Hi,

The most commonly used isotherm for metal adsorption is the Freundlich equation:

x/m = Kc^1/n

in which x is the mass of metal adsorbed, m is the mass of adsorbent (biosorbent) added, K and n are fitting parameters (adsorbent, matrix, temperature, ... dependent; this is an empirical equation, these parameters do not have any physical meaning) and c is the equillibrium metal concentration.

If you take the logarithm at both sides, you will obtain a linear equation log(X/m) = log(K) + 1/n log(c)

Just take a couple of beakers (e.g. 5), fill them with metal contaminated sample and add different amounts of your biosorbent to each of them. (1 is a blank without any adsorbent). As such, the ratio metals adsorbed/adsorbent added (x/m) changes. Stirr for about 24h (or let overnight) and if you then measure your metal concentrations, you will obtain 5 different concentrations (the highest will be the blank one of course). By considering your blank, you can calculate x/m (mg metal adsorbed per mg adsorbent added). Calculate log(x/m) and log(c) for each beaker.

Then make the following graph: plot log (c) against log (x/m). The linear relationship you will obtain will have 1/n as slope and log(K) as intercept. By adding a trend line or using a solver, you can obtain the values of these parameters.

Regards,

Wim

Dear Rahman

You can find more detail about PFO and PSO parameters in my papers (links below) :

www.aidic.it/cet/13/35/143.pdf

www.aidic.it/cet/14/40/040.pdf

All parameters were determinated by a simple linear regression.

Good luck

Dear Prof. Salim Bousba,

Thanks for your papers. I have gone through your papers, really interesting!!! However, for a clear understanding of the result interpretation, can you please share me your raw experimental data on the basis of which you had calculated different parameters of PFO, PSO and made figures.

With sincere regards.

Hi Rahman,

It is really not that difficult. For PSO, you should plot t/q(t) vs t, q is the amount adsorbed (mg/g) and t is time. If it results in straight line, the slope gives you equilibrium capacity ans the intercept can be used to calculated rate constant. You can see the simplification here.

 https://www.researchgate.net/publication/263968794_Solventless_extractionstripping_of_phenol_using_trioctylphosphine_oxide_impregnated_hollow_fiber_membranes__Experimental__modeling_analysis

Calculating PFO parameters are little different because you can not isolate equilibrium capacity (qe) from amount adsorbed (qt) in the equation. It gives you a relation between ln(qe-qt) and t. So, you need to assume a value of qe (trial and error, keep changing the value close to the experimental qe) and look for the best regression coefficient when you fit in the experimental data. Hope this helps!

Article Solventless extraction/stripping of phenol using trioctylpho...

Dear Dr. Md. Sayedur Rahman

Hello and I hope every thing is fine with you. You can go to the my profile in Researchgate and download my papers. Many of the have various kinetic and isotherm model. You can so enrich yours. If you had any question, please contact me.

Best wishes

Hi, yours question is really the most problematic one for the beginners. I have faced similar problems at the start however once you read few papers these calculations become very easy. Please formulate the equation in  excel and next time you wont face problem. You can view my papers in my research gate profiles and still if you have any query please feel free to write to me for clarifications.

OK, take the experimental data and replace it in the linearized rate law equations of pseudo-first and pseudo-second order. But in my opinion, it is not possible that the same process satisfies  pseudo-first and pseudo-second order. You will agree with me that its not possible. Anyway, answer to your question is that you raplace the data in different rate law equations and calculate the value of R2 and explain the data.

You can refer to many articles written by me and others in standard journals.

Dear Profs., scholars and well wishers,

Thank you very much for your kind inputs towards my problems. I become enriched through your scholastic discussion. 

However, my special thanks and gratitude goes to Dr Prashant Praveen @ NUS, who actually demonstrated me a closer insight into solving the problem practically with a set of example data, which was very much effective for my learning of the critical things regarding the PFO and PSO kinetic models. 

I wish to the Almighty for all of your sound health and long lives.

With sincere regards,

M. S. Rahman.

Anyone have the calculations for Pseudo first order reaction in excel, Could you please send it to me? I want compare my calculations with.

Hello Sayedur,

Check out my papers below which will explain everything in detail:

1.  Zaynab Aly, Adrien Graulet, Nicholas Scales, Tracey Hanley, “Removal of aluminium from aqueous solutions using PAN-based adsorbents: characterisation, kinetics, equilibrium and thermodynamic studies, Journal of Environmental Science and Pollution Research (2014) 21: 3972-3986. DOI 10.1007/s11356-013-2305-6.

Z. Aly and V. Luca, “Uranium extraction from aqueous solution using dried and pyrolyzed tea and coffee wastes”, J. Radioanaly. Nucl. Chem., Volume 295, issue 2, (2013) pages 889-900. DOI: 10.1007/s10967-012-1851-6.

