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Langmuir 1991, 7, 3118-3126
Isosteric Heats of Multicomponent Adsorption: Thermodynamics and Computer Simulations Fokion Karavias and Alan L. Myers* Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104 Received April 5,1991. In Final Form: July 2, 1991
Heats of adsorption were obtained by molecular simulations for the binary mixtures CzH4402, C02CHI, and ~ - C ~ H I O - C adsorbed ~ H ~ on zeolite 13-X.These simulations provide the first comprehensive sets of data on adsorption isotherms and isosteric heats of single gases and their mixtures. The simulations agree very well with experimental isotherms for the amount adsorbed, but individual heats of adsorption have not been measured calorimetrically and the calculation of mixture heats from adsorption isotherms using the methods of thermodynamics is impractical. Heats of adsorption from gas mixtures are predicted from single-gas heats of adsorption by use of two thermodynamic models based on an ideal adsorbed solution (IAS). The IAS heats are in excellent agreement with results from molecular simulations at low surface coverage. At high surface coverage, the predictions agree with the simulations for binary mixtures that form ideal solutions (C2Hd-CO2). However for a highly nonideal, azeotropic mixture such as i-C4H10C2H4 in zeolite X, IAS predictions of isostericheats are poor. For adsorption on energetically heterogeneous surfaces, the excess enthalpy is exothermic and individual heats of adsorption are always greater than those predicted for an ideal solution.
Introduction The heat of adsorption of a single gas can be obtained directly from calorimetric measurements. Alternatively, isosteric heat can be determined indirectly from thermodynamics by differentiating adsorption isotherms with respect to temperature.' Heats of adsorption of the components of a mixture are very difficult to measure calorimetrically and, as far as we know, have never been studied experimentally in conjunction with mixed-gas adsorption equilibria. It is impractical to calculate heats from equilibrium binary isotherms, since data measured along loci of constant (P,yl),(P,T),and (T,yJ are needed.2 At present, one must make some ad hoc assumption, for example, that the mixture heats are constant or that they are equai to the single-gas values at the same bulk-phase pressure. Thermodynamically consistent isosteric heats for binarygas adsorption can be obtained from molecular simulations of systems in which the intermolecular potentials are precisely defined. New theories can be developed and tested based on insights and data obtained from such studies. This work has two purposes: (1)to derive the thermodynamic equations connecting heats of adsorption for the components of a gaseous mixture to heats of adsorption of the pure gases; (2) to generate isosteric heats from computer simulations of single- and binary-gasadsorption in zeolite cavities of type X. Thermodynamics of Isosteric Heats Thermodynamic System. The thermodynamics of multicomponentgas adsorption has been covered very well by Sircar;2 some of the fundamental equations and concepts in that work are summarized in this section. A Gibbs dividing surface that separates the gas phase from the adsorbed phase is shown schematically in Figure 1. A second dividing surface separates the adsorbed phase from the solid phase, and it is assumed that the solid is inert and nonvolatile. By inert we mean that the chemical (1) Talu, 0.;Myers,A. L. AZCHE J. 1988,34, 1887-1893. (2) Sircar, S. J . Chem. SOC.,Faraday Trans. I 1985,81, 1527-1540.
Ub, Vb, nib
BULK GAS PHASE
Ua, Va, np
ADSORBEDPHASE
SOLID ADSORBENT
Figure 1. Schematic representation of Gibbs dividing surface separating bulk phase from adsorbed phase.
