Article pubs.acs.org/IECR
Concept of Ideal Adsorbed Solution using Gibbsian Surface Excess Framework Shivaji Sircar* Department of Chemical and Biomolecular Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, United States ABSTRACT: A concept of creating an ideal multicomponent- gas adsorbed phase by mixing pure gas adsorbed phases of the components at constant surface potential and temperature using the Gibbsian Surface Excess (GSE) framework of adsorption is proposed. The GSE provides the only unambiguously measured experimental quantity to describe the extent of adsorption. The actual amounts adsorbed can be estimated from the GSE only by making extraordinary assumptions about the volume and structure of the adsorbed phase. The correlations between the multicomponent and pure gas GSE isotherms generated by the proposed concept closely resemble those for the classic Ideal Adsorbed Solution Theory (IAST), which was developed by using the actual amounts adsorbed as the primary variables. The proposed correlations can be obtained by simply substituting the actual amount adsorbed of components in the corresponding IAST correlations by the GSE of the components. All pure and multicomponent gas adsorption data published in the literature are GSE, and they are directly used in the IAST by loosely assuming that they are actual amounts adsorbed. Thus, it appears that the proposed concept of this work has been inadvertently used in all past works which use IAST to estimate multicomponent adsorption isotherms from pure component isotherm data.
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gas mixture at P, T, and yi are given, respectively, by na0 i (= Va0ρa0) and nai (= Vaρaxai ), where xai (= nai /na) is the mole fraction of component i in the adsorbed phase and na (= ∑inai = Vaρa) is the total amount adsorbed. The quantities V a0 , V a , ρ a0 , ρ a , and x ia cannot be experimentally measured since they depend on the unknown location of the Gibbs interface dividing the bulk and the adsorbed phases. In addition, the location of the interface depends on P, T and yi. Thus, the actual amounts adsorbed (na0 i and nia) cannot be experimentally measured. The sheer conceptual beauty and strength of the Gibbsian surface excess framework of adsorption is that the excess quantities do not depend on the location of the Gibbs dividing surface, and they can be unambiguously estimated by monitoring the changes in the bulk gas phase properties (P, T, and yi) during the experimental protocol designed (mentioned earlier) for measuring the GSE.1−3 It follows from eqs 1 and 2 that extraordinary assumptions regarding the adsorbed phase volume, density, and composition are necessary for estimation of actual amounts adsorbed from the experimentally measured surface excesses. Such assumptions are, however, speculative and can be misleading. a0 The equations, however, indicate that (a) nm0 i ∼ ni for a pure a gas if ρ ≫ ρ (low gas pressure and high adsorption capacity) a a a and (b) nm 1 ∼ n1 for a binary gas if ρ x1 ≫ ρy1, which is satisfied a when ρ ≫ ρ and the selectivity of adsorption of component 1 over component 2 [S12 (= na1y2/na2y1)] > 1, or when y1 → 0 and a m a S12 ≫ 1.0. It is important to note that nm 2 ≠ n2 even if n1 ∼ n1. a The difference between nm and n calculated by assuming that i i
INTRODUCTION The Gibbsian Surface Excess (GSE) is the true experimental variable in quantifying the extent of equilibrium adsorption from a pure gas or a multicomponent gas mixture.1−3 All conventional macroscopic experimental techniques for measurement of adsorption equilibria such as gravimetric, volumetric, column dynamic, total desorption, closed loop recycle, isotope exchange, etc. directly yield GSE as functions of gas phase pressure (P), temperature (T), and composition (yi = mole fraction of component i).1−3 The GSE is defined as follows: Pure Gas i: ⎡ ρ ⎤ nim0 = V a0(ρa0 − ρ) = nia0⎢1 − a0 ⎥ = [nia0 − V a0ρ] ⎣ ρ ⎦ (1)
Mixed Gas: ⎡ ρy ⎤ nim = V a(ρa xia − ρyi ) = nia⎢1 − a i a ⎥ = [nia − V aρyi ] ρ xi ⎦ ⎣ (2)
