7542
J. Phys. Chem. B 1999, 103, 7542-7550
Hysteresis and Isotherm Equations in Gas-Solid Adsorption from Maximum Entropy Production Stefano A. Mezzasalma* Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden UniVersity, P.O. Box 9502, 2300 RA Leiden, The Netherlands ReceiVed: February 11, 1999; In Final Form: May 11, 1999
Hysteresis and isotherm equations in gas-solid adsorption have been investigated by using a variational condition derived from the constraint of maximum irreversible entropy production. Functional formulation for the adsorption entropy has been stated through a useful formulation for the isotherm equations. The ideal adsorption law was rewritten in the integral form involving the sum between a continuous function and a Dirac distribution located at the concentration value where fluid condensation is expected. The real adsorption law was related to the broadening of the impulsive term that is driven by the entropy production. To generalize the entropy definition to a functional dependence in the adsorbate concentration, the sum among the ideal continuous law, a delta convergent sequence and a perturbation which is sensitive to the broadening degree, was considered in the integral equation for the real adsorption isotherm. Application of the Euler-Lagrange theorem pointed out two different behaviors that can be identified with ascending (adsorption) and descending (desorption) curves of a cycle at a constant temperature. They can be approximated to formally equivalent expressions which involve the sum between the ideal law in the gaslike regime and two liquidlike contributions respectively related to the Kelvin equation and a modified Freundlich adsorption. Accordingly, the difference between real and ideal isotherms has been introduced as a quantity independent of the adsorption law in the low concentration regime. A phenomenological description of some hysteresis loops and experimental data is presented. The theoretical procedure pointed out here is quite general and implicitly suggests a possible relationship with other hysteresis mechanisms (i.e., magnetic).
Introduction Adsorption of gas onto solid surfaces involves many practical applications. Recall, for instance, statics and dynamics of wetting,1 materials chemistry, processing and optimization of specific surface areas and porosity,2-3 purification of gases and liquids,4-5 chromatography,6 membrane technology,7 catalysis,8 etc. On the other side, hysteresis and isotherm equations in solid-gas adsorption have been a subject of active theoretical research. We remind the first capillary condensation theory of Zsigmondy,9 who used the Kelvin equation in 1907 to explain condensation and hysteresis loops, the modern molecular theory of capillarity,10 the ink-bottle theory of Kraemer,11 which takes into consideration geometrical irregularities of the pore distribution in a solid body, the well known contact-angle hysteresis with related topics1,12,13 (see, for instance, surface roughness and heterogeneity, deformation, pinning of a solid-liquid-gas interface, etc.), which is based on the difference between the so-called advancing and receding contact angles, density functional theories of adsorption14 (DFT), applications of the thermodynamic Saam-Cole approach,15 microscopic and lattice models,16,17 etc. However, while the literature is wide and quite scattered, the comparison between theoretical predictions and experimental results is not yet completely straightforward.13-16 In this paper, a condition of maximum irreversible entropy production will be applied in the framework of a variational procedure where the isotherm equations are usefully represented according to a delta convergent sequence of ordinary functions. Formal developments (rather elaborate and thus isolated in Appendix A-E) lead in the end to a simple approximated relation for a degenerate isotherm equation couple which can
account for isotherm equations and different hysteresis loops. Although it is known specific entropic contributions can be responsible for contact-angle hysteresis,12 the proposed theoretical study should be regarded as a general description in terms of irreversibility. Influence of temperature,18 complex recursive mechanisms,19 solid surface dissolution20 and adsorption dynamics and kinetics21 will not be dealt with in this study. The method proposed here can be applied to any hysteresis phenomenon, where some extreme conditions are expected to take place, and draws a possible relationship between gas-solid adsorption and other hysteresis loops (i.e., magnetic). Formulation of the Isotherm Equations Consider the adsorption process of a gaseous specie onto the surface of a porous solid body.22,23 Let c ∈ [0, 1] be the gas concentration in the bulk phase (expressing the ratio between adsorption and bulk pressures, i.e., c ) p/p*) and c ) c0 be the concentration value at which fluid condensation takes place. Denote with ϑ ∈ [0, 1] the fractional surface coverage and with ϑ(c) the isotherm equation that rules all implied gas-surface mechanisms.24 When an adsorption-desorption cycle is completed in the whole concentration range, one can usually observe different ascending and descending curves ϑ versus c, whereas (i) a rapid increase in the adsorbed amount and (ii) no hysteresis phenomena should be expected for an ideal solid body (see schemes in Figure 1a,b).11,24 To derive a relation between ideal and real adsorptions, the law sketched in Figure (1a) can be described as a Heaviside step function, H(c - c0), added to a continuous contribution θ(c). In integral form, this means (see Figure 2a)
10.1021/jp9905225 CCC: $18.00 © 1999 American Chemical Society Published on Web 08/13/1999
Equations in Gas-Solid Adsorption
J. Phys. Chem. B, Vol. 103, No. 35, 1999 7543 where, in principle, λ can be an arbitrary function of the gas concentration and where one expects ν (g 2) be related to the criticality order in the neighborood of c0. We will hereinafter define the condition x˜ f 0 (or c = c0) as the asymptotic limit. Accordingly, on the basis of eqs 2-4, the problem is equivalent to determine a proper analytical form λ ) λ(c, x˜ ; ν), which becomes singular when the asymptotic limit is approached. To this end, consider the set consisting of λ and its derivatives λ(q) ≡ dqλ/dcq (we conventionally define λ ≡ λ(0)), and consider a set of functions such that ψq ) ψq(λ(q)) and ψq * ψq(λ(s)) when q * s. For some N ∈ ℵ, the positions N-1
λ f ΛN )
λ(q) ∏ q)0
(5)
and N-1
ψ f ΨN )
Figure 1. Scheme of (a) ideal and (b) real adsorption cycles performed at a constant temperature value.
