erros em experimentos de adsorção
- 1. Hydrometallurgy 105 (2011) 314–320 Contents lists available at ScienceDirect Hydrometallurgy journal homepage: www.elsevi e r.com/locate/hydromet Common data analysis errors in batch adsorption studies Mohammad I. El-Khaiary ⁎, Gihan F. Malash Chemical Engineering Department, Faculty of Engineering, Alexandria University, El-Hadara, Alexandria 21544, Egypt a r t i c l e i n f o a b s t r a c t Article history: Received 16 March 2010 Received in revised form 29 October 2010 Accepted 2 November 2010 Available online 21 November 2010 Keywords: Adsorption Regression Isotherm Linearization Kinetics Mechanism Many models exist for describing the experimental results of batch adsorption which are used in research to study equilibrium, kinetics, and mechanisms of adsorption. In the process of statistically analyzing the experimental data, the adsorption literature contains errors that render the results unreliable. These errors include incorrect application of theoretical models and also incorrect application of statistical analysis. Some errors are so abundant in the adsorption literature that they have actually gained credibility and mistakenly taken for granted that these are sound scientific practices. This article highlights some common errors in adsorption data analysis that are frequently found in the literature and provides suggestions for more sound practices. © 2010 Elsevier B.V. All rights reserved. 1. Introduction In the course of data analysis, statistical methods are applied to fit experimental data to models and to assess statistical inferences. There is an abundance of software for statistical calculations which come as stand-alone commercial programs, as part of commercial spread-sheets or as shareware and freeware. However, the availability of good software does not ensure a correct statistical analysis. Pre-requisites for the correct use of software are a familiarity with the basics of statistics; a knowledge of the assumptions and limitations of each statistical technique; and the ability to choose the appropriate method of analysis. Uncritical use of statistical software may lead to applying the techniques incorrectly, or even using statistical methods in cases where they are not appropriate. In the field of batch adsorption research, certain errors in data analysis keep showing up over and over again. They show up in top ranking journals and are so abundant in the adsorption literature that they have actually gained credibility. New comers to adsorption research may take it for granted that these are sound scientific practices and spread the errors even more. The objective of this article is to highlight some of these errors and to provide suggestions for more sound practices. For clarity, a few specific examples are taken as prototypes from the literature. In the choice of these examples, articles were chosen to represent several journals and authors from different countries. 2. Linearization The misuse of linearization is probably the most common error in data analysis. It originatedmany decades agowhen computerswere not yet available and it has not lost its popularity today. There are problems associated with trying to linearize an inherently nonlinear equation by use of various transformations. Themain issue when transforming data to achieve a linear equation is knowledge of the error-structure of the data and how this structure is affected by transformation. When the errors are additive on the dependent variable (Y) and satisfy the usual assumptions of normality and homo-skedasticity (equal variance) throughout the range of the data, then transforming the dependent variable with a nonlinear function can destroy the assumed distribu-tional properties. However, when the original error-structure does not satisfy these assumptions, by judicious choice of a transformation, the model can in some cases be transformed to satisfy these assumptions. For example, if the variance of errors increaseswith increasing values of Y, a square root or log transformation will often help, but if the variance of the errors decreaseswith increasing Y, then the square or exponential transformation will often be appropriate. In the absence of data about the error-structure,which is the case inmost adsorption studies, there is no point in applying linearization. Typical examples are adsorption isotherms and adsorption kinetic models. These equations are nonlinear, i.e. the observed response (de-pendent variable) does not depend linearly on the independent variable. The transformed (linearized) response function is used for the quantita-tive evaluation of the parameters by linear regression. For example, the nonlinear form of the well-known Langmuir (1916) isotherm is: qe = qmKaCe ð Þ= 1 + KaCe ð Þ ð1Þ ⁎ Corresponding author. Fax: +20 3 5921853. E-mail address: elkhaiary@gmail.com (M.I. El-Khaiary). 0304-386X/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.hydromet.2010.11.005
