Improper Estimation of Thermodynamic Parameters in Adsorption Studies with Distribution Coefficient K_D (q_e/C_e) or Freundlich Constant (K_F): Considerations from the Derivation of Dimensionless Thermodynamic Equilibrium Constant and Suggestions

Abstract

Adsorption processes often include three important components: kinetics, isotherm, and thermodynamics. In the study of solid–liquid adsorption, “standard” thermodynamic equilibrium constant K Eq o ; dimensionless) plays an essential role in accurately calculating three thermodynamic parameters: the standard Gibbs energy change (∆G°; kJ/mol), the standard change in enthalpy (∆H°; kJ/mol), and the standard change in entropy [∆S°; J/(mol × K)] of an adsorption process. Misconception of the derivation of the K Eq o constant that can cause calculative errors in values (magnitude and sign) of the thermodynamic parameters has been intensively reflected through certain kinds of papers (i.e., letters to editor, discussions, short communications, and correspondence like comment/rebuttal). The distribution coefficient (KD) and Freundlich constant (KF) have been intensively applied for calculating the thermodynamic parameters. However, a critical question is whether KD or KF is equal to K Eq o . This paper gives (1) thorough discussion on the derivation of thermodynamic equilibrium constant of solid–liquid adsorption process, (2) reasonable explanation on the inconsistency of (direct and indirect) application of KD or KF for calculating the thermodynamic parameters based on the derivation of K Eq o , and (3) helpful suggestions for improving the quality of papers published in this field.

Revisiting the calculation of thermodynamic parameters of adsorption processes from the modified equilibrium constant of the Redlich–Peterson model

“…This is because they directly applied the distribution constant K D ( q e / C e ) as in Eqn (14) for calculating the thermodynamic parameters. The problem with applying K D for such calculations has been discussed in the literature 18 . Other authors reported low magnitudes of Δ G °, such as 1.32 kJ mol −1 for adsorbing phosphate by ground burnt patties 87 .…”

“…The derivation of the standard thermodynamic equilibrium constant (; dimensionless) for the adsorption process in solid–liquid phases has been recently discussed by Lima et al 17 and Tran 18 . is simply introduced as Eqn (14).…”

Yesterday, Today, and Tomorrow. Evolution of a Sleeping Beauty: The Freundlich Isotherm

“…As explained by Tran, this parameter is not a true adsorption equilibrium constant, and thus it cannot be applied for calculating thermodynamic adsorption parameters. However, by combining eq with eqs and , we can express this parameter as follows: K normalF ( q max , α , m , s ) = q max · A ( α ) · m − α / 2 true( s K true) α with K = 10 –2 bar kg –1/2 m 2 for gas adsorption and K = 4 × 10 –4 (mol/L) kg –1/2 m 2 for solution adsorption, when the pressures are expressed in units of bar and the concentrations are expressed in units of mol/L.…”

“…It is somewhat puzzling to note that, almost simultaneously, Radian et al studied such LSERs for a case involving 30 organic pollutants (including mainly pesticides and pharmaceutical compounds) adsorbed onto a clay–polycation nanocomposite, whose adsorption behavior was described by the Freundlich equation written in the following way: C ads = K normalF × C eq n and introducing an adsorption coefficient K D at a specific concentration as K normalD = C ads C eq = K normalF × C eq n − 1 However, unfortunately, the correlation applying the Abraham solvation parameters model to the corresponding K D data (calculated at a very low concentration) is not satisfying; the authors explained that this can probably be because the LSER method was initially created for adsorption by a partitioning mechanism (where a linear Freundlich trend is observed, with n ≈ 1), although this is not the case in their study where 0.2 ≤ n ≤ 1.55 . As for us, according to a recent discussion paper, we consider that the K D parameter is not appropriate for the estimation of such correlation; moreover, with the units selected by the original authors ( C ads in mmol/kg; C eq in mg/L), K D should be presented in the cumbersome units of mmol L/(mg kg)!…”

“…If C°i s expressed in units of mol L −1 , the formula becomes The Freundlich Constant K F . As explained by Tran, 46 this parameter is not a true adsorption equilibrium constant, and thus it cannot be applied for calculating thermodynamic adsorption parameters. However, by combining eq 11 with eqs 29 and 33, we can express this parameter as follows:…”