339 0.992015 58.55884 0.076508 0.952495 53.94964 0.096387 0.832165 57.5799 0.08021 0.832165 0.348739 11.70886 0.863749 1.890776 0.001411 2.860092 0.997383 1.078035 11.2793 0.911797 47.34901 0.089889 0.998151 49.88849 0.076367 0.992081 48.70118 0.081748 0.971251 49.18961 0.079398 0.971251 0.351064 9.934011 0.934775 1.248642 0.021127 2.967657 0.999903 0.896011 10.24195 0.968524 39.48787 0.081893 0.996292 42.16696 0.06628 0.981185 40.78734 0.073192 0.937442 41.67071 0.068614 0.937442 0.33745 8.464142 0.916021 1.412123 0.007872 1.939367 0.999733 0.838676 8.492755 0.949419 37.05689 0.059074 0.996886 39.12086 0.050677 0.986198 37.87298 0.055317 0.952426 38.54971 0.052685 0.952426 0.37478 6.306979 0.939711 1.428039 0.005578 1.419762 0.999234 0.538395 8.359262 0.970996 Temkin

Redlich-Peterson

Freundlich

Langmuir-4

Langmuir-3

Langmuir-2

zinc 303 333 323 lead 313

and Qcalc is the equilibrium capacity obtained from the isotherm model, Qe is the equilibrium capacity obtained from experiment, and Qe is the average of Qe. Isotherms equations can either be used in their original forms (which are generally nonlinear), or they can be converted to a linear form and then the coefficient of determination is calculated. Both of these methods are described below, and they have been used to obtain parameters for different isotherms in order to find the best suited isotherm. 3.1. Linear Method. This method is frequently used where all the isotherms are linearized, and then, the method of leastsquares is employed to obtain the best-fit parameters for each isotherm. Freundlich, Redlich-Peterson, and Temkin isotherms are easily linearized. Langmuir isotherm can be linearized in four different ways as shown in Table 1, and the parameters were calculated for all the forms. It is obvious that due to the transformations into different linear forms, the error structures are altered in different ways in all the forms and, therefore, the results obtained would be different for all of them. Langmuir-1 and Langmuir-2 forms showed a clear advantage over Langmuir-3 and Langmuir-4 forms as shown by their higher coefficients of determination leading to minimal deviations from the fitted equation and the best error distribution.13 Figures 1a-d to 3a-d show the four linear Langmuir isotherms with the experimental data for the sorption of copper, lead, and zinc onto pyrophyllite at various temperatures, viz. 303, 313, 323, and 333 K. The parameters, Qs, saturated monolayer sorption capacity, and Ka, sorption equilibrium constant, and values of Langmuir constants were calculated for all the forms of isotherms and listed in Table 2. The values of coefficient of determination, R2, obtained for Langmuir-1 at all the four temperatures indicate that sorption of copper strictly follows the Langmuir isotherm followed by lead and zinc. An interesting observation from the results was that for copper and lead, the isotherms were a better fit at lower temperatures and the coefficient of determination values decreased at the higher temperature ranges. For zinc, all the isotherms were a better fit at higher temperature ranges which is evident from the increasing values of the coefficient of determination with temperature. It is evident that transformation of the nonlinear Langmuir isotherm equation to linear forms has significantly affected Langmuir parameter calculation. All the four forms of Langmuir isotherms have been compared for the sorption of copper, lead, and zinc onto pyrophyllite at 303 K as shown in Figures 4-6. It appears that Langmuir-1 isotherm provided a better fit to the experimental data in all the three cases of copper, lead, and zinc sorption. The plot of log(Qe) versus log(Ce) was plotted, and the parameters were calculated after regression for analyzing the applicability of the Freundlich sorption isotherm. The data are tabulated in Table 2. When the coefficient of determination of

313

(1)