- 2. M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320 315 where the independent variable Ce is the equilibrium concentration (mg/L), the response qe is the amount adsorbed at equilibrium (mg/g), qm is qe for complete monolayer adsorption capacity (mg/g), and Ka is the equilibrium adsorption constant (L/mg). This nonlinear form can be mathematically manipulated and linearized to at least three linear forms as shown in Table 1 (El-Khaiary, 2008). Likewise, Table 2 (El-Khaiary et al., 2010) shows four linearized forms of the widely used pseudo-second-order kinetic model of Ho (2004) which has the following nonlinear form q = q2e kt 1 + qekt ð2Þ where k is the rate constant of pseudo-second-order adsorption (g/mg min), qe is the amount of solute adsorbed at equilibrium (mg/g), and the dependent variable q is the amount of solute adsorbed at time t (mg/g), the independent variable. These linearized forms are used extensively in adsorption literature. The structure of experimental error is transformed along with the data (in some cases leading to loss of homo-skedasticity), and a basic assumption of the least squares method (i.e. that the independent variable has only a negligible error) is sometimes violated. Moreover, the statistical tests used to check the goodness of fit will often not detect that the parameters are biased. Tables 1 and 2 present the effects of different linearizations to the models of Langmuir and Ho. The effect of linearizing Ho's equation on the accuracy of parameter estimates is graphically demonstrated in Fig. 1. This figure was constructed by applying linear and nonlinear least squares regressions to published experimental kinetic data (Kononova et al., 2007) to estimate the kinetic parameters, qe and k, then using the estimated parameters to plot Eq. (2) with the untransformed data. It is clear that linear regression of the three linearized forms of Ho's equation produced parameter estimates (and consequently curve fittings) that vary wildly from each other. It can be seen that nonlinear regression produced a better fit that is closer to the data points. This should not be surprising as there is a large body of literature that warns from the use of linearization in adsorption studies (El-Khaiary, 2008; Ho et al., 2005; Kumar, 2006; Bolster and Hornberger, 2007; Kundu and Gupta, 2006; Hamdaoui, 2006; Badertscher and Pretsch, 2006; Crini et al., 2008; Tsai and Juang, 2000). In spite of warnings in the literature, modeling using linearized equations have been published in literally thousands of adsorption papers (Liu et al., 2010b; Yuan et al., 2010; Zhang et al., 2009; Rengaraj et al., 2007; Wan Ngah and Fatinathan, 2010; Hubicki and Wołowicz, 2009; Unuabonah et al., 2008; Acharya et al., 2009; Nadeem et al., 2009; Kamal et al., 2010; Akperov et al., 2009; Abd El-Ghaffar et al., 2009; Cox et al., 2005; Atia, 2005; Agrawal and Sahu, 2006). 3. Abuse of R2 and model comparison A common practice in research is to fit the experimental data to several models, then perform some kind of test to compare and decide which model fits the data better. Based on the choice of the “best model”, conclusions about the intricate mechanism of the system are often available. The most popular tool for model comparison is the coefficient of determination, R2. It is the square of Table 1 Linearized forms of Langmuir isotherm. Type Linearized form Plot Effects of linearization LR I Ce qe = 1 Kaqm + 1 qm Ce Ce qe vs:Ce Ce in both dependent and independent variables, leading to spurious correlation The error distribution of the dependent variable, Ce/qe, is different from both the error distributions of Ce and qe Reversal of relative weights of data points because of 1/qe in dependent variable LR II 1 qe = 1 qm + 1 Kaqm 1 Ce 1 qe vs: 1 Ce Distortion of relative weights of data points because of 1/qe and 1/Ce in dependent and independent variables Independent variable is 1/Ce, leading to distortion of error distribution LR III qe Ce = Kaqm−Kaqe qe Ce vs:qe qe in both dependent and independent variables, leading to spurious correlation The relative weights of data points are distorted because the independent variable is qe/Ce The error distribution of the dependent variable, qe/Ce, is different from both the error distributions of Ce and qe Table 2 Linearized forms of the pseudo-second-order kinetic model. Type Linearized Form Plot Effects of linearization Linear 1 t q = 1 kq2e + 1 qe t t/q vs. t Reversal of relative weights of data points because of 1/q in the dependent variable t in both dependent and independent variables, leading to spurious correlation Linear 2 1 q = 1 qe + 1 kq2e 1 t 1/q vs. 1/t Reversal of relative weights of data points because of 1/q in dependent variable Independent variable is 1/t, leading to distortion of error distribution Linear 3 q = qe− 1 kqe q t q vs. q/t qin both dependent and independent variables, leading to spurious correlation The presence of q in the independent variable (q/t) introduces experimental error, violating a basic assumption in the method of least squares 1/t in independent variable, leading to distortion of error distribution Linear 4 q t = kq2e −kqeq q/t vs.q qin both dependent and independent variables, leading to spurious correlation The presence of q in the independent variable introduces experimental error, violating a basic assumption in the method of least squares 1 2 3 4 5 6 12 10 8 6 4 2 time, min q, mg Ag/g Experimental — — N o n l in e ar — — Lin 1 —. .— Lin 2 ........ Lin 3 ---- Lin 4 Fig. 1. Comparison of the pseudo-second-order parameters estimated by linear and nonlinear regressions for the adsorption of silver thiocyanate complexes on anion exchanger AV-17-10P.
