Langmuir 1995,11, 3223-3234 3223

Equation for Adsorption fkom Gas Mixtures Narasimhan Sundaram

Pall Corporation, 6301 49th Street North, Pinellas Park, Florida 34665

Received March 29, 1995. In Final Form: May 17, 1995@

A multicomponent adsorption equation is developed from a modified Dubinin isotherm for the pure components. This isotherm equation satisfies thermodynamic consistency requirements while being easy to use and predicts binary and ternary mixture data from pure component parameters. A description of the effect of coadsorption of bulk gas on a trace adsorbate is also possible. Further, successful mixture predictions from the isotherm show that it significantly reduces hitherto reported large, apparent nonidealities which are believed to be associated with energetic heterogeneity in adsorption.

Introduction

The description of adsorption from gas mixtures is an important practical problem. Adsorbed solution theory alleviates some of the problems by providing a thermo- dynamic framework for predicti0ns.l However, require- ments of thermodynamic consistency, accuracy of pre- dictions, and simplicity have limited the number ofuseful expressions available since Markham and Benton ex- tended the Langmuir isotherm to mixtures.2 Ideal ad- sorbed solution (IAS) theory provides useful equations when assumptions are made about the molecular volumes or sorption capacities of individual components, e.g., refs 3 and 4. In general the method prescribes numerical manipulations5 and is especially so when energetic heterogeneity must be ~ons idered .~ ,~

A key aspect in the application of IAS is the existence of a Henry’s law limit for the pure component isotherm or data. This ensures a finite spreading pressure, which must be nonzero. As discussed in ref 8, errors in the Henry’s law regime greatly amplify in spreading pressure calculations and can render the IAS predictions inad- equate. The discrepancy may be due to a poor fit of low pressure data or the nonexistence of a Henry’s law limit in the isotherm chosen for the extrapolation. An example of the former is the Langmuir isotherm which, while possessing proper limits, is oRen simply not flexible enough to accurately describe data. The Dubinin-Raduskevich (DR) equationg is an example of a popular isotherm which, however, does not possess a finite nonzero Henry’s law limit.

Recently a modification of the Dubinin isotherm was proposed,1° which allowed incorporation of the Henry’s law limit. It is the purpose of this paper to present a simple extension of this equation to mixtures and to show that the resulting multicomponent isotherm satisfies all thermodynamic requirements. Comparisons to IAS theory using the same pure component isotherms show the simplicity of the extension. Predictions of multicomponent equilibria from the extension using only pure component

@ Abstract published in Advance ACS Abstracts, July 15, 1995. (1) Myers, A. L.; Prausnitz, J. M. AIChE J. 1966, 1 1 , 121. (2) Markham, E. C.; Benton, A. F. J. Am. Chem. SOC. 1931,53,497. (3) Ruthven, D. M.; Loughlin, K. F.; Holborow, K. A. Chem. Eng. Sci.

(4) Jaroniec, M.; Toth, J. Colloid Polym. Sci. 1976,254, 643. (5) O’Brien, J. A,; Myers, A. L. Ind. Eng. Chem. Process Des. Dev.

(6) Valenzuela, D. P.; Myers, A. L.; Talu, 0.; Zweibel, I. MChE J.

(7) Kapoor, A.; Ritter, J. A.; Yang, R. T. Langmuir 1990, 6, 660. (8) Richter, E.; Schutz, W.; Myers, A. L. Chem. Eng. Sci. 1989,44,

(9) Yang, R. T. Gas Separation by Adsorption Processes; Butter-

(10) Sundaram, N. Langmuir 1993,9, 1568.

1973, 28, 701.

1985,24, 1188.

1988, 34, 397.

1609.

worths: Boston, MA, 1987.

0743-74631951241 1-3223$09.OQ/Q

parameters are compared to binary and ternary data from the literature to display the flexibility ofthe new equation. We begin with spreading pressure calculations for the pure component isotherms.

Dubinin-Raduskevich (DR) Isotherm The Dubinin-Raduskevich (DR) isothermg which is

based on the Polanyi potential theory is written as

(1) p = p eb[-ln(e)lm SAT

The Dubinin-Astakhov (DA) isotherm is an extension of eq 1 where the index 112 on the right-hand side of eq 1 is replaced by a parameter, Unp. In eq 1, Pis the equilibrium pressure, PSAT is the limiting pressure, and D is a temperature-dependent energy parameter which is nega- tive. The dimensionless loading 8 in eq 1 is defined by

8 = n/[1000 eL(r) Wdmol wtl = n/n, (2)

where @dTl is the density of the adsorbed phase in g/cm3, Wo is a parameter representing the pore volume in cm3/g, mol wt is the molecular weight of the adsorbate, and n is adsorbate loading in molkg. Equation 1 may also be inverted and written as

e = e - [ in (~ /~~ , ) /W (3)

The spreading pressure, defined by the integral

(4)

can be evaluated for eq 1 as

Inspection of eqs 1 and 3 reveals incorrect limiting behavior in the limit of zero pressure and loading,* namely, lim- (BIP),++ = 0. Both equations have finite limits a t satura- tion.

Modified DR Isotherm A modified DR isotherm relies on truncation of the

logarithmic series inherent in eq 1. As proposed in ref 10, this is written as

(6)

where K is a finite integer and H may be regarded as a

8 eHIXi=lK (l-WiIm = pSAT

0 1995 American Chemical Society

3224 Langmuir, Vol. 11, No. 8, 1995

series in inverse temperature, written in a first ap- proximation as D - EIRT, where D and E in addition to Wo are all temperature-independent parameters.

Equation 6 generates no singularities a t the origin. It provides a good fit of a wide range of pure component experimental data and allows a calculation ofthe isosteric heat in a simple way.1° It may not be inverted like the DR isotherm in eq 3. It may be used to describe microporous, type I adsorption especially where energetic heterogeneity is important. This is possible due to the complex dependence of the isosteric heat on loading available in eq 6 and implicit in the DR development. In calculations for the spreading pressure analytic expres- sions could be evaluated for eq 6 only for K = 1 and K = 2. These are

Sundaram

We note that -H = -(D - EIRn is positive for all the pure component correlations in ref 10. Equations 7 and 8 display the positive and finite character of the spreading pressure for no/nm 5 1. In eqs 7 and 8, no is a variable. At saturation, for example, when no = n,, eq 7 gives nAJ RT = n, ( 1 - (2H13)) and eq 8 gives d R T = n, (1 - 0.759H).

IAS Procedure for Binary Mixtures For the present we restrict ourselves to binary mixtures

and ideal solutions. We illustrate the IAS procedure using K = 1 for which eq 6 is

(9)

For two components we have two simultaneous equations in LAS from (a) equality of spreading pressures (in this case from eq 7)

n20 - - 3 n2m n2m

(b) mixing at constant T and spreading pressure

(11)

Here nIo and nZo are pure component loadings which generate the same spreading pressure as that of the mixture. In IAS n1,Zo (n1,nZ) from eqs 10 and 11 are used to generate P1,z0 using eq 9. Finally

(12) PM,$ = P yi = Pi0 xi

is used to calculate the gas phase composition, yi, and the total pressure P for each pair of nl, n2 values. For K = 2, eq 6 becomes

(13)

Then using eq 13 with eq 11 and the equality of spreading pressures given by eq 8, eq 12 gives the IAS solutions. The equations for K = 2 are formidable in their algebraic complexity. Ae is seen, even for K = 1 any analytic solution for n1,Zo (nl, nz) is well nigh impossible. Consequently we solve eqs 10 and 11 simultaneously using a nonlinear root finder on the IMSL, NEQNF, for different sets of (nl, nz) or alternately, adsorbed phase composition and total moles adsorbed (xi, ntotal). For K 2 3 no analytic expression like eqs 7 and 8 may be written for the spreading pressure. This complicates the application of IAS since the IAS procedure requires the simultaneous solution of sets of equations in n1,2O (nl, n2).

