Net, excess and absolute adsorption in mixed gas adsorption

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1 dsorpton (27) 23: DOI.7/s Net, excess and absolute adsorpton n mxed gas adsorpton Stefano Brandan Enzo Mangano Mauro Lubert Receved: 25 ugust 26 / Revsed: 3 January 27 / ccepted: February 27 / ublshed onlne: 24 February 27 The uthor(s) 27. Ths artcle s publshed wth open access at Sprngerlnk.com bstract The formulaton of a thermodynamc framework for mxtures based on absolute, excess or net adsorpton s dscussed and the qualtatve dependence wth pressure and fugacty s used to hghlght a practcal ssue that arses when extendng the formulatons to mxtures and to the Ideal dsorbed Soluton Theory (IST). Two mportant conclusons are derved: the correct fundamental thermodynamc varable s the absolute adsorbed amount; there s only one possble defnton of the deal adsorbed soluton and whchever startng pont s used the same fnal IST equatons are obtaned, contrary to what has been reported n the lterature. Keywords dsorpton equlbra Net adsorpton bsolute adsorpton Ideal adsorbed soluton theory Lst of symbols c Gas phase concentraton of component (mol m 3 ) c Concentraton of component at nfnte pressure (mol m 3 ) f Fugacty of component n gas mxture (ka) f Fugacty of pure component at the reference state (ka) G Gbbs energy (J) Gbbs energy of the sold wthout the adsorbate (J) G Electronc supplementary materal The onlne verson of ths artcle (do:.7/s ) contans supplementary materal, whch s avalable to authorzed users. * Stefano Brandan s.brandan@ed.ac.uk Scottsh Carbon Capture and Storage, School of Engneerng, The Unversty Ednburgh, The Kng s Buldngs, Mayfeld Road, Ednburgh EH9 3FB, UK K Dmensonless Henry law constant ( ) M S Mass of sold (kg) n Moles adsorbed of component (mol) n abs bsolute adsorbed amount of component (mol) n ex Excess adsorbed amount of component (mol) n net Net adsorbed amount (mol) n S Moles of sold (mol) n Tot Total moles n the system (mol) ressure (ka) q dsorbed phase concentraton of component (mol m 3 ) q abs bsolute adsorbed concentraton of component (mol m 3 ) q net Net adsorbed concentraton of component (mol m 3 ) q t Total adsorbed phase concentraton (mol m 3 ) R Ideal gas constant (J mol K ) S Entropy (J K ) T Temperature (K) V N Volume not accessble (m 3 ) V S Volume of sold, ncludng mcropores (m 3 ) x Mole fracton n adsorbed phase ( ) y Mole fracton n gas phase ( ) z Compressblty factor ( ) Greek letters φ Fugacty coeffcent of component n gas mxture ( ) μ Gas phase chemcal potental of component (J mol ) μ Chemcal potental of adsorbate (J mol ) μ S Chemcal potental of sold (volume bass) (J m 3 ) μ S Chemcal potental of sold wthout adsorbate (J m 3 ) η C Reduced densty at close packng ( ) Vol.:( ) 3