The variables nm0 and nm i i are, respectively, the specific equilibrium Gibbsian surface excess (m mol/g) of pure gas i at P and T, and that for component i of a gas mixture at P, T, and yi. The variables Va0 and Va are, respectively, the specific Gibbsian adsorbed phase volume (cm3/g) of the pure gas and the mixed gas system. ρa0 and ρa are, respectively, the equilibrium adsorbed phase density (mol/cm3) for a pure gas at P and T and that for a mixed gas system at P, T, and yi. The variable ρ is the density of the gas phase at P, T, and yi. The actual specific amount of component i adsorbed (mmol/g) in equilibrium with a pure gas at P and T and in equilibrium with a © XXXX American Chemical Society
Received: Revised: Accepted: Published: A
August 30, 2016 October 27, 2016 October 27, 2016 October 27, 2016 DOI: 10.1021/acs.iecr.6b03337 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research Va is defined by the total pore volume of the adsorbent or ρa is equal to the liquid density of the adsorbate shows that the difference is small when the gas pressure is low and the component is very selectively adsorbed. The difference can be very large when the gas pressure is high and the component is less selectively adsorbed.3,4 Thus, in general, equality between the Gibbs excess and actual amount adsorbed cannot be granted.
Thus, a relationship between P*i for different pure gases for equal spreading pressures (π0i = π0j = π*) at T can be obtained by integration of eq 6 as Aπ* = RT
IDEAL ADSORBED SOLUTION THEORY (IAST) Myers and Prausnitz proposed an elegant concept for estimating multicomponent gas adsorption equilibria from the corresponding pure gas adsorption equilibria at the same temperature by assuming that the mixed gas adsorbed phase is produced by ideal mixing of corresponding pure gas adsorbed phases at a constant spreading pressure (π) and temperature (T).5 They used the solution thermodynamics model of adsorption employing the actual amounts adsorbed of components (nai ) as the primary extensive variables to describe the adsorbed phase. The intensive variables were T, π, and xai (= nai /∑inai ). The adsorbent was assumed to have a specific surface area of A (m2/g). It was proposed that the change in the molar Gibbs free energy for creating an ideal, mixed adsorbed phase [Δg = g − ∑ixai g0i ] by mixing pure gas adsorbed phases at constant T and π can be described by5
1 = na
∑ i
xia nia *
constant π and T
constant T
(7)
(8)
However, eq 5 is valid only when γai = 1. Thus, estimation of a multicomponent adsorption isotherm for a nonideal adsorbed phase (γai ≠ 1) using eqs 5 and 8 may be questionable.
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CONCEPT OF IDEAL ADSORBED SOLUTION USING THE GIBBSIAN SURFACE EXCESS FRAME WORK Sircar developed a thermodynamic framework for describing adsorption of pure and multicomponent gases on a porous or a nonporous adsorbent in which the extensive variables were the Gibbsian surface excess (GSE) of component i (nm i ) defined by eqs 1 and 2.4 A detailed theoretical and experimental study of pure and multicomponent gas heats of adsorption using the GSE framework was also reported.9 The intensive variables for these works were the surface potential of the adsorbed phase (φ), the system temperature (T) and the fractional surface m m m m m excess of component i (xm i = ni /n , n = ∑ini , ∑ixi = 1). An a excess Gibbs free energy (g , cal/g) for this model was defined as4
where g(T, π, xai ) is the molar Gibbs free energy of the mixed gas adsorbed phase at T, π, and xai , which is in equilibrium with a mixed gas phase at P, T and yi, and g0i (T, π) is the molar Gibbs free energy of a pure gas adsorbed phase of component i at T and π which is in equilibrium with a pure gas phase at pressure P*i and temperature T. The expression for Δg given by eq 3 was proposed by drawing an analogy between Δg (= RT∑iyi ln yi) for creating an ideal gas mixture (mole fraction of component i = yi) at P, T by mixing pure gases at P and T or Δg (= RT∑ixi ln xi) for creating an ideal liquid mixture (mole fraction of component i = xi) at P, T by mixing pure liquids at constant P and T. The IAST yields the following relationships between the pure gas and the mixed gas equilibrium isotherm properties when the adsorbed phase is thermodynamically ideal.5 Pyi = Pi*xi a