Multicomponent Extension It is desirable to have an analytic equation for the

description of adsorption from a mixture since this greatly facilitates calculations for designs and especially where the isosteric heat of adsorption may be important.l1-l3 From the IAS development it can be seen that no such equation may be written given the complexity of the spreading pressure equations. It is, however, possible to write a multicomponent extension by inspection of eq 6. We first illustrate this for K = 1, namely, eq 9, and a binary mixture,

Similar equations for K = 2 may be written at once using eq 13. Further, the equations for mixtures with more than two components may also be written down quite easily, for any value of K. However, before comparing these isotherms with data or IAS solutions such as from eq 12 they must be checked for proper limits and thermodynamic consistency. These rules are well sum- marized in ref 14. We will use the simplest case, namely K = 1 and a binary mixture to begin this analysis.

Gibbs Isotherm over the Entire Composition Range

This test is an important one since it reveals the conditions that must exist between pure component adsorption parameters if the model is to be thermody- namically consistent.'J5 The test requires the evaluation of the Gibbs isotherm

For the form of eqs 14 and 15 it is simpler to use

(16)

(11) Sircar, S. Langmuir 1991, 7, 3065. (12) Sircar, S. Ind. Eng. Chem. Res. 1991,30, 1032. (13) Sircar, S.; Myers, A. L. Surf. Sci. 1988,205, 353. (14) Talu, 0.; Myers, A. L. MChE J. 1988, 34, 1887. (15) Sircar, S.; Myers, A. L. Chem. Eng. Sci. 1973,28, 489.

Equation for Adsorption from Gas Mixtures Langmuir, Vol. 11, No. 8, 1995 3225

d In P, d In n, dn, + - dn, (17)

which becomes

H1n1

n1 n9

To show that the integral of the right-hand side of eq 18 is independent of path, two alternate two-step paths may be used.16 These are for path 1

step 1: integrate over dn, from 0 to nlu with n, = 0

step 2: integrate over dn, from 0 to nZu U with n, = n,

and for path 2

step 1: integrate over dn, from 0 to nzu with n, = 0

step 2: integrate over dn, from 0 to n p with n, = nzu

The integrations must give the same values where nlu, nzu designate upper limits and are arbitrary points. Defining x = nlu/nl, and y = n2U/n2m for simplicity integration of path 1 results in

n: + H+ [(2 + x ) ~ -

2 + 3 x G - 3 x 6 1 + nzu + H2n2m [(2 - 2x + y l J i - 3 C - y - 2(1 - x)&l 3

(19)

Integration of path 2 results in

2 + 3y- - 3y-I + nlu + H1nlm [(2 - 2y + x ) G - 2(1 - y > G I 3

(20)

which equations also permit direct calculation of the spreading pressure of the mixture a t the nIu, nZu point (K = 1). After some algebra it can be shown that eqs 19 and 20 are identical only when the condition

Hlnlm = Hznzm (21)

is satisfied. Equation 21 is also true whenK > 1 and also when the general parameter np in eq 1 is the same for both components. This condition between the pure component adsorption parameters may be viewed in two ways. Consider first the multisite Langmuir (MSL) model,17 written for a pure component as

(16) Levan, M. D.; Vermeulen, T. J . Phys. Chem. 1981, 85, 3245 .

e t a l - elt

P =

and for component 1 in a binary as

(22)

(23) 61 P, =

t , q i - e, - where

Here Vis the saturation capacity, t is the number of sites on which adsorption occurs, and C is a constant. In this model parameters ti and ni, are temperature independent. Thermodynamic consistency requires that Vbe the same for all components in the mixture.ls This is seen from eq 24 to be the condition

This is analogous to eq 21, where, however, Hi, ni, are temperature dependent. Consequently isosteric heats of adsorption will depend on loading in the present model whereas the multisite Langmuir model is energetically homogeneous. Variation in K also provides greater flexibility for the proposed multicomponent extensions, which may now be written for the general eq 6 as

The other interpretation of eq 21 is obtained by considering the DR isotherm at a fxed temperature. Then parameter D of eq 1 may be replaced by the conventional ,8 of the Dubinin notationg and in eqs 14 and 15 are analogous to the parameters Further, in the DR notation n,, = W d v , where WO, the pore volume of the adsorbent, is a constant. With this understanding of H and n,, eq 21 now gives

(27)

which equation has been the empirical basis for extrapo- lating the DR isotherm across components, i.e., attempting to get a universal i ~ o t h e r m . ~

Selectivity in the Henry’s Law Limit and Continuity

Clearly the pure gas isotherms corresponding to eqs 14 and 15 possess finite and positive Henry’s law limits.1° For the binary mixture under consideration (K = 11, selectivity S 1 2 is written as

As the total pressure P approaches zero, so do individual adsorbed phase loadings nl, nz and the limiting selectivity is seen to be independent of composition. At a k e d pressure P, since eqs 14 and 15 are functions of nl, n2, the generation of selectivity plots versus composition requires

(17) Nitta, T.; Shigetomi, T.; Kuro-oka, M.; Katayama, T. J . Chem.

(18) Golden, T. C.; Sircar, S. AZChE J . 1994,40, 935 . Eng. Jpn. 1984, 17, 39.

3226 Langmuir, Vol. 11, No. 8, 1995 Sundaram

~~ ~

solid lines -- Eqs. 14, 15 lines with dots -- IAS with Eq. 9

0.0 0.2 0 . 4 0.6 0.8 1 .o x i

~~

solid lines -- Eqs. 14, 15 lines with dots -- IAS with Eq. 9

N

L : 0.6 n

0.4

- I I

I ptotal I 0.1 i o.2 1 0.0 ! I I

0.0 0.2 0.4 0.6 0.8 1 .o x l

Figure 1. (a)x-y curves and (b) binary selectivity Snvariation withxl at constant pressurephM using parameters in eq 30 for eqs 14, 15, and 28. IAS with eq 9.

the solution of simultaneous nonlinear algebraic equa- tions. Equations 14 and 15 also collapse to pure-compo- nent forms when either nl and n2 is zero or when either y1 or yz is zero. This is also true for the general eq 26. Thus the proposed extension possesses the feature of continuity.

