Hi
A set of adsorption data yields a S shape that fits a cubic formula exactly. While numerous adsorption isotherms exist for this sort of adsorption, they all agree on fitting the data to a model that diverges as y grows; in other words, that function lacks a horizontal asymptote where qm can be found. In the case of Langmuir model for example, the function donvege as Ce increases and a horizontal asymptote resemble qm.
So the question is, how can we calculate qm from such S shape adsorption isotherms? What, more significantly, what is the PHYSICAL MEANING of an adsorption in which qe grows as Ce increases? How could we explain this in terms of adsorption-desorption behaviors
Please support your answer with good references
Dr. Ismail Badran
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 (still requires an adsorption cross section for the adsorbate).
I have read the Inglezakis paper sever times, along with so many other references
What you said actually makes sense, I am confident now that such behavior, qe keeps increasing with ce, is explained by different mechanism. The multilayer one is most plausible since it is an aqueous solid adsorption.
I was also thinking of the adsorption-desorption rates. Could it be that the adsorption rate becomes higher at higher Ce, so desorption becomes slower? And the opposite is true at lower Ce's
Another interpretation, which I used in my last paper, and it is widely accepted in the S shape isotherm community, is the multi stage mechanism. Where the adsorption mechanism itself changes gear as Ce increases. So after the first inflation point in the S curve, another mechanism kicks in with different parameters and so on.
Cheers