Most people in industry study degradation kinetics in pseudo-first order conditions. For academic purposes look at the order of reaction in "Physical Organic Chemistry" by Calvin D. Rithcie (Marchel Dekker Inc.).

In fact, to study any industry reaction by increased the used material compared to one active material and shifted the reaction kinetically to pseudo- first order..

Kinetic model is not a single equation. Kinetic model is a set of elementary chemical reactions with values of reaction rate constants. This set of reaction quantitatively describes the dependence of concentration of all species as a function of time (kinetic curves). In some cases it 's possible to derive the reaction rate law, which might be a pseudo-first order. Commonly, the digital integration is used to calculate kinetic curves. There is no a simple protocol how to establish the reaction mechanism

Respected Yurii V Geletii, Luma M. Ahmed, Andrei Blasko I am studying metal uptake kinetics by growing mushrooms on compost. When I establish first order euqation and plot a graph log(C0/Ct) vs t, it gives R2 values nearly .94-.96. What does it means? Should I try pseudo-first order equation?

You are applying the methods of chemical kinetics to describe a complex system. The question is not in choosing an equation but in explaining what process is described by this equation.

What is the difference between the first and pseudo-first order equation in your case?

Dear Yurii V Geletii Sir,

I have reviewed a lot of literature and found that fungal mycilia of mushrooms secrete enzymes that binds with heavy metals and further these ligands are transported to the mushroom bodies and hence removed from the compost as nutrients.

I want to apply kinetics on removal of heavy metals from compost by growing mushrooms.

What is your valuable advise for me?

If I apply kct=ln(C0/Ct) and plot a curve of ln(C0/Ct) vs t, I get R2 value of .90-.94. Is it valid?

Based on you data, kinetics is definitely not exponential. The curve has an induction period about 30 min. At time > 30 min you don't have enough points to make any conclusion. The induction period suggests the autocatalytic process.

Yurii V Geletii Sir, My time is in days and initial 20-25 days are not considered in the data because during cultivation of my mushroom species this period is defined as spawn running period and do not have significant metal removal. So, should I consider the whole time period i.e. 0-60 days or only 30-60?

Below is the attached mechanism of metal removal, please help me understanding it.

Thank you very much for your kind cooperation.

From chemical kinetics point of view, you are dealing with two parallel process:

growing and dying of mushrooms, which changes the number of active sites

metal accumulation in active sites.

I believe, that a kinetic model can be created, but You can't describe the whole process by a single equation. I guess you have to consider 0-90 days

Yurii V Geletii Sir,

The mushrooms were harvested when they matured. A total 3 crop cycles were harvested and their were no more mushroom production after 60 days because of the aging of used spawn and the life cycle was completed.

Thank you.

So, you have even more complex system, which includes growing and aging of spawn, growing and harvesting of mushrooms, accumulation of metals in growing mushrooms. I believe that it would be possible to build a kinetic model for such system, but probably you do not have required skills in kinetics.

Yurii V Geletii Sir, you are right. If you could collaborate with me in this project?

I only want to apply kinetics in my data for removal of heavy metals from compost in terms of initial metal concentration and final metal concentration.

Dear Pankaj, I'm too busy with several projects to accept your proposal to collaborate.

Keep in mind, that for kinetic studies it's useless to collect only the initial and final concentrations. You need at least 5 kinetic curves with at least 10 points on each curve.

Yurii V Geletii Sir, I got you. I am reading some good books to clear my basics about chemical kinetics. Thank you very much for your valuable suggestions and help.

Regards

Pankaj

First of all, you have to decide whether you want to determine the form of the kinetic equation/s (of course along with the value of its/their parameters), or on the approximation of the kinetic data using selected algebraic equations. Unfortunately, those who deal with sorption processes choose the second solution. Regards

Dear Yurii,

This time you did a really good educational work. I am afraid, however, that this will not change much, as evidenced by the above question. There were many similar questions on RG.

Regards,

Respected Mirosław Grzesik Sir,

I have discussed with one of my university colleague who is expert in kinetics. She suggested me simply to calculate rate constant value "k" for my data and apply first order rate law. However, she suggested me to discuss the effect of initial concentration and temperature, separately.

I read some publications like:

Article Experimental and kinetics study for phytoremediation of suga...

Article Assessment of phyto-kinetic removal of pollutants of paper m...

Where, they applied some kinetic models on heavy metal and physico-chemical parameters.

Hello Pankaj;

I tried to follow the previous discussion on your RG question. As Dr. Yuri quoted, first thing you need to do is to know the objective of your kinetic model. From your messages I understand that you want to know the time dependence of the metal uptake by the fungi. In reaction kinetics this is called the time-course of your process. If you follow this approach what you will get is a rate constant, as you commented in your last message.

In this regard, the problem that I see in your data treatment is that you plot a logarithmic function of concentration in terms of time. This is not correct. Besides this, the way you treated your data (logarithms) clearly does not fit a linear curve. First thing you need to do is to plot Ct vs time. In this way you will see plainly how Ct changes with time. By doing this you will be able to see the actual behaviour of your fungi, and perhaps will decide what parts of your data you will include in your time-course model. For example, if there is an induction period, or a type of lag-phase period, you may have to decide to neglect this period, or to use a specific mathematical model such as a logistic equation.