2 57 dsorpton (27) 23: η Reduced densty of adsorbed phase at close packng C ( ) ρ S Sold densty ncludng mcropores (kg m 3 ) Ψ Reduced grand potental (mol m 3 ) Introducton In general t s not possble to measure gas adsorpton drectly and ths has led to the use of excess (see for example Srcar 999) and net adsorpton (Gumma and Talu 2), whch are vald optons when reportng expermental data but not to develop a thermodynamc framework (Myers and Monson 24) whch has to be based on absolute adsorpton. Talu (23) has recently developed the use of net adsorpton wthn a thermodynamc framework and has derved a verson of the Ideal dsorbed Soluton Theory (IST) wthout the need to convert ths to absolute adsorpton. We have recently dscussed the defntons of adsorpton for pure gases (Brandan et al. 26) n mcroporous solds and n ths contrbuton we extend ths to the case of mxed gases. Ths further analyss hghlghts some mportant qualtatve characterstcs whch can be used to show that the IST formulaton of Talu (23) s nconsstent. The second part of ths contrbuton wll therefore try to resolve ths nconsstency and show that there s only one possble defnton of an deal adsorbed phase. 2 Defntons of net, excess and absolute adsorpton for mxed gases We consder here the system to be that of a rgd mcroporous crystal as assumed by Myers and Monson (24) as adopted by Brandan et al. (26). We are nterested n adsorbents for separaton processes, where the mcropore volume s well defned and the adsorpton on the external surface s neglgble by comparson. We are defnng a fxed volume, V S, whch comprses the porous sold and the mcropore volume. We can defne the total number of moles n the system as n Tot n + n S () of the system wth a concentraton at equlbrum wth the adsorbed phase that would occupy the volume of the system. n net Tot nabs Tot V S c ( ) n V S c n net (3) In the case of a mxture t s a bt more complcated to defne the excess amount adsorbed, snce dfferent molecules may access dfferent portons of the mcropore volume. Nevertheless, f one can defne a reference non accessble volume for the excess adsorbed amount, then excess adsorpton can be defned. Ths s obtaned by subtractng the moles that would be n a flud at the same pressure and temperature of the system wth a concentraton at equlbrum wth the adsorbed phase that would occupy the accessble volume of the system. n ex Tot nabs Tot ( ) V S V N c [ n ( ) ] V S V N c n ex (4) In Eqs. 3 and 4 the total concentraton can be wrtten n terms of the compressblty factor, z, whch s equal to one for an deal gas. c z (5) We can now defne the adsorbed phase concentratons by dvdng the number of moles by the volume q abs nabs Tot n q Tot V S V S and the equvalent net and excess concentratons q net Tot nabs Tot V S c q q ex Tot nabs Tot ε (8) V m c q ε m c S s dscussed by Brandan et al. (26) the correct lmt at low pressure s gven by Henry s law and for mxed gases q abs Tot q net Tot K c ( K ) c c (6) (7) (9) () where the suffx ndcates an adsorbate and S s the sold. In absolute adsorpton we smply remove the sold and defne n abs n Tot Tot n S n (2) In net adsorpton we subtract from ths the moles that would be n a flud at the same pressure and temperature 3 q ex Tot ( ) K ε m c whle at nfnte pressure q abs η C V S V N η C V S e the saturaton capacty of the mcropores and c q abs Sat () (2)

3 dsorpton (27) 23: q net ( ) η C V S V N c < η C V S ( ) η q ex C ε η m c < C (3) (4) Therefore qualtatvely the absolute adsorbed amount wll ncrease monotoncally to the saturaton capacty. The net and excess adsorbed amounts wll ntally ncrease and then go through a maxmum and the correct lmt at nfnte pressure wll be negatve and fnte f an equaton of state s used whch has the correct lmt of nfnty for the compressblty factor (Brandan and Brandan 27) and hence a fnte densty. For the excess adsorbed amount, one alternatve approach to defnng the accessble and non-accessble volumes s the use of very hgh pressure adsorpton data as dscussed by Malbrunot et al. (997), who carred out experments up to 5 Ma. These authors suggested that the deal method accordng to the Gbbs surface defnton would be to measure the adsorbent densty for each gas wth the gas tself, but ths may not be practcal. Whle they recognse that ths approach s mpractcal (t would mply measurng sotherms up to 5 Ma or smlar pressures), t should be clear that ths method would lead to accessble volumes that are dependent on the guest molecule and therefore not vald for gas mxtures. Furthermore, even the accessble volume defned by ths method wll stll yeld a negatve excess sotherm as shown by Eq. (4), snce the negatve excess at nfnte pressure s obtaned regardless of the porosty, ε m, that one defnes from the accessble volume. t best the excess amount at nfnte pressure can only be exactly f one defned the accessble volume, not as the true accessble volume but as the larger volume that