ψq (λ(q)) ∏ q)0
(6)
transform eq (2) into the following:
ϑ(c) )
(1 + ΨN) + π-1/2ΛNe-(x˜ Λ ) ) dc ∫0c (dθ dc N
ν
(7)
provided with (see eq 3)
lim PΛN ) δ(x˜ )
(8)
ΛNf∞
Equation 7 can be associated with a physical constraint and, to this end, we are going to develop a variational condition of maximum entropy production under validity of the asymptotic limit. The state which is stable with respect to arbitrary small changes in the variable c will point out the broadening condition compatible with maximum adsorption entropy. The Maximum Entropy Production Figure 2. Scheme of the transition from (a) ideal to (b) real behaviors described as a broadened Dirac distribution.
ϑ(c) )
+ δ(x)) dc ∫0c (dθ dc
(1 + ψ(λ)) + Pλ(x˜ )) dc ∫0c (dθ dc
(2)
ψ accounting for a variation of θ, and with
lim Pλ ) δ(x˜ )
λf∞
Pλ ≡ π-1/2 λe
∫ ln c dn
(9)
where n ) ∑ϑ is the amount adsorbed, ∑ its maximum limit value, and R is the universal gas constant.23 A quite general extension of eq 9 to a functional formulation in the variables c and ϑ can be obtained by expressing dn via eq 7, introducing two linear applications, i.e., c f ξ and x˜ f χ, and defining a specific surface entropy sN ≡ S(ΨN,ΛN)/∑R 26
sN )
(1 + ΨN) + π-1/2 ΛNe-(χΛ ) ) ln ξ dξ ∫ (dθ dξ N
ν
(10)
Equation (10) can be interpreted as an Nth order functional
(3)
for some displacements x˜ . To choose the mathematical form of Pλ, recall the delta function can be represented as the limit of a sequence of ordinary functions, for instance, limlf∞ λe-(λx˜ )2 ) π-1/2δ(x˜ ).25 This suggests to adopt the more general relationship given by -(λx˜ )ν
S)R
(1)
where ϑ(c) ) ∫ dθ ≡ θ(c) if no transitions in the adsorbed gas amount occur, and where δ(x) denotes a Dirac distribution that is centered at the critical concentration value, i.e., x ≡ c - c0. A first, general representation of the real adsorption law can therefore be written as (see Figure 2b)
ϑ(c) )
We start from the definition of uncompensated heat24 associated with a hysteresis loop, ∆Sirr ) R I ln cdn, and consider a functional defined as:
(4)
sN ≡
∫ σ({ψq}, {λ(q)}; ξ) dξ
(11)
∫ σ(ΨN, ΛN; ξ) dξ
(12)
or, symbolycally
sN ≡
which can be studied variationally by using the Euler-Lagrange constraint for the annulment of the first-order functional derivative27
7544 J. Phys. Chem. B, Vol. 103, No. 35, 1999
∂σ ∂λ
N
d
∑ q)1 dξ
)
Mezzasalma
( ) ∂σ
it is possible to show (see Appendix B)
(13)
∂λ(q)
Equation (13) will be applied twice under the validity of the asymptotic limit. The first time we will obtain the stable solution ΨN ) Ψ/N (see eq 6)
δsN )
sN [Ψ/N
+ δΨN] -
while the behavior ΛN ) condition
Λ/N
sN [Ψ/N]
)0
(1 + Ψ*) + π-1/2Λ*e-(χΛ*) ) dξ ∫0ξ (dθ dξ ν
where it is meant
Ψ/N
f Ψ* and
Λ/N
/ ΨN,i
(
∂λ
) ∑( ) ∂ ln ψq
N-1
ΨN
∂λ(q)
q)0
ν
φ′ - φ′′
(17)
)( )
∂ψ* ∂λ*
∂ ln ψ* N N ν ψ* ) Pνe-(λ/ χ) λ(2ν+1)N (18) * ∂λ*
which is equivalent to solve
