Both linear as well as non-linear isotherm models can be used to characterize the adsorption process. From the study you can find out by comparing which curve linear or non linear fit to the experimental data that you have obtained and estimate isotherm parameters.

I had conducted 244 experiments in a span of almost 2-3 years as a part of my PhD work before 15 years. Linear and non linear analysis had shown that the results were not statistically significant. Error analysis had shown error up to 25%

I had then gone for mathematical model. It is always advisable to go for mathematical model rather than regression analysis in adsorption study

Best way is to fit your data and get the best fit curve. The nature of curve will help you to select best adoption isotherm. Fit the Isotherm and get square of error (R2) to know the efficacy of fitting.

To me i prefer nonlinear models to describe the isotherm. of course i agree with Dr Sheth the accurate two models can describe the isotherm study but i will discuss in my own opinion why I prefer nonlinear model:

1- You can plot Different model at the same plot (Ce, Qe) no need to plot minimum 3 or 4 plot to do that.

2- All of your points present in your curve without any exception.

3- Your constants can be automatically calculated from the minimum of SSD Summation Square Diviation.

4- you can apply different errors of your models such as (The Sum of the Squares of the Errors (ERRSQ), A Composite Fractional Error Function (HYBRD), Derivative of Marquardt’s Percent Standard Deviation (MPSD), The Average Relative Error (ARE), The Sum of the Absolute Errors (EABS) and Chi error) and sometimes you can describe your model according to the summation of all errors.

but

What kind of model you should describe (you have more than 15 isotherm model?

How to control your data especially constants?

if you have linear and nonlinear model at the same time how can you make a comparison between them?

Thanks

Dear Ahmed,

For me non linear regression is more presice because linear regression is based on single iteration while non linear is based on more than one iterations.

More iterations favor accuraty ans good prediction of constants.

One can not compare "accuracy and good prediction" considering the number of iterations. Here the most important is the correct form of the equation / mathematical model. In addition, numerical calculations are much easier in the case of linear equations than non-linear ones. Non-linearity is also associated with problems such as stiffness, local minima and numerical correlation between values of model parameters, ill posed task, ambiguity etc. etc. Regards,

Dear Miroslaw,

I want to know if the number of iteration do not increase the presicion of an answer obtained by calculation

Dear Christian,

I literally referred to what you wrote about linear and non-linear regression. Hence, the first sentence applies only to these two methods. So again. It is impossible to compare the "accuracy and good prediction" of both methods from the view point of the number of iterations, because the first one gives an exact solution and the second is approximate. And furthermore, only the second is an iterative method If on the other hand you would like to know my opinion about the relation between the number of iterations and "accuracy and good prediction", I can say that it is neither simple nor obvious - as increase -> better. The validity hierarchy is as follows: the correct model, the correct numerical method, a good starting point, and only at the end the number of iterations. It is enough to mistake the value of the starting point and instead of the global minimum, the method will stop at the local minimum. Increasing the number of iterations is not good here. The result will be invalid for both 100 iterations as for 1000 iterations. Regards,

A few remarks about the linearity and non-linearity raised in your question.

Algebraic equations can be linear and can also be non-linear. Some of algebraic nonlinear equations can be transformed into a linear form. We say about such equations that they are linear with respect to parameters. Here, I would like to emphasize that the equations used to describe the adsorption isotherms beyond one (Henry's equation) are non-linear. The plot of adsorption isotherm is a curve line with the possibility of inflection points and sometimes discontinuities. Sometimes such a line can also take the shape of a hysteresis loop. Can experimental points taking such shapes be directly described by means of a linear equation? Of course not. It is, however, possible to describe a part of such data using the linear function. It is also possible to transform some of these equations to a linear form, replacing the original variables with new ones that usually have no physical meaning. This is particularly evident in the case of BET, GAB, Huttig’s and simplified form of LGD equations, where the new variables are a function of both the original independent and dependent variables. As a consequence, statistical analysis used in the linear regression method can not be used here.

You also asked about linear and nonlinear kinetic models and about fitting measures of mathematical models to experimental data. I have repeatedly referred to the issue of kinetics in this forum, but when it comes to measures of fitting, I prefer the average relative error and the weighted average relative error. Regards,

Dear Dr. Mirosław Grzesik Thank you very much for respectable answer and I'd like to know what is the maximum benefit that we can get from Isotherm constants values.

