Langmuir - Science topic

The material is a hydrogel.

Thank you very much, Sir Rafael Vargas-Bernal . My adsorption is not about gas but on aqueous solution. Still, thanks sir for your insights.

Tania Akther don't bother to let the results affect you. however, this tells you about the inconsistency in data points used for the plotting of your data, as it may come from poor reproducibility and repeatability studies of the baseline data for the effects of concentration. You may need to go back and also see if you used non-linear extrapolations that gave you these judgments, then you can feel free to accept the variation in the results. However, to add to it it is inconsistently unacceptable to use R-squared to decide the adsorption process of the mechanism. Have a great day

I guess there is a mistake; the Freundlich exponent cannot be negative. As correctly noted by Bairam Lohan, this means when concentration of a sorbing species increases, its sorbed concentration decreases. Generally speaking, it is thermodynamically forbidden (if any other variables are maintained constant). Perhaps, there is a confuse: if sorption data were plotted in log log scale, to linearized the Freundlich model, the intercept (log Kf) may become negative

What about the removal percentage? Is it going down continuously?

If so, keep increasing the concentration and see what is going on.

What is the specific surface area of your adsorbent?

@Belbasee AnkiL, what is the full meaning of bbb?

From the D-R model, u can consider physical adsorption or chemisorption; when E<8 kj/mol is physisorption, 8 kj/mol<E<16 kj/mol is chemisorption. and also you can find these references will help you

1-Influence of Synthesis and Functionalization Procedures of Fe3O4 NPs by Mono- and Diamino Silane Coupling Agents on the Adsorption Efficiency of Anionic Dyes

2-Cutting-edge in the green synthesis of MIL-101 (Cr) MOF based on organic and inorganic waste recycling with extraordinary removal for anionic dye

Hello, very interesting question. What does "the Langmuir bonding energy coefficient (Kl) shows the impossible negative value" mean? Foremost, I would check how this "negative" coefficient was obtained. I hope it was not done by the data linearization (and a respective Langmuir model linearization) but through the direct non-linear data fit. It appears that with the rise of As (which one?) sorption isotherms go up (like in BET? as mentioned above by Jürgen Weippert?). It is not necessarily multilayer adsorption; such BET-like isotherms or the Freundlich models with exponent exceeding 1 may reflect lateral interactions in 'a monolayer" which enhance sorption with rise of As concentration or cooperative changes in a sorbent matrix (like in protein molecules where several molecules or ions interact stronger than each one taken separately). Both mechanisms lead to the Freundlich exponent exceeding 1. However, I understood that the major issue is why it happened at in particular high concentration of the competing anion (silicate?). There might be very many mechanisms difficult to address without knowing details but foremost it is important to know what was pH at high concentrations of "a competitor'? was it constant compared to other systems? pH may affect As speciation, soil organic matter dissolution, concentration of dissolved organic matter in solution phase, ionization of surface groups and precipitation of some components (capable of involving other components due to entrapment). I hope pH was always controlled and the same in the experiments which could help in the identification of mechanisms

The very high concentration of another ion that can compete with the adsorbate in the adsorption process can cause this problem. I am also trying to decipher the reason.

Thank you, and Aida Fekaouni , for your responses. My experiments were conducted with a fixed adsorbate concentration of 1000 ppm, varying the adsorbent mass. I conducted trials with additional dopings in the material under the same concentrations and obtained consistent results, except for this particular sample (67) that I cannot explain. The adsorbent used is calcium phosphate, and the adsorbate is anionic organic dye.

Mehmet Can Dal, Sorry, sir, to screw you a little. I got my answers based on your clue, which I was searching for. I have already followed you. Thank you very much for your help and support.

Eugenio Rhoads

You're correct that using models like the Freundlich and Langmuir equations to estimate soil capacity for phosphorus (P) fixation can be accurate but may be complex for farmers to implement directly. Fortunately, there are more practical and farmer-friendly approaches available:

1. Soil Testing: Regular soil testing is a simple and effective way for farmers to assess soil fertility, including nutrient availability and fixation potential. Soil testing laboratories can provide farmers with comprehensive reports that include recommendations for nutrient management based on the specific characteristics of their soils.

2. On-Farm Trials: Conducting on-farm trials allows farmers to observe the performance of different management practices, including fertilization strategies, on their own fields. By comparing the results of different treatments, farmers can gain practical insights into how their soils respond to different inputs and make informed decisions about nutrient management.

