Both ways are correct.

The first one expresses the equilibrium constant as the partition coefficient between two phases: adsorbate concentration at the interface (Co - Ce) and solute concentration bulk solution in equilibrium (Ce). If you can measure Ce, knowing Co it is possible to get the equilibrium constant.

The second one assumes the adsorption process can be modeled according to the Langmuir theory. Applying a linearization transformation, the hyperbolic relationship in Langmuir model (qe vs. Ce) is converted to a linear relationship (1/qe vs. 1/Ce), similar to that often seen in enzyme kinetics. From linear regression it is possible to obtain the equilibrium constant.

Once you measure Ke at different temperatures, it is possible to apply the van 't Hoff relationship and estimate the enthalpy and entropy of adsorption.

Beware:

1. Very often adsorption processes are not reversible process, and equilibrium models are not appropriate.

2. Linearization transformations violate the assumptions considered in linear least-squares regression analysis. Therefore, non-linear least-squares regression analysis must be performed.

3. Application of the van 't Hoff equation is rather imprecise, since: usually the equilibrium constant changes very little over the practical temperature range, and the enthalpy is often considered constant (zero adsorption heat capacity).

Thank you Adrian for your answer.

Good one@ *Adrian Velazquez-Campoy* you've answered correctly. This may help also..

Dear Saddam M. Al-Mahmoud,

in the van't Hoff equation, K is a dimensionless value with the definition of thermodynamic equilibrium constant. Thus, 1st, you cannot the Langmuir constant without making it dimensionless with precise correction as the following references discussed:

[78] H.N. Tran, S.-J. You, A. Hosseini-Bandegharaei, H.-P. Chao, Mistakes and inconsistencies regarding adsorption of contaminants from aqueous solutions: a critical review, Water research, 120 (2017) 88-116.

[79] F.-D. Kopinke, A. Georgi, K.-U. Goss, Comment on “Mistakes and inconsistencies regarding adsorption of contaminants from aqueous solution: A critical review, published by Tran et al.[Water Research 120, 2017, 88–116]”, Water research, 129 (2018) 520-521.

[80] X. Zhou, X. Zhou, The unit problem in the thermodynamic calculation of adsorption using the Langmuir equation, Chemical Engineering Communications, 201 (2014) 1459-1467.

[81] E.C. Lima, A. Hosseini-Bandegharaei, J.C. Moreno-Piraján, I. Anastopoulos, A critical review of the estimation of the thermodynamic parameters on adsorption equilibria. Wrong use of equilibrium constant in the Van't Hoof equation for calculation of thermodynamic parameters of adsorption, Journal of Molecular Liquids, 273 (2019) 425-434.

[82] A. Fenti, P. Iovino, S. Salvestrini, Some remarks on “A critical review of the estimation of the thermodynamic parameters on adsorption equilibria. Wrong use of equilibrium constant in the Van't Hoof equation for calculation of thermodynamic parameters of adsorption”-Journal of Molecular Liquids 273 (2019) 425–434, (2019).

[83] E.C. Lima, A. Hosseini-Bandegharaei, I. Anastopoulos, Response to “Some remarks on a critical review of the estimation of the thermodynamic parameters on adsorption equilibria. Wrong use of equilibrium constant in the van't Hoff equation for calculation of thermodynamic parameters of adsorption-Journal of Molecular Liquids 273 (2019) 425–434.”, Journal of Molecular Liquids, 280 (2019) 298-300.

[84] S. Salvestrini, V. Leone, P. Iovino, S. Canzano, S. Capasso, Considerations about the correct evaluation of sorption thermodynamic parameters from equilibrium isotherms, The Journal of Chemical Thermodynamics, 68 (2014) 310-316.

2nd, the distribution coefficient is good as it is dimensionless. But is it the same as the reaction equilibrium constant? I am not sure so.

THx.

Akbar Samadi All equilibrium constants (the partition or distribution coefficient is a particular type of equilibrium constant) are dimensionless once a certain standard state is defined