Determination of thermodynamic parameters from Langmuir isotherm constant-revisited

    Determination of thermodynamic parameters from Langmuir isotherm constant-revisited Partha S. Ghosal, Ashok K. Gupta PII: DOI: Reference:

S0167-7322(16)32378-9 doi: 10.1016/j.molliq.2016.11.058 MOLLIQ 6605

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

22 August 2016 16 November 2016 17 November 2016

Please cite this article as: Partha S. Ghosal, Ashok K. Gupta, Determination of thermodynamic parameters from Langmuir isotherm constant-revisited, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.11.058

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ACCEPTED MANUSCRIPT Determination of thermodynamic parameters from Langmuir isotherm constant-Revisited

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Partha S. Ghosal, Ashok K. Gupta*

Indian Institute of Technology Kharagpur, 721 302, India

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Email: [email protected]; [email protected]

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Environmental Engineering Division, Department of Civil Engineering,

*Corresponding Author

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Dr. Ashok K. Gupta Professor

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Environmental Engineering Division, Department of Civil Engineering, Indian Institute of Technology Kharagpur, 721 302, India

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Ph:+91-3222-283428; Fax:+91-3222-255303 E-mail: [email protected]

List of abbreviations: ∆G0: Gibbs free surface energy change, ∆H0: Change in standard enthalpy, ∆S0: Change in standard entropy, qe: Amount of adsorbate adsorbed per unit weight of adsorbent at equilibrium, Ce: Concentration of solute at equilibrium, qmax: Maximum monolayer adsorption capacity, b: Adsorption constant related to binding energy or affinity, R: Universal gas constant, T: Temperature, Keq: Equilibrium constant, Kaeq: Apparent equilibrium constant, aA: Activity of solute, e: Fraction of surface covered at equilibrium, e: Activity coefficient, Cr: Concentration of reference state, MA: Molar weight of solute, Z: Charge of the ion, µ: Ionic strength of the solution.

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ACCEPTED MANUSCRIPT Abstract

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An analytical approach for estimation of the thermodynamic parameters from the Langmuir isotherm constant has been introduced in the present paper. The concept of the

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thermodynamic equilibrium constant for the Langmuir isotherm based adsorption process was

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critically analysed. The study was attributed to the fact that the use of the numerical value of the Langmuir isotherm constant (b) in the van’t Hoff equation is supported strictly for the solute

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concentration presented in mol/L in the non-ionic solution or the dilute solution of ionic species.

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Otherwise, necessary modifications of b must be conducted to adopt b as the thermodynamic equilibrium constant. A critical review of some related studies on thermodynamics of adsorption

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was performed in this paper, which unequivocally demonstrated the improper estimations of the

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thermodynamic parameters are in practice. The nature of the reaction, spontaneity, etc. is

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incorrectly appraised in many literature. The present study attempted to represent correct estimations of the thermodynamic parameters appraised by many published literature by the proposed approach and exhibited significant deviation from the original works. The influence of activity on thermodynamic equilibrium constant for few ionic solutions has also been estimated. It may be ascribed to the fact that the activity of the solute on the thermodynamic equilibrium constant has more influence at the higher concentrations than at the lower concentrations.

Keywords: Langmuir Adsorption Isotherm, Thermodynamics, van’t Hoff Equation, Equilibrium Constant, Activity.

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ACCEPTED MANUSCRIPT 1. Introduction

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Adsorption is a widely adopted, cheaper and feasible technology in the field of water treatment. A wide variety of the organic and inorganic pollutant had been separated either from

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the aqueous or from the other solvent phases by the adsorption technique [1–10]. An adsorption

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system must follow an established relationship between the sorbate in sorbent phase and the solute phase at equilibrium. The adsorption equilibrium has been well defined with various

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isotherm models relating to the amount of the solute adsorbed per unit mass of the sorbent (qe)

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and the concentration of solute in the solvent phase (Ce) [11]. In this direction, the Langmuir model had been widely used irrespective to the assumptions, originally adopted to define the

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adsorption isotherm.

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The thermodynamic parameters, i.e., Gibbs free surface energy change (∆G0), change in

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standard enthalpy (∆H0) and change in standard entropy (∆S0) define the feasibility of the adsorption process. The van’t Hoff equation had been widely used for the determination of those parameters. The equilibrium constant of the adsorption process is the prime requirement in this equation. The constant in van’t Hoff equation has been determined by adopting various considerations, i.e., the isotherms constant from Freundlich model, distribution coefficient, and other approaches [12–19]. Amongst, the use of the Langmuir isotherm constant with or without modification had frequently been adopted [20–24]. A pronounced non-uniformity and diversification in the determination of the thermodynamic equilibrium constant is the significant gap area in this field. Furthermore, the determination of ∆G0 must be conducted from unit less equilibrium constant according to the reaction thermodynamics. Eventually, the substantial use of the Langmuir isotherm constant, which has a unit of L/mol or L/mg or L/g, etc. clearly

3

ACCEPTED MANUSCRIPT contradicts its consequence in this field. A uniform formulation of the derivation of van’t Hoff constant is required to assess the adsorption thermodynamics.