3. Soil Amendments: Applying soil amendments such as lime or gypsum can help reduce phosphorus fixation in soils with high levels of aluminum or iron oxides. These amendments can help improve soil pH and cation exchange capacity, making phosphorus more available to plants.

4. Precision Agriculture Technologies: Advancements in precision agriculture technologies, such as remote sensing, soil mapping, and variable rate application, enable farmers to manage their fields more efficiently and effectively. By accurately targeting inputs based on site-specific soil and crop conditions, farmers can optimize nutrient use and minimize the risk of phosphorus fixation.

5. Integrated Nutrient Management: Adopting an integrated approach to nutrient management that combines organic and inorganic sources of nutrients can help reduce phosphorus fixation and improve soil fertility over the long term. Practices such as crop rotation, cover cropping, and organic matter addition can enhance soil health and nutrient cycling, reducing the need for synthetic fertilizers.

By combining these practical approaches with scientific knowledge and expertise, farmers can effectively manage phosphorus fixation and optimize nutrient use in their agricultural systems. Agricultural extension services and agronomic advisors can also play a valuable role in supporting farmers in implementing best management practices tailored to their specific circumstances.

In kinetics modeling (such as PFO or PSO), the theoretical values of Qe (theo.) and ki are determined through nonlinear regression analysis and then compared with experimental data. Typically, it's standard procedure to first measure equilibrium time before delving into concentration dependence or isotherm studies. However, some authors have mistakenly considered experimental values as Qe (theo.), which is incorrect. For further insights, you can refer to the following articles:

Elias Mosaffa thank you so much sir

Just because something is published, it does not mean it is correct. There is a lot of nonsense out there.

This paper refers a "linear langmuir adsorption isotherm" - which does not exist. You can compute a linearized form of an adsorption isotherm - which is also unnecessary, as you can compute the parameter by non-linear regression.

If you want to calculate the separation factor, you can use IAST where you can define the selectivity as S=(Yi/Xi)/(Yj(Xj) for each cTotal or you go with the linear region of the isotherm. The linear region is called "Henry regime". If you want to define a separation factor in the Henry regime, you just need to calculate the quotient of the Henry constants. There RL definition is useless.

If you use C0 for a caluclation, you also must use the amount of adsorbent. Otherwise the results would not be comparable.

Adsorption isotherms are per definition an equilibrium isotherms.

If you have fundamental questions about adsorption I recommend a text book (preferably by a serious publisher), not a paper from a non scopus/webofscicnece indexed journal.

I would also say that the lack of adequate fitting is a sufficient reason to omit the Langmuir model: neither maximal capacity nor sorption affinity make sense, and further only contradictions will be expected, such as temperature dependence of the maximal capacity and inability to account for competitive effects. Indeed, there are plenty of other models to be used. Nevertheless, if the Freundlich model stands against the statistical check, i.e., it fits the experimental data, the RMSD is at the level of the data scatter, and there are no obvious trends in the distribution of residuals (i.e., deviations of experimental values from the theoretical ones), I would definitely consider it useful. If the physical system allows considering a multiplicity of sorption sites acting independently, and each site may be assumed to follow the Langmuir model, then, the Freundlich mode with an exponent less than 1l has a precise interpretation, i.e., an exponential distribution of sorption sites by energy. In order to perform the thermodynamic analysis, it may also be used by using a (concentration-dependent distribution coefficient) which will define the free energy of a transfer from a reference state to a sorbed state. If the exponent is greater than one, than, we deal with cooperative interactions where BET will be a part of the story....

RL indicates the shape of the isotherms to be either unfavourable (RL > 1), linear (RL = 1), favourable (0 < RL <1) or irreversible (RL = 0).

Of cource it could be better to fit with langmuir mathematical model regarding this particular case.

The adsorbent is just proficient in the adsorption of methylene blue hence the data achieve adhere to both isotherm

Hello, Vincent,

1. Yes, you can use a buffer, but one should consider the potential interferences of buffer ions. For example, buffer anions could form complexes with metal cations of interest both in solution and in an adsorbed phase thus affecting the metal adsorption affinity. On the other hand, if the intention is to maintain a constant pH, how may a pH adjustment per se affect a surface? I might think that the adjustment changes the composition of a background ionic composition, and to avoid that, a background electrolyte is useful.