- 3. 316 M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320 the correlation coefficient, r, between the dependent variable and the regression-predicted value of the dependent variable. In other words, R2 is the fraction of the total variance of the dependent variable that is explained by the model's equation. Mathematically, it is defined as: R2 = 1− SSE SStot ð3Þ where SSE is the sum-of-squared deviations of the points from the regression curve and SStot is the sum-of-squared deviations of the points from a horizontal line where Y equals the mean of all the data points. 3.1. Abuse of R2 In spite of the apparent simplicity of interpreting R2, it is not always suitable for evaluating the goodness of fit and comparing models. The most obvious issues with R2 are: • R2 is sensitive to extreme data points, resulting in misleading indication of the quality of fit. This is further complicated when the data points are subjected to linear transformations; existing extreme-points may disappear and new extreme-points may be created. • R2 is influenced by the range of the independent variable. R2 increases as the range of independent variable increases and decreases as the range decreases. • R2 can be made artificially large by adding more parameters to the model. In other words, R2 value increases with the decrease in degrees of freedom for error. The first two issues with R2 can be avoided by fitting the data to the model without any transformations and by examination of extreme-points, while the third issue is more subtle and is discussed further in Sections 3.1 and 3.2. 3.1.1. Spurious goodness of fit concluded from high R2 values when the fitting has a small degree of freedom This is especially common in adsorption studies when it comes to estimating thermodynamic parameters and it is also found in other cases such as isotherm and diffusion calculations (Wan Ngah and Hanafiah, 2008; Abd El-Ghaffar et al., 2009; Unuabonah et al., 2008; Agrawal and Sahu, 2006). To illustrate, Fig. 2 is reproduced from a recent article (Unuabonah et al., 2008) where three data points were used to plot the natural logarithm of b, the Langmuir constant related to energy, against the reciprocal absolute temperature, 1/T, to calculate ΔH and ΔS from the slope and intercept of a straight line according to the linearized equation: ln b = ΔS Rc −ΔH RcT ð4Þ where Rc is the universal gas constant (8.314 J/K mol), ΔH is the enthalpy change (J/mol) and ΔS is the entropy change (J/K mol). Linear regression analysis was performed and some of the results are shown in Table 3. The linear plot in Fig. 2 seems good, with little deviation of the points from the regression line, and also the value of R2 (which represents the proportion of the variation in log b that can be accounted for by variation in 1/T) is 0.9415, which is relatively high and may suggest an acceptable fit. However, the numbers in Table 3 tell another story. The estimated slope is highly uncertain because zero is within its 95% confidence interval, and consequently, there is no evidence that the slope is different from zero. Therefore, the apparent variation of ln b with changes in 1/T may be due to random variation (noise). By calculating the 95% confidence interval of ΔH it turns out to be from −22.5 to +50.8 kJ/mol, a very wide range that renders the estimated value of ΔH (+14.1 kJ/mol) virtually useless. The insignificance of the slope is further corroborated by the result of t-test, the significance level of this test is 0.1283 which means that there is a 12.83% probability that the apparent slope is caused by noise, and since 0.1283 is much larger than the conventional 0.0500 cut-off value, the hypothesis that the slope is zero is not rejected. The same discussion applies to the statistical insignificance of the intercept, and accord-ingly, any conclusions based on the values or signs of ΔH and ΔS are unsupported. The previous analysis does not prove that ln b is independent from 1/T, it only shows that fitting the straight line to three data points did not give enough evidence – from a statistical point of view – to support a hypothesis that there is a linear correlation between the two variables. 