Isothermal-Isobaric Integral Test and the X-Y Intersection Rule

This test is also an evaluation of the Gibbs isotherm using the mixture isotherm but a t a constant pressure. Since the proposed isotherm has been shown to satisfy the Gibbs isotherm generally, giving the condition in eq 21, it also satisfies the isothermal-isobaric integral test. The integral to be evaluated is

J‘ (3 - ;) dy, O Y1 (29)

and is compared with the difference of the pure component spreading pressures, each evaluated up to the same, constant pressure ptotsl, for example using eqs 9 and 7 when K = 1 for a binary mixture. The loadings to be used in these equations are found from the pure gas isotherms with the pressure set equal to ptoa . Since the proposed isotherm may not be inverted to a function in pressure,

1 .o

0.8

0.8

0.4

0.2

0.0 0.0 0 . 2 0 . 4 0.6 0.8 1 .o

0.4 - ntotal - 3.8 0.3

0.2

0.1 ntoul I 2 I

ntotal = 1 0.0

0.0 0.2 0.4 0.6 0.8 1 .o X I

Figure 2. (a) x-y curves and (b) ptotsl variation with X I at constant nt . ,~ using parameters in eq 30 for eqs 14 and 15.

the test may be shown to be satisfied using a numerical integration technique for eq 29. This test has been per- formed on the extension and reiterates the consistency of the model which has been generally established by eqs 19-21.

The x-y intersection rule which may be regarded as a necessary but not sufficient condition for thermodynamic consistency permits rapid evaluation of the consistency of data which may be insufficient to accurately evaluate the integral in eq 29. The rule states that at constant pressure and temperature, the x-y curves of thermody- namically consistent theories or data intersect at least once in the region 0 < x 1. As such it is a practical way to implement the isothermal-isobaric integral test and serves to discard theories or data whose x-y curves do not intersect the x-y curve from a thermodynamically con- sistent model. An example of such a model is the IAS solution with any pure component isotherm. This inter- section rule also applies to the selectivity-composition curves.

Features of Proposed Mixture Isotherm Figure l a shows the x-y curves at two pressures for a

representative set of pure component parameters given by

H, = -1, H, = -2.5, n,, = 10, n,, = 4, PsATl = 1, PSAT, = 0.6047 (30)

Equation for Adsorption from Gas Mixtures

1.5-

1.0-

0.5 -

Langmuir, Vol. 11, No. 8, 1995 3227

. * e 0 -

I /-- '. z

' \

-- - -c--

_e - -

- 0 n

- L

U-- ". .

, I I 0.0 0.2 0.4 0.6 0 . 8 1 .o

X l

:

0.0 0.2 0 .4 0.6 0 . 8 1 .o X l

0.8 ''-1 (b)

0.6 -

0.4 -

0.2 -

ntotal- 1 - 0.0 0.2 0.4 0.6 0.8 1 .o

x l

Figure 3. (a) x-y curves and (b) ptoa variation with XI at constant ntotal using parameters in eq 31 for eqs 14 and 15.

which were chosen to satisfy eq 21. Units for n are mollkg and for PSAT are kPa. We are considering the simplest form of eq 26, namely eqs 14 and 15 for the binary mixture (K = 1) and eq 9 for the pure components, only for the purposes of illustrating some features of the proposed isotherm. The pure gas isotherms for these parameters intersect once, at around P = 0.13 Wa, with component 1 having a lower Henry constant but a higher saturation capacity than component 2. This implies that over some range of pressure an azeotrope is expected. Figure la displays the x-y intersection rule at the two pressures p t O a = 0.1 and 0.5 Wa with the IAS solution and eqs 14 and 15. It also shows the formation of an azeotrope using the proposed mixture isotherms at higher pressures. Since the curves move across thex = y line the binary selectivity is expected to change from being selective for component 2 at low pressure to being selective for component 1 at higher pressure. This is shown in Figure l b where the binary selectivity S12 moves across unity as the pressure increases. The proposed model intersects the IAS selec- tivity-composition curves and also shows the selectivity reversal associated with the azeotrope at the higher pressure. The selectivityreverses, i.e., reduces below unity for pbta1 = 0.5 kPa at x1 = 0.731. While IAS may not describe such nonidealities, theories such as HIAS6 can.

Figure 2a shows the x-y curves from eqs 14, 15, and 30 at fixed nTOTAL where once again the azeotrope appears a t the higher concentration. The corresponding variation

1.8 -

1.4-

1.0 - O3 - 0.6 --

ptotal I 1 O3 - ptotal I 1

0.6 --

_ . _ , . , . , . I . , . ,

0.0 0.2 0.4 0.6 0.8 1 .o x1

....................................................................... CI K = 19 in Eqs. 6 and26

system parametem from Eq. 32 dashed lines .- pure component

fixed ptotal values marked 4.5 spreading pressures '

f 4.0 '\

_.", . , . , . , . , . , 0.0 0.2 0.4 0.6 0.8 1 .o

X l A Y

Figure 4. (a) x-y curves, (b) ntotal variation with XI for K = 1 (eqs 14 and 151, and (c) mixture spreading pressure variation withxl for K = 19 (eqs 6,261 at constant p t o ~ using parameters in eq 32.

inptotal withx1 is shown in Figure 2b. These figures show essentially the same behavior predicted by the multi- component form of the isotherm based on statistical mechanic^.^ Indeed if we use the parameters

3228 Langmuir, Vol. 11, No. 8, 1995 Sundaram

Table 1. Pure Component Parameters from Equation 6 Using K = 19

Data of Costa et al. (19811, Hydrocarbons on Activated Carbon, T = 293.15 K unconstrained 1 constrained 2 constrained 3

Hi nima P S A T ~ ~ erroF Hi nima PSAT~* erroF Hi nima P S A T ~ ~ error CH4 -0.3624 34.941 889.89 0.00116 -1.3729 362.53 67982.1 0.0018 -1.4371 385.226 81470.8 0.00179 CzH6 -2.957 187.22 11220.5 0.0437 -2.8504 174.741 8476.05 0.04446 -2.957 187.22 11220.5 0.0437 CzH4 -2.6727 186.36 12426.6 0.01634 -2.6727 186.36 12426.6 0.01634 -2.7667 200.09 16174.6 0.0165 C3Hs -5.4022 258.12 46119.9 0.25056 -3.2481 153.346 974.768 1.2515 -3.5975 153.887 1241.5 0.77823

Data of Danner and Wenzel(1969), Oz-Nz-CO on 1OX Sieve, T = 144 K constrained 2 constrained 3 unconstrained 1

nimd P s A T ~ ~ error Hi nimd P S A T ~ ~ errof Hi nimd P s A T ~ ~ errof Hi 0 2 -1.2111 125.4 1202.18 0.2019 -2.0533 301.318 14704.3 0.47804 -2.1621 336.01 20522.9 0.48307 Nz -4.8313 128.06 7301.9 0.05254 -4.8313 128.06 7301.9 0.05254 -5.204 139.599 14619.3 0.07513 CO -5.4754 132.68 2756.5 0.347 -4.8123 128.56 1555.42 0.54181 -5.4754 132.68 275%5 0.347

Data of Reich et al. (19801, Hydrocarbons on Activated Carbon, T = 301.4 K unconstrained 1 constrained 2 constrained 3

H, nbma PSAT? errorf HI nima PSAT? errorf H , nlma PSAT? error CH4 -1.3314 4.2439 3945.99 0.05844 -2.0367 7.5297 24046.7 0.35206 -1.9383 6.73027 17355.3 0.33211 CzH6 -2.9555 5.1889 2077.69 0.07678 -2.9555 5.1889 2077.69 0.07678 -2.5476 5.12062 1598.79 0.39082 CzH4 -2.509 5.1994 2108.57 0.06448 -2.7994 5.47824 3096.58 0.12151 -2.509 5.1994 2108.57 0.06448