2. Regarding varying solution concentration vs solid to solution ratio, I vote for the first one, and always avoid varying sorbent to solution ratio. There are several reasons for that: 1) a so-called dosage effect when changes in sorbent to solution ratio affect an equilibrium adsorption isotherm determined at a given dosage for different solute concentrations. If this effect is present, by any reason, then, varying sorbent to solution ratio for a given initial solute concentration produces the solution/sorbed concentration pairs belonging to apparently different isotherms; 2) in some cases, specifically, in the work with environmental and geological materials (e.g., soils, sediments, coal, rocks), sorbents may release soluble material capable of complexing with solutes in solutions (e.g., metals). The concentration of released soluble material will vary when a sorbent to solution ratio varies; 3) a kinetic of attainment of equilibrium may change depending on a sorbent to solution ratio, which needs to be reexamined for each such a ratio; 4) in some cases, varying solute concentrations for a given sorbent to solution ratio allows reaching a broader range of equilibrium solution concentrations that when maintaining a given initial dissolved concentration and varying a sorbent to solution ratio.

Not easy to reply without further details on the data.

To start: here are two helpful references on adsorption:

Inglezakis et al., 2018. Microp. & Mesop. Mat. 272, 166-176 (review on S-Shaped sorption)

Tran et al., 2017. Wat. Res. 120, 88-116. (review on Retention data handling and classical pitfalls)

qe always grows with Ce, because of mass action law.

The absence of plateau for qe may represent multi-layer retention, or other mechanisms (precipitation for solutes, condensation for gases)

Then you cannot define a maximum sorbed quantity: qmax.

If you still want to assess a "site capacity" a mechanistic model with more details will be required. As for example with BET equation wchich lead to a specific surface area (which requires an adsorption cross section for the adsorbate).

Dear Dhirar Ben Salem, let me cite: "For n > 1,

Eq. (3) may be identified as describing a cooperative reaction

between a sorption site and n sorbate molecules, such that b is

the equilibrium constant of this reaction. In this case, Eq. (3) coincides

with the Hill model characterizing cooperative interactions

of ligand molecules with a macromolecule (Cantor and Schimmel,

1980)."

It's possible by gradually calculation the no. Of moles that covers a mono layer of the solid sorbent, then convert it to no of molecules which is directly related to the no. Of active sites.

Well, you would have to ask yourself what's the scientific value of comparing a number from a fitting fit to a number from a poorly fitting fit?

Are you allowed to share your graph and a coarse description of your substrate, maybe someone has an idea for a better model which would provide more valid data?

For low pressures/concentrations, it would be expectable that all three are similar: Freundlich can be gained via Taylor expansion from the others and Temkin is in the derivation essentially Langmuir augmented by lateral interaction, see Desjonqueres/Spanjaard: Concepts in Surface Physics (Springer).

If Temkin and Langmuir match also at higher pressures, that means that the lateral interaction between adsorbate molecules is very weak.

So, your question title suggests you see a lack of calculation possibilities at all while your description is more of a complaint about the evaluation some people choose. So, what is your actual focus?

Dear Seyed Faridin Rafie;

Did you work at low concentrations?

At low concentrations, the adsorption occurs by the first order and proceeds linearly. isotherms coincide. Langmuir model saturation is reached. In the Freundlich model, adsorption continues even though the affinity decreases. Differences occurs between the two different models

It is Ok, to the best of my knowledge, to determine adsorption isotherms at a controlled pH (and an ionic strength, if important), a given dosage and with varied concentrations of a sorbate. Obviously, a selected dosage has to guarantee that the meaningful part of a sorbate is removed from solution during adsorption, and the remaining concentration is measurable, e.g., a 20-80% removal range. However, there may be also "a dosage effect" representing either interactions between sorbent particles/surfaces or "a sorbent dissolution" (a release of some material) such that the measured adsorption isotherms become unpleasantly "dosage-dependent". I would add more: if (and it is always so), the final use of the adsorption isotherm data will be applying the equilibrium physico-chemical models and the interpretation of model parameters [there are almost no papers without that], the adsorption kinetics have to be examined. Moreover, adsorption kinetics can be dependent on a specific batch parameters such that the tests are desirable at different solute concentrations. And, finally, even adsorption kinetics do not necessarily show that a full equilibrium is reached. The desorption tests if show the lack of hysteresis will convincingly prove that the equilibrium models are applicable. Unfortunately, a vast majority of solution adsorption studies deals with the data for which no real proofs for attainment of equilibrium are provided or, which is worse, a hysteresis is clearly demonstrated, and the equilibrium models are used without a shadow of doubts. Which suggests future problems in understanding and model interpretations.

You can use the prominent adsorption isotherm models namely Adam-bohart, Thomas, Yoon-Nelson, Clark model which basically designed according to the continuous column adsorption experiments indeed.