( ) ( )
∂ψ* ∂ψ* 2 N ) P* ψ1-N λ(2ν+1)N e-(λ/ χ)ν φ* * * ∂λ* ∂λ*
(19)
with P* = Pν/Nφ′′ and φ* ) φ′/φ′′. This gives the asymptotic solution couple specified by
{
P* N ν φ* - ψ1-N e-(λ/ χ) λ(2ν+1)N * * ∂ψ* φ* ) P ∂λ* i * 1-N -(λN/ χ)ν (2ν+1)N ψ e λ* φ* *
( )
{
If one defines
P* φ* Kν,i ) P * φ2*
and
IνN )
∫0λ
*
{
(i ) 1) (20) (i ) 2)
(25)
Λ/ν 2 =
(ξ1 + ξ ln2 ξ + 2ν χ- 1) νχ1
(26)
ν-1
Development of IνN and ∂IνN/∂λ* according to eqs 22, 25, 26 allows to write the final solution couples (see Appendix D)
Ψ/i ) Kν × 1 ν 1/ν ν ξ+ ln ξ 2ν-1 ν-1 1 -2-1/ν Γ ξχ ν 2+1/ν
{
(
) {(
2ν+1
) ( )}
ν
e[(2ν-1)/ν][(ν-1)/ν] (i ) 1) (i ) 2) (27)
where Γz˜ denotes the Gamma function (Euler’s second kind integral) evaluated in z ) z˜, while /ν
Λi )
{
1 × νχν-1
ξ(ln ξ)2 - 1 1 2 2ν - 1 + + + ξ ξ ln ξ ξ ln ξ(1 + ξ ln ξ) χ 1 2 2ν - 1 + + ξ ξ ln ξ χ
(i ) 1) (28) (i ) 2)
by which two different adsorption laws ϑi hold. They can be approximated to the same formal behavior provided the involved algebraic coefficients take different values (see Appendix E). Starting from eq 16 expressed as
ϑi ) θi +
∫ Ψ/i dθ + π-1/2 ∫ Λ/i e-(Λ χ) * i
ν
dξ
(29)
leads in the end to the following relationship
(i ) 1) (21) (i ) 2)
N)ν
(24)
(i ) 2)
ξ (ln ξ)2 - 1 ξ ln ξ(1 + ξ ln ξ)
/ν νχν-1 (Λ/ν 1 - Λ2 ) ∼
τi ) Gi(c - c0)qi + Li ln
x-(2ν+1)N e-(χx
(i ) 1)
*
with
where φ′ ) d(dθ/dξ(ln ξ))/dξ, φ′′ ) φ′ + (dθ/dξ) ln ξ, and where, asymptotically, one finds Pν = -π-1/2ν2χ2ν-1 ln ξ. Mathematical details are reported in Appendix A. Equation 17 admits the mean convergent solution specified by ΨN ) ψN* (λ*) and ∂ ln ψ*/∂λ* ) ∂ ln ψk/∂λ(k)
(
= Kν ×
∂IνN
where Kν ) ξPν(dϑ/dξ)-1. Substituting eqs 24 into the adsorption isotherm in eq 7 and using the variational criterion in eq 15 return the functions Λ/i ≡ λN*,i, which satisfy (see Appendix C)
f Λ* when N f +∞.
) Pν e-(ΛNχ) Λ2ν+1 N
(23)
(i ) 2)
(ξ + ln1ξ) ( ∂λ )
(16)
After developing eqs 14, 15, one obtains a relationship for ΨN that is given by
∂ψ0
(i ) 1)
NIνN
Variational Procedure and Isotherm Equations
φ′ - φ′′
∂IνN ∂λ*
and, if all second derivatives are neglected, one has
will be determined from the
The last stage of calculations will consist of the evaluation of the global adsorption isotherms by considering the solutions of eqs 14, 15 in the original eq 7, i.e.,
{
( )
KνN,2
KνN,1 IνN
(14)
δsN ) sN [Ψ/N(Λ/N + δΛN)] - sN [Ψ/N(Λ/N)] ) 0 (15)
ϑ)
/ N ΨN,i ≡ ψ*,i )
c c0
(30)
with τi ≡ ϑi - θi, and where Gi, Li, and qi are phenomenological coefficients. Results and Discussion
dx
(22)
The obtained isotherm equations (ϑi) consist of one term, which is defined in the low gas concentration regime (θ), and of two contributions that take into account the liquid state of the adsorbed layer (∼ cq and ln c). The behavior of eq 30 is
Equations in Gas-Solid Adsorption
Figure 3. Behavior of the obtained adsorption isotherm ϑ ) ϑ(c).
