What are the adsorbate and adsorbent?

The adsorbate is PNP while the adsorbent is cellulose based polymer

Mayby try Kiselev equation. This theory applies uniform adsorption on the surface and takes into account the interactions of the adsorption layer.

The S-like shape of the isotherm is treated very well with Fowler-Guggenheim or Frumkin type of isotherm

ʘ* С= (1- ʘ)* K*exp(A ʘ)

This is three-constant equation (constants K, A and the monolayer capacity that implicitly intergrated in the equation for determination of ʘ)

The S-shape isotherm or sigmoidal curve usually occurred for non-polar organic compounds. The best model fit for this isotherm is Sigmoidal Langmuir:

Q/Qmax =L*C/(1+L*C+S/C)

For further information in this regards, I strongly suggest you refer to this review paper:

Limousin, G., Gaudet, J.P., Charlet, L., Szenknect, S., Barthès, V. and Krimissa, M. (2007) "Sorption isotherms: A review on physical bases, modeling and measurement." Applied Geochemistry 22(2), 249-275.

When the isotherm is plotted as ads. vs conc and has an S-shape, then the adsorption is cooperative (positive cooperativity). There are two simple isotherm equations that can handle this: the Frumkin-Fowler-Guggenheim equation (see the comment by Valentin) and the Hill equation. The FFG equation is an extension of the Langmuir equation with lateral interaction (attraction in your case) and the Hill equation is an extension of the Langmuir equation with cooperativity.

The Langmuir isotherm itself does not account for cooperativity/lateral interaction and will never provide an S-shaped isotherm when plotted as ads. vs eq. conc. (only fixed S-shape for ads vs log c or ln c; probably Babakhani confuses this). FFG eq. can be found in many textbooks, the hill equation in biochemistry textbooks. For complicated s-shapes extended Hill type equations are leo available.

Maybe BET isotherm

Dubinin-Serpinski equation usually fits very well S-shaped isotherms

I suggest you to have a look at the review article recently published in chemical engineering journal. New insight into adsorption isotherms

Chemical Theory, proposed by Talu and Meunier (1998)

You should have a look at the thermodynamic approach to sorption isotherms developed by Adolphs and Setzer (J. Adolphs, M.J. Setzer, J. Colloid Interf. Sci. 207 (1998) 349. and J. Adolphs, Applied Surface Science, 253, Issue 13, 30 April 2007, Pages 5645-5649) The treatment was developed for gas sorption on solids but it might also apply for other S-shaped isothermes. The approach provides meaningful figures.

Use a hyperbolic model for n different absorption sites. See, for example, Fragata et al, J. Phys. Chem. B 2007, 111, 3315-3320.

The shap of the isotherm depends on how you plot your isotherm.

1. Linear adsorption scale and linear pressure/concentration scale: Shape indicates that the attraction to the surface increases with increasing coverage. The first option for this behaviour is lateral attraction between the adsorbed molecules and the Fowler-Guggenheim (Frumkin) equation provides a simple description (lateral interaction increases linerly with coverage). Other equations that describe the S-shape are equations that take the lateral interaction into account assume that this leads to association between the adsorbate molecules. This type has been described by GU et al for surfactant binding. When the initial affinity is low and the association is strong, a S-shaped isotherm results. When  both  the initial affinity and association are strong the isotherm shows a second adsorption step.

2. On a lin ads vs. log pressure/ concentration scale most isotherms have an S-shape. In this case the slope of the isotherm after the initial part tells what happens: decreasing slope, heterogeneity or lateral repulsion (like with ion binding); higher slope Langmuirian, lateral attraction or association.

Lower slope: Heterogeneity: Langmuir Freundlich isotherm

increasing slope: Langmuir, Fowler-Guggenheim, Gu

This is a good analysis of various adsorption theories, and for example Langmuir adsorption (likely the best !). However. a critical and difficult problem is the meaning of n, the structure-function behind it. Or. in other words, the physics of structure-function. Anyhow, this matter deserves more deep discussions.

Have a pleasant day. Mario

Chapman

A very interesting paper on this subject was recently published by Christoph Buttersack, dealing especially with Type IV and V isotherms. However, as it has been said, the fitting itself is seldom sufficient without thermodynamic justification of the occurring processes.

pubs.rsc.org/en/content/articlelanding/2019/CP/C8CP07751G