You need some particular parameters which help to interpret your experimental data set undoubtedly.

Alternatively, you can use the Dubinin-Radushkevich (D-R)model to find qm.However, it is a very complex and difficult model to calculate.

Dear Dounya,what exactly are you asking, I don't understand

You might double ckeck your fitting using pyGAPS. It is an open source software by Paul Iacomi.

If you have more detailed questions, feel free to send me a DM.

So, which C0 should we use?? as the experiment for the isotherm have multiple C0´s as the second question asks. Thanks

I would think one should be careful with using the Langmuir constants per se, since they are always experimentally determined, depending on concentration (or pressure) units (derived from the concentration scale used for solutions or partial pressures in the gas phase). Thus, -RTlnK as an experimentally determined value is a solute concentration unit-dependent; changing the concentration (and the reference state), the absolute value of the Langmuir affinity (as well as the free energy change for the selected reference states) will change, in contrast to the adsorption enthalpy. However, generally, I assume, it is a separate story to obtain, based on the Langmuir affinity, per se, the enthalpy change, since the entropy changes associated with the interactions with a sorption site are not necessarily easily accountable. Certainly, it is a problem for liquid phase adsorption, and/or for less well-defined materials, anyway. Hence, I consider for myself two points helping in evaluating how strong or weak the Langmuir- model-based interaction is: (1) obtaining enthalpies, if possible, from the temperature dependence, which will give a direct idea about energies that may be compared with many other data and calculations, (2) in terms of chemistry, the Langmuir model is just a simple 1:1 complex formation; formally and physically, it is the same (or essentially similar) process controlled by a stoichiometric reaction. In such a case, I compare the Langmuir affinity with any other quantified and characterized complexes formed in solutions or at the surfaces; for example, the equilibrium constant about 1 L/mol will be well comparable with equilibrium constants determined for middle-scale strength hydrogen bonding; it would give a some idea. When talking about the metal adsorption, their interactions with a surface are often interpreted in terms of chemical speciation models; the Langmuir affinity could be compared with the similar constants determined for the complexes studied in solutions. It may give a more broad perspective for ranking and comparing.

Dear colleagues

I think will definitely be able to find information in the famous and huge Lenin State Library in Moscow…

In particular, the works on line.

I have friends who have active library cards - I will ask.

I will contact you soon

The LB and SAM films are quite similar in structure, although the former is deposited in the form of a close-packed monolayer from the water interface, while the latter is self-organized on the metal surface from a solution, for example, long-tailed thiols (SAM) on the surface of a silver or gold substrate. In fact, the LB film can be considered as a SAM, since the molecules in the Langmuir film are self-organized in the process of deposition of a monolayer on a solid surface. An example is the self-organization of fatty acid molecules on the surface of mica during the deposition of a monolayer. Since SAM molecules have long tails, they can form LB films on the same substrates (silver, gold) SAM LB. But the quality of SAM LB (packing density) will be much higher than that of a monolayer of SAM from solution. We have developed a method for horizontal deposition of a monolayer on a solid surface. In this case, the deposition of an LB monolayer can be considered as a SAM monolayer, since, as a rule, the processes of self-organization of molecules in a monolayer are realized during the formation of LB films.

Sachin Liyanage, You did not specify the unit of constant K in given formula. Heats of physisorption are of the order of several kJ/mol and there may be mistake here. The heat of adsorption is not the activation energy of adsorption.

When observing the 2 graphs it appears that the BET one fits well.

The Langmuir one usually recommanded for monolayer adsorption is not taking in to account porous distribution all over the surface area on the solid material.

Adnan Irshad, Equilibrium is the limit state for time that is approaching infinity. So neither time nor contact time can be a parameter. In addition, there are no linear versions of the Langmuir isotherm. Instead, there are versions implicit in q like the one you are using. As for the surprising results, you'd better check the data. To use the Langmuir isotherm, the equilibrium curve should resemble the saturation curve.

The Langmuir (L), LF-Freundlich (LF) and Fowler-Guggenheim (FG) models are based on localised adsorption. In principle, the maximum adsorption is, in this case, determined by the number of adsorption sites. The Hill-deBoer model is for mobile adsorption, so based on different premisses than L, FL, FG. Here the theoretical maximum is, in principle, determined by the size of the molecules and the specific surface area. The two maxima should not be confused. With the FG model the lateral interaction parameter can only be obtained when the maximum adsorption is known.