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In the present paper, the relevance of adopting the Langmuir isotherm constant for the

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thermodynamic study is analysed. In this connection, some literature is revisited as typical examples and the corresponding inadequacy are identified. An analytical method for the

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estimation of the constant in the adsorption equilibrium has been proposed to overcome the

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ambiguity in the determination of the equilibrium constant. The proposed methodology was adopted to appraise the thermodynamic parameters by recalculating them from some of the

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existing studies. The influence of the activity of ionic solution on the thermodynamic equilibrium constant was also addressed. Overall, the present study attempts to enlighten the proper

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estimation of the thermodynamic parameter from the Langmuir isotherm.

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2. Isotherm and thermodynamics 2.1. Langmuir isotherm

The Langmuir adsorption isotherm model was developed for the adsorption of gas on a solid adsorbent. The assumptions of the isotherm model are monolayer surface coverage, identical and equivalent surface sites with equal sorption activation energy of each molecule resulting in homogeneous adsorption and no transmigration or interaction between the adsorbed species in the plane of the surface [7,11,12,25]. The mathematical expression of Langmuir isotherm is as follows:

qe 

q max bC e 1  bC e

(1)

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ACCEPTED MANUSCRIPT where, qe represents the amount of the adsorbate adsorbed per unit weight of the adsorbent at equilibrium (mol/g) and Ce is the concentration of the solute at equilibrium (mol/L).

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The parameters, qmax and b, are the Langmuir constants. qmax is represented as the maximum

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monolayer adsorption capacity and b is related to the binding energy or affinity parameter of the adsorption system. In the literature, Ce is usually represented as mg/L and qe is in mg/g for the

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adsorption of the solute from aqueous solution on a solid medium. Accordingly, the qmax and b

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are defined with the unit of mg/g and L/mg, respectively.

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2.2. Thermodynamics of adsorption

The thermodynamic parameters of the adsorption, i.e., (∆G0, ∆H0 and ∆S0), evoke the

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spontaneity and feasibility of the adsorption process as well as the influence of temperature on

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adsorption. Generally, the determination of the thermodynamic parameter has been performed

ln K eq  

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from the van’t Hoff equation as follows [26]: H 0 S 0  RT R

(2)

where, R is the universal gas constant (8.314 J/mol/K), T is the temperature (K) and Keq is the equilibrium constant. The plot of lnKeq against 1/T provides the ∆H0 and ∆S0 as the slope and intercept, multiplied by R. The ∆G0 can be obtained from any of the following equations [26]: G 0  H 0  TS 0

(3)

G 0   RT ln K eq

(4)

The ∆H0 and ∆S0 are independent of temperature. The positive and negative values of ∆H0 represent endothermic and exothermic reaction, respectively. A positive value of ∆S0 reflects the affinity of the sorbent towards the sorbate, the increased randomness in the solid5

ACCEPTED MANUSCRIPT liquid interface, increased the degree of freedom of the sorbate and more favourable condition for the occurrence of the adsorption process. However, a negative ∆S0 implicates a lesser active

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interface of the solid-liquid system causing a reduction in adsorption [27]. The negative and positive values of the ∆G0 reflect the spontaneous and non-spontaneous adsorption process,

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respectively. Eventually, the thermodynamic feasibility of a reaction depends on ∆G0. The

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correct estimation of the thermodynamic parameter is essential in delineating the adsorption

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process.

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3. Discussion

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3.1. Langmuir isotherm constant and equilibrium constant

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The Langmuir isotherm model was derived for the gas-solid interface with the assumptions as discussed in Section 2.1. However, the model has been widely applicable in the

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liquid-solid interface as well. The fitting of the Langmuir isotherm was observed for a neutral or charged particle from the aqueous or other solution, although the assumptions of the Langmuir isotherm might not have strictly followed. The main discrepancy can be identified as the existence of heterogeneity in the adsorbent surface, which leads to an island-type isotherm [7,28]. The restriction of monolayer coverage or non-interaction among the sorbed species is also not strictly followed in many cases. Nevertheless, the Langmuir isotherm constants were widely used to depict the adsorption capacity and thermodynamic energy parameters in several studies. According to the concept of Langmuir isotherm model, an adsorption equation can be represented as follows [25]:

A (aq)  M (s)  M  A(ad )

(5)

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ACCEPTED MANUSCRIPT where, A is the adsorbate and M is the vacant site of adsorbent. M-A is representing the adsorbed site on the adsorbent. The equilibrium constant, Keq can be defined as follows:

M  A AM 

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K eq 

(6)

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The molar concentration of the adsorbate and the adsorbent should be represented in

aM  A a A aM

(7)

