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
Introduction
It is spread in the literature different ways how to calculate changes in Gibb's free energy (ΔG), changes of enthalpy (ΔH°), and changes in the entropy (ΔS°) in the adsorption process using the Van't Hoff equation's, as depicted below.
The Van't Hoof equation measures the changes in the equilibrium constant with variations of the temperature, as depicted on in Eq. (1) [1,2].whereR is the universal gas constant (8.314 J K−1 mol−1), T is the absolute temperature (Kelvin), and Ke° is the thermodynamic equilibrium constant [1,2].
Considering the 3rd principles of the Thermodynamics (Eq. (2))
Combining the Eqs. (1), (2), it leads to Eq. (3).
By constructing a plot of Ln(Ke°) versus 1/T, from the intercept is calculated the change in entropy (ΔS°) and by the slope, it is possible to calculate the change in enthalpy (ΔH°) [1,2]. The main point of discussion of this paper is how the equilibrium constant (Ke°) is calculated [3]. Several authors even do not comment on their papers distinctly, how this value was calculated [[4], [5], [6], [7], [8], [9]].
The correct way to calculate the equilibrium constant for adsorption system is to obtain isotherms of adsorption at different temperatures ([[10], [11], [12], [13], [14], [15], [16], [17]],) and making the nonlinear fitting of the isotherms [3,17]. From the best-fitted model at the different temperatures, the equilibrium constant is obtained one for each isotherm at a given temperature [18,19]. This equilibrium constant obtained in the isotherms (usually expressed in L mg−1) must become dimensionless for being applied in the Vant' Hoff equation [1,2].
In this sense, it is recommended to use Eq. (4) below described for calculating the thermodynamic parameters by the Vant' Hoff equation (Eq. (3)).where γ is the coefficient of activity (dimensionless), [Adsorbate]° is the standard concentration of the adsorbate (1 mol L−1) [2] and K°e is the thermodynamic equilibrium constant that is dimensionless. It is calculated by converting the units of Kg (the best isotherm model fitted, such as K of the Liu equilibrium constant, or K of the Sips isotherm model or KL, the Langmuir equilibrium constant), that are given initially in L mg−1 into L mol−1 [[15], [16], [17],20]. This conversion is obtained by the multiplication of the value of K (L mg−1) by 1000, to convert L mg−1 into L g−1 and subsequently making the multiplication of this result by the molecular weight of the adsorbate (g mol−1) multiplied by the unitary standard concentration of the adsorbate (1 mol L−1) and making the division by the activity coefficient (dimensionless) [2,21]. For this calculation, it is considered that the adsorbate solution is very diluted to consider that the activity coefficient is unitary [2,21]. The parameter K°e becomes dimensionless after making these calculations.
The principles of chemical equilibrium described in the textbooks of General Chemistry [22] and Physical-Chemistry [1,2,21] should be considered for establishing the equilibrium of adsorption.
Considering the chemical equilibrium of an adsorbate by one adsorbent as give below:
The chemical potential (μ) for this equilibrium system could be considered as [1,2]:where μAdsorbent is the chemical potential of the adsorbent, μAdsorbate is the chemical potential of the adsorbate, μAdsorbent-Adsorbate is the chemical potential of the Adsorbent-Adsorbate. The μ° is the standard chemical potential. Being μ°Adsorbent the standard chemical potential of the adsorbent; μ°Adsorbate is the standard chemical potential of the adsorbate and μ°Adsorbent-Adsorbate is the standard chemical potential of the adsorbent-adsorbate.
The [Adsorbate]°, [Adsorbent(s)]° and [Adsorbent-Adsorbate(s)]° are the standard concentrations of the adsorbate, adsorbent and adsorbent-adsorbate, respectively, whose concentration by definition is 1 mol L−1 at standard conditions [1,2,21]. The standard state of pure solids and liquids is defined as the state with the pressure of 1 bar and at one temperature of interest (usually 298 K) [21].
The free Gibbs energy of the adsorption process will be given by the equation below:
Substituting Eqs. (6), (7), (8) onto Eq. (9), it gives:where Q means:
Being μ°Adsorbent-Adsorbate − (μ°Adsorbent + μ°Adsorbate) = ΔG°ad, Eq. (10) becomes:
By definition at the equilibrium ΔGad = 0, and Q becomes K [1,2,21]:
Moreover, considering all standard concentrations of 1 mol L−1, and the concentrations of Adsorbent, Adsorbate, and Adsorbent-Adsorbate are also expressed in mol L−1, the equilibrium constant will be dimensionless. [2,21,22].