In practice, isotherms often show a pseudo-maximum determined by a limited concentration range or lateral repulsion between the adsorbate molecules (see also the comment by Sidi Souvi).

See also my review on adsorption (from solution) in

Advances in Colloid and Interface Science 280 (2020) 102138;

You could measure your system's adsorptions isotherms to find which model isotherms fit the best (Langmuir, Freundlich, etc.). Each adsorption model provides an equation for calculating your equilibrium constant K which you can use if the model reproduces your adsorption isotherms well

The negative values of intercept can be possible depending upon your starting point of applied concentration.

For example,, if your first Ce is near 0 there will be chances of positive Kl while if you started your work from a higher concentration (let's suppose 10 mg L-1) then you will end up Ce a little bit away from 0 which may cause intercept (Value of Y when X=0) to be negative. It's just the way you have made your graph and presented data.

You can explain the negative results while keeping in mind this point.

There are tonnes of paper in which they have reported negative intercept and even slope in some cases (even Qmax that can be of discussion).

Find the following once for your references

Erum Ayub, Simply put, the term linear fit refers only to the fit of linear functions to given data, and the term nonlinear fit refers to the fit of nonlinear functions to given data. Unfortunately, the term linear fit is also often mistakenly applied to various graphic methods, in particular to the H-W like plot method used, inter alia, for the Langmuir isotherm. Fitting, however, does not concern the original equilibrium data (C_e, q_e), but nonlinear transformed data (C_e, C_e/q_e). Therefore, the term linear fit is a far-reaching shortcut here. What is better? Nonlinear methods are better because graphical methods modify the error distribution and the statistical distribution of new variables. But it should be remembered that graphical methods allow for easier, much faster, but less accurate estimation of approximating function parameters than nonlinear methods.

Sneha Shajan, For a type IV isotherm, the Langmuir isotherm (equation) is unsuitable, hence the negative value of q_max. Try to use multi-parameter isotherms such as BET or GAB. Or, you may need a more complicated function. Obviously negative q_max makes no physical meaning and I would avoid interpreting such a value.

Dear prof for answer plz check these articles

Figured it out! Selected the wrong model but got it to work. Graphpad has a predesignated "Langmuir" model but there's the specific vs total binding models that require some finangling to get it to work. The graphs look very sharp.

You can use different initial concentrations, e.g. by diluting samples, as well as you can change the concentration of the adsorbent. Forget about time here, it has to be long enough to get closer to the equilibrium state

It seems that such information can be found in the manual for a given measuring instrument. I would also not confuse measurement methods with individual isotherms used to describe the equilibrium data.

Hello

I think this article will be helpful

In order to understand which adsorption isothem fits better with your adsorption process, you need more than R2 values. First of all, you should calculate RL value for Langmuir isotherm to see if it is applicable with your process, then you should calculate 1/n value for Freundlich isotherm to see if it is applicable for your process. If RL and 1/n values are below 1, it clearly indicates that the adsorption process is compatible with both Langmuir and Freundlich isotherms. Being compatible with Langmuir model usually means that the adsorption process may have a chemical character and being compatible with Freundlich means it has a physical character. You should also calculate adsorption energy by using D-R isotherm to ultimately determine if the process is a physical or a chemical adsorption.

Manisha Agrawal Thank you for sharing.

Hi,

The rationale you presented is not correct for IAST selectivity calculation. Two possibilities:

A) You calculate the selectivy from the Langmuir Equation, which is a constant and does not vary with the component pressures: S = (q1*b1)/(q2*b2)

B) You want to calculate IAST selectivities, which are not explicit and require iterations. The rationale for IAST calculations is presented in the chapter 5.3 in Do, D.D. (1998) Adsorption Analysis: Equilibria and Kinetics. Imperial College Press, London.

It will be good if we go through the nonlinear fitting to get more accuracy

hello friend. langmuir give you information about adosrption in homogeneous surface materials while freundlich for heterogenous materials. to better choose the model. check the non linear regression and stastical errors analysis. i attach you the below article which can help you.

Try to calculate R-squared in a few other software solutions, or check it value by using some on-line services. Here is an one neat R-squared calculator: https://agrimetsoft.com/calculators/R-squared%20correlation

I believe, this approach can help you to figure where you doing a mistake....

Visit also the following useful RG link:

Maurizio Cuomo, The Langmuir and Freundlich isotherms are not concerned with the adsorption kinetics but with equilibrium. Moreover, your question is very general. To start talking about kinetics, it is necessary to perform kinetic tests and determine the dependence of the adsorbate concentration on time. Experiments are usually performed for selected adsorbate and adsorbent concentrations, different temperatures and pH. The next stage may be a quantitative description of the obtained kinetic curves, but one must be able to use numerical methods.