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K eq 

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terms of activity and is as follows:

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where, aA, aM, aM-A denotes the activities of A, M and M-A respectively. From the concept of the Langmuir isotherm regarding surface coverage rate of adsorption and desorption

(9)

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d   K des a A dt des

(8)

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d  K adsa A (1   ) dt ads

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reaction at equilibrium are as follows [25,28]:

where,  is the fraction of the surface coverage, Kads and Kdes are the rate constant of adsorption and desorption, respectively. At equilibrium, the summation of rate of adsorption and desorption is zero and the apparent equilibrium constant (Kaeq) is as follows [25,28]: K aeq 

K ads e  K des a A 1   e 

(10)

where, e is the fraction of the surface coverage at equilibrium. Now, assuming the equal activity coefficient of sorbate at adsorbed and vacant site of adsorbent and the dependency on activity on surface coverage, the following relationship can be established as Eq. (11):

e aM  A  aM (1   e )

(11)

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ACCEPTED MANUSCRIPT From the Eqs. (7), (10) and (11), it can be attained the applicability of Kaeq as Keq. According to the concept of surface coverage, the Langmuir equation [Eq. (1)] can be re-

bC e 1  bC e

(12)

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e 

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written as follows:

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The concentration of the solute is considered in Eq. (12) as mol/L. Stated that the fraction

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of surface covered is the ratio of the sorbent adsorbed in equilibrium (qe) to the maximum capacity of adsorbent (qmax), i.e.,

qe qmax

MA

e 

(13)

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where, qe and qmax are in mol/g. The isotherm constant b from Eq. (12) can be written as

e

C e 1   e 

(14)

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b

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follows:

The activity of solute in solution at equilibrium is defined as follows [29]: aA   e

Ce Cr

(15)

where, e is the activity coefficient (unit less), Cr is the concentration of reference state where the ions and molecule concentration is typically 1 mol/L [29]. Substituting aA from Eq. (15) to Eq. (10), Kaeq can be written as follows: K aeq 

e

Cr

(16)

C e 1   e   e

From Eq. (14) and Eq. (16), the relationship between the Langmuir constant and the apparent equilibrium constant can be written as follows:

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ACCEPTED MANUSCRIPT K aeq  b

Cr

(17)

e

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Now, the activity coefficient is considered only for ionic adsorbate. It is taken as unity for the dilute solutions. The activity differs from the molar concentration when the concentration of

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ionic solute increases. If the solution possesses multiple ionic solutes, the resultant concentration

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is significantly influenced by activity, common ion effect, etc. The charge electrolyte with

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considerable concentration should be represented by the activity with proper estimation of the activity coefficient. This can be ascribed to the fact that for non-ionic species or ionic substance

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in dilute solution the equilibrium constant can be written as follows:

K aeq  bC r

(18)

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Hence, Kaeq is numerically equal to b and unit less. This observation is in agreement with

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the work of Liu (2009) [30]. However, the above consideration holds good for molar

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concentration of solute (mol/L). In most of the adsorption study, the Langmuir isotherm was derived with a solute concentration in mg/L and the adsorption capacity in mg/g. In congruence with Eq. (14), where Ce is now represented as mg/L and qe and qmax is in mg/g, the Eq. (10) shall be modified. The value of Ce has to be converted from mg/L to mol/L. Assuming the molar weight of A is MA in mg/mol and considering the molar weight of the unreacted sorbent and the sorbent after adsorption are MM and MM-A mg/mol, respectively, the Eq. (10) can be re-written as follows:

e K aeq 

M M A  e Ce  1   e  M A  M M

(19)

  

Considering, MM  MM-A, the Eq. (19) can be written as follows:

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ACCEPTED MANUSCRIPT K aeq 

M A e  e Ce 1   e 

(20)

bM A

e

(21)

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K aeq 

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Eq. (20) is subsequently implies the following equation:

ignored and the Eq. (21) can be written as follows:

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K aeq  bM A

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In the case of non-ionic solute or dilute ionic solution, the effect of activity can be

(22)

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Liu (2006) had also reported similar observation [31]. The apparent equilibrium constant can be obtained from the Langmuir isotherm constant, b, by multiplying it with the molar weight

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of the adsorbate in mg in case of solute concentration is expressed as mg/L. For example, if the

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adsorption of fluoride is considered in dilute solution and the equilibrium fluoride concentration

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is represented as mg/L, a value of 18998.4 shall be used as MA. If the electrolyte solution is not dilute, the effect of the activity coefficient must be incorporated in the equilibrium constant for accurate results. The activity coefficient can be obtained from Guntelberg approximation from Dedye-Huckle relationship (applicable up to ionic strength of 0.1 M) or Davis relationship (applicable up to ionic strength of 0.5 M) as provided in Eq. (23) and (24), respectively [29].

log   0.5Z 2



(23)

1 

   log   0.5Z 2   0.2  1     

(24)

where,  is the activity coefficient, Z is the charge of the ion and µ is the ionic strength of the solution and is given by Eq. (25) as follows [29]:

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ACCEPTED MANUSCRIPT n

  0.5 Ci Z i 2

(25)

i

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where, Ci and Zi are the molar concentration and charge of ith ion, respectively. In the case of adsorption of single or multiple solutes in synthetic water, the ionic strength of solution

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will depend on the precursor salts used to prepare the standard solution. The influence of activity

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coefficient for different equilibrium concentration of fluoride and hexavalent chromium, as

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example, had been shown in Table S1 and S2 in the Supplementary Material (SM), respectively. It explicitly represented that the variation of the solute concentration influences the activity

MA

coefficient.