The adsorption is a heterogeneous equilibrium [2,22] where the adsorbate is present in a fluidic phase (aqueous solution, or gas) and the adsorbent will be present in a solid phase. Usually, the concentration of the solid phase {[Adsorbent] and [Adsorbent-Adsorbate]} is considered constant, because the number of moles of the solid phase divided by the volume of the solid phase is practically constant, because the solid phases did not alter their volume during the process of the adsorption [2,22]. In this sense, Eq. (14), becomes [1,2,22]:
A constant K being multiplied and divided by a constant (the concentrations of adsorbent and adsorbent-adsorbate) will originate a new constant (Ke), as [1,2,21,22]:
Ke is the equilibrium constant of an ideal solution, whose numeric values is already considered the concentration of the solid phases [1,2].
The adsorption of an adsorbate by a solid adsorbent is far away of an ideal solution [23], the equilibrium constant should be expressed regarding activity, instead of concentration [2].
The activity of the adsorbate is defined as [2,21]:where [Adsorbate] is the molar concentration of the adsorbate (mol L−1); and γ is the coefficient of activity of the adsorbate (dimensionless) [2].
Substituting Eq. (17) onto Eq. (16), it becomes:
Note Eq. (18) is the same Eq. (4) given above.
Therefore, from the best isotherm model fitted at different temperatures [24,25] it is possible to obtain the thermodynamic equilibrium equation that is dimensionless, after some little arithmetic's.
In a recent review article, [26], also support that the use of Eq. (20) given above. In that situation, the best isotherm model that was adjusted to the equilibrium data was the Langmuir model. Therefore, the KL (L mol−1) was used in the Van't Hoof equation, in order to estimate the thermodynamic parameters.
Note that the equilibrium constant utilized in the Vant' Hoof equation used in Eq. (21), is the same as defined in Eq. (20) of this paper. Also, the use of Eq. (21) was already reported by [17].
Section snippets
Other equations of equilibrium constant used in the adsorption literature
In the adsorption literature, several expression of equilibrium constants is utilized [27].
The most common found equilibrium constants reported in the literature, are expressed in equations below:Kd is the distribution constant (L g−1), qe is the sorption capacity (mg g−1) at the equilibrium, and Ce is the equilibrium concentration (mg L−1) [5,6,[28], [29], [30]]. By using this equation Kd presents the unit of L g−1, considering that qe is expressed in mg g−1, and Ce is expressed in mg L
Defining the correct equilibrium constant for use in the Vant' Hoof equation
Recently [38] published a paper using Avocado seed activated carbon as an adsorbent for 3-aminophenol. It will be shown calculations of thermodynamic parameters using different the equilibrium constants described on Eqs. (20), (23), (24), (25). In Fig. 1 and Table 1 is presented the experimental points of initial adsorbate concentration (Co), the concentration at the equilibrium (Ce), sorption capacity at the equilibrium (qe). These are the experimental data. Also, the next columns of Table 1
Conclusion
This work showed that accurate estimation of thermodynamic parameters for adsorption systems needs a correct equilibrium thermodynamic constant (Ke°) to be used in Van't Hoof equation. For this purpose, after deriving the correct thermodynamic equilibrium constant from isothermal studies, considering the principles of chemical equilibrium, information was provided about the most propagated wrong constants used as the thermodynamic equilibrium constant in the literature (Kc and Kd). Then, using
Acknowledgments
The authors thank the National Council for Scientific and Technological Development (CNPq, Brazil) for financial support.
References (57)
- et al.
Dynamic and thermodynamic mechanisms of TFA adsorption by particulate matter
Environ. Pollut.
(2017) - et al.
Adsorption thermodynamics and kinetics of yohimbine onto strong acid cation exchange fiber
J. Taiwan Inst. Chem. Eng.
(2017) - et al.
Adsorption kinetics, thermodynamics, and equilibrium of a-toluic acid onto calcium peroxide nanoparticles
Adv. Powder Technol.
(2016) - et al.
Removal of fluoride from water through bacterial-surfactin mediated novel hydroxyapatite nanoparticle and its efficiency assessment: adsorption isotherm, adsorption kinetic and adsorption thermodynamics
Environ. Nanotechnol. Monitor. Manage.
(2018) - et al.
Ethylenediamine-functionalized cubic ZIF-8 for arsenic adsorption from aqueous solution: modeling, isotherms, kinetics and thermodynamics
J. Mol. Liq.
(2018) - et al.
Green synthesis of iron nano-impregnated adsorbent for fast removal of fluoride from water
J. Mol. Liq.
(2015) - et al.
Uptake of propranolol on ionic liquid iron nanocomposite adsorbent: kinetic, thermodynamics, and mechanism of adsorption
J. Mol. Liq.
(2017) - et al.
Artificial neural network modelling of amido black dye sorption on iron composite nanomaterial: kinetics and thermodynamics studies
J. Mol. Liq.
(2018) - et al.
Equilibrium, thermodynamics, and mechanisms of Ni2+ biosorption by aerobic granules
Biochem. Eng. J.
(2007) - et al.
Review- biosorption isotherms, kinetics, and thermodynamics
Sep. Purif. Technol.
(2008)