illustrated in Figure 3 when neither of two members on the right dominates on the other in the positive neighborood of c ) c0, namely, when O (Li/Gi) = O (qcq0). While the ideal gaslike contribution determines the adsorption law in c e c0, the other addenda describe the region c g c0 and, in this framework, do not depend on the analytical form of θ ) θ(c). This would suggest to regard the displacement (τi) introduced in eq 30 as a quantity not sensibly influenced by the first specific adsorption regime. At first account, the aforesaid results are consistent with the relationship for the chemical potential of the adsorbed surface layer,28 i.e., µA = µG0 + RT ln(p/p*) + µ˜ , where µG0 and µ˜ are, respectively, the chemical potential of the ideal gaseous phase at the standard pressure (p ) p*) and a correction which describes the effect of the layer thickness (i.e., the number of multilayers adsorbed). It should be observed that the logarithmic contribution coherently recalls the Kelvin equation11 (relating the surface tension γ, the contact angle θSL, the molar volume of the adsorbed molecule V, the curvature of the solid surface Rm, and the layer thickness δ according to ln(p/p*) ) (2γ cos θSLV)/(Rm-δ), while the power law of the concentration resembles a modified version of the Freundlich law, often used to represent experimental plots of gas-solid adsorption measurements.22
J. Phys. Chem. B, Vol. 103, No. 35, 1999 7545 As concerning hysteresis, Figure 4a-d shows some examples of loops (belonging to the classification scheme specified by de-Boer29) that can be represented in the plane (c, τ) by using the isotherm class in eq 30. Notice the theory also predicts the particular case of a cycle with no hysteresis (see the end of Appendix E). Figure 5a-d reports instead a description of previous isothermal adsorption data29,30 of some polar sorbates on (a) anthracite, (b) silica gel, (c) tetramethylammonium, and (d) natural montmorillonite, and it is possible to observe the agreement is quite satisfactory in all cases. The best fit functions returned the coefficients c0 ) 0.23, 0.45, 0.35, and 0.49, which agree with the observed experimental values for the gas concentration where hysteresis arises.29,30 Unfortunately, as often occurs in adsorption problems,31,32 the heuristic character of the other experimental coefficients allows only to verify the formal agreement between theoretical solution and experiment. In order to elucidate the nature of the constant coefficients (see, for instance, the criticality parameter ν in Eq. 8), a detailed numerical analysis of eq 29, namely, the exact representation of the general adsorption law, is required as well as an opportune analysis of experimental data. These are interesting issues left for future work. Conclusions 1. A variational criterion of maximum entropy production has been formulated to investigate isotherm equations and hysteresis in gas-solid adsorption. Transition from ideal to real behaviors has been interpreted as a broadening process of a Dirac distribution driven by irreversibility. The proposed theory can be employed to describe any hysteresis loop where the expected extreme constraints can be formulated variationally. 2. The Euler-Lagrange theorem yields a degenerate solution couple which can be approximated to a new isotherm class (say, ∼ θ + ln c + cq) provided the algebraic coefficients assume different values (i.e., different adsorption and desorption curves). It consists of the sum of three terms: the ideal law in the low
Figure 4. (a-d) Examples of hysteresis described according to ϑi ) ϑi(c) in eq 30, where i ) 1, 2 denote respectively the ascending (adsorption) and descending (desorption) curves.
7546 J. Phys. Chem. B, Vol. 103, No. 35, 1999
Mezzasalma
Figure 5. Phenomenological description of adsorption data of (a) benzene (C6H6) on a steam-activated anthracite coal charcoal29 (L1 ) 0.044, G1 ) 0.0225, q1 ) 9, L2 ) 0.05, G2 ) 2.5, q2 ) 0.004, c0 ) 0.23); (b) ethanol (C2H5OH) on silica gel30 (L1 ) 0.46, G1 ) 0.25, q1 ) q2 ) 150, L2 ) 0.6, G2 ) 0.2, c0 ) 0.45); (c) benzene on tetramethylammonium montmorillonite29 (L1 ) 0.09, G1 ) 0.8, q1 ) 18, L2 ) 0.33, G2 ) 0.49, q2 ) 10, c0 ) 0.35), and (d) methanol (CH3OH) on natural montmorillonite29 (L1 ) 0.15, G1 ) G2 ) 0.4, q1 ) 15, L2 ) 0.45, q2 ) 9, c0 ) 0.49). Subscripts 1 and 2 as in Figure 4.
{( )( )
dθ ∂ψ0 Ψ(N,0) + dξ ∂λ
gas concentration regime (θ), a Freundlich-like isotherm (∼ cq), and a contribution related to the Kelvin equation (∼ ln c). 3. The displacement from real to ideal adsorptions has been introduced as a quantity characteristic of the hysteresis region and independent of the isotherm equation in the low concentration regime. It has been applied to describe loops of different shape and previous gas-solid adsorption measurements. 4. A description of adsorption laws and loops based only on irreversible entropy production has been proposed. It traces a possible correspondence with other known hysteresis mechanisms (i.e., magnetic).
∂σ ) ∂λ
Acknowledgment. The author thanks Ger Koper and the whole Colloid and Interface Science Group (LIC) for helpful discussions. This work has been supported by the Marie-Curie TMR Contract n. ERBFMBICT-98-2918 under the European Community. The author is also indebted to Marco Capurro and Paolo Cirillo for continuous supports in his scientific work.