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3.2. Adsorption thermodynamics from Langmuir isotherm

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The determination of the thermodynamic parameters is dependent on the equilibrium

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constant used in the van’t Hoff equation. As mentioned in Section 3.1, if an adsorption isotherm is defined by Langmuir model, the values of Keq in Eqs. (2) and (4) can be substituted with the values of Kaeq as derived in Eqs. (17), (18), (21) and (22) as per applicability. In fact, the Langmuir constant b can be represented as follows: ln b 

 H RT

(26)

Eq. (26) is similar to Eq. (2). Summarising, van’t Hoff equation for adsorption in solid aqueous interface can be re-written in light of Langmuir isotherm model as follows: Case-1: If Ce is represented as mol/L, ln

b

e



H 0 S 0  RT R

(27)

Case-1a: In case of non-ionic solute or ionic solute in a dilute solution,

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ACCEPTED MANUSCRIPT H 0 S 0 ln b    RT R

(28)

bM A

e



H 0 S 0  RT R

(29)

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ln

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Case-2: If Ce is represented as mg/L,

H 0 S 0  RT R

(30)

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ln bM A  

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Case-2a: In case of non-ionic solute or ionic solute in a dilute solution,

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3.3. Application of Langmuir isotherm in adsorption thermodynamics A diversified approach for using the Langmuir isotherm constant for determination of the

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thermodynamic parameter had been observed in many literatures. Some researchers have used b

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in van’t Hoff equation, wherein Ce is isotherm relationship is expressed in various ways, i.e.,

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mg/L, mmol/L, µmol/L etc. [32–34]. The arrived values of thermodynamic parameters were erroneous unless the proper conversion factor of Langmuir constant is performed as discussed in Section 3.1. Some researchers have multiplied b by 55.5 and developed the following equation as follows [35,36].

G 0  RT ln(55.5  b)

(31)

This multiplication was used to nullify the unit of b (L/mol) with 55.5 mol of water per L of aqueous solution. However, Albadarin et al. (2012) used the unit of b in L/mg [36]. Some researchers had multiplied b (L/g) by 1000 for the same reason. The concept of multiplication of b and mol/L or g/L of water to cancel out the unit of equilibrium constant is ambiguous. Daifullah et al. (2007) had attained the equilibrium constant by multiplying b and qmax [37]. For a

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ACCEPTED MANUSCRIPT low value of e or the lesser capacity of the adsorbent, Eq. (14) may be re-written as Eq. (32) by considering (1-e) as 1:

e

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b

Ce

(32)

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Comparing Eq. (13) and Eq. (32), the following Eq. (33) may be written as follows:

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qe  qmax bCe

(33)

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where, qmaxb is a constant. Hence, a linear relationship qe and Ce has arrived from Eq.

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(1) for bCe1. However, qmaxb cannot be used as apparent equilibrium constant from Langmuir isotherm model. The erroneous calculation of equilibrium constant may lead to arrive

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at wrong conclusions, such as apparent non-spontaneity of the adsorption process, in some

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studies [38–40]. Ghosal and Gupta (2015) had mentioned the various approaches for the determination of thermodynamic equilibrium performed in a number of studies. The authors had

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identified the drawback associated with the inadequate theoretical background or improper applications of the methods studied and proposed a formulation for adopting the thermodynamic equilibrium constant when the adsorption is defined with the Freundlich or linear isotherm [41]. Few literatures mentioned the concerns associated with the thermodynamic calculation from Langmuir isotherm. However, they are not comprehensive [30,31,42] and some of them have conceptual error [42]. Some typical references for the determination of the thermodynamic parameters calculated from the Langmuir isotherm has been depicted in Table 1. The estimation of thermodynamic constant from Langmuir constant b is computed by different methods, such as b, 55b, qmb, 1000b and so on (Table 1). Nevertheless, those conversions are inappropriate. The approaches clearly represent an ambiguity in estimation of the thermodynamic parameters from the Langmuir adsorption isotherm.