Data of Talu and Zweibel(19861, COZ-H~S-C~HE on H-Mordenite, T = 303.15 K unconstrained 1 constrained 2 constrained 3

HL nlma PSAT? er ros Hi nrma PSAT? errold H, nlma P S A T , ~ error COz -3.4751 3.3036 1401.22 0.07887 -1.5996 2.15 89.2195 7.6871 -3.8041 3.9066 3185.29 0.09173 HzS -5.2638 2.8233 394.966 0.07675 -1.5776 2.18 13.232 7.1733 -5.2638 2.8233 394.966 0.07675 C3H8 -3.3015 1.0417 72.0168 3.9295 -3.3015 1.0417 72.0168 3.9295 -5.6753 2.61859 18262.4 7.34

a In mol/kg. In mmHg. Number of data points used: CH4 = 10, C & j = 13, CzH4 = 12, C3H6 = 11. In cm3/g (STP). e Number of data points used: 0 2 = 14, Nz = 17, CO = 17. f Number of data points used: CHd = 25, CzH6 = 16, CzH4 = 16. g Number of data points used: COz = 41, HzS = 22, C3H8 = 34. In kDa.

which reverse the saturation capacities while maintaining the same Henry's constants for components 1,2, we obtain parts a and b of Figure 3 from eqs 14 and 15 which show no azeotropes or selectivity reversals and where the total pressure increases with composition at all appropriate values of ntotal.

To illustrate further features of the isotherm extension, we use the parameters

to generate parts a and b of Figure 4 whereptotal is fxed. Once again pronounced azeotropic behavior occurs at the higher pressures. A further signature of nonideality is the appearance of maxima in the total amount adsorbed with composition. Parts a and b of Figure 4 (K = 1) show that these maxima appear even when selectivity reversal does not occur (e.g. a t ptotal = 1 W a ) as noted in ref 19. The mixture spreading pressures also display maxima and may exceed the saturation spreading pressure of component 1 at higher pressures as shown in Figure 4c, for K = 19.

Comparison with Experimental Binary and Ternary Data

In comparison of predictions of mixture data available in the literature, pure component regressions at a fixed temperature were first obtained using eq 6. Since some of the data sets required more flexibility, the integer K was set at 19. This is arbitrary and can be changed depending on the robustness of the regression package.

(19) Talu, 0.; Zweibel, I. AZChE J. 1986, 32, 1263.

For this value of K, the regression parameters were then checked with eq 21. While this condition is approximately obeyed for some of the datasets, it cannot be guaranteed. Therefore some form of constraints have to be imposed on the parameters.

There may be methods of simultaneously regressing individual pure gas data such that eq 2 1 is obeyed; however in this paper the regressions were performed by simply fixing the product (Hini,,,). That is, if the condition in eq 21 was not satisfied when unconstrained, then the regression was again performed for one of the pure components in which the condition in eq 21 for the product (Hini,,,) obtained from the other component was forced on the parameters. A similar procedure to use the multi- component Toth isotherm, by forcing the saturation capacity of each component to a common value was used in ref 11. This served to generate parameters satisfying consistency, albeit at the cost of some accuracy in the pure component fit. This was performed for each pair in a binary and of the resulting two regressions, the regression with lower error was chosen for prediction and comparison with data. Errors are the product of variance and the number of data points. Pure component param- eters and errors in the regressions are given in Table 1. In Table 1, set 1 denotes parameters obtained with no constraints, whereas sets 2 and 3 denote those parameters obtained by forcing (Hini,) of one of the components to satisfy eq 21, obtained from the other component. The Simplex based algorithm MINSQ was the regression package used which has the ability to find the best minima even from poor initial guesses. The parameters in Table 1 were then used in eq 26 with K = 19. Figure 5a shows the comparison of the predictions from eq 26 and the binary data from ref 20 for hydrocarbons on activated carbon.

Equation for Adsorption from Gas Mixtures

Table 2. Prediction of Ternary Mixture Data from Equation 28 Using K = 19

Data of Costa et al. (1981) for Activated Carbon, T = 293.15 K Methane (1)-Ethylene (21-Ethane (3)

Using Parameter Set 2 from Table 1

Langmuir, Vol. 11, No. 8, 1995 3229

Pbtal (mmHg) y1 YZ y3 x1 x2 x3

Po 75 0.95 0.035 0.015 0.377 0.347 0.276 Db 75 0.95 0.035 0.015 0.376 0.309 0.315 P 75 0.921 0.033 0.046 0.256 0.21 0.535 D 75 0.921 0.033 0.046 0.276 0.239 0.484 P 75 0.899 0.085 0.016 0.254 0.555 0.191 D 75 0.899 0.085 0.016 0.275 0.48 0.244 P 75 0.79 0.113 0.08 0.124 0.345 0.531 D 75 0.79 0.113 0.08 0.161 0.417 0.422 P 75 0.685 0.25 0.065 0.091 0.621 0.288 D 75 0.685 0.25 0.065 0.113 0.582 0.305 P 75 0.71 0.138 0.152 0.088 0.309 0.604 D 75 0.71 0.138 0.152 0.113 0.293 0.594 P 75 0.633 0.23 0.137 0.071 0.453 0.477 D 75 0.633 0.23 0.137 0.091 0.445 0.464

errors % average absolute = 12.1 14.63 10.19 11.47

Ethylene (1)-Ethane (2)-Propylene (3) Using Parameter Set 2 from Table 1

P,,, ("Ha) YI YZ Y3 21 2 2 x3

P 75 D 75 P 75 D 75 P 75 D 75 P 75 D 75 P 75 D 75 P 75 D 75 P 75 D 75 P 75 D 75

0.889 0.889 0.865 0.865 0.81 0.81 0.734 0.734 0.604 0.604 0.573 0.573 0.394 0.394 0.332 0.332

0.11 0.11 0.124 0.124 0.175 0.175 0.245 0.245 0.387 0.387 0.388 0.388 0.583 0.583 0.656 0.656

0.001 0.001 0.011 0.011 0.014 0.014 0.021 0.021 0.009 0.009 0.039 0.039 0.022 0.022 0.012 0.012

0.808 0.8 0.66 0.629 0.569 0.559 0.463 0.455 0.416 0.397 0.285 0.291 0.213 0.226 0.196 0.187

0.172 0.18 0.162 0.174 0.209 0.205 0.262 0.251 0.454 0.437 0.324 0.324 0.533 0.483 0.657 0.583

0.02 0.02 0.178 0.197 0.222 0.236 0.275 0.294 0.13 0.166 0.391 0.385 0.254 0.291 0.146 0.231

errors in x , % average = 6.99 3.36 5.66 11.96

a P, x values predicted using eq 26. D, experimental data for y values.

Table 2 shows comparisons of ternary predictions with the data of ref 20. Parts b and c of Figure 5 show binary data from ref 21 for Molecular Sieve 1OX. Table 3 shows a comparison of selected predictions for adsorption on activated carbon with the data of ref 22. Further predictions for this dataset are in Table 8. Figure 6 shows comparisons with the data of ref 19 for mixtures of C02- H2S-C3& on H-mordenite.