so that (under the constraint dχ/dξ ) 1)
Appendix A: Derivation of the Basic Eq 17 for ΨN To carry out the relationship for ΨN, one has to work out eq 13. Accordingly
∂σ ) ∂λ
{( )( )
}
ν dθ ∂ΨN ∂ + π-1/2 (ΛN e-(ΛNχ) ) ln ξ (31) dξ ∂λ ∂λ
which, after developing the last derivative at the right side and introducing the notations
ΨN ≡ ψ(q)Ψ(N,q) becomes
ΛN ≡ λ(q)Λ(N,q)
}
ν
π-1/2 Λ(N,0) e-(ΛNχ) [1 - ν(ΛNχ)ν] ln ξ (33) Similarly, for any q ) 0, ...N - 1, one has
∂σ (q)
)
∂λ
{( )( )
dθ ∂ψq Ψ(N,q) + dξ ∂λ(q)
}
ν
π-1/2 Λ(N,q) e-(ΛNχ) (1 - ν(ΛNχ)ν) ln ξ (34)
( ) {( ) ( )
dθ ∂ψq d ∂σ ) Ψ(N,q) + (q) dξ ∂λ dξ ∂λ(q) 1 d2θ ∂ψq Λ(N,q) -(ΛNχ)ν ν e [1 ν(Λ χ) ] Ψ(N,q) + + N 1/2 2 (q) ξ π dξ ∂λ Λ(N,q) d -(ΛNχ)ν {e [1 - ν(ΛNχ)ν]} ln ξ (35) π1/2 dξ
} {( ) ( ) }
and
d
∑ qg1 dξ
( ) { ( )} ∑ ∂σ
∂λ(q)
)
d
dξ
ln ξ
dθ dξ
Ψ(N,q)
qg1
pν
(32)
∑
qg1
where
∂ψq
+ ∂λ(q) ν Λ(N,q) e-(ΛNχ) (36)
Equations in Gas-Solid Adsorption
pν ) -π-1/2
J. Phys. Chem. B, Vol. 103, No. 35, 1999 7547
{
}
1 - νΛνNχν + νχ ln ξ [ν(ΛνNχν - 1) - 1] ΛνN ξ (37)
Combination of eqs 33, 36 gives us
{
ν dθ (N,0) ∂ψ0 Ψ + Λ(N,0) π-1/2 e-(ΛNχ) (1 dξ ∂λ ∂ψq d dθ (N,q) νΛνNχν) ln ξ ) ln ξ + qg1 Ψ dξ dξ ∂λ(q)
-
}
(
)∑
ν
e-(ΛNχ) pν
∑qg1 Λ(N,q)
(38)
ν
( ) ( ) [( ∂ψq
∑qg1 Ψ(N,q)
ν
-e(ΛNχ) Ψ(N,0)
Appendix B: Derivation of Ψ/i from Eqs 20 (i ) 1). As Ψ/N ≡ ψN* and Λ/N ≡ λN* , the first case reduces to
( )
dθ d dθ ln ξ + {( ) ln ξ} ∂I ( dξ) dξ dξ ln ξ (41) ( d dθ ∂λ ) ln ξ (dξ{(dξ) })
( ) ( )( )
∂IνN ν ∂ Kν ln ξ + π-1/2 {ΛNe-(χΛN) } ln ξ ∂λ* ∂λ*
∂σ dθ ) ∂λ* dξ
[(dθdξ) K λ
Ψ/N = Kν
(
)( ) ∂ΙνΝ ∂λ*
1 +ξ ln ξ
(42)
if Kν is introduced (see eq 24) and the second derivatives are neglected. (i ) 2). The second of eq 24 can be obtained by direct integration of the corresponding eq 20. In fact, one has
P* -(λN/ χ)ν (2ν+1)N 1 dΨN* ) e λ* dλ* N φ*
(43)
so that, by developing P* and φ*
ΨN* π-1/2ν2χ2ν-1 ln ξ )N d dθ ln ξ dξ dξ
{( ) } ΨN*/N
/ N
ν
λ(2ν+1)N dλ* ≡ Kν,2IνN * (44)
in the asymptotic limit, it is first
+
]
while the other q addenda are
( )
π-1/2 NλN-1 *
({ dθdξ) K e ν
-(χΛN)ν
}
ln ξ +
d -(χΛN (1 - νχνλνN {e * ) ln ξ} (50) dξ )ν
or, after some elaborations: ν
( )
( )
e(χΛN) d ∂σ d dθ ) λ(2ν+1)N K * ln ξ dξ ∂λ(q) dξ ν dξ * 1 dθ 4π-1/2 Nχλ(ν+1)N-1 - νχλνN K + + λ(2ν+1)N * * * ξ ln ξ dξ ν
(
)(
)
(1 - νχνλνN π-1/2 N λN-1 * * ) (51) Combining eqs 49, 51 yields
{
(dθdξ) K - λ
}
d dθ K + dξ ν dξ -1/2 π-1/2 N (1 - νχνλνN NχλνN * ) + 4π * ) 1 dθ λ2νN+1 K π-1/2 N (1 - νχνλνN - νχλνN * * * ) (52) ξ ln ξ dξ ν
λ2νN+1 *
ν
(
∫ e-(λ χ)
(2ν+1)N ν *
π-1/2 N (1 - νχνλνN λN-1 * * ) (49)
νN *
(48)
Accordingly, since Λ/N ≡ λN*
(40)
leading to
For evaluating useful to notice33
/ ν N
d d ∂σ ) λ(2ν+1)N * dξ ∂λ(q) dξ *
2
(47)
(1 + KνIνN) + π-1/2 Λ/N e-(χΛ ) ) ln ξ dξ ∫ (dθ dξ
ν
Neglecting the term on the left side in the limit of large N and using the definitions of IνN in eq 22 yield the first of eqs 23:
-π-1/2ν2χ2ν-1
sN )
) e-(χΛN) ln ξ
/ P* /2ν+1 -(χΛN* )ν 1 ∂ΨN ) φ*Ψ/(N-1)/N Λ e N N ∂λ φ* N
( )
Appendix C: Derivation of Λ/i from Eqs 13, 15
and adopts the corresponding isotherm law in eq 7, the functional sN reads
+
where Pν ) pν - π-1/2 lnξ(1 - νΛνN χν). An algebraic rearrangement then transforms eq 39 into eq 17.