303

- Qe)2

333

calc

323

∑ (Q

copper

- Qe)2 +

313

calc

303

∑ (Q

- Qe)2

T (K)

calc

Isotherm

∑ (Q

Table 2. Isotherm Parameters Obtained Using the Linear Method

R ) 2

323

In this study, the coefficient of determination, R2, was used to test the fitting of equilibrium isotherms12 for a given set of experimental data in a particular temperature range:

Qs (mg/g) Ka (dm3/mg) R2 Qs (mg/g) Ka (dm3/mg) R2 Qs (mg/g) Ka (dm3/mg) R2 Qs (mg/g) Ka (dm3/mg) R2 1/n kF (mg/g)(dm3/mg)(1/n) R2 g B (dm3/mg)g A (dm3/g) R2 aT RT/bT R2

3. Results and Discussion

Langmuir-1

333

initial lead, copper, and zinc ion concentrations. Equilibrium was reached within 30 min in most of the cases.

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40.08508 0.132794 0.998242 41.99345 0.109037 0.987796 41.33901 0.115513 0.964473 41.74473 0.111409 0.964473 0.273178 12.0561 0.911352 1.209383 0.033109 3.357182 0.999995 1.996759 7.602935 0.94021

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Figure 4. Langmuir isotherm obtained for sorption of Cu(II) on pyrophyllite at 303 K using the linear method. Figure 7. Isotherms for the sorption of Cu(II) onto pyrophyllite using the linear method.

Figure 5. Langmuir isotherm obtained for sorption of Pb(II) on pyrophyllite at 303 K using the linear method.

Figure 6. Langmuir isotherm obtained for sorption of Zn(II) on pyrophyllite at 303 K using the linear method.

the Freundlich isotherm was compared with those obtained from other isotherms, viz. Langmuir-1, the Freundlich isotherm was found to be a less suitable fit for the experimental data. The coefficients of determination were much higher for Langmuir-1 and Langmuir-2. However, when compared with Langmuir-3

and Langmuir-4, the results were inconclusive on account of coefficients of determination that did not show a consistent pattern. The Temkin isotherm was generally a slightly better fit than the Freundlich isotherm. The parameters for the Temkin isotherm were calculated by plotting Qe versus ln(Ce). The only 3-parameter isotherm employed to fit the experimental data was the Redlich-Peterson isotherm in the linear form of Table 1. The parameters A, B, and g, and the coefficient of determination, R2, were calculated for the entire data set. The R2 value was invariably the highest in the case of the Redlich-Peterson isotherm as compared to any other isotherm, making it the best fit isotherm. From the values tabulated in Table 2, it can be observed that the “g” values are not close to unity. Hence, the isotherm is more similar to the Freundlich isotherm than the Langmuir isotherm. However, the Freundlich isotherm alone was not better suited than Langmuir, as inferred earlier from their corresponding values of R2. This could be attributed to the fact that for the calculation of parameters of Redlich-Peterson, a trial and error method was employed for obtaining one of the parameters and then the other parameters were obtained by the linear regression analysis. The comparison of theoretical Langmuir isotherm, the empirical Freundlich isotherm, and Redlich-Peterson and Temkin isotherms with experimental data for sorption of copper ions onto pyrophyllite at 303 K has been shown in Figure 7. It is observed that the Redlich-Peterson and Langmuir isotherms approach each other even though these isotherms have high values of R2. 3.2. Nonlinear Method. The linear regression methods, in general, have problems associated with their error structure. This variation in error distribution has been attributed to different axial settings which transform the dependent variables to different axial positions.14-16 Hence it is advisible to employ nonlinear regression techniques which are free from such axial distribution errors. For a nonlinear method, the isotherm parameters were determined using the optimization procedure employed by the solVer add-in with Microsoft’s spreadsheet program, Microsoft Excel. The coefficients of determination, R2, values were maximized by allowing the values of different parameters for all the isotherms to change according to the optimization procedure. Table 3 shows the values of param-