In the scientific literature one can find information about the physical meaning of the parameters of the equations used to describe the adsorption isotherms. In addition, individual equations refer to specific behaviors or mechanisms. And these are the benefits. Unfortunately, numerical calculations are usually necessary, not only in the case of nonlinear equations but also linearized ones (linear regression), and then it is difficult to obtain an unambiguous result. Regards,

Sir, I am also facing a lot of difficulty regarding the same issue. I have found that if we apply different form of linear equation even for the same isotherm model such as Langmuir adsorption isotherm, the maximum adsorption capacity changes drastically. For example in my case, I found the Qo to be 119 mg/g and 140 mg/g when different linear equations of the same Langmuir adsorption isotherm were applied. Although here in the same post, many experts researchers have expressed their views of true solution, I am facing difficulty in understanding the difference between the curve fitting and regression as said by Dr. R S Singh. Also said by Dr. K N Sheth, to do mathematical modeling instead of regression. I want to know how to apply mathematical modeling in adsorption of heavy metals to get isotherm which reliably agrees with experimental data?

After the transformation of the equation/function, new variable/s must be introduced. Without this, the transformed equation/function will remain nonlinear. And this is the main reason for the differences.

What do you mean "mathematical modeling instead of regression"? Regards,

Hello everyone,

I have used the non-linear and linear kinetics and adsorption study in my investigation but I am unable to understand the purpose of both study. what is the purpose of non-linear and linear fitting? how to compare the nonlinear and linear study? Please give me your suggestion regarding this question.

To answer briefly the questions, "linear study" by which you probably understand graphical methods are easier to use than non-linear methods. Unfortunately, the probability of obtaining falsified results is much higher in the case of graphical methods than in the case of nonlinear methods.

Often the nonlinear correlation to be fitted to data can be 'somehow' linearized, as a first stage. Linearized correlation and the corresponding linearized plot are often quite convenient to qualitatively evidentiate scattering around the trendline and to emphasize major effects and the physical meaning of the correlation parameters. Estimates of the parameters derived by least squares after the linearized correlation (modified from a former nonlinear correlation) can possibly be refined by iterative nonlinear least-squares regression; to find unbiased least-squares estimates for the original correlation. It may be advisable to compare both the linearized and the nonlinear correlations.

The following RG discussions seemingly can be found of some interest, concerning to your query:

https://www.researchgate.net/post/How_can_I_find_an_specific_equation_that_can_describe_the_relationship_between_X_and_Y_in_the_attached_file

https://www.researchgate.net/post/How_to_perform_nonlinear_data_fitting_for_a_given_equation_and_data

https://www.researchgate.net/post/How_to_generate_a_non_linear_curve_fitting_function_with_multiple_variables_using_Origin

https://www.researchgate.net/post/How-can-I-fit-an-exponential-curve-of-the-form-y-AexpBx-C

https://www.researchgate.net/post/How_choose_determine_initial_values_of_parameters_in_curve_fitting

https://www.researchgate.net/post/How_to_fit_viscosity_data_using_Vogel-Tamman-Fulcher_VTF_equation

https://www.researchgate.net/post/How_to_estimate_initial_values_in_non_linear_models

https://www.researchgate.net/post/How_it_can_be_possible_to_fit_the_four-parameter_Fedlund_model_by_only_3_PSD_points

https://www.researchgate.net/post/How-to-evaluate-the-fitting-results-when-I-use-OriginPro-to-do-curve-fitting

These videos can answer your questions: https://www.youtube.com/channel/UC5MiS3nMoji95OZ7uE1V8qw/playlists?view=50&sort=dd&shelf_id=6

This video shows how to fit Jovanovic isotherm model nonlinear equation easily using originpro. #aminulcheminnovation #chemcolony #aminulsir #aminulchem https://youtu.be/1w7kNbDabyg

Nonlinear approach is better than linear ones for many reasons. To learn more please watch https://youtu.be/NAgntHhAutk

Non-linear models are better than linear, error functions and other parameters are calculated automatically by the softwares (Origin, Kyplot etc.)

I think it is better to use Non-linear to see details about equilibrium conditions from graph easily. But for calculation of each constant (k1, k2, kid, KL, kf ) and others can be calculated from linear graph draw as Dr Carlos said