(0) use non-linear fitting, not the linear fits to get meaningful parameters

IDM: the data you have is close to the equilibrium, thus the typical IDM behavior (q ~ t^0.5) does not apply (the eq. above - with "offset", i.e. C>>0 - may be fitted but I do not think, that it has any value close to equilibrium). IDM models close to equilibrium follow the exponential decay law (Boyd equation), just like almost all other equations (with the exception of SOE/PSOE).

From your kinetic data it seems that (almost) all action happens at t 0.9 (with some exceptions for highest concentrations of Cd, Cu on MX). I know that it is vary difficult to get any kind of reasonable mesurements in solutions at such short contact times (even mixing may take take more time). If you would use much higher concentrations (and/or smaller granules), you should be able to see some of the initial changes.

(1) plots are not linear or multi-linear, so you may get some range of numerical parameters by using various data sections

(2) IDM should work near t=0 (relative adsorption uptake, F=qt/qe < 0.3), however, this W-M is empirical equation so everything is possible (I am skeptical)

(3) I would not try to interpret your data by using W-M eq.

(4) It would be very bad to remove 50% of data to get nice linear fit! (for 2 points you allway get R2=1). What I see from log-log plot and data table, you have 1 "offending"/outlier point (Cini=30). Perhaps you should first check raw data for some simple error like swapped digits - like 172 instead of 127 - or repeat this measurement. If it is removed the isotherm data should look reasonable (slightly non-linear in log-log plot). Data should be fitted by using nonlinear LSQ fitting (weighted or at least adjusted to the observed data deviation model - e.g. (a) deviations of Q independent on Q or C (fit Q) or (b) deviations of Q proportional to Q - fit logQ. However, the initial slope in log-log coordinates is 2.34 whereas for isotherms based on Langmuir model (without lateral interactions) slope should be 1 or >>1 (and (1/n)

I would like to add to my earlier comments, that the mechanism of sorption here may be quite different than the Langmuir one, i.e. monolayer physical adsorption on non-heterogeneous surface, no lateral interactions - or partly compensated by energetic heterogeneity (see http://guide.adsorption.org/ ). Similar equation corresponds to ionic exchange (on single group). Introducing surface heterogeneity (e.g. Generalized Langmuir, Langmuir-Freundlich, Langmuir-Freundlich, Tóth for quasigaussion energy distributions, UNILAN etc.) cannot improve your fit, because all such equations corresponds to slower increase of adsorption in comparison to Langmuir (slope in log Q ~ log C is ALWAYS lower than 1 - quite different from your data). You could try Frumkin isotherm (like Fowler-Guggenheim, but for ions) - the log-log slope may be >1. Effect of pH may be included in Langmuir model too (multicomponent Langmuir with possible electrostatic interactions) with or without energetic heterogeneity. Moreover, the temperature effect is included in the model through the definition of equilibrium constants. As opposed to GL, GF, LF, Tóth equations, Freundlich and Redlich-Peterson (aka. Radke-Prausnitz) are empirical equations and it is difficult to discuss the meaning of their parameters. BTW, RP was obtained as a harmonic sum of Freundlich and Henry terms (1/Q.RP = a/Q.F + b/Q.H), producing nice physical behavior in the Henry region (low conc./pressure) and Freundlich beyond that.

There are 2 main problems:

1. Your data sets are not sufficient to fit multiparameter equations (number of data points, narrow data range). In order to get reliable isotherm parameters (not only nice data fit), the data range should be 2 or 3 orders of magnitude for concentrations with range center corresponding to 0.5Qmax=Qmono; e.g. for KC=0.1 to 10 i.e. 2 orders of magnitude you get Q/Qmono = 0.09 to 0.9. However, for heterogeneous surfaces you need even wider concentration range, i.e. for LF/Sips (Q/Qm = (KC)^m / [1 + (KC)^m] and heterogeneity parameter m=0.5 you get corresponding conditions (KC)^m = 0.1 to 10 or equivalently KC = 0.01 .. 100). So you need more data points (then it will be reasonable to remove some outliers) and wider data range (then parameters will be reliable). 

2. The log-log slope of your data (slope >> 1) suggests that your uptake model is wrong - perhaps it is surface preciptiation or sth else, but not adsorption?

And stick with non-linear fitting - just like Langmuir iso. or SOE/PSOE kinetics have 3 or 4 linear plots, each of those gives quite different parameters - there is no justification to chose one over another (one favors fit in low concentrations, another one near equlibrium data etc.), unless error analysis is performed. Non-linear fit may be easily obtained in Excel or (OpenOffce/LibreOffice) by using free Solver add-in.

BTW, W-M equation is not a model but an approximation of IDM behavior (and very limited in scope) - the IDM model describes entire kinetic curves from t=0 to infinity (see http://kinetics.adsorption.org/). 

Best regards

AWM

I am very thankful .. Thank you very much for your efforts and i really appreciate that you try to help me ... Dr. Adam and Dr. Ioannis .. I am going to do that this week:

Will work on higher concentrations (Maybe 70, 100 ppm) ... but not so high concentrations as 200, 300 , 500 ppm because i want to apply this study on wastewater .. and before there was M.Sc. discussion about removal of heavy metals from wastewater and the examiner said it is not preferable to work on so high concn, because concentrations in wastewater almost does not exceed 10 ppm  that's why i worked on low concentrations but obviously at 10 ppm and 20 ppm results does not help to interpret the adsorption and apply intra-particle diffusion model or Boyd equation.

* Dr. Adam, The log-log slope of my data (slope >> 1) but when i removed the concentrations below 30 ppm .. slope became 0.245 which give me an evidence that if i worked on higher concentrations for example (70 ppm and 100 ppm in addition the current concentrations 30, 40, 50 ppm), i will obtain better and reliable results and will be able to interpret the data obtained beside slope is expected to be lower than 1.

* I tried to remove 10 and 20 ppm in all sheets (MX and SX adsorbents) , and in the attached excel file in Sheet "3- SX - Cadmium"  as example you can review it and notice the difference but the problem still exist in sheet "4- SX - Copper" :(  ... Equilibrium (Dose was 1.0 g) but i am thinking to repeat the work (effect of concentrations) in this sheet with lower dose ( 0.2 g or 0.4 g) >> what's your opinion?

*  About Frumkin isotherm .. to be honest i don't know about it but i will read about it and if there is an ability, will try to apply it.

*  I am sorry but i don't understand what do you mean Dr. Adam by this:

In order to get reliable isotherm parameters , the data range should be 2 or 3 orders of magnitude for concentrations with range center corresponding to 0.5Qmax=Qmono; e.g. for KC=0.1 to 10 i.e. 2 orders of magnitude you get Q/Qmono = 0.09 to 0.9. However, for heterogeneous surfaces you need even wider concentration range, i.e. for LF/Sips (Q/Qm = (KC)^m / [1 + (KC)^m] and heterogeneity parameter m=0.5 you get corresponding conditions (KC)^m = 0.1 to 10 or equivalently KC = 0.01 .. 100).

(orders of magnitude, LF/Sips, KC^m !! ) please illustrate these to me with example from my results.

* You advised me Dr. Adam to work on more data points ... i worked on 1, 3, 5, 7, 10, 15, 20, 30 mins .. In your opinion what are the data points that i can add and work on them to obtain reliable parameters ?

Intraparticle diffusion kinetic model is explained herein: https://youtu.be/iwi1c_-_qyQ