potential of the solid is unchanged by isothermal adsorption (except by a negligible amount V8 dP).The inertsolid approximation is useful for molecular simulations in which the atoms composing the solid are fixed in space and pairwise interactions between molecules of the gas and atoms of the solid are included in the energy of the adsorbed phase Ua. Values for extensive variables (U,V, ni) refer to unit mass of solid adsorbent (m, = 1). Let n; be the absolute amount adsorbed of ith component. The experimental variable for measuring the amount of adsorption from a single-gas or multicomponent mixture is the surface excess: nr = ni - Vpib (1) where pp is the molar density of i in the bulk gas phase, ni is the total amount of i introduced to the system, and V = vb + Va is the dead space accessible to the gaseous molecules. The superscripts a and b stand for the adsorbed and bulk phases, respectively. The excess thermodynamic properties Ze (enthalpy, entropy, Gibbs free energy, grand potential, etc.) are defined in accordance with the Gibbs model of Figure 1 as follows: z" = Vypaza - pbzb) (2) where za and zb are molar thermodynamic properties of
0743-7463/91/ 2407-3118$02.50/0 0 1991 American Chemical Society
Langmuir, Vol. 7, No. 12, 1991 3119
Isosteric Heats of Multicomponent Adsorption the adsorbed and gas phases, respectively. Equation 2 applies to He, Se, Ge, Qe, etc., but Ve = 0. The integrated forms of the fundamental equations for the adsorbed and gas phases are
(3) The sum -(PVa + IIA) has been replaced by the grand potential of the adsorbed phase Qa, because for microporous adsorbents any breakdown into volume ( P V a ) and volume ( Va), area (IIA)terms is arbitrary. Pressure (P), spreading pressure (II), and area (A) are not well-defined for microporous adsorbents like activated carbon or zeolites. The spreading pressure can be calculated from the pressure tensor for a planar surface but not for a curved one. However, the grand potential Qa is a well-defined thermodynamic variable whether the surface is planar or curved. For experiments and molecular simulations of adsorption in micropores, the value of Qa = - P V a - IIA is calculated from thermodynamics by eq 5 below. For the bulk phase, the grand potential contains only one term: Qb = -Pbvb. The Gibbs equations for the adsorbed and bulk gas phases are
The equilibrium conditions for minimum energy Ua+ Ub at constant S a S b , constant n: + np, and constant Qa + Qb are equality of temperature (T"= Tb = T )and equality of chemical potentials (p! = pp = pi) in the adsorbed and gas phases. Contrary to the case for vapor-liquid equilibrium between bulk phases, there is no requirement of pressure equality in the adsorbed and gas phase and in general P z Pb. The Gibbs adsorption equation follows from eqs 2 and 4:
+
-dQe = Se d T + c ( n f dpi)
(5)
i
Enthalpy (Ha= Ua + P V a ) and Gibbs free energy (Ga = Ha- TSa)in the adsorbed phase are defined by the usual bulk-phase relations. It follows from the above equations that
IT = Ge + TS" (7) Differentiating, making use of eq 5, and dropping the term (P- P)Va, which is usually very small dGe = -Se d T +
c(pi
d(IIA) = S' d T + c ( n f dpi) i
If the adsorbent is not inert, by definition its chemical potential changes upon adsorption. Bering et ala4showed that from a thermodynamic viewpoint the equations describing adsorption in inert and noninert adsorbents are equivalent. In the case of adsorption in a noninert solid, eqs 5 and 10 contain another term: -d(Qe+ (p, - p : ) ) = S' d T + c ( n f dpi)
(11)
i
d(IIA - (p, - p : ) ) = Se d T + c ( n f dpi)
(12)
i
where p, and pi refer to the chemical potential of the solid adsorbent with adsorbed gas and in the pure state, respectively. Thermodynamic measurements yield the terms within the braces. Heats of Adsorption. The isosteric heat of adsorption is defined as the difference of partial molar enthalpy in the gas phase and the excess partial molar enthalpy in the adsorbed phase: (13) Although qst is traditionally called the isosteric heat of adsorption, eq 13 shows that it is actually the heat of desorption. A second heat of adsorption in common use is the differential heat of ad~orption,~ defined by (qsJi I hp - hf
(qd)i E
Since hf
N
up - uf
(14)
uf (qd)i
= (qst)i - pbup
(15)
where up is the partial molar volume of i in the bulk gas phase. For a perfect gas ( ~ & i = (q,Ji - R T (16) To simplify notation, from here on the symbol q refers to the isosteric heat qst. The excess partial molar enthalpy in the adsorbed phase is related to the chemical potential by the Gibbs-Helmholtz equation:
hf = - F ( d ( p i / n / W n ;
(17)
Adopting the perfect-gas reference state p*, the chemical potential is pi = pi + R T In V J P ) (18) and P" is the perfect-gas reference pressure (1 atm). Combination of eqs 13, 17, and 18 yields the Clapeyron equation for mixtures:
dnf)
i
1
Equations 1-9 define the thermodynamic system for adsorption. For molecular simulations that generate absolute quantities for the adsorbed phase, eqs 5-9 are written for absolute values (Qa = ne,n' = nf, etc.). The excess grand potential from eq 2 is Qe = -HA - (P- Pb)Va. For microporous adsorbents, the volume of the adsorbed phase (Val is usually very small; the volume term can be dropped and eq 5 can assume the form used in adsorption
where (hp- h;,') is enthalpy departure from the perfectgas state. For the special case of adsorption from a pure perfect gas at pressure P q = R p ( d In PIdT),,
(20)
Isosteric Heat of Multicomponent Adsorption. The last step in the development is to connect mixture heats (3) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989. (4) Bering, B. P.; Myers, A. L.; Serpinskii, V. V. Dokl. Akad. Nauk
USSR 1970,193, 119-122.