ni a0d ln P
Pyi = Pi*xiaγia
(3)
i
Pi*
Equations 4, 5, and 7 can be used to estimate nai for a given set of P, T, and yi from the corresponding pure gas adsorption isotherm of component i [ na0 i (P, T)] at temperature T. Equation 7 can be used to obtain the correlations between P*i for different components at constant π and T. The IAST has been used by numerous authors for predicting binary adsorption isotherms. Several compilations of these studies can be found in the literature.6,7 The quality of prediction can be good to fair to poor depending on the characteristics of the adsorbate molecules and the adsorbent.6,8 It is a common practice to modify eq 4 by introducing the concept of adsorbed phase nonideality using an activity coefficient (γai ) for component i in the adsorbed phase for a case where IAST fails to estimate xai so that5
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Δg = RT ∑ xia ln[xia]
∫0
Mixed Gas: g m = V a(ρa g a − ρg );
(4)
Pure Gas i: gi m0 = V a0(ρa0 gi a0 − ρgi 0)
constant π and T
where ga and g are, respectively, the molar Gibbs free energies of the adsorbed and the equilibrium bulk gas phases of a mixed 0 gas system at P, T, and yi. The variables ga0 i and gi are, respectively, the molar Gibbs free energies of the adsorbed and the equilibrium bulk gas phases of a pure gas (component i) system at P, T. The isothermal (constant T) thermodynamic relationships based on the Gibbsian surface excess framework, which are relevant for this work, are reproduced below.4 Mixed Gas:
(5)
where the variable nai *(π,T) is the actual amount of pure gas i adsorbed at pressure Pi* and temperature T. It should be noted that the derivation of eq 5 invokes the concept of a surface area of the adsorbent (A. m2/g) and a specific area per mole of adsorbate in the ideal adsorbed phase (pure and mixed). There is no change in the specific area of the adsorbate during ideal mixing of the adsorbed phases at constant T and π.5 The following Gibbs adsorption isotherm equation for a pure gas can be integrated to obtain the spreading pressure (π0i ) of pure gas i at a P and T: A d πi 0 = ni a0RT d ln P
constant T
(9)
(a) g m = φ +
∑ μi 0 ni m i
(b) dg m =
∑ μi dni m i
(c) dφ = −∑ ni dμi m
(6)
i
B
(10) DOI: 10.1021/acs.iecr.6b03337 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research ⎡ 1 =φ*⎢ m + ⎢⎣ n
Pure Gas: (a) gi m0 = φi 0 + μi 0 ni m0 0
m0
(c) dφi = −ni dμi
(b) dgi m0 = μi 0 dni m0
0
∫0
P
Δ = RT ∑ xim ln[xi m]
∫0
P
Equations 15 and 16 can be combined to get ⎡ 1 φ*⎢ m − ⎢⎣ n
g̅
φ = m + n
∑ i
ximμi ;
constant T
constant T , yi
gi̅
i
⎛ Py ⎞ xim ⎤ ⎥ + RT ∑ xim ln⎜ i m ⎟ = 0 nim0 ⎦⎥ ⎝ Pi*xi ⎠ i
(17)
Equation 17 is a characteristic property (identity) of the mixing process for creation of an ideal mixed gas adsorbed phase from pure gas adsorbed phases at constant φ* and T using Gibbs excess as primary variables. The identity is automatically satisfied for any combination of φ* and xm i when 1 = nm
∑ i
xim nim0
or
∑ i
nim nim0
=1
constant φ* and T (18)
Pyi =1 Pi*xim
(12)
m0
=
φi0 nim0
+ μi0
or
Pyi = Pi*xi m
constant φ* and T (19)
It should be noted that the derivation of eq 18 does not invoke the concept of a surface area for the adsorbent. Thus, there is no apparent physical implication like that associated with eq 5. Equations 12, 18, and 19 can be simultaneously used to estimate the equilibrium surface excess of component i of a multicomponent gas (nm i ) at P, T, and yi using pure gas surface excess isotherms nm0 (P, T) of component i. Thus, the concept i of ideal adsorbed phase using GSE as primary variables, which is proposed in this work, is described by the set of eqs [12, 18, and 19] which is analogous to the set of equations [4, 5, and 7] describing the IAST developed in terms of actual amounts adsorbed as primary