In addition to this, if you want to further kinetically characterize your biological process, you have to carry out additional experiments to study the dependance of the metal intake rate on the initial concentration. You need to carry out experiments with a set of initial concentrations and determine the intake rate for each concentration. This is regularly performed using the method of the initial rates. Plotting this data, and applying basic rate equations you may be able to obtain the order of the reaction. However, if the behavior of the reaction rate in terms of C0 cannot be adjusted by these simple rate equations , then you will need to explore more complex models perhaps using a mathematical software (there are some good ones available for free).

The must-read reference for the theory on this subject is the book by O. Levenspiel, Chemical Reaction Engineering. 3rd Edition, John Wiley & Sons. A good physical chemistry text book will do.

Hope this helps. Regards.

Respected Rogelio Valadez-Blanco Sir,

Firstly, thank you very much for your kind and valuable reply of my question. You just simplified my understanding to this.

For your concern, I just grown my mushrooms on paddy straw and determined the metal concentration (% dm) at different intervals. However, I used industrial effluents treatment to enhance the nutrient level of the substrate, and hence the initial metal concentration was varied in different treatments. As the metal concentration was increased the removal efficiency (R) was also decreased which I calculated by this formula;

Re=(C0-Ct)C0*100

Where C0 is the initial metal concentration and Ct is the final metal concentration after t time while excluding 0-10 days where there was no uptake.

Further, the role of temperature of cultivation chamber was also studied.

If I maintain the temperature 30-38 then R was maximum. Below and above this range the R decreased.

Le Chatelier's Principle suggest that the R is inversely proportional to the initial metal concentration.

The attached figure shows a plot Ct vs t.

Sir, I am indeed of a person who could collaborate with me in my project for developing such a mathematical model.

Thank you again.

Hello Pankaj,

I am glad that my input was useful for your research. We can keep in touch and see if there is something else I can help.

Regards.

Dear Pankaj, You asked me a question, so I'll try to write something. You must decide what your goal is. If you only care about the approximation of a few experimental points using, one of the known classes of functions, eg power, exponential, logarithmic, hyperbolic etc. then you can do it. In your case, I would use the quadratic equation. R2 will certainly be bigger than in the case of the linear function. ....... But what for? Both in the first and in the second case, the equation parameters will have no physical meaning!

Therefore, as proposed by Yurii (and me too), an experiment should be performed for several initial substrate/adsorbate concentration coi. Three are enough. Then check if the points ( (coi-cij) /coi, tij) are on one line. If so, the process/reaction rate is proportional to the substrate concentration, i.e. the process is of first order - the process/reaction, not the equation or kinetics. In your case, however, as Yurii suggests (and me too), you are probably dealing with an auto-catalytic mechanism. Regards,

Dear Mirosław Grzesik Sir,

Thank you very much.

Panjak,

I want to echo what others have said before. You need to decide what it is you are trying to model. An empirical model describing metal uptake rates by your mushrooms could be useful and is possible for your system. I think that a mechanistic model (one that explains the elementary steps of uptake) is not possible with your system. Metal ion capture, transport, and uptake involving large ligands is a very complicated process with many many elementary steps. I study relatively simple ligand exchange systems (2 ligand and 1 or 2 metal ions). If I want probe the mechanistic detail of one of my reaction systems, I need to repeat my experiments many dozens of times while varying reaction conditions (pH, ligand concentrations, etc.). Using living organisms, real soils, industrial effluents only complicates this process.

Respected Nathan Boland Sir,

Could you suggest me names of some empirical models describing metal uptake rates by mushrooms?

Thank you !

Pankaj,

I would suggest you the simplest chemical model.

The maximum number of binding sites in mushrooms per volume of soil is No = a* Wt, where Wt is the total weight of mushrooms at time t, a is a conversion coefficient. Mt is the number of metal atoms in soil at time t. The number of absorbed metals is X = Mo-Mt. The number of available sites N = a* Wt – Mo + Mt. Wt is the experimental dependence of the total mushroom weight on time.

"Chemical reaction"

M +N = X; Equilibrium constant K

From this “reaction” you can obtain the following equation:

Mt = K*X/(Mt*Nt) = K* (Mo-Mt)/{(a* Wt – Mo + Mt)*Mt}

This equation can be solved digitally with respect to Mt at known Mo and a. K can be found from fitting experimental dependence of Mt against time.

As you see the very simple model results in the rather complicated mathematical equation.

Good luck, Yurii

PS. You must check the derivation, I could make a mistake.

Respected Yurii V Geletii Sir,

Thank you again for helping me to understand it in such easy way. I will go through and try to implement it.

Unless you already know the mechanism of a reaction, consider a rate equation to be just the empirical fit of your data. In that case a simple power law fit of your data should be sufficient as long as you don't stray too far in operating conditions.