You can compare the different types of isotherms -

Akito, please set a lower limit '<0' for the non-lienar fit. And, non-linear fit will give you the values of K_L and Q_Max directly, unlike linear fit where you have to calculate these two values separately from slope and intercept. Hence, non-linear fit is more accurate.

Another thing, as written in my previous comment, if your curve is exactly like a limiting curve, i.e. rapid change of y axis initially and gradually leading to very faint change w.r.t. x axis, you will never ever get negative K_L and Q_Max. Not only for Langmuir, for the parameters derived from other adsorption isotherm models too. Those models have been designed, based on this phenomena particularly.

Hi @Phat Tan Vo@, even of so, and depending on the complexity of the processes and experimental way you do that, I would be very cautious with non-isothermal kinetics. The use of formal equations will need plenty of the data across of the temperature rise. When using thermal or calorimetric methods, there are continuous scanning curves of a heat flow or mass changes vs. temperature. If adsorption is studied when continuously scanning temperature, it could be similar. However, if the effect of temperature on adsorption is determined at selecting only several (e.g., 2-5) temperatures, it will hardly be enough to go this way when the empirical kinetics is unknown. A common way to recover rate constants from adsorption kinetics measured at each temperature and applying the Arrhenius analysis would be more confident, to predict the effect of temperature

Hello Jona Vin

Did you find a way to solve the ion exchange Column equations using Matlab?

When using adsorption isotherm models (linear or non-linear form), we want the experimental points to be as close as possible to a model.

The best choice is nonlinear equation. You can try with Origin software to do it. In the Md. Aminul Islam youtube channel you will find how to do it.

Dear Dr. Aditya,

1. Prepare different dye concentrations

2. Treat it using optimum pH, nano dose, stirring, etc.

3. Measure the final dye concentration Ce.

4. Calculate Qe based on dose and volume used in the batch experiment.

5. Draw the linear or non linear relations.

6. Ensure most of point used present in the linear curve.

7. Measure intercept and slope.

8. Calculate calculated Qe based on previous data to ensure the reality of results.

9. Discuss the adsorption of dye onto nano surface based on highly identity model (The best R^2).

Yours'

Ahmed

Rui Daniel de Oliveira

If we talk about Langmuir-Blodgett films, we mean a monomolecular layer of amphiphilic molecules (classical). However, a mixture of amphiphilic molecules with semi-hydrophilic nanoparticles is also used now. However, it is impossible to prove that nanoparticles are on the surface and that the film is uniform. You can get a layer of polystyrene nanoparticles only in the "gaseous" part of the pressure-area curve. With further compression of the layer after contact of polystyrene nanoparticles on the surface, polystyrene nanoparticles will rise from the surface. They are not fixed in water as they are hydrophobic. It is possible to advise you to "sew" hydrophilic groups to the surface of polystyrene nanoparticles or to coat it with a layer of higher carboxylic acids, but this may interfere with the solution of your problem.

It does give ample idea about nutrient behavior under controlled conditions

but relating with field conditions , becones a formidable task to have reproducible results...

When testing new sorbent material, you can use for kinetic modelling : the pseudo second order and intraparticle diffusion. For equilibrium modeling use freundlich, Dubinin raduskevich and sips. For both calculation use the non linear equation for the model.

I suggest you read Biological Wastewater Treatment. Principles, Modelling and design. 2nd edition. Chapter 17 and 18 Biofilm reactors

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

This video shows how to fit Jovanovic isotherm model nonlinear equation [qe = qm (1- exp-K* Ce)] easily using originpro.

The video shows you how to fit equilibrium isotherm adsorption data to nonlinear Langmuir and Freundlich model using OriginPro easily. #aminulcheminnovation #AminulSir

Check these videos

The Sips model is a hybrid model combining both Langmuir and Freundlich models. Sips model can describe the homogeneous or heterogeneous systems. when ns = 1 the Sips model becomes the Langmuir model, on the other hand, sips becomes the Freundlich model at low concentration.

Muzafar Khalifa, In a word, R² is your objective function which you maximize using the internal Excel optimization procedure. There is no problem to find R² for each of the two approximating functions separately and then, for example, calculate the mean. Instead of R², you can use SSE or ARE as a measure of fit and look for the minimum instead of maximum.