variables. The mathematical forms of these two sets of equations are nearly identical. The set of equations proposed in this work can be obtained by simply replacing actual amounts adsorbed as primary variables in IAST by the corresponding GSE as the primary variables. . It should be pointed out here that all experimental pure and multicomponent gas equilibrium adsorption isotherm data published in the literature actually report GSE of adsorbates as functions of P and T (pure gas) or P, T, and yi (mixed gases). However, the GSE is often loosely referred to as amounts adsorbed. These data are then directly used to estimate multicomponent adsorption isotherms from pure gas isotherms using IAST. It appears that these exercises inadvertently used the ideal adsorption solution concept in GSE framework, which is presented in this article.
(13)
A new variable [gm̅ = gm/nm] is introduced where gm̅ (cal/ mol) is the excess Gibbs free energy for the mixed gas adsorption system at P, T, and yi per unit amount of total equilibrium surface excess at P, T, and yi. The corresponding m0 m0 variable for a pure gas i is gm0 i̅ (= gi /ni ). Thus, these new variables represent the excess molar Gibbs free energy (cal/ mol) analogue in the Gibbsian framework of adsorption. Equations 10a and 11a can be rearranged to get m
∑
and
nim0 dP P
nm dP P
(15)
(16)
i
Mixed Gas: ϕ(P) = −RT
i
⎛ Py ⎞ xim ⎤ ⎥ + RT ∑ xim ln⎜ i ⎟ nim0 ⎦⎥ ⎝ Pi* ⎠ i
The gas phase is assumed to be ideal in the formulation of eq 15. It is now proposed analogous to the IAST concept [eq 3] that the change (Δ) for an ideal mixing is given by
(11)
Equations 10 and 11 were derived rigorously using the classic Gibbs surface excess model of adsorption and the bulk gas thermodynamic properties without making any simplifying assumptions.4 The variables φ and φ0i , are respectively, the surface potential m of the mixed gas adsorbed phase defined by nm i (or xi ) and T, and that of pure gas adsorbed phase of component i defined by nm0 and T. The variable μi [= μi*(T) + RT ln Pyi] is the i chemical potential of component i in an ideal gas phase at P, T, and yi which is in equilibrium with an adsorbed phase at φ, T 0 and nm i . Similarly, μi [=μ* i (T) + RT ln P] is the chemical potential of pure ideal gas i at P (= Pi*) and T which is in equilibrium with a pure gas adsorbed phase at φ0i , T and nm0 i . The variable μ*i is the standard state (P = 1 atm, T) chemical potential of pure gas i. Equations 10c and 11c are the Gibbs adsorption isotherms in the Gibbsian surface excess framework. They can be integrated to obtain φ0i (P) and φ(P) as follows: Pure Gas: φi 0(P) = −RT
∑
(14)
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FORMATION OF AN IDEAL MULTICOMPONENT GAS ADSORBRD PHASE BY MIXING PURE GAS ADSORBED PHASES AT CONSTANT φ AND T The IAST proposes the formation of a mixed gas adsorbed phase described by π, T and nai (or xai ) by ideal mixing of pure gas adsorbed phases of the components described by π0i , T and 5 na0 i . A similar concept is applied here using the Gibbsian surface excess framework for creating a multicomponent gas m adsorbed phase characterized by φ, T, and nm i (or xi ) by mixing pure gas adsorbed phases of the components characterized by 0 φ0i , T, and nm0 i at constant surface potential (φ = φi = φ*) and T. m m0 The change [Δ = [gm̅ (φ*, T, xm i ) − ∑ixi g ̅ ] in the excess Gibbs free energy per mole of total surface excess in the adsorbed phase due to an ideal mixing process described above can be obtained by combining eqs 10, 11, and 14 as
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CONCLUSIONS The ideal adsorbed solution concept of creating a multicomponent gas adsorbed phase by mixing the corresponding pure gas adsorbed phases at constant surface potential and temperature is proposed employing the Gibbsian framework of C
DOI: 10.1021/acs.iecr.6b03337 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
(10) Sircar, S. Excess Properties and Column Dynamics of Multicomponent Gas Adsorption. J. Chem. Soc., Faraday Trans. 1 1985, 81, 1541.
adsorption. The concept can be used to estimate multicomponent GSE isotherms from pure gas GSE isotherms, which are the true experimental variables. The governing equations of the proposed concept relating pure and multicomponent GSE isotherms are analogous to those of the IAST and they become the same when the GSE is equal to the actual amount adsorbed. It was demonstrated earlier that the GSE framework of pure and multicomponent gas adsorption can satisfactorily describe all practically useful aspects of adsorption technology such as (a) equilibria and kinetics on homogeneous and heterogeneous adsorbents, (b) isosteric heats of adsorption, (c) heat capacity of the adsorbed phase, (d) various thermodynamic consistency tests for pure and binary gas adsorption systems, (e) component mass and overall heat balances in an adsorbent column by using the conventional differential mass and enthalpy balance equations across a differential section in the column, and by invoking the relationships between the total adsorbate mass and enthalpy per unit mass of adsorbent in the column and the corresponding GSE for mass and enthalpy, to fully describe nonisothermal column dynamics,10 and hence, (f) formulation of a numerical adsorptive process model for gas separation.3,9,10 The present study shows that the GSE model can also be used to describe a thermodynamically consistent correlation between mixed gas and pure gas GSE isotherms analogous to IAST. Consequently, the author’s view is that there is no need to estimate actual amounts adsorbed from GSE by making extraordinary, nonverifiable assumptions about the structure and properties of the adsorbed phase, which can be very misleading. The GSE, which can be experimentally measured unambiguously, provides a framework which is capable of meeting all basic and practical needs for characterizing pure and multicomponent gas adsorption systems and modeling adsorptive gas separation processes.
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AUTHOR INFORMATION
Corresponding Author
E-mail: *
[email protected]. Notes
The author declares no competing financial interest.
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REFERENCES
(1) Gibbs, J. W. The Collected Works of J. W. Gibbs; Longmans and Green: New York, 1928. (2) Defay, R.; Prigogine, I.; Bellmans, A.; Everett, D. H. Surface Tension and Adsorption; John Wiley: New York, 1966. (3) Sircar, S. Gibbsian Surface Excess for Gas Adsorption − Revisited. Ind. Eng. Chem. Res. 1999, 38, 3670. (4) Sircar, S. Excess properties and Thermodynamics of Multicomponent Gas Adsorption. J. Chem. Soc., Faraday Trans. 1 1985, 81, 1527. (5) Myers, A. L.; Prausnitz, J. M. Thermodynamics of Mixed gas Adsorption. AIChE J. 1965, 11, 121. (6) Valenzuela, D.; Myers, A. L. Gas Adsorption Equilibria. Sep. Purif. Rev. 1984, 13, 153. (7) Wu, C.-W.; Sircar, S. Comments on Binary and Ternary Gas Adsorption Selectivity. Sep. Purif. Technol. 2016, 170, 453. (8) Sircar, S. Influence of Adsorbate Size and Adsorbent Heterogeneity on IAST. AIChE J. 1995, 41, 1135. (9) Sircar, S.; Mohr, R.; Ristic, C.; Rao, M. B. Isosteric Heat of Adsorption: Theory and Experiment. J. Phys. Chem. B 1999, 103, 6539. D
DOI: 10.1021/acs.iecr.6b03337 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
