نقدی بر تئوری مدل ایزوترم جذب لانگمویر در محیطهای آبی

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 ? for parameter[AS]. ? is a fraction (percentage) of the surface covered (occupied) by adsorbate. Therefore, (1-?) 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.

The desorption rate is proportional to the occupied site of the absorbent surface.

And using the principle of equilibrium in adsorption, the rates of adsorption and desorption become equal. ?r.?_ads=?r.?_des
Therefore, relation (3) is obtained to be as follows:
(3)
By defining ? as: with q_e representing the amount of adsorbate on the absorbent surface (milligrams per gram) and q_max representing the maximum capacity of the absorbent (milligrams per gram) for a layer, equation (3) is written as follows:
K_a C_e ?((q_max-q_e ))?((q_max ) )?=K_d [q_e?q_max ] (4)
So that:
?r.?_ads= K_a C_e ?((q_max-q_e ))?((q_max ) )?
?r.?_des= K_d [q_e?q_max ]
By deleting q_max from both sides of equation (4), equation (5) is obtained:
K_a C_e (q_max-q_e )=K_d q_e
If K_a to K_d ratio is called K_l, Langmuir model can be written as follows:
q_e=q_max (K_L C_e)/(1+ K_L C_e ) (6)
Equation (6) is classic Langmuir equation. In fact, boundary conditions are considered well in the form (6) of the model, because when equilibrium concentration (C_e) tends to zero, the equilibrium concentration in solid phase (q_e) also becomes zero, and when equilibrium concentration tends towards infinity, the equilibrium concentration in solid phase will be equal to the maximum capacity of the absorbent.
The inefficiency of equilibrium models, especially Langmuir and Freundlich models, has been investigated previously. Here, even if it is assumed that the total absorption takes place within the system’s equilibrium area (the basis of the available isotherm models) and the equilibrium concentration is considered as the independent variable, Langmuir model suffers from some fundamental theoretical problems and is not able to explain adsorption behavior. Some of these shortcomings are mentioned in the following:
As shown in equation (6), when C_e tends toward infinity, the amount of q_e reaches to its maximum amount i.e., q_max, but this does not hold true for relation (5) which is one of the assumptions of the model. In relation (5), when C_e tends toward infinity, the left side of the equation will be equal to the product of “zero multiplied by infinity” which is mathematically ambiguous. This means that, when the concentration of the dissolved substance [A] tends to infinity in relation (2), vacant surfaces or vacant sites [S] become zero. Though this relation is understandable in terms of equilibrium relation and Langmuir monolayer assumption, it is mathematically unexplainable. In this case, of course, the right side of the equation (5) does not become equal to the left side of the equation.
Using the principle of adsorption-desorption rate equality is basically wrong, because
based on definition, in chemistry, the reaction rate depends on concentration changes versus time. The reaction rate formula is written as r= d[A]/dt, but in equilibrium state, changing time makes no sense. In adsorption isotherm tests, all parameters including pH, temperature, the solution volume, the mass of absorbent, and contact time are considered to be constant and only 3 parameters undergo changes including 1) the initial concentration C_0 as an input to the system, 2) the amount of adsorption in solid phase, X_e (shown in mass unit as q_e) and 3) the remaining concentration (C_e) in the liquid phase (equilibrium pressure in the gas phase); the second and third variables are both considered as the outputs of the adsorption system.