40.90936 0.121146 0.999926 0.249347 13.09387 0.999606 1.204012 0.034125 3.38116 1 1.988845 7.604771 0.999749 38.60451 0.097854 0.999875 0.268591 11.02439 0.999568 1.277019 0.018807 2.455299 0.999972 1.350372 7.55347 0.999713 38.93644 0.05487 0.999505 0.356516 6.899109 0.999059 1.738125 0.001252 1.270445 0.999838 0.460933 9.025784 0.999356 34.8069 0.045613 0.99952 0.37952 5.336286 0.999268 1.455044 0.003919 1.079613 0.999619 0.371553 8.207124 0.999461 87.92116 0.050362 0.998549 0.506527 9.321701 0.999334 0.516374 198.0913 1995.586 0.999318 0.669302 17.32039 0.998483 74.73204 0.047582 0.999334 0.482055 8.247338 0.999588 0.520689 46.72572 390.4803 0.999587 0.478478 16.38683 0.999288 66.52934 0.042401 0.999291 0.476005 7.095797 0.999428 1.334818 0.011591 2.044175 0.992805 0.403555 14.90193 0.999247 38.23324 0.10193 0.999987 0.266969 11.08847 0.999795 1.09301 0.055303 3.195351 1 1.477208 7.379144 0.999897 53.31844 0.101758 0.99965 0.305505 13.51972 0.999103 1.44661 0.009826 3.355311 0.999916 1.069897 11.28601 0.999406 47.97187 0.0864 0.999936 0.317905 11.13736 0.999575 1.209622 0.025881 3.078115 0.999996 0.893764 10.244 0.999799 40.1793 0.077466 0.999885 0.306562 9.454125 0.999572 1.334818 0.011591 2.044175 0.99999 0.835921 8.494935 0.999747 37.34006 0.058024 0.999923 0.345504 7.024353 0.999697 1.287218 0.011333 1.550989 0.999977 0.537707 8.359135 0.999848 Temkin

Redlich-Peterson

lead 313 303 333 323 copper 313 303 T (K)

Qs (mg/g) Ka (dm3/mg) R2 1/n kF (mg/g)(dm3/mg)(1/n) R2 g B (dm3/mg)g A (dm3/g) R2 aT (dm3/mg) RT/bT (mg/g) R2 Freundlich

• Linear regression analysis is an inappropriate technique to determine the best-fit isotherm to the experimental data. • For linear forms of the Langmuir isotherm, Langmuir-1 was the best suited. • Redlich-Peterson and Langmuir isotherms represent the sorption of lead, copper, and zinc green onto pyrophyllite

isotherm

4. Conclusions

Table 3. Isotherm Parameters Obtained Using the Nonlinear Method

eters obtained for all the isotherms at all the temperatures for all the metal ions, obtained by nonlinear optimization. When compared with the parameter values generated by the linear method, it was found that, for the Langmuir isotherm, the values generated by the linear method (considering all the forms) and the values obtained by the nonlinear method were almost the same. However, Langmuir-1 was found to match closest to the nonlinear estimates. For the Redlich-Peterson isotherm, there were differences between the values generated by the linear and nonlinear methods. This could be accounted for by the differences in efficiency of an iterative procedure employed in a nonlinear method and the restrictions of a linear regression. However, in both the cases, the Redlich-Peterson isotherm had the highest coefficients of determination for all data sets across all temperature ranges and, therefore, was the most suitable isotherm for predictions. The comparison of the theoretical Langmuir isotherm, the empirical Freundlich isotherm, and Redlich-Peterson and Temkin isotherms with experimental data for sorption of copper ion onto pyrophyllite at 303 K has been shown in Figure 8. In the case of the Freundlich isotherm, the nonlinear method had very high coefficients of determination as compared to the linear method. In the cases where the R2 values differed, the parameter values between the linear and nonlinear methods were also different. The Temkin isotherm showed a remarkable similarity between linear and nonlinear parameters. All parameter values were almost the same in both the cases. However, the coefficient of determination values generated in the case of the linear method were much less compared to the R2 values obtained by the nonlinear method, with a slight change in the parameters. This shows the inherent advantage of the nonlinear method over the linear method.