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ACCEPTED MANUSCRIPT Table 1 Thermodynamic parameters from Langmuir isotherm constant in existing literature

(Unit)

1

Fluoride (mg/L)

Original paper T(ºC)

b

Kaeq

CeO2/Mg-

25

0.0567

Not

∆G0 (kJ/mol) -2.62

Fe LDH

35

0.0540

clear

-2.27

(un-

45

0.0515

20

0.272

30

0.247

40

0.200

60

0.147 0.032

-1.93

modified)

3

Chromium

Magnetic

10

(mg/L)

CoFe2O4/

30

MgAl-

50

4

2,4,6-Tri

AC CE P

LDH

nitro-phenol (mg/L) 5

Copper (mg/L)

6

Methylene blue (mg/L)

-6.617

[43]

-13.21

-0.022

[36]

-7.18

-0.021

[44]

15.65

0.026

[38]

4.24

0.0156

[32]

-6.264

52b

0.024

-5.763 -1.14 -0.71

0.022

-0.28

30

0.04

Carbon

40

0.06

7.32

50

0.06

7.56

2

0.197

Not

-0.067

10

0.141

clear

-0.193

20

0.170

-0.349

30

0.137

-0.506

40

0.100

-0.662

10

3.65

Not

-17.0

20

3.81

clear

-17.7

30

4.08

-18.5

40

5.25

-19.4

Bentonite

∆S0 (kJ/mol/K) -0.035

-6.595

Activated

Tree fern

∆H0 (kJ/mol) -13.0

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(mg/L)

55.5b

MA

Dolomite

D

Chromium

TE

2

Ref.

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No

Adsorbent

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Pollutant

SC

Sl.

b

8.05

14

10-3

9.21

0.092

[45]

ACCEPTED MANUSCRIPT 10

0.0017

Not

-1.065

(mg/L)

CO3 DLH

30

0.0022

clear

-1.632

50

0.0024

20

0.621

(mg/L)

bacterial

25

0.836

0.433

cellulose

35

1.345

-0.735

b. Lead

20

1.001

-0.002

(mg/L)

25

1.489

-0.986

35

1.978

c. Cadmium

20

1.106

(mg/L)

25

1.813

-1.474

35

2.763

-2.602

1.01

qm in

-2.840

1.34

mg/g

-3.064

20

(mg/L)

sawdust

30

b. Nickel (mg/L)

c. Chromium (mg/L)

10

b

43.16

0.145

[39]

57.65

0.197

71.76

0.247

4.331

0.024

0.876

0.013

0.908

0.008

-48.6

-0.151

[37]

36.331

0.142

[47]

SC

1.161

-1.746

MA

-0.245

1.79

-3.330

20

0.21

-2.898

30

0.28

-3.030

40

0.36

-3.156

20

0.30

-1.295

30

0.37

-1.359

40

0.26

-1.446

AC CE P

40

[20]

PT

Amino-

oak

0.028

RI

a. Copper

a. Copper

6.98

-2.199

NU

9

Mg-Al-

D

8

Fluoride

TE

7

Fluoride

KMnO4-

25

0.27

(mg/L)

modified

45

0.05

-0.9

activated

55

0.03

0.9

bqm

-3.6

[46]

carbon 11

Methylene

Activated

32

0.054

Not

-7.289

blue (mg/L)

carbon

40

0.066

clear

-8.063

50

0.087

-8.618

15

ACCEPTED MANUSCRIPT 0.244

-11.506 106 b

Neutral Red

Spent

20

0.182

-29.50

(mg/L)

cottonseed

30

0.186

-30.57

hull

40

0.205

-31.83

4.67

cotton

10

0.027

1000

-7.73

(mg/L)

pretreated

30

0.017

b

-7.10

with

60

0.009

NU

Waste

20

0.049

Black 5

biomass

30

0.066

L/m-

-10.6

40

0.071

mol

-11.1

0.141

bqm

-1.392

Chestnut

20

(mg/L)

shell

30

Fluoride

18

0.114

-0.876

0.091

-0.298

30

0.067

Not

-2.452

modified

45

0.11

clear

-3.853

CeO2/

60

0.153

-5.255

Al2O3

75

0.164

-6.656

composite

90

0.308

-8.058

Methylene

Fabricated

20

0.860

Not

0.118

blue (mg/L)

porous

40

1.007

Clear

-0.356

functional

50

1.273

-0.831

carbon

60

1.787

-1.305

Malachite

Activated

30

0.036

green

carbon

45

0.041

8.61

60

0.042

8.75

20

0.0041

(mg/L) 19

-9.5

Plasma

(mg/L)

17

AC CE P

40 16

D

Copper

TE

15

b as

MA

Reactive

(L/mmol)

Methylene

a. Parsley

-17.43

-0.003

[49]

13.4

0.079

[50]

-17.423

-0.055

[51]

25.86

0.093

[52]

2.95

0.047

[53]

5.68

-9.20

[40]

8.58

0.032

[54]

-6.03

chitosan 14

[48]