Coadsorption of Bulk Gas with a Trace Component

In addition to the above comparisons eq 26 was also tested for use in describing the effect of coadsorption of a bulk carrier gas on a trace adsorbate. This problem has recently been discussed18 with reference to Freon-12 adsorption on BPL carbon. However their pure component data were not presented in tabular form. To perform calculations using eqs 6 and 26, their fit of data was used,

(20) Costa, E.; Sotelo, J. L.; Calleja, G.; Marron, C. AZChE J. 1981,

(21) Danner, R. P.; Wenzel, L. A. AZChE J . 1969, 15, 515. (22) Reich, R.; Ziegler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process

27, 5.

Des. Dev. 1980, 19, 336.

1 .o

0.8

0.6 c r

0.4

0.2

A CHl( l ) -CM6(2) L3

0 CH4(1)-C2HU2) U2

C?M[l).WH6(2) XZ

+ CWl ) -C3H6(2 ) 13

A CZHl((lCZ74612) X2

lines .. predinicns from Eq. 26 symbola-.darpolCottamal.(1gBl) Hydrccarbonr on Activated Carbon 263.16 K. 75 mm Hg, X - s e t used lrom Table 1

0 .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 . , . , . I . , . I . , . , . , . , .

x i

1 .o

0.8

0.6

c r

0.4

0.2 lines .. predictions from Eq. 26 symbols .. daa of Dannar and Wenzel 11969) 02.CO.N2 on 1OX. 144 K. 760 mmHg P =ret used from Table 1

0.0 1

0.0 0.2 0.4 0.6 0 8

X l

130 - 2

(C) lim .. prodi*lcna from Eq. 26 symbols. data 01 Danner and Wenzsl(1969) 02, CO, N2 on lox. 144K, 760 mm Hg # I am used from TsMo 1

115 -

N2-CO. 113 0

100 1 . 1 . I . , . r 0.0 0.2 0.4 0.6 0.8 1 .o

X l

Figure 5. Comparison of predictions from eq 26 with binary data of (a) Costa et al. (1981) for activated carbon and (b, c) Danner and Wenzel(1969) for 1OX molecular sieve.

which was obtained using the MSL mode1.l' This fit is quite good and serves our purpose of simulating the unavailable data. The pure component parameters obtained for T = 310.9 K are given in Table 4. Table 5 gives the predicted Henry's constant in comparison with the experimental values. These constants are obtained from eq 26 in the following manner. Equation 26 written

3230 Langmuir, Vol. 11, No. 8, 1995

Table 3. Prediction of Binary and Ternary Mixture Data from Equat ion 26 Using K = 19

DataD of Reich et al., (1980) for Activated Carbon, T = 301.4 K Methane (1)-Ethane (2)-Ethylene (3)

Using Parameter Set 2 from Table 1

Sundaram

~~

pb 124.1 0.200 0.192 0.608 0.019 0.332 0.649 2.4628 Dc 124.1 0.200 0.192 0.608 0.019 0.278 0.703 2.5830 P 131.7 0.230 0.520 0.250 0.019 0.755 0.226 2.6463 D 131.7 0.230 0.520 0.250 0.021 0.747 0.232 2.7990 P 339.2 0.624 0.174 0.202 0.112 0.514 0.374 2.9475 D 339.2 0.624 0.174 0.202 0.156 0.469 0.375 3.159

errors in nl, n2, n3 (%) (overall = 10.72) 16.28, 6.89, 9.00

Ethane (1)-Ethylene (2) Using Parameter Set 2 from Table 1

Ptotal y1 y2 x1 nl n2 Ntotai

P 137.894 0.24 0.76 0.336 0.926 1.829 2.755 D 137.894 0.24 0.76 0.281 0.84 2.148 2.988 P 308.193 0.24 0.76 0.33 1.192 2.422 3.614 D 308.193 0.24 0.76 0.29 1.116 2.733 3.849 P 737.043 0.24 0.76 0.32 1.469 3.093 4.562 D 737.043 0.24 0.76 0.286 1.35 3.369 4.719 P 1341.02 0.24 0.76 0.316 1.611 3.483 5.094 D 1341.02 0.24 0.76 0.275 1.479 3.899 5.378 P 1981.54 0.24 0.76 0.312 1.708 3.758 5.466 D 1981.54 0.24 0.76 0.272 1.615 4.323 5.938 P 217.873 0.472 0.528 0.584 1.94 1.382 3.322 D 217.873 0.472 0.528 0.559 2.008 1.585 3.593 P 549.508 0.472 0.528 0.575 2.483 1.833 4.317 D 549.508 0.472 0.528 0.533 2.442 2.14 4.582 P 1132.8 0.472 0.528 0.57 3.328 2.513 5.842 D 1132.8 0.472 0.528 0.533 2.868 2.512 5.38 P 144.099 0.682 0.318 0.773 2.293 0.673 2.965 D 144.099 0.682 0.318 0.762 2.489 0.778 3.267 P 346.114 0.682 0.318 0.767 2.983 0.905 3.888 D 346.114 0.682 0.317 0.748 3.049 1.027 4.076 P 692.917 0.682 0.318 0.762 3.499 1.091 4.59 D 692.917 0.682 0.318 0.746 3.633 1.237 4.87 P 1367.91 0.682 0.318 0.757 4.133 1.325 5.458 D 1367.91 0.682 0.318 0.727 4.076 1.53 5.606

errors in nl, n2 (%) (overall = 8.87) 6.41 11.33

P, x’s, n’s, Ntotd predicted using eq 26. D, data for €‘total, y’s. Ptotai in kPa, n1, n2, Nbbi in mol/kg.

for a binary with component 1 being the trace adsorbate permits the total pressure to be written in the Henry limit as

a Ternary data taken from O’Brien and Myers (1985).

Once this nonlinear equation is solved for n2, eq 26 written for component 1 in the Henry limit using the value of n.2 from eq 33, gives the Henry constant as

Ternary mixtures require two nonlinear equations to be solved, for nz and n3, given the gas phase composition of the bulk gas. As Table 5 shows, predictions from eq 26 for the mixtures are quite good, although the pure gas Henry constants show some discrepancy with the simu- lated data. This may be due to the use of data obtained from a fit since tabular data were unavailable. Further PSAT in eq 6 is necessarily bounded at maximum coverage so using the simulated data from eq 22 (which is unbounded at saturation) may also generate some error.

Table 6 gives parameters for the adsorption of pure components COz and CHI on BPL carbon from ref 18, at

0.0 0 . 2 0 . 4 0.6 0 . 8 1 .o x i

we¶. podldonr Ifan Eq. 26

- 303.15 K. 8.13 kPa * c -

2.5

2.0 1

0 . 0 0 .2 0 . 4 0.6 0 . 8 1 .o

x1

- hz5(1 )-C02(2) QI W-Mordwrile - 303.15 K. 15.55 kPa

13 p 1.8

1.0

0.8

0.6

0.4

0.2

0.0

c

0.0 0.2 0.4 0.6 0 . 8 1 .o Xl

Figure 6. Comparison of predictions from eq 26 with binary data ofTalu and Zweibel(1986) for (a), (b), (c), COZ-C~HB-HZS mixtures on H-mordenite. Number refers to parameter set of Table 1 used with eq 26.