3.2. Comparing models that have different degrees of freedom For a fixed sample size, increasing the number of regression parameters leads to a decrease in the degrees of freedom, and almost universally decreases SSE. The value of R2, as calculated from Eq. (3), has no consideration for the degrees of freedom. Consequently, models with more regression parameters will tend to have higher R2 values. Therefore, the goodness of fit cannot be based solely on SSE (and R2) but must also include a penalty for the decrease in the degrees of freedom. It is customary in batch adsorption studies to fit the equilibrium uptake data to several isotherms, then to use R2 to compare the goodness of fit and select the best isotherm model. With the best isotherm supposedly identified, conclusions are usually presented regarding the homogeneity of the adsorbent surface and the mechanism of adsorption. However, a common pitfall is that some studies use R2 to compare isotherms that have two, three, and four parameters (Nadeem et al., 2009; Lu et al., 2009; Gunay et al., 2007; Chan et al., 2008; Wang et al., 2005; Debnath and Ghosh, 2008). Akaike's Information Criterion (AIC) (Burnham and Anderson, 2002) is a well established statistical method that can be used to compare models. It is based on information theory and maximum 0.0031 0.00315 0.0032 0.00325 0.0033 0.00335 -5.3 -5.4 -5.5 -5.6 -5.7 -5.8 1/T (K-1) ln b Fig. 2. Plot of ln b vs. 1/T for cadmium adsorption onto unmodified kaolinite clay. Table 3 Linear regression results of the experimental data shown in Fig. 2. Intercept 95% confidence limits of intercept Slope 95% confidence limits of slope Probability value for slope (t-test) R2 −0.12 −14.30 to 14.07 −1698 −6106 to 2711 0.1283 0.9415
- 4. M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320 317 Table 4 Estimated parameter-values for Langmuir, Freundlich, and Redlich–Peterson isotherms. Langmuir isotherm Freundlich isotherm Redlich–Peterson isotherm qm (mg/g) likelihood theory, and as such, it determines which model is more likely to be correct and quantifies how much more likely. For a small sample size, AIC is calculated for each model from the equation: AIC = N ln SSE N + 2Np + 2Np Np + 1 N−Np−1 ð5Þ where N is the number of data points, and Np is the number of parameters in the model. AIC values can be compared using the Evidence ratio which is defined by: Evidence ratio = 1 e−0:5Δ ð6Þ where Δ is the absolute value of the difference in AIC between the two models. This comparison method is illustrated using isotherm data from a recent study (Gunay et al., 2007) where several two and three-parameter isotherms were fitted to the data. The two-parameter models are Langmuir and Freundlich (1906) isotherms, while the three-parameter model is Redlich and Peterson (1959) isotherm. Freundlich isotherm : qe = Kf C1 = n e ð7Þ Redlich–Peterson isotherm : qe = KRCe 1 + aRCβ e ð8Þ The results of nonlinear regression, published in the study, are presented in Table 4. The study concluded, on the basis of R2 comparison, that the three-parameter isotherm is a better fit. However, AIC would be a more sound method to compare the goodness of fit to Langmuir and Redlich–Peterson isotherms. Accordingly, AIC values were calculated for Langmuir (1.521) and Redlich–Peterson (6.330) isotherms. Having a smaller AIC value suggests that Langmuir isotherm is more likely to be a better fit. The Evidence ratio of 11.07 means that it is 11.07 times more likely to be the correct model than the Redlich–Peterson isotherm. 4. Incorrect application of models 4.1. Incorrect application of Webber's pore-diffusion model Webber's pore-diffusion model (Weber and Morris, 1963) is commonly used in adsorption studies. It is defined by the equation: q = kit0:5 + c ð9Þ where ki (mg/g min0.5) is the pore-diffusion parameter, and c (mg/g) is an arbitrary constant. It can be seen from Eq. (9) that if pore-diffusion is the rate limiting step in the adsorption process, then a pore-diffusion plot (q vs t0.5) is expected to be a straight line with a slope that equals ki. In practice, things are not that simple because pore-diffusion plots often show several linear segments. It has been proposed that these linear segments represent pore-diffusion in pores of progressively smaller sizes (Ho and McKay, 1998; Allen et al., 1989, 2005; Koumanova et al., 2003; Cheung et al., 2007). Eventually, equilibrium is reached