Elsevier

Journal of Molecular Liquids

Volume 273, January 2019, Pages 425-434
Journal of Molecular Liquids

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

https://doi.org/10.1016/j.molliq.2018.10.048Get rights and content

Highlights

  • Determination of thermodynamic parameters by using Van't Hoof equation.

  • The correct equilibrium constant for determination of thermodynamic parameters of adsorption.

  • Use of nonlinear equilibrium isotherms at several temperatures to obtain the K.

Abstract

In the adsorption literature, the Van't Hoff equation is used in different manners without any criteria about the concepts of physical-chemistry of equilibrium for calculation of thermodynamic parameters of adsorption. Indeed, the equilibrium constant (K) should be dimensionless for being used in the Van't Hoff equation. However, this is not a simple adjustment of units, as being spread in the literature, to become K dimensionless. In this paper, it will be calculated the equilibrium constants using numeric examples and show the flaws of the thermodynamics calculations, when the value of K is wrongly calculated, and what are the expected results of the changes in enthalpy (ΔH°) and changes in the entropy (ΔS°) that are spread in the literature.

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].ΔG0=RTLnKe°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))ΔG0=ΔH0T.ΔS0

Combining the Eqs. (1), (2), it leads to Eq. (3).LnKe0=ΔH0R.1T+ΔS0R

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)).Ke0=1000.Kg.molecular weight of adsorbate.Adsorbate°γwhere 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:Adsorbents+AdsorbateaqdesorptionadsorptionAdsorbentAdsorbates

The chemical potential (μ) for this equilibrium system could be considered as [1,2]:μAdsorbentAdsorbate=μAdsorbentAdsorbate°+RTLnAdsorbentAdsorbatesAdsorbentAdsorbates°μAdsorbent=μAdsorbent°+RTLnAdsorbentsAdsorbents°μAdsorbate=μAdsorbate°+RTLnAdsorbateAdsorbate°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:ΔGad=μAdsobentAdsorbateμAdsorbent+μAdsorbate

Substituting Eqs. (6), (7), (8) onto Eq. (9), it gives:ΔGad=μ°AdsorbentAdsorbateμ°Adsorbent+μ°Adsorbate+RTLnQwhere Q means:Q=AdsorbentAdsorbatesAdsorbentAdsorbates°Adsorbents.AdsorbateAdsorbents°.Adsorbate°

Being μ°Adsorbent-Adsorbate − (μ°Adsorbent + μ°Adsorbate) = ΔG°ad, Eq. (10) becomes:ΔGad=ΔGad°+RTLnQ

By definition at the equilibrium ΔGad = 0, and Q becomes K [1,2,21]:ΔGad°=RTLnK

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].K=AdsorbentAdsorbatesAdsorbentAdsorbates°Adsorbents.AdsorbateAdsorbents°.Adsorbate°

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]:K.AdsorbentsAdsorbents°AdsorbentAdsorbatesAdsorbentAdsorbates°=Adsorbate°Adsorbate

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=Adsorbate°Adsorbate

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]:Activityadsorbate=Adsorbate.γwhere [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:Ke°=Adsorbate°γ.Adsorbate=Keγ

Note Eq. (18) is the same Eq. (4) given above.Ke0=1000.Kg.molecular weight of adsorbate.adsorbate°γ=Adsorbate°γ.Adsorbate1000.Kg.molecular weight of adsorbate.adsorbate°γ=Keγ=Ke°

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.lnKLγe=ΔH0RT+ΔS0R

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=qeCeKd 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.

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