T = 303 K. Table 7 gives the binary mixture predictions using eq 26 for these data, also at T = 303 K. As seen from Tables 5 and 7, in all the cases studied, eq 26 performs as well or better than the MSL model17 and permits variation of isosteric heat with loading, which is unavail- able in the homogeneous MSL model. Pure component saturation spreading pressures are also unbounded in the

Equation for Adsorption from Gas Mixtures Langmuir, Vol. 11, No. 8, 1995 3231

Table 4. Parameters for Pure Component Adsorption Dataa on BPL Carbon of Golden and Sircar (1994) at T = 310.9 Kb parameters from eq 6 with K = 19

unconstrained 1 constrained 2c Henry constants, (mol/kg)/atm Hi ni, PSAT~ erroP Hi nim PSAT; errold dataa set 2 set 1 ....

F-12 -3.2313 3.41 0.877 3.7977 -3.2313 3.41 0.877 3.7977 630 1710 1710 Nz -0.9729 3.4626 60.94 0.25134 -1.5159 7.268 364.6 0.42719 0.27 0.3404 0.355 COz -1.9542 9.61 80.06 4.6625 -1.2368 8,909 38.0 15.443 2.4 2.408 3.945 CH4 -1.6705 6.4 97.95 3.6711 -1.7217 6.399 102.94 3.7245 1.05 1.592 1.5193

a Obtained from multisite Langmuir (MSL) model fit of data by Golden and Sircar (1994). F-12, Freon 12.' ni, in molkg, PSAT~ in atm. Parameters satisfy consistency, eq 21. Number of data points used from MSL simulation fit: F12 = 35, N2 = 35, COz = 99, CHI = 74.

Table 6. Predicted Henry Constant from Equation 26 for Trace Freon-12 Adsorbate in Binary Mixtures on BPL Carbon at T = 310.9 K

Henry constant for Freon-12, (mol/kg)/atm eq 26, K = 19 from Golden and Sircar (1994)

bulk gas P w , atm set 1 set 2" data Nitta eqb IAS' Nz 7.1 118 3 19 310 298 250 coz 7.1 67.78 54.5 46 40 20 coz 2.4 166.3 162 220 151 80 C O Z - C H ~ ~ 7.1 77.05 73.7 70 64 37

Parameters from Table 4 that satisfy consistency, eq 21. Multisite Langmuir (MSL) model, eq 23. Using data, extrapolated to zero loading limit of spreading pressure integral, eq 4. Ternary mixture [F12(1)-C02(2)-CH4(3)1 with yz = 0.43, y3 = 0.57.

Table 6. Parameters for Pure Component Adsorption Dataa on BPL Carbon of Golden and Sircar (1994) at T = 303 Kb parameters from eq 6 with K = 19

unconstrained 1 constrained 2" Henry constants, (mol/kg)/atm Hi ni, PSAT; erroTd Hi nim PSAT~ errold dataa set 2 set 1

COz -1.8227 9.51 58.07 9.97496 -1.8227 9.51 58.07 9.97496 2.88 5.07 5.07 CH4 -2.0819 7.4 174.85 7.3671 -2.277 7.6125 227.296 8.4237 1.29 2.44 2.13

Number of data points used from MSL simulation fit: COz = 20, CH4 = 84. a Obtained from multisite Langmuir (MSL) model fit of data by Golden and Sircar (1994). ni, in mol/kg, PSAT~ in atm. Parameters

satisfy consistency, eq 21.

MSL model, as opposed to those obtained with eq 6, for example, eqs 7 and 8.

Discussion and Conclusions

It is noted that all the regressions were performed at a fured temperature although eq 26 permits the incor- poration of a temperature dependence. Ternary predic- tions for the data of ref 20 are quite good as measured by the errors in Table 2. For the data of ref 22 compared in Table 3, the error in individual amounts adsorbed is also quite low. For the binaries containing methane in these data, however, parameter sets 2 and 3, which obey eq 21, give a large error in the methane (component 2) loading, shown in Table 8. These particular data, i.e., binaries containing methane, are also poorly described by both IAS-HIAS theories6 and by a potential theory approach used in ref 22.

As seen in parts a and b of Figure 6, the effect of using parameters with the consistency condition is a correction in the right direction, although the prediction of data is quite poor for both the data sets. However, both figures qualitatively show the presence of nonidealities with the thermodynamically consistent parameter sets, an azeo- trope around x1 = 0.79 in Figure 6a and a maxima in Figure 6b for ntotal with set 2, which are confirmed by the data.

Table 5 also shows that the constrained parameter set 2, which satisfies eq 21, gives significantly improved mixture Henry constants for Freon-12. As an aside it is noted that Table 8 shows that the unconstrained param- eter set 1 predicts the high coverage data of ref 22

surprisingly well, especially the drop in X I with pressure which is not prddicted well by eq 26 or by the IAS, HIAS theories. The pure component fit of eq 26 for methane in the data of ref 22 and for propane in the data of ref 19 is not adequate. This may account for some of the discrep- ancy evident from the errors in Table 8 and Figure 6a,b, especially, in the location of the azeotrope and in the magnitude of the amount adsorbed, although these are qualitatively predicted by eq 26. The error may also depend on the nature of the nonidealities for the particular system: although the H-mordenite data were chosen especially to study heterogeneity, HIAS theory too, does not adequately describe these data suggesting that they possess further nonidealities from other sources. Indeed, the intransigence of these data has necessitated the use of some form of lateral interaction with an HIAS theory to describe the e q ~ i l i b r i a . ~ ~ Further, binary information was required in their correlations as also in the spreading pressure dependence (SPD) theory.lg

Apparent Activity Coefficients

It is possible to calculate apparent activity coefficients for those data which are predicted satisfactorily by eq 26. The debate over the source of nonidealities in adsorption equilibrium has been ongoing ever since a possible explanationz4 for the large, negative deviations from ideality observed by ref 20 for hydrocarbons on activated

(23) Karavias, F.; Myers, A. L. Chem. Eng. Sci. 1992,47, 1441. (24) Myers, A. L. AIChE J. 1983,29, 691.