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3120 Langmuir, Vol. 7, No. 12, 1991
of adsorption with values for single gases. From this point on let n = ne for simplicity in notation. Let $ = -Oe/RT; +has the units of specific amount adsorbed (mol/kg). We make this connection by writing the fugacity f i in the mixture in terms of a standard-state fugacity (measured at the same values of T and $ as the mixture:' fi
= (Tixi
24
t
I
q1-
(21)
where xi = ni/n, and n = Cini is the total amount (excess) in the adsorbed phase. At this point we introduce a key assumption: that the adsorbed phase is an ideal solution and therefore yi = 1. According to the theory of ideal adsorbed solutions, the total amount adsorbed obeys the equation 1 4 1 1 . . .
0.0
The superscript O refers to the standard state for the pure adsorbate. From calculus, the partial derivative needed in eq 19 is
,
,
,
,
I
,
,
.
,
,
,
,
,
,
,
,
1 .o
0.2
0.4 0.6 0.8 Mole fraction of the adsorbed phase, x1
Figure 2. Prediction of eqs 36 and 37 for simplified IAS model: q? = 25, qy = 18. Constant pressure and temperature.
where
(3)a$
TS
(*) dT (23) n j
Constant x means that the composition of the adsorbed phase is fixed. Using eq 21, the two partial derivatives of the fugacity are
(a In fi/an,, = (a In (/an,
(24)
(a ln f i / w ) T j = (a ln ( / W I T
(25)
For a perfect gas, the partial molar enthalpy in the bulk phase hi = h; = hi and Pyi = (xi, where yi is the mole fraction of the gas phase. Thus
and The last derivative in eq 23 is written as a ratio of derivatives:
and Equation 30, which connects mixture heats qi to singlegas heats q;, applies to a perfect-gas phase and an ideal adsorbed solution (IAS). Superscripts O identify the singlegas reference states measured at a common value of $. Equation 30 has the limits required by thermodynamics. For pure component i
If qp is constant (independent of coverage) for the single gases, then qi = and the mixture heat qi = q; is also constant.
iji is the molar integral isosteric heat of adsorption, eq A2, and hi is the molar integral enthalpy of the bulk phase, eq A8. Substitution of eqs 27 and 28 into 26, followed by substitution of eqs 24-26 into eq 23, followed by substitution of eq 23 into eq 19 yields
A Simplified IAS Theory for Isosteric Heats For binary adsorbed solutions that are nearly ideal, the isosteric heat of adsorption increases (or decreases) monotonically from the value for the pure gas (q:) to the valueatinfinitedilution (ql). Thisbehaviorisexemplified by Figures 2 and 5. From eq 30, the isosteric heats a t infinite dilutions of component 1 in component 2 and of component 2 in component 1 are n:(q; -
= ni(qi - $t
ni(q,"-
0
0
= nl(ql - ~
0
1 )
(32)
To understand the meaning of these equations let the isosteric heats of the pure gases be represented by linear equations: i
(29)
qi = qy
+ Cln;
qi = qp + C,ni
(33)
Langmuir, Vol. 7,NO.12, 1991 3121
Isosteric Heats of Multicomponent Adsorption
for the other component. This behavior is observed at low coverage for the ideal CzH4-CO2 mixture, Figure 5. Excess Isosteric Heat of Multicomponent Adsorption. The excess partial molar enthalpy of adsorption is related to the excess Gibbs free energy, eq 6, through the Gibbs-Helmholtz equation:
lo
I. 0
,
,
,
,
,
,
1
,
, 2
' 3
--
By use of eq 13, the isosteric heat can be substituted for partial molar enthalpies:
,
4 5 6 Molecules per Cavity
7
8
9
10
Figure 3. MC results for isosteric heat as a function of coverage inzeolite X at 298.15 K (0) COz,(V)CzH4, ( 0 )C H I ,(A)i-C4H10. q y is the isosteric heat at the limit of zero coverage (Henry's law region of the isotherm). Although the assumption of linear single-gas isosteric heats is a crude approximation, it is usually valid at low and medium coverage, even for adsorption on highly heterogeneous surfaces, Figure 3. From eqs 33 and A2, the molar integral isosteric heats are
(34) The standard states refer to a fixed value of spreading pressure, for which the np are approximately equal at finite coverage and exactly equal at the limit of zero pressure. Equation 32 reduces to q; = s;
+ q; - 4;
q; = 4; + q: - 4;
where qid is the isosteric heat calculated by the ideal adsorbed solution theory, eq 30. An excess heat of adsorption is defined as
8" = &q; i
= &(qi
- qid)
i
Subtraction of eq 39 from 40 following substitution of eq 41 yields the Gibbs-Helmholtz equation for the excess heat of adsorption:
(35)
Combination of eqs 33-35 with n; = ni and the assumption of constant spreading pressure at isobaric conditions yields
Then, the isosteric heat of component i can be calculated as a linear function of the adsorbed-phase mole fraction:
+
Equation 39 holds for both real and ideal adsorbed solutions. In the case of ideal adsorbed phases
(37) q; = q; (qp - q;)xi Equations 36 and 37 may be used instead of eq 30 for the calculation of isosteric heats of multicomponent adsorption. Both models are based upon the ideal adsorbed solution assumption. Equations 36 and 37 are significantly simpler than eq 30, but they involve some additional assumptions: (a) Isosteric heats of pure gases are linear in surface coverage. (b) At constant Pand T, the spreading pressure of the binary adsorbed phase is also constant. (c) At a fixed value of the spreading pressure, np for all components are equal. (d) For isobaric adsorption, the isosteric heat varies linearly from the value for the pure gas to the value at infinite dilution. Equations 36 are physically reasonable: the average value of the infinite-dilution heats is approximately equal to the average of the isosteric heats of the single gases. The difference of the infinite-dilution heats is approximately equal to the difference of the isosteric heats at the limit of zero coverage because surface interactions dominate over cooperative interactions. In the limit of zero coverage q; = q; = q y as required by Henry's law. Figure 2 shows the prediction of eqs 36 and 37. At constant pressure, the lines are parallel. If the isosteric heat at infinite dilution is higher than the value for the pure gas for one component, then the isosteric heat at infinite dilution is lower than the value for the pure gas
where AGe = Ge - Gesid is the excess Gibbs free energy of mixing, a measure of the ideality of the adsorption system. For adsorption on energetically inhomogeneous surfaces, AGe is usually negative and increases as a function of temperature (at sufficient high temperatures all adsorbed solutions are ideal and AGe = 0). In this case, the excess heat of adsorption 8" is always positive.
Statistical Mechanics of Isosteric Heats The statistical mechanics of adsorption has been thoroughly surveyed by Nicholson and Par~onage.~ In this section, we discuss the calculation of isosteric heats from computer simulations of adsorption in microporous adsorbents. In the grand canonical ensemble the chemical potentials are independent variables and adsorption isotherms are obtained directly from simulations, in contrast to other ensembles (i.e., canonical, microcanonical) in which this is not possible. In inhomogeneous systems, the grand potential of the adsorbed phase Qa contains both volume ( P V a ) and area (HA) terms: = Pv"+ rIA =kTlnE (43) where E is the partition function in the grand canonical ensemble. The mean number of molecules of component i is (nq)= k T a( In2 e) (44) -Qa
'pi
T,F~+~
The (nq)are calculated as averages in the grand canonical ( 5 ) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: New York,1982.
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3122 Langmuir, Vol. 7,No. 12, 1991
ensemble. The pressure and the composition of the gas phase corresponding to a given value of the chemical potentials pi are calculated from an equation of state for the bulk fluid. The number of adsorbed molecules calculated from a computer simulation is the absolute amount of the ith component in the adsorbed phase, not the surface excess nr. A t low pressures, nr N n;. However, at high pressures the absolute amount adsorbed is in general not equal to the surface excess. Unlike experiments, surface excess cannot be evaluated easily from computer simulations of adsorption in microporous adsorbents because the volume of the system is an ill-defined property. In certain regions of space within, for example, a zeolite the solid-fluid potential energy will be essentially positive infinite, in others it will be negative. The boundary surfaces on which it is zero are a crude measure of the shape and the volume accessible to the incoming molecules, but there is no exactly defined volume to which they can be said to have access.6 Therefore, all the properties calculated from computer simulations are absolute quantities for the adsorbed phase and not surface excess properties. Although the determination of the dead space volume in a microporous cavity is somewhat ambiguous, the surface excess may be calculated from the amount adsorbed by eq 1. Similar differences are encountered in the experimental determination of the dead space volume by helium measurements. The isosteric heat of adsorption was defined in eq 13 as a difference of partial molar properties. If absolute quantities are used, then qi = h! - hy Since Va
