A critique on the theory of langmuir adsorption isotherm model in aquatic environments

One of the most important goals of absorption studies on aqueous solutions is determination of the maximum adsorption capacity. During the past hundred years, a number of models have been offered in this area. But none of them have an important mathematical theory, except Langmuir Isotherm model and also to some extent the model by Brunner et al. (1938). Isotherm models have two aspects in common with Langmuir model: 1) the model’s independent variable is equilibrium concentration of solution (C_e) and 2) the maximum capacity of the absorbent is only related to the equilibrium adsorption isotherm (q_e) part.
For the first time, Langmuir wrote equilibrium relationship between two different phases. Though Langmuir tests were performed for gas adsorption at first, this theory was generalized to include equilibrium adsorption between two liquid and solid phases later. In fact, Langmuir equilibrium relationship is between vacant sites of absorbent surface (S), occupied surface sites (AS) and the solution concentration (A) which is written in relation (1). He believed that adsorption occurs only in one layer. In the other words, when the absorbent surface is completed with a single layer of absorbed material (adsorbate), adsorption would stop and adsorption capacity will reach to its maximum threshold. Langmuir equilibrium reaction is as follows:(1)
Where, K_a and K_d are adsorption and desorption equilibrium constants, respectively. Therefore, according to the principle of equilibrium in chemistry, relation (1) is written as relation (2):K_a [A][S]= K_d [AS] (2)
Where, K_a [A][S] is the absorption rate of reaction (〖r.〗_ads) and K_d [AS] is the desorption rate of reaction (〖r.〗_des); they become equal at equilibrium state of the system.
Parameter [A] shows the equilibrium concentration of adsorbate which is expressed in terms of moles per liter (or milligrams per liter), and [AS] & [S] are quantities related to the absorbent surface expressed in terms of moles per square centimeter (or milligrams per square centimeter). Langmuir introduced symbol θ for parameter[AS]. Θ is a fraction (percentage) of the surface covered (occupied) by adsorbate. Therefore, (1-θ) or [S] is a fraction (percentage) of the absorbent surface which is vacant. On the other hand, if [A] is shown by symbolC_e, considering the previously mentioned concepts and relations (1) and (2) we will arrive at:The adsorption rate is proportional to the equilibrium concentration and the vacant site of the absorbent surface.