3232 Langmuir, Vol. 11, No. 8, 1995

Table 7. Prediction of Binary Mixture Data on BPL Carbon of Golden and Sircar (1994) from Equation 26

UsingK=lBatT=303K

COz (l)-CHr (2) Using Parameter Set 2 from Table 6

Sundaram

P 1 0.000 0.000 0.000 0.9081 5.4034 0.9081 Db 1 0.000 0.000 0.000 0.96 0.96 P 1 0.258 0.4733 0.5459 10.2926 0.6076 4.4627 1.1536 D 1 0.258 0.4377 0.495 0.636 1.131 P 1 0.478 0.7062 0.9499 4.0546 0.3952 9.7714 1.345 D 1 0.478 0.7333 0.99 0.36 1.35 P 1 0.748 0.888 1.387 0.5718 0.175 8.8657 1.562 D 1 0.748 0.879 1.395 0.192 1.587 P 1 1,000 1.000 1.7497 3.5958 0.000 0.7497 D 1 1.000 1.000 1.815 0.000 1.815 P 21.3 0.000 0.000 0.000 4.1714 9.238 4.1714 D 21.3 0.000 0.000 0.000 4.596 4.596 P 21.3 0.258 0.5565 3.0461 16.0436 2.4279 5.3756 5.474 D 21.3 0.258 0.5326 2.625 2.304 4.929 P 21.3 0.478 0.7738 4.9062 5.8518 1.4339 31.3104 6.3401 D 21.3 0.478 0.8093 4.635 1.092 5.727 P 21.3 0.748 0.9201 6.6128 8.3174 0.5745 11.3279 7.1872 D 21.3 0.748 0.9221 6.105 0.516 6.621 P 21.3 1.000 1.000 7.8173 1.5899 0.000 7.8173 D 21.3 1.000 1.000 7.695 0.000 7.695 P 35 0.000 0.000 0.000 5.0467 2.8726 5.0467 D 35 0.000 0.000 0.000 5.196 5.196 P 35 0.258 0.5736 3.7819 20.6336 2.8108 6.4713 6.5927 D 35 0.258 0.5429 3.135 2.64 5.775 P 35 0.478 0.7869 5.936 7.2443 1.6074 44.0342 7.5434 D 35 0.478 0.8322 5.535 1.116 6.651 P 35 0.748 0.926 7.7729 8.4086 0.6211 56.8537 8.394 D 35 0.748 0.9477 7.17 0.396 7.566 P 35 1.000 1.000 8.9565 2.9483 0.000 8.9565 D 35 1.000 1.000 8.7 0.000 8.7

average errors % 2.95 7.46 16.33 6.56

&tal in atm; nl, nz, Ntotal in molflrg. a P, XI, n’s, Nbtal predicted using eq 26. D, data for Ptotai, y’s

carbon was pointed out. Reference 25 suggested that low coverage data from this set maybe predicted by IAS. Other relevant work includes refs 6, 12, 19, 23, 26, and 27.

Since the binary and ternary predictions of eq 26 were especially good for the data of ref 20, activity coefficients from these data were calculated using

D

(35)

where Pio(n) was calculated from the pure gas isotherms, eq 6, at the pure gas loadings that generated the same spreading pressure as the mixture. For K L 2 the solution is necessarily numerical. For K = 19 used in the regressions the numerical calculations were performed using the routine IVPRK on the IMSL.

Figure 7 shows that eq 26 removes much of the apparent nonideality in these data, as foreseen by Myers.z4 The activity coefficients now are also only slightly asymmetric about the equimolar concentration unlike those derived in ref 20 from their binary data. As has since been pointed o ~ t , ~ J ~ , ~ ~ these derivations did not account for nonideality in the mixing equation, eq 11. Also, activity coefficients derived from eqs 6, 26, and 35 possess correct limits as the total pressure approaches zero or the mole fractions xi, yi approach zero or unity.

On the basis of the quality of predictions in Figure 5b, the same calculations were performed for the 1OX sieve

(25) Kaul, B. K. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 711. (26) Dubinin, M. M.; Jakubov, T. S.; Jaroniec, M.; Serpinsky, V. V.

(27) Kaminsky, R. D.; Monson, P. A. Langmuir 1994, 10, 530. Pol. J . Chem. 1980,54, 1721.

Table 8. Effect of Thermodynamic Consistency in Parameters on Predictions of High Coverage Data of

Reich et al. (19801, T = 301.4 Ka

Ethylene (1)-Methane (2) Using Parameter Sets 3 and 1 from Table 1

3 130.999 0.740 0.260 0.9679 2.2682 0.0751 2.3433 1 130.999 0.740 0.260 0.9489 2.2176 0.1195 2.3370 D 130.999 0.740 0.260 0.9640 2.3647 0.0883 2.4530 3 344.735 0.740 0.260 0.9634 3.2770 0.1244 3.4013 1 344.735 0.740 0.260 0.9330 3.1594 0.2270 3.3864 D 344.735 0.740 0.260 0.9320 3.3589 0.2451 3.6040 3 682.575 0.740 0.260 0.9594 4.0406 0.1708 4.2114 1 682.575 0.740 0.260 0.9171 3.8304 0.3464 4.1768 D 682.575 0.740 0.260 0.9210 4.0045 0.3435 4.3480 3 1225.88 0.740 0.260 0.9552 4.6163 0.2163 4.8326 1 1225.88 0.740 0.260 0.9061 5.1539 0.5338 5.6877 D 1225.88 0.740 0.260 0.9120 4.6840 0.4520 5.1360 3 2000.84 0.740 0.260 0.9513 5.0899 0.2608 5.3507 1 2000.84 0.740 0.260 0.8777 4.4702 0.6229 5.0931 D 2000.84 0.740 0.260 0.8910 5.1063 0.6247 5.7310

errors for complete data, % parameter set 3 7.75 3.52 36.9 6.29 parameter set 1 4.93 12.36 10.83 7.95

Ethane (1)-Methane (2) Using Parameter Sets 2 and 1 from Table 1

~~

2 129.62 0.733 0.267 0.9788 2.6339 0.0569 2.6909 1 129.62 0.733 0.267 0.9590 4.5740 0.1089 2.6829 D 129.62 0.733 0.267 0.9740 2.8178 0.0750 2.8930 2 350.94 0.733 0.267 0.9740 3.6190 0.0966 3.7155 1 350.94 0.733 0.267 0.9416 3.4791 0.2157 3.6948 D 350.94 0.733 0.267 0.9420 3.6813 0.2267 3.9080 2 684.644 0.733 0.267 0.9698 4.2754 0.1333 4.4087 1 684.644 0.733 0.267 0.9237 4.0314 0.3328 4.3641 D 684.644 0.733 0.267 0.9250 4.1810 0.3390 4.5200 2 1234.15 0.733 0.267 0.9650 4.7520 0.1722 4.9242 1 1234.15 0.733 0.267 0.9075 4.9507 0.5046 5.4553 D 1234.15 0.733 0.267 0.8940 4.7051 0.5579 5.2630 2 2005.67 0.733 0.267 0.9604 5.1090 0.2106 5.3196 1 2005.67 0.733 0.267 0.8762 4.4432 0.6270 6.0708 D 2005.67 0.733 0.267 0.8720 5.0672 0.7438 5.8110

errors for complete data, % parameter set 2 7.78 3.25 46.2 6.25 parameter set 1 5.07 11.52 17.24 6.94

a 3,2, and 1 are XI, n’s, and Nbtal predicted using eq 26. D, data for Phltal, y’s. Pbta1 in kPa; nl, n2, Ntotal in molkg.

data of ref 21. These data were shown to require some form of interaction parameters in mixture predictionz5 and the IAS method could not be applied. Equation 26 predicts these data very well for 02-CO and 02-NZ mixtures, though poorly for the Nz-CO mixture. Thus the apparent activity coefficient calculated using eq 26 and eq 35 is reliable for the first two systems.