323

Figure 8. Isotherms for the sorption of Cu(II) onto pyrophyllite using the nonlinear method.

Langmuir

333

303

313

zinc

323

333

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well. • Both Langmuir-1 and Redlich-Peterson have R2 values very close to unity and either can be used, depending on the number of parameters. • The nonlinear method is better than the linear method. • Redlich-Peterson is the most suitable isotherm for the sorption of lead, copper, and zinc onto pyrophyllite. • For lead and copper, isotherms are a better fit at lower temperatures, and for zinc, the better fit is at higher temperatures. Acknowledgment The authors are grateful to the Director, A.M.P.R.I., Bhopal, for his kind permission and encouragement for the present research work and to publish this paper. Literature Cited (1) Vasanth Kumar, K. Comparative analysis of linear and non-linear method of estimating the sorption isotherm parameters for malachite green onto activated carbon. J. Hazard. Mater. 2006, B136, 197–202. (2) Allen, S. J.; Gan, Q.; Matthews, R.; Jhonson, P. A. Comparison of optimized isotherm models for basic dye adsorption by kudzu. Bioresour. Technol. 2003, 88, 143. (3) Freundlich, H. M. F. Uber die adsorption in Iosungen. Z. Phys. Chem.-Frankfurt 1906, 57A, 385. (4) Ho, Y. S. Selection of optimum sorption isotherm. Carbon 2004, 42, 2115. (5) Ho, Y. S. Second order kinetic model for the sorption of cadmium on tree fern: A comparison of linear and non-linear methods. Water Res. 2006, 40, 119–125.

(6) Ho, Y. S.; Porter, J. F.; McKay, G. Equilibrium isotherm studies for the sorption of divalent metal ions onto peat: Copper, nickel and lead single component systems. Water Air Soil Pollution 2002, 141, 81–86. (7) Kinniburgh, D. G. General purpose adsorption isotherms. EnViron. Sci. Technol. 1986, 20, 895. (8) Langmuir, I. The constitution and fundamental properties of solids and liquids. J. Am. Chem. Soc. 1916, 38, 2221–2295. (9) Myers, R. H. Classical and Modern Regression with Applications. PWS-KENT 1990, 444, 297–298. (10) Prasad, M.; Saxena, S. Attenuation of Divalent Toxic Metal Ions Using Natural Sericitic Pyrophyllite. J. EnViron. Manag. 2008, 88, 1273– 1279. (11) S, S.; Amritphale, M.; Prasad, S. Saxena & Navin Chandra.‘Adsorption behavior of lead ion on pyrophyllite sutface’. Main Group Metal Chem. 1999, 22, 557–565. (12) Ratkowski, D. A. Handbook of Nonlinear Regression Models; Marcel Dekker: N.Y., 1990. (13) Redlich, O.; Peterson, D. L. A useful adsorption isotherm. J. Phys. Chem. 1959, 63, 1024. (14) Temkin, M. I. Adsorption equilibrium and the kinetics of processes on non- homogeneous surfaces and in the interaction between adsorbed molecules. Zh. Fiz. Chim. 1941, 15, 296–332. (15) Vasanth Kumar, K.; Sivanesan, S. Prediction of optimum sorption isotherm: Comparison of linear and non-linear method. J. Hazard. Mater. 2005, B126, 198–201. (16) Vasanth Kumar, K.; Sivanesan, S. Comparison of linear and nonlinear method in estimating the sorption isotherm parameters for safranin onto activated carbon. J. Hazard. Mater. 2005, B123, 288–292.

ReceiVed for reView August 27, 2009 ReVised manuscript receiVed January 13, 2010 Accepted January 26, 2010 IE9013343