If the adsorption of arsenic on soil is primarily physical, an increase in temperature might reduce the bonding energy (Kl decreases) but could increase the adsorption intensity (1/n increases) due to increased mobility and diffusion.

If the adsorption is primarily chemical and endothermic, an increase in temperature might increase the bonding energy (Kl increases), while the adsorption intensity (1/n) could decrease if higher temperatures disrupt the surface heterogeneity or affect the adsorption sites differently.

In addition to Jajati Mandal 's answer, please be aware of the following:

The Langmuir isotherm can actually be derived from statistical thermodynamics using e.g. a grand canonical ensemble, so it is actually based on microscopic parameters.

Freundlich is a purely empirical parametrization and does not provide an "intrinsic" temperature dependency. There may be trends published in papers, but they may strongly vary from system to system, so there is way less systematization behind it.

Hello, Arkaprava Roy,

An interesting question. Please, let me share my vision.

1) The simple answer – yes. "The Langmuir constant related to bonding energy (Kl) and Freundlich exponent (1/n)" can "show opposite trends with temperature". They should show opposite trends. When adsorption enthalpy is negative, the Langmuir constant (i.e., an affinity, an equilibrium constant) decreases with the temperature. In the use of the Freundlich model associated with the consideration of multiple sorption sites, each one is characterized by its own negative adsorption enthalpy. The whole isotherm becomes more linear with the rise of the temperature which will make itself evident in increasing Freundlich exponents becoming close to one. So, there are indeed opposite trends. These opposite trends are natural outcomes of adsorption thermodynamics. Importantly, I do not think that the Freundlich exponent is a measure of "adsorption intensity" (as indicated in the question).

2) The long answer: I would foremost ask why even to contrast the Langmuir model vs the Freundlich model? We can hardly use them indiscriminately simultaneously for the same experimental data series. Where does this question come from? If, in terms of adequacy of the experimental data description, the Langmuir model fits better and provides an adequate data description, then, there is no place to think about the Freundlich model and its exponent. The latter is irrelevant. When we talk about adequacy, it is not r2, but residual deviations properly analyzed for the lack of systematic trends in deviations identified via non-linear fitting procedures. If the Freundlich model fits better, providing an adequate description for the data, and the Langmuir model does not fit, then, the Langmuir model is out of interest. If both models provide compatible extents of an adequate description...then, there is no question, again. We will choose a model with a simple physico-chemical meaning. :-)

3) In a broad interest, how to interpret the temperature dependence of the Freundlich model exponent? If we think that there is a variety of independent sorption sites such that each kind of them interacts with sorbate species via the Langmuir model mode, and the whole experimental sorption isotherm follows the Freundlich model, then, the Freundlich model is not indeed empirical. Its exponent will have a simple meaning. The Freundlich exponent is a measure of the exponential distribution of sorption sites by their energy (free energy) (see the classic study by Sips, apparently, 1948; he dealt with purely site energy distribution, but similarly site free energy distribution may be used). Hence, if the Freundlich exponent changes with temperature (and we are convinced that the Freundlich model is adequate to represent experimental data in a solute concentration/vapor pressure range), then, we may rationalize this change as weakening of sorbate-sorption site interactions with temperature, differing for different sorption sites.

4) The bottom line: are the adsorption isotherm data well-fitted at different temperatures by the Freindlich model such that the Langmuir model shows poor behavior? I would not even think to go into the Freundlich exponent interpretations. One should use the concept of isosteric enthalpies, and using the Freundlich models fitted to the data at different temperatures, to examine the plot of ln(solution concentration) vs 1/T for each selected sorbed concentration (fixed for different temperatures). The slope (derivative) will produce the isosteric enthalpy. The concept of isosteric enthalpies is strict thermodynamically and will allow us to calculate how the adsorption enthalpy depends on loading/sorbed concentration etc, i.e., to test sorption site heterogeneity. Thus, having the Freundlich models determined at different temperatures, the strict measure of adsorption intensity may be obtained with isosteric enthalpies/heats, without dealing with the Freundlich exponents.

Jajati Mandal da, thank you!

The second case in your answer is perfectly describing our observation.

I am wondering, what could be the reason behind the decreased surface heterogeneity with increasing temperature? We have carried out our experiment with a soil which is rich in amorphous Fe, Al and Mn oxyhydroxides, clay (largely smectitic and illitic) and organic matter. Is it that increasing temperature is having its effect on organic matter? Or is it changing the composition of secondary minerals? or something else?

Dear Professor Mikhail Borisover ,

Thanks for your answer, it gave me a lot new perspectives.

It would be great if you can share relevant research paper(s), which have dealt with the important things that you discussed in your reply.

Thanks :)

Hello, Arkaprava Roy,

1) foremost, I would recommend the original paper by Sips (attached), and in particular the first two pages there (albeit the history described in the three paragraphs of the introduction is also of very interest). This gives some perspective to what the Freundlich exponent means (presuming that there is a variety of independent sorption sites differing by energy).

2) I could suggest my paper which is related to essentially different problem (albeit related to soil organic matter), but in this paper, instead of site-energy distribution, site-free energy distribution is used. So, mostly two paragraphs around eq. (12) are relevant

3) regarding isosteric enthalpy I suggest looking any textbook on adsorption (adsorption thermodynamics). Determination of isosteric adsorption enthalpies is done without any assumptions on a specific adsorption model; it requires only adsorption isotherm data at different temperatures and a some reasonable way to approximate isotherms. However, an important note: if the focus on adsorption of ions, with involvement of ion exchange and comlexed mechanisms, a caution is needed to check the applicability of both "simple" isotherms and isosteric heat.

Mikhail Borisover Dear Professor,

Thank you!

Dear Dr. S. Kalidhasan

Thanks for your lucid answer!

Any research paper to back your answers would be very helpful for me to dive deeper.

Regards