SC

Lac dye

RI

substrate 13

0.117

PT

12

60

b

8.47

bqm

16

-0.69

-1.04

40

0.0050

-1.35

50

0.0050

-1.67

60

0.0051

-1.95

b.Cucum-

20

0.0573

-4.51

ber peels

30

0.0453

-3.50

40

0.0397

-3.29

50

0.0280

-2.30

60

0.0267

c. Water-

20

0.0448

melon

30

0.0344

-0.31

seed hulls

40

0.0264

-0.64

50

0.0327

-1.04

0.0301

-1.50

Chromium (mg/L)

Algal

20

AC CE P

20

TE

60

biomass

50

0.02

-21.5

-0.059

RI

0.0047

NU

SC

30

-2.21 0.09

MA

stalks

D

blue (mg/g)

PT

ACCEPTED MANUSCRIPT

1000b

0.03

-7.3

11.58

0.039

7.1

0.049

[55]

-8.7

The subsequent rectification in the light of this present study has also been shown (Table 2) narrating the importance of the present work. The revised values of the thermodynamic parameters exhibited drastic change compared to that reported in the original studies. Moreover, the reported results demonstrated incorrect inference of the spontaneity, nature of the reaction, and thermodynamic feasibility of the process in many cases. Some typical example may be noticed from Table 2, such as studies represented in Sl. No. 4, 8, 10, 17, etc. claimed the nonspontaneity of the reactions, although the reactions are spontaneous in reality. Furthermore, some endothermic reactions are represented as exothermic by the inappropriate estimation of ∆H0, such as the studies represented in Sl. No. 9c., 19c. The value of ∆S0 is often reported erroneously 17

ACCEPTED MANUSCRIPT as negative, which in turn attributed to the lesser feasibility of adsorption. However, ∆S0 of those cases are positive (Sl. No. 1, 2, 3 and so on). Overall, the adsorption process is incorrectly

PT

assessed through the inappropriate estimation of thermodynamic parameters in several studies.

Remarks

Proposed modification b#

Error due to improper calculation of

25

0.0567

equilibrium constant. The b should be

35

0.0540

converted by Eq. (22). Here, MA is

45

0.0515

-18.206

NU

0.272

16999

-23.652

30

0.247

15436

-24.236

40

D

0.200

12499

-24.820

60

0.147

12530

-25.988

Error due to multiplication with 52.

10

0.032

1663.90

-17.387

The b should be converted by Eq. (22).

30

0.024

1247.93

-18.107

Here, MA is equal to 51997 mg/mol.

50

0.022

1143.93

-18.827

Error due to non adjustment of unit of

30

0.04

7898

-22.766

pollutant to mol/L. b should be

40

0.06

11847

-24.067

converted by Eq. (22). Here, MA is

50

0.06

11847

-25.369

Error due to improper calculation of

2

0.197

12518.56

-21.492

equilibrium constant. The b should be

10

0.141

8959.986

-21.822

converted by Eq. (22). Here, MA is

20

0.170

10802.82

-22.234

equal to 63546 mg/mol.

30

0.137

8705.802

-22.645

40

0.100

6354.6

-23.057

10

3.65

1167453

-32.762

TE

Here, MA is equal to 51997 mg/mol.

4

978.39

20

Error due to multiplication with 55.5. The b should be converted by Eq. (22).

3

1025.89

-17.752

equal to 18998 mg/mol. 2

1077.19

AC CE P

1

∆G0 (kJ/mol) -17.299

Kaeq

SC

T

No.

MA

Sl.

RI

Table 2 Rectification of calculation of thermodynamic parameters using data from Table 1 ∆H0 (kJ/mol) -3.788

∆S0

-6.358

0.058

-7.192

0.036

16.666

0.130

-10.170

0.041

8.427

0.146

(kJ/mol/K)

0.045

equal to 197450 mg/mol. 5

6

Error due to non adjustment of unit of

18

3.81

1218629

-34.218

converted by Eq. (22). Here, MA is

30

4.08

1304988

-35.673

equal to 319850 mg/mol.

40

5.25

1679213

-37.128

Error due to improper calculation of

10

0.0017

32.297

-8.230

equilibrium constant. The b should be

30

0.0022

41.796

converted by Eq. (22). Here, MA is

50

0.0024

45.595

a. Error due to improper calculation of

20

0.621

equilibrium constant. The b should be

25

0.836

53126.46

converted by Eq. (22). Here, MA is

35

1.345

85469.37

-29.079

32462.07

-25.813

20

1.001

207407

-29.989

equilibrium constant. The b should be

25

1.489

308520

-31.055

converted by Eq. (22). Here, MA is

35

1.978

409841

-33.185

c. Error due to improper calculation of

20

1.106

124326

-28.761

equilibrium constant. The b should be

25

1.813

203801

-30.002

converted by Eq. (22). Here, MA is

35

2.763

310591

-32.486

a. Error due to improper calculation of

20

1.01

64181

-26.955

equilibrium constant. The b should be

30

1.34

85151

-28.620

converted by Eq. (22). Here, MA is

40

1.79

113747

-30.284

b. Error due to improper calculation of

20

0.21

12325

-22.953

equilibrium constant. The b should be

30

0.28

16434

-24.438

converted by Eq. (22). Here, MA is

40

0.36

21129

-25.923

20

0.30

15598

-23.741

equal to 63546 mg/mol.