Contact me again incase of problems.

I have confusion to compare experimental and calculated values for qe and k in pseudofirst order and second order please clarify......

or 

how can I calculate theriotically....qe(cal)

pls refer these papers

Dutta, A., Zhou, L., Castillo-Araiza, C., De Herdt, E. (2016). Cadmium(II), Lead(II) and Copper(II) biosorption on Baker’s yeast (Saccharomyces cerevesiae). Journal of Environmental Engineering, 142 (9), art.nr. C6015002, 1-7.

Dutta, A., Diao, Y., Jain, R., Rene, E., Dutta, S. (2016). Adsorption of Cadmium from aqueous solution onto coffee grounds and wheat straw: Equilibrium and Kinetics Study. Journal of Environmental Engineering, 142 (9), art.nr. C4015014, 1-6.

A bit late than never.

It starts with the selection of a measure of the adjustment of the proposed equation to the kinetic data and the method of numerical minimization of the measure in the function of unknown values of the equation parameters. It is convenient to use the Levenberg-Marquardt procedure here. This procedure specifies the form of the fit measure. Most often it is the sum of the squared differences. But it can also be an average relative error. In the case of chemical reaction kinetic equation is assigned to the set of kinetic curves. In contrast, in the case of batch adsorption, kinetic curves are approximated, each separately, using common functions, such as power, exponential, hyperbolic or logarithmic functions. Above all, however, the use of outdated graphical methods should be avoided here, and the approximating functions should be in an explicit form in relation to the measured value. Regards,

I recommend Prashanthakumar's answer

Best Regards

The kinetic experiments are conducted by starting with different initial concentration (Co). With respect to time t, measure the concentration (Ct), then calculate the adsorption capacity at any time t (qt). Also after the equilibrium time, measure equilibrium concentration (ce), then calculate equilibrium adsorption capacity (qe).

Now we have the set of values of qt v/s t.

From the pseudo first order kinetic model,

ln(qe-qt)=lnqe - kf t

plot a graph of ln(qe-qt) v/s t, find the value of first order rate constant kf.

Then from the pseudo second order kinetic model,

t/qt=(1/ks(qe)**2)+(t/qe)

plot a graph of t/qt v/s t, then find the value of ks.

Pseudo first-order kinetic model as given by Lagergren is ln (qe-qt)=lnqe-k1*t. You can calculate only k1 from this equation because qe is known to you from batch adsorption studies. qe is equilibrium adsorption capacity (or adsorption capacity at equilibrium time), which is known to us through experiments.

It is worth noting here that the so-called PFO and PSO are in fact empirical functions. The first is the exponential function, also used in an implicit (relative to the dependent variable) logarithmic form. The second is a hyperbolic function, also often used in implicit form, resulting from the Hanes transformation. It should also be added that the use of these functions to describe the kinetic data of batch adsorption is purely data archiving without any scientific value.

Pseudo second order kinetic model.a complete step by step process...

https://youtu.be/7Y7BdUeBzkA

Pseudo first order kinetic model.a complete step by step process..

https://youtu.be/gonP5o9R3XY

Read my article

Sorption Kinetics, Thermodynamics and Regeneration for Lipid Feedstock Deacidification Using a Mixed-Bed Ion-Exchange Resin

Article Sorption Kinetics, Thermodynamics and Regeneration for Lipid...

You will find answers to a number of your questions

Asif Ayub, Have you really seen these two demos on YT. If so, you should be aware of the biased selection of kinetic data for the only true thesis that PSO is the best. This is identical to the 1998 and 1999 work by Ho and McKay on which the authors probably relied. It seems, the trash can is the best place for both demos.

Please read my article for details " Sorption Kinetics, Thermodynamics and Regeneration for Lipid Feedstock Deacidification Using a Mixed-Bed Ion-Exchange Resin "

https://onlinelibrary.wiley.com/doi/full/10.1111/jfpe.12056

Asif Ayub When some of your Qt are bigger than your Qe, you cannot get a log of a negative value. What do you do in this case

Prashant Praveen I have a question. You say that for PSO qe is the slope but other sources say it is 1/slope. Can you explain that?

And is K2 still slope^2/intercept?

Thanks

Josephine Getz There must be some confusion. Qe can not be the slope and I haven't indicated that either.

Prashant Praveen ok then I misunderstood this wrong "the slope gives you equilibrium capacity and the intercept can be used to calculated rate constant"

Qe is 1/slope right? and K is slope^2/intercept?

The graph of pseudo first order and pseudo second order were plotted log (qe-qt) and t/qt versus t then the value of qe, K1and K2 were calculated from the intercept and slope that were obtained from the graph.

How can we plot linear pseudo first order [ln(qe-qt) vs t] for more than at one species at once?, For eg. I want to linear plot for pseudo first order Pb and Cr at the same graph, how to do this?