Figure 8a shows that the apparent nonideality of these systems is greater than for the hydrocarbon mixtures- activated carbon systems in Figure 7a. This larger nonideality is attributed to enhanced lateral interactions in zeolites which are pronounced at the higher coverages (>50% for these particular data). Equation 26 with eq 21 being simply the rigorous extension of eq 6, has no overt lateral interaction term, which enhances the negative deviation for these data.z8

In Figure 8b are shown the activity coefficients extracted from eq 35 for the ethane-ethylene data of ref 22 and the HzS-CO~ data on H-mordenite of ref 19. Both these sets are predicted reasonably well by eq 26 as seen from Figure 6c and the errors in Table 3 and also by the IAS-HIAS

(28) Ritter, J. A. AIChE National Meeting, Miami, Nov 1991.

Equation for Adsorption from Gas Mixtures Langmuir, Vol. 11, No. 8, 1995 3233

0.65 -

0.80 - 0.75 -

solid llnes ... CH4(1)42H4(2) on Anivated Carbon dashed lines. CH4(l).C2H6(2) on Activated Carbon

(a)

" _ " " I . I 1 I . 1 ' I ' I

0.0 0.2 0 .4 0.6 0.6 1 .o

X l

1.000

0.995

0.990

0.985

' C2H6 C3H6 \

sdi lines -. CZH4(1)-C2H6(2) a+ CZH4(1 )-C3H6(2) dashed lines. CZHql)-C3H6(2) , . , . , 1 , . , ,(bJ I 293.15 K. 75 M Hg

0.0 0.2 0.4 0 . 6 0 . 8 1 .o X I

Figure 7. Apparent activity coefficients calculated from eq 35 using predictions from eq 26 for (a), (b), hydrocarbon mixture data on activated carbon of Costa et al. (1981).

method.6 Consequently, apparent activity coefficients reflect the expected, reduced degree of nonideality.

summary In summary, when eq 21 is satisfied, eq 26 is the rigorous

multicomponent extension of eq 6 which obeys the Gibbs isotherm, possesses features which are desirable and required in a mixture adsorption equation,14 and suc- cessfully predicts a wide range ofbinary and ternary data. The condition for thermodynamic consistency is easy to impose on pure gas data. It is considerably more difficult to impose or justify a condition of equal saturation capacities that would render the multicomponent Lang- muir or Toth models thermodynamically consistent.

Forcing this condition of thermodynamic consistency actually generates improvements in some of the predic- tions, e.g., in Table 5 and seen in the qualitative behavior in Figure 6a,b. This is a reiteration of the practical consequence of thermodynamic consistency, ignoring which can lead, for example, to incorrect extractions of activity coefficients from data.z4 An interpretation of the previously empirical rule for extending the DR isotherm to a universal isotherm is obtained from the consistency condition, eq 21.

It is also interesting to see that, indeed, large negative deviations from ideality are substantially reduced as in Figure 7a while using a simple yet thermodynamically consistent equation. From a comparison of figures such

1 .c

0.9

0.8

0.7

solid lines -02(1)-N2(2) on lox dashed lines .02(1).CO(2) on l o x 144 K. 760 mmHg

\

\

-.., , . , . 8 . 1 . I '

0.0 0.2 0 .4 0.6 0 .8

1.01

1 .oo

0.99

0.96

0.97

0.96

x i

oiid lines ... CZH6(1)-CZH4(2) on Activated Carbcn. 301.4 K, 150 kPa ashsd lines. HZS(l)-C02(2) on H-Mordenile. 303.15 K, 15.55 KPa

C2v4 I-----,, _----

-\, **' /'<.

.** / *

0- -. >2H6 -.

0 '\

/ /' '\\

/ 0 '.

H2S /' 8

/ 8

I , '. a-32 /

/ I

1 /

I I

3

0.2 0.4 0.6 0.8 1 .o X l

Figure 8. Apparent activity coefficients calculated from eq 35 using predictions from eq 26 for binary data on (a) 1OX molecular sieve of.Danner and Wenzel(1969) and (b) activated carbon of Reich et al. (1980) and H-mordenite of Talu and Zweibel(1986).

as Figures 7a and Ba, which reveal what may be termed a "residual" nonideality, contributions from heterogeneity and other sources such as adsorbate-adsorbate interac- tions may be distinguished for experimental data. Equa- tion 26 permits a strong dependence of the isosteric heat on loading and therefore it is not surprising that it accounts for heterogeneous interactions quite well.

For pure-gas isotherms which have steps, or for systems with inadequate data where eq 21 cannot be enforced, HIAS theory coupled with lateral interactionsz3 offers a rigorous and preferable approach, inspite of the concomi- tant numerical complexity. For type I systems, however, which are generally observed in practical applications of adsorption, such as PSmSA and where design calcula- tions must be facilitated, eq 26 appears to be a suitable equation for predicting and describing adsorption from gas mixtures.

Glossary

Ci

D

E

H

constant in multisite Langmuir model, eqs 22 and

dimensionless energy parameter in DR isotherm,

energy parameter in modified DR isotherm, eq 6,

dimensionless energy parameter in modified DR

23

eq 1

kJ/mol

isotherm

3234 Langmuir, Vol. 11, No. 8, 1995 Sundaram

i , ii

K

mol wt n no

nio

Nto ta l , &tal

nP

P Ptotal ,

PtOtd

p i

P m L , i

P i "(n) PSAT W R T R s 1 2

ti

T V

Vmi

WO

integer counter for series expansion terms, e.g.,

integer, constant in modified DR isotherms, eqs

molecular weight of adsorbate, eq 2, g/mol moles adsorbed, mol/kg moles adsorbed defining spreading pressure, eqs

moles of i giving same spreading pressure as

total moles adsorbed in a mixture, mol/kg

parameter in Dubinin-Astakhov isotherm equa-

pressure or total pressure, Wa, etc. total pressure in a mixture, kPa, etc.

pressure of component i in multicomponent

pressure of component i in binary mixture, eqs

pressure of component i in equilibrium with nio,

saturation pressure, kPa, etc. spreading pressure, eq 16, mol/kg gas constant, 8.314 JImo1.K selectivity for a binary mixture, defined in eq 28 constant in multisite Langmuir model, eqs 22 and

temperature, K constant in multisite Langmuir model, denoting

pore volume constant in DR isotherm pore volume in eq 2, cm3/g

in eqs 6, 26, etc.

6 and 26

7 and 8, moVkg

mixture, mol/kg

tion

mixture, Wa, etc.

14 and 15, Wa, etc.

eqs 12 and 35

23

XY

Xl

Y1

Greek Letters

P parameter in DR isotherm Yi

mixture, eq 35 Y1"

dilution, eq 35 e dimensionless loading, eq 2 JCSPR

e L ( r )

Subscripts

192 i j m SAT

Superscripts

dimensionless variables defined for eqs 19 and

mole fraction of component i in adsorbed phase

mole fraction of component i in gas phase of a

20

of a mixture

mixture

apparent activity coefficient of component i in

activity coefficient of component 1 at infinite

spreading pressure, eq 16, mol/kg density of adsorbed phase, eq 2, g/cm3

components in a binary mixture components i, j in a multicomponent mixture denotes maximum value for loading, eq 2 denotes saturation value for pressure

0 properties at the standard state ofequal spreading pressures

U upper limit of spreading pressure integral for eqs 19 and 20

Acknowledgment. The author wishes to acknowledge R. Sundaram without whom this work could not have been conceived or written. LA9502498