P* ∂IνN ) 2 ≡ φ ∂λ*
(46)
If one considers the simpler case represented by (all subscripts like i and *, i will be omitted in the following)
Λ(N,0) Pν (39)
Ψ/N
( )
Ψ/N ) KνIνN
{ ( )}]
)
(45)
Pν dϑ -1 Kν Γ2+1/ν ξχ-2-1/ν ≡ Γ2+1/ν χ-2-1/ν ν dξ ν
+ ΛNqν )
∂λ(q) dθ dθ d ln ξ + ln ξ dξ dξ dξ
∂ψ0 ∂λ
1 Γ χ-(2ν+1)/ν νN [(2ν+1)N+1]/νN
where Γz˜ is the Gamma function evaluated in z˜ ) z (notice, as (2ν+1)/ν e [(2ν+1) N+1]/νN e 2(ν+2)/ν, its contribution is always finite). Thus, at large N values and neglecting all second derivatives, the solution takes the form given by
Ψ/N )
or, equivalently
-e(ΛNχ) ΨN
lim IνN )
λ*f∞
2νN+1 *
)[
]
( )
and
π-1/2 N λ2νN+1 *
1 d dθ νN Kν - ν2λνN × * ) 1 + νχλ* dξ dξ ξ ln ξ dθ -1/2 1 - νχλνN N λ2νN+1 Kν (53) * +π * dξ
(
)
(
(
)
)
7548 J. Phys. Chem. B, Vol. 103, No. 35, 1999
Mezzasalma
∫χRξ ln ξ dξ
If the asymptotic limit is concerned, i.e., O (λr*) , O (λs*) if r < s, from eq 53 the following relationship holds:
Λν*
≡
(
{(
)})
1/2
1 1 dθ π Kν d = ν-1 ln + ξ ln ξ dξ dξ ν νχ
λνN *
and equates
(54)
and so the expression of λ*,2 in eq 26 once the derivative at the right side has been calculated (remember that (π1/2Kν/ν)(dθ/ dξ) ) -νχ2ν-1 ξ lnξ > 0). In a similar way, the aforesaid procedure applied to Ψ/1 yields
Λν*
(
{(
))})
1/2
1 1 1 dθ π Kν d = ν-1 ln ξ+ + ξ ln ξ dξ dξ ν ln ξ νχ
(
(55)
ξ 1 (χR+1 ln ξ - Q{R + 1}) + × 1+R R+1
)
∫
( Q{R + 1} dξ -
{
[ξ(R + 1) + ξ0] ln ξ (R + 1)(R + 1)
∂IνN * ν = Λ/-(2ν+1) e-(χΛ1) 1 ∂λ*
(56)
To this end, one can conveniently deal with F ) (ξ(ln ξ)2-1)/ (ξ ln ξ(1+ξ ln ξ)) in eq 25 as a small perturbation. Accordingly, after observing Λ/2 ) Λ/1(F ) 0), one obtains
( )(
1 - 2ν 1 νχ νχν-1
1/ν
)
2ν - 1 χ
1/ν-1
-
(
)
1/ν
(
1 / Λ ν 2
)
(57)
ξ2
) }
{[
2ξ20 (R + 1)(R + 2)(R + 3)
[
1/ν
(R + 3)2
+
]}
3R3 + 12R2 + 9R - 4 χR+1 + 2 2 2 (R + 1) (R + 2) (R + 3) 2ξ30 Q{R} (67) (1 + R)(2 + R)(3 + R)
so that:
2ν+1
ν
e[(2ν-1)/ν][(ν-1)/ν] χ2ν+1
2ξ30
Appendix E: Calculation of Adsorption Isotherms from Eqs 16 In order to develop the isotherm equations when the asymptotic condition is adopted, it is first useful to take into consideration the integrals that are involved. The simplest is R+2
χ χ + ξ0 ξχ dξ ) R+2 1+R
with ξ0 ) ac0. The second has the form
χR-r (-1)rξr0 + (-1)RξR0 ln ξ R-r r)1
R-1
Q{R} )
∑
(61)
(69)
If R ∈ ℵ, let the decomposition R ) [R] + q, where [R] ∈ ℵ and q ∈ R+\ℵ, it is possible to write
Q{ R} ) χqQ{[R]} - qQ{[R] + q - 1}
1+R
(68)
To evaluate the contribution coming from Q{R}, consider first R ∈ ℵ. This means
(60)
∫
ξ20
∫χ ξ ln ξ dξ = (1 + R) (2 + R) (3 + R)Q{R}
){(2ν ν- 1) (ν -ν 1)}
R
]
ln ξ + -
R2 + R - 4 + ξ0 ξ (R + 2)2(R + 3)2
R
Ψ/1 ) 1 ln ξ
(66)
where P{f} denotes the primitive of the function f, and can be expressed as
ξ0
and to express Ψ/1 as
(
(65)
∫χRξ2 ln ξ dξ ) ξ∫χRξ ln ξ dξ - ∫P{χRξ ln ξ}dξ
* ν
Kν ξ +
(64)
The third integral is given by
(58)
Λ/-(2ν+1) e-(χΛ1) = 1 ν 1/ν ν 2ν+1 [(2ν-1)/ν][(ν-1)/ν]ν 2ν+1 e χ (59) 2ν - 1 ν - 1
)(
(R + 1)(R + 2)
0 Q{R} ∫χRξ ln ξ dξ = (R + 1)(R + 2)
This allows to write eq 56 as
{(
ξ20 Q{R}
and, since O(Q{R}) < R + 1 (see the following eqs 69-76), we have
/ ν so that, since Λ/ν 2 = (2ν-1)/χ asymptotically, it turns out Λ1 / / - Λ2 = -Λ2/ν, or
Λ/1 = 1 -
}
χR+1 +
2ξ0ξ ξ2 + + (R + 3) (2 + R)(3 + R)
)
1 2ν - 1 νχ χ
(63)
ξ + (R + 2)2
(R + 1)2(R + 2)2
The equations for Ψ/2 and Λ/1,2 follow respectively from eq 46, eq 54, and eq 55. For determining Ψ/1, it is necessary to calculate the derivative which appears in eq 42 by employing the asymptotic expression of Λ/1:
Λ/1 = Λ/2 +
-
(R2 + R - 1)
ξ0
Appendix D: The Stable Solutions Ψ/i and Λ/i
∫χR+1 ln ξdξ)
where the notation Q{y} stands for the primitive of the function χy/ξ. Developments of the sum in eq 63 lead to
which is equivalent to the first of eqs 26.
( )
(62)
(70)
whose degree can be further lowered by other integrations per parts and by applying eq 69. Thus, taking into account the dominant terms, i.e.,
Equations in Gas-Solid Adsorption
J. Phys. Chem. B, Vol. 103, No. 35, 1999 7549
[R]-1 [R]-1 Q{[R]} = (-1)[R]ξ[R] ξ0 χ + O(χ2) 0 ln ξ + (-1) (71)
implies
ξ[R]-1 0 Q{R} = (-1)[R]-1 χq+1 + (-1)[R]ξ[R] 0 Q{q} (72) q+1 The primitive function included in the second term on the right follows from a complex representation that involves a Gauss hypergeometric function33 of the form 2F1(1, q + 1; q + 2; - χ/ξ0). However, for the purpose of this study, it is sufficient adopting the behaviour Q{q} = χq/q, and the following:
Q{R} = so that
{
IR,β,γ(R) ≡
∫Ψ/2 dθ ) gν,2 I2ν-3-1/ν,1,1 ) Gν,2χq
ν,2
gν,2 ) π-1/2νΓ2+1/ν
ξ2+[2ν-3-1/ν] 0 q[ν-1-1/(2ν)](2ν-1-1/ν)
∫
π-1/2 Λ/2 e-(χΛ2)ν dξ ) lν,2
β)γ)1
χq χq
(74)
∂I
τi ) Gi(c - c0)qi + Li ln
(75)
when i ) 1. As concerning the first integral contribution, one has
)
∂I
∫Ψ/1 dθ ) π-1/2ν2 ∫ξχ2ν-1 ∂λνN* dξ + ∫ξ2 ln ξχ2ν-1 ∂λνN* dξ
and, once the derivative on the right is evaluated from eqs 5659, we can write
∫Ψ/1 dθ = gν,1 I4ν,2,1 ) Gν,1χq
ν,1
(77)
where
e[(2ν-1)/ν][(ν-1)/ν]
ν
and
ξ3+[4ν] gν,1 0 q(1+4ν)(1+2ν)(3+4ν)
are constant coefficients. On the other hand, the second contribution on the right of eq 75 gives
∫Λ/1 e-(χΛ ) dξ = lν,1∫dξχ * ν 1
where
lν,1 ) so that one finds
ν-1 e (2ν-1 ν ) ( ν ) 1/ν
(82)
[(2ν-1)/ν][(ν-1)/ν]ν
(78)
c c0
(83)
Gi, Li, and qi being phenomenological coefficients. Last, notice 2ν that the limit of large ν values yields Gν,1 ∼ ξ4ν 0 /ν, Gν,2 ∼ ξ0 /ν, Q -1/2 2/e Lν,1 ∼ Lν,2 ≡ L ) π e , and thus the same expression ϑQ for both ascending and descending curves
(76)
2ν+1
(81)
which suggests the suitable heuristic dependence that follows by choosing the arbitrary linear maps as ξ ) χ ) c: * ν i
1/ν
∫dξχ ) Lν,2 ln χ
τi ) Gν,iχqν,i + Lν,i ln χ
∫Ψ/i dθ + π-1/2∫Λ/i e-(χΛ ) dξ
ν ν {(2ν-1 ) (ν-1 )}
*
with lν,2 ) (1 - 1/ν)lν,1 and Lν,2 ) (1 - 1/ν)Lν,1. Finally, one has the equation form given by
β ) 2, γ ) 1
Consider the total isotherm equation:
Gν,1 )
× gν,2
Then
q(1 + R) (2 + R) (3 + R)
gν,1 ) π-1/2ν2
(80)
where
Gν,2 )
β ) 1, γ ) 0
(
(79)
with Lν,1 ) π-1/2lν,1. As concerning the isotherm equation when i ) 2, the previous mathematical procedure shows in the end that the following relation holds:
(73)
q
q(1 + R) (2 + R) 2ξ3+[R] 0
τi ≡ ϑi - θi )
* ν
and
χq ξ[R] 0
∫χRξβ(ln ξ)γ dξ =
1 1+R χ 1+R 2 ξ2+[R] 0
∫
π-1/2 Λ/1 e-(χΛ1) dξ ) Lν,1 ln χ
lim ϑi ) ϑQ ) θ + LQ ln χ
νf∞
(84)
provided ξ0 e 1. References and Notes (1) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 289. (2) Huo, Q.; Margolese, D. I.; Stucky, G. D. Chem. Mater. 1996, 8, 1147. (3) Kaneko, K. J. Membr. Sci. 1994, 96, 59. (4) Keller, G. E. Chem. Eng. Prog. 1995, 11, 56. (5) Deng, X.; Yue, Y.; Gao, Z. J. Colloid Interface Sci. 1997, 192, 475. (6) Helfferich, F. G. J. Chromatogr. A 1997, 768, 169. (7) Munoz-Aguardo, M.-J.; Gregorkievitz, M. J. Colloid Interface Sci. 1997, 185, 459. (8) Mehandjiev, D.; Bekyarova, E.; Khristova, M. J. Colloid Interface Sci. 1997, 192, 440. (9) Zsigmongy, A. Z. Anorg. Chem. 1907, 71, 356. (10) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, 1982. (11) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans: London, 1966. (12) Andrade, J. D. Surface and Interfacial Aspects of Biomedical Polymers; Plenum Press: New York, 1985; Vol. I. (13) Andersen, J. V. Mod. Phys. Lett. B 1996, 10, 359. (14) Ravikovitch, P. I.; O’Domhnaill, S. C.; Neimark, A. V.; Schuth, F.; Unger, K. K. Langmuir 1995, 11, 4765. (15) Inoue, S.; Hanzawa, Y.; Kaneko, K. Langmuir 1998, 14, 3079. (16) Marini Bettolo Marconi, U.; Van Swol, F. Phys. ReV. A 1989, 39, 4109. (17) Donohue, M. D.; Aranovich, G. J. Colloid Interface Sci. 1998, 205, 121.
7550 J. Phys. Chem. B, Vol. 103, No. 35, 1999 (18) Morishige, K.; Shikimi, M. J. Chem. Phys. 1998, 108, 7821. (19) Mezzasalma, S. A. Chem. Phys. Lett. 1998, 107, 9214. (20) Mezzasalma, S. A. J. Chem. Phys. 1997, 107, 9214. (21) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995. (22) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1991. (23) Parfitt, G. D.; Rochester, C. H. Adsorption from Solution at the Solid/Liquid Interface; Academic Press: London, 1983. (24) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (25) Roos, B. W. Analytic Functions and Distributions in Physics and Engineering; John Wiley Sons: New York, 1969. (26) Chattoraj, D. K.; Birdi, K. S. Adsorption and the Gibbs Surface Excess; Plenum Press: New York, 1984.
Mezzasalma (27) Moiseiwitsch, B. L. Variational Principles; John Wiley Sons: London, 1966. (28) Ponec, V.; Knor, Z.; Cerny, S. Adsorption on Solids; Butterworth: London, 1974. (29) Flood, E. A. The Solid-Gas Interface; Marcel Dekker, Inc.: New York, 1967; Vol. II. (30) Brunauer, S. The Adsorption of Gases and Vapours; Oxford University Press: London, 1945. (31) Ruthven, D. M. Principle of Adsorption and Adsorption Processes; John Wiley and Sons: New York, 1984. (32) Beruto, D.; Mezzasalma, S.; Oliva, P. J. Colloid Interface Sci. 1997, 186, 318. (33) Gradshteyn, I. S.; Ryzhik, I. M. Tables of Integrals, Series and Products; Academic Press: New York, 1973.