TE

AC CE P

equal to 207200 mg/mol.

D

b. Error due to improper calculation of

0.052

38.344

0.219

32.435

0.213

44.000

0.248

21.806

0.166

20.556

0.148

-5.217

0.063

-10.238

-26.907

MA

8

6.614

-9.279

NU

equal to 18998 mg/mol.

PT

20

RI

7

pollutant to mol/L. b should be

SC

ACCEPTED MANUSCRIPT

equal to 112411 mg/mol. 9

equal to 63546 mg/mol.

equal to 58693 mg/mol. c. Error due to improper calculation of

19

equilibrium constant. The b should be

30

0.37

19238

-24.373

converted by Eq. (22). Here, MA is

40

0.26

13518

-25.005

Error due to improper calculation of

25

0.27

5129.5

-20.744

equilibrium constant. The b should be

45

0.05

949.9

converted by Eq. (22). Here, MA is

55

0.03

569.9

Error due to improper calculation of

32

0.054

equilibrium constant. The b should be

40

0.066

21110

converted by Eq. (22). Here, MA is

50

0.087

27826

-28.328

MA

ACCEPTED MANUSCRIPT

11

78043

-30.546

20

0.182

52558

-26.448

equilibrium constant. The b should be

30

0.186

53713

-27.505

converted by Eq. (22). Here, MA is

40

0.205

59200

-28.561

Error due to improper calculation of

10

0.027

13402

-22.377

equilibrium constant. The b should be

30

0.017

8438

-22.740

converted by Eq. (22). Here, MA is

60

0.009

4467

-23.285

Error due to improper calculation of

20

0.049

49300

-26.407

equilibrium constant. The b should be

30

0.066

66300

-27.784

converted L/mol from L/mmol by

40

0.071

70900

-29.161

Error due to improper calculation of

20

0.141

8953

-22.178

equilibrium constant. The b should be

30

0.114

7239

-22.365

converted by Eq. (22). Here, MA is

40

0.091

5778

-22.553

30

0.067

1273

-18.079

D

Error due to improper calculation of

AC CE P

43.292

0.221

4.504

0.106

-17.241

0.018

13.943

0.138

-16.683

0.019

21.008

0.129

RI

-24.337

0.244

equal to 288780 mg/mol. 13

17271

60

TE

12

-0.131

-16.794

-26.111

equal to 319850 mg/mol.

-59.981

-18.769

NU

equal to 18998 mg/mol.

SC

10

PT

equal to 51996 mg/mol.

equal to 496380 mg/mol. 14

multiplying 1000. 15

equal to 63546 mg/mol. 16

Error due to improper calculation of

20

2090

-20.014

converted by Eq. (22). Here, MA is

60

0.153

2907

-21.949

equal to 18998 mg/mol.

75

0.164

3116

-23.884

90

0.308

5851

-25.819

Error due to non adjustment of unit of

20

0.860

275198

pollutant to mol/L. b should be

40

1.007

322025

-33.348

converted by Eq. (22). Here, MA is

50

1.273

407041

-34.859

equal to 319850 mg/mol.

60

1.787

571572

-36.370

Error due to non adjustment of unit of

30

0.036

13136

pollutant to mol/L. b should be

45

0.041

14961

-25.329

0.042

15326

-26.729

a. Error due to non adjustment of unit

20

0.0041

1301

-17.598

of pollutant to mol/L. b should be

30

0.0047

1513

-18.337

converted by Eq. (22). Here, MA is

40

0.0050

1586

-19.076

50

0.0050

1606

-19.815

60

0.0051

1618

-20.554

b. Error due to non adjustment of unit

20

0.0573

18327

-23.909

of pollutant to mol/L. b should be

30

0.0453

14489

-24.168

converted by Eq. (22). Here, MA is

40

0.0397

12698

-24.427

equal to 319850 mg/mol

50

0.0280

8956

-24.686

60

0.0267

8540

-24.946

c. Error due to non adjustment of unit

20

0.0448

14329

-23.018

of pollutant to mol/L. b should be

30

0.0344

11002

-23.562

converted by Eq. (22). Here, MA is

40

0.0264

8444

-24.106

equal to 319850 mg/mol

50

0.0327

10459

-24.650

60

0.0301

9627

-25.194

20

0.02

1040

-16.922

D

TE

equal to 319850 mg/mol.

Error due to non adjustment of unit of

21

13.943

0.151

4.354

0.093

4.055

0.074

-16.317

0.026

-7.078

0.054

10.634

0.094

RI

SC

60

equal to 364119 mg/mol.

20

-30.326

-23.929

converted by Eq. (22). Here, MA is

19

PT

0.11

NU

18

45

AC CE P

17

equilibrium constant. The b should be

MA

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT pollutant to mol/L. b should be

50

0.03

1560

-19.744

converted by Eq. (22). Here, MA is

PT

equal to 319850 mg/mo *The effect of activity coefficient has not been considered due to data inadequacy.

SC

RI

3.4. Influence of activity

The electrolyte solution with a higher concentration is influenced by the activity

NU

coefficient and accordingly the equilibrium constant will also change. The influence of the activity is rarely considered while calculating the equilibrium constant from the Langmuir

MA

isotherm for using it in van’t Hoff equation, as per the best of the knowledge of the authors. For the precise estimation of the equilibrium constant considering the influence of the activity

follows:

 0 C0   e Ce X V



a0  ae X V

AC CE P

q ae 

TE

D

coefficient, the Langmuir isotherm should be plotted as qae versus ae, where qae is calculated as

(34)

where, C0, a0 and 0 are the initial concentration, activity and activity coefficient at the initial stage, respectively. X is the mass of adsorbent at a volume of V of the solution. The units of the activity and adsorption capacity are mol/L and mol/g respectively. Nevertheless, 0 and e cannot be taken as equal as the initial and equilibrium concentration; they will vary considerably. The Langmuir constant shall be attained from this plot at different temperature. In this case, the Langmuir constant is now numerically equal to the equilibrium constant. In this present study, an attempt to envisage the methodology for the estimation of the influence of activity had been presented in Table 3 for the reference studies (Sl. No. 1 and 2 of Table 2). The activity coefficient for the average concentration had been taken due to non22

ACCEPTED MANUSCRIPT availability of data. It is also assumed that NaF and K2Cr2O7 were used to prepare the synthetic fluoride and hexavalent chromium standard solution, and no other ions were present at the

PT

equilibrium. Although the influence of activity had a lesser effect in this concentration range

RI

(Table 2 and Table 3) higher values of initial concentrations, higher ionic charged solute [Table S1 and S2 in SM, framed using Eqs. (23) and (25)] and estimation of other ions in the solution

SC

may induce a lesser value of activity coefficient and thereby greater influence on the equilibrium

NU

constant.

Average

No.

Ce (mg/L)

1

60

ae (mg/L) 56.43

0.0567

1145.21

35

0.0540

1090.78

-17.909

45

0.0515

1040.18

-18.368

20

0.272

16999.02

-23.808

30

0.247

15436.61

-24.181

40

0.200

12499.28

-24.553

60

0.147

9186.97

-25.297

25

2

30

AC CE P

TE

0.9406

0.832

24.96

Proposed modification ∆G0 (kJ/mol) -17.450

T

b

D

e

Sl.

MA

Table 3 Effect of activity on thermodynamic parameters (using the data from Table 2)

Kaeq *

∆H0 (kJ/mol) -3.789

∆S0

-12.901

0.037

(kJ/mol/K)

0.046

* Kaeq is calculated from Eq. (26)

4. Conclusions The determination of the thermodynamic equilibrium constant from the Langmuir isotherm concept has been analysed. The applicability of the Langmuir isotherm constant (b) in the van’t Hoff equation was assessed for the neutral species, dilute and concentrated ionic solutions. The deductions may attribute to the effective use of b as the thermodynamic

23

ACCEPTED MANUSCRIPT equilibrium constant for the non-ionic or dilute solution of the ionic solute presented as mol/L. Otherwise, necessary modifications on b, as listed in this paper, should be employed to adopt b

PT

as the equilibrium constant. The influence of activity on the equilibrium constant for the ionic

RI

solute with a higher concentration was also estimated. The issues associated to improper estimation of the equilibrium constant in various published documents was addressed and

SC

rectified by the present concept. The present study unequivocally ascribed to the fact that an

NU

appropriate estimation of equilibrium constant may explore the adsorption thermodynamics with the meaningful conclusions. In this direction, the present paper has tremendous significance for

AC CE P

TE

D

MA

appraising the proper methodology for the determination of thermodynamic parameters.

24

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PT

[1]

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SC

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RI

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NU

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MA

[3]

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TE

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PT

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RI

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SC

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NU

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MA

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D

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TE

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AC CE P

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NU

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MA

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TE

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PT

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SC

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NU

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MA

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AC CE P

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PT

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Graphical abstract

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ACCEPTED MANUSCRIPT Highlights

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Applicability of Langmuir constant as thermodynamic equilibrium constant. Inappropriate use of Langmuir isotherm constant in van’t Hoff equation. Formulation for estimation of thermodynamic parameters for different solutes. Novel method to incorporate activity in equilibrium constant from Langmuir model. Rectification of thermodynamic parameters in published papers by proposed method.

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