and q stops changing with time; and a final horizontal line is established at qe. It follows from the previous discussion that it would be a good practice to examine pore-diffusion plots and decide how many linear segments exist. When a group of points are identified as belonging to a linear segment, linear regression can then be applied to these points and the corresponding ki is estimated. In some cases the linear segments are strikingly obvious, but in others they are obscured and/ or a group of points may form a curved segment. What a researcher may do when faced with uncertainty in identifying segments is a matter of judgment. The linear segments can be either chosen visually, or determined numerically by piecewise linear regression (PLR) (Malash and El-Khaiary, 2010). Some common errors frequent-ly occur in the application of Webber's pore-diffusion model, these errors are discussed next. 4.1.1. Extending the linear regression of pore-diffusion plots to include points after equilibrium After equilibrium is reached q remains constant and the data points represent a horizontal line. If the data points after equilibrium Ka (L/mg) R2 SSE Kf n R2 SSE KR (L/g) aR (L/mg) β R2 SSE 129.7 0.134 0.993 3.20 33.1 3.28 0.934 8.98 22.1 0.242 0.921 0.995 2.34 Fig. 3. Pore-diffusion plots for the removal of Pd(II) complexes from NaCl–HCl solutions containing 100 μg/cm3 Pd (II) determined for the weakly basic Amberlyst A 21. 10 mg/L 20 mg/L 2 3 4 5 6 7 8 9 7 6 5 4 3 t 0.5 (min0.5) q (mg/g) ······· Line in original publication —— Suggested linear segments Fig. 4. Pore-diffusion plots for the adsorption of chromium(VI) onto activated carbon for different initial feed concentrations.
- 5. 318 M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320 1.2 1 0.8 0.6 0.4 0.2 are lumped with pre-equilibrium data (Bhattacharyya and Gupta, 2008; Hubicki and Wołowicz, 2009) to make one regression line, then the quality of fit will seem poor. For the purpose of this illustration, Fig. 3 is a partial reproduction of a published pore-diffusion plot (Hubicki and Wołowicz, 2009). In this plot the last four data points to the right are at (or near) equilibrium; and they obviously don't belong to the same straight line with the rest of the points. By excluding the first point to the left and fitting the data by PLR, the linear segments plotted in solid lines were obtained. The details of PLR analysis is presented elsewhere (Malash and El-Khaiary, 2010). 4.1.2. Ignoring the presence of linear segments Fig. 4 shows a published pore-diffusion plot (Acharya et al., 2009), where it is easy to visually separate the data points into segments. Clearly the estimated slopes, and consequently ki values, differ greatly when segmentation is applied. Many papers (Abd El-Ghaffar et al., 2009; Atia, 2005; Debnath and Ghosh, 2008) present unsegmented pore-diffusion plots, the result is either a faulty estimate of ki or a wrong conclusion that the pore-diffusion model does not apply to the system. 4.1.3. Segmenting the data and discarding the first linear segment Another common practice is to detect and acknowledge segments, then automatically dismiss the first segment(s) as a period where film-diffusion is controlling the rate of adsorption (Sarkar et al., 2003; Kumar et al., 2005; Liu et al., 2010a). In most cases this practice is associated with a common misconception of Boyd's diffusion models, which will be discussed later in Section 4.2. A typical case is shown in Fig. 5. Here the published study (Kumar et al., 2005) passed a pore-diffusion line through two points only just prior to equilibrium, arguing that the data that precede these two points are in a film-diffusion controlled period. This argument was based on the observation that the first linear segment does not have a zero in-tercept. However, the first (and sometimes the only pre-equilibrium) segment of a pore-diffusion plot does not necessarily need to have a zero intercept. A zero intercept of the first linear segment that starts from t=0 would imply that pore-diffusion is rate controlling throughout the entire adsorption period. That would be a special case, possibly when the system is very vigorously agitated so that the resistance in the boundary layer is negligible at all times. Moreover, the first data point in this study was taken after 5 min; and during these 5 min 18% and 35% of qe were adsorbed for the initial concentrations of 20 and 60 mg/L, respectively. During this 5 min period anything could have happened, maybe there are more linear or curved segments, it is simply unknown because there is no data. It is not correct to extrapolate a pore-diffusion line and base conclusions on the extrapolation. In addition, even if strong evidence exists against the pore-diffusion hypothesis, one cannot automatically conclude that film-diffusion is in control, other mechanisms may be in control, such as the rate of chemical reaction. By analyzing the data in Fig. 5 by piecewise linear regression, the linear segments plotted in solid lines were obtained. The numerical values of regression parameters (in case of initial concentration 20 mg/L) are listed in Table 5. It can be seen that the confidence interval of the intercept of the first segment embraces zero, thus the intercept is not significantly different from zero. The break point is the point where two linear segments meet. By defining break-time as the time a break point occurs, it is noticed that the first linear segment ends at a break-time of 23.2 min. These results are very different from those presented in the original study, and are based on chemical and statistical theories. 4.2. Incorrect application of Boyd's diffusion models In 1947 Boyd et al. published their legacy series of papers, where they presented theoretical models for ion-exchange that simulate equilibrium (Boyd et al., 1947a), kinetics (Boyd et al., 1947b), and non equilibrium conditions (Boyd et al., 1947c). The adsorption commu-nity found that these kinetic models also apply to adsorption systems and Boyd's diffusion models have been applied in numerous adsorption studies. However, a distorted version of Boyd's pore-diffusion model is circulating the literature and was used in many recent research papers. 4.2.1. Boyd's diffusion models If diffusion inside the pores is the rate limiting step, the following equation was derived (Boyd et al., 1947b): F = 1− 6 = π2 ∞ Σ n=1 1 = n2 exp −n2Bt ð10Þ where F is the fractional attainment of equilibrium, at different times, t, and Bt is a function of F F = qt = qe ð11Þ where qt and qe are the dye uptakes (mg/g) at time t and at equilibrium, respectively, and B is defined as: B = π2Di = r2 o : ð12Þ From Eq. (10), it is not possible to estimate directly the values of B for each fraction adsorbed. Reichenberg (1953) managed to obtain the 0 2 4 6 8 0 t 0.5 (min0.5) q (mg/g) ······· Line in original publication —— Suggested linear segments 60 mg/L 20 mg/L Fig. 5. Pore-diffusion plots for the adsorption of methylene blue onto fly-ash for different initial feed concentrations. Table 5 The results of piecewise linear regression analysis of the kinetic data shown in Figs. 5 and 6. The values in parentheses represent the 94% confidence interval. Slope of first segment Intercept of first segment Slope of second segment Intercept of second segment Break time (min) Pore-diffusion plot (Fig. 5) 0.164 (0.144–0.183) 0.002 (−0.069–0.070) 0.119 (0.084–0.155) 0.215 (0.015–0.416) 23.2 Boyd plot (Fig. 6) 0.044 (0.035–0.054) −0.096 (−0.022–0.032) 0.097 (0.074–0.120) −0.135 (−2.11–0.596) 23.9
- 6. M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320 319 following approximations by applying the Fourier transform and then integration: for F values N 0:85 Bt = −0:4977− lnð1−FÞ ð13Þ and for F values b 0:85 Bt = q ffiffiffi π p − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π− π2F =3 2 : ð14Þ In order to apply this model to experimental data, the right-hand sides of Eqs. (13) and (14) are calculated from the available q vs. t data and a knowledge of qe. The resulting Bt values are then plotted against t (Boyd plot). If the plot is linear, the slope is equal to B and it can be concluded that pore-diffusion is the rate controlling step. The effective diffusion coefficient, Di, (cm2/s) can be calculated from Eq. (12). Linear segments can also be encountered in Boyd plots and in such cases every segment is analyzed separately to obtain the corresponding diffusion coefficient. 4.2.2. Distorted Boyd's diffusion models The following distorted equation of Boyd's pore-diffusion model is found in many recent publications: for all F values Bt = −0:4977− lnð1−FÞ: ð15Þ Possibly because a 1947 publication is not available in many libraries and databases, many studies copied Eq. (15) from each other (Behera et al., 2008; Acharya et al., 2009; Ofomaja, 2010; Kamal et al., 2010; Liu et al., 2009; Sarkar et al., 2003) , including studies by one of the present authors (El-Khaiary, 2007). The use of Eq. (15) for all values of F leads to erroneous values of Bt when F is less than 0.85, the magnitude of error increases as F becomes smaller. This is graphically illustrated in Fig. 6 where Boyd's pore-diffusion model is applied to the same kinetic data of Fig. 5, the numerical results of piecewise linear regression are presented in Table 5. Although the points clearly show two linear segments, the original study passed a single straight line through all the points, resulting in a poor fit and an intercept far from zero. Accordingly, the original published study considered this as an affirmation of its previous conclusion obtained from Fig. 5, confirming that pore-diffusion does not control the rate of adsorption in the time period from 5 to 40 min. Conversely, the results obtained from the correct model of Boyd, by acknowledging the presence of segmentation, lead to the opposite conclusion. Interestingly, the break point in Fig. 6 is 23.9 min, a remarkably close value to the 23.2 min obtained from Fig. 5. 5. Discussion and conclusions Statistical analysis and hypothesis testing are universally accepted as the basic fundamentals of experimental science, and accordingly, research papers are supposed to present conclusions that are supported by sound statistical tests. Excellent textbooks are now freely available online (Motulsky and Christopoulos, 2004; NIST, 2010). The misuse of linearization (linear transformation) is probably the most common error in batch adsorption literature. In the past, researchers needed to transform their data into a form suitable for simple linear regression, but in the present age computers and software are easily available to do this instead. In order to justify a transformation two conditions should be met: 1. The error-structure of the experimental data is known to violate some assumptions of the least squares method. 2. A specific transformation is expected to change the error-structure to better satisfy these assumptions. If these conditions are not met, then there is no point in linearizing the data. 3 2.5 2 1.5 1 0.5 0 -0.5 ······· Line in original publication —— Suggested linear segments - - - - Linear segments from distorted Boyd equation Bt from correct Boyd equation Bt from distorted Boyd equation 0 10 20 30 40 Time (min) Bt Fig. 6. Boyd plot for the adsorption of methylene blue onto fly-ash and initial feed concentration 20 mg/L. R2 is generated in the regression output of virtually all spread-sheets and statistical software. An issue with R2 is that its value can be artificially large when a model has a small degrees of freedom for error. Therefore, one should not rely solely on R2 in assessing the goodness of fit. The significance of estimated regression parameters should also be tested with conventional statistical tests. In addition, R2 is often incorrectly used for comparing models that have different degrees of freedom. Akaike's Information Criterion is very easy to compute and provides a sound basis for comparing such models. Webber's pore-diffusion model is often abused in batch adsorption studies. This is mainly manifested in the disregard of segments or mismanagement of segmented data. It is recommended that pore-diffusion plots are examined carefully and segments, if present, identified numerically by PLR. It would also be beneficial to have as many kinetic data points as possible if Webber's model is a candidate for data analysis. This would ensure having a reasonable number of points in each segment and thus obtaining statistically significant estimates of the diffusion parameters. A distorted version of Boyd's pore-diffusion model is widely spread in the literature. Using this distorted model leads to wrong estimates of Bt from the beginning of adsorption up to 85% attainment of equilibrium; consequently, it leads to wrong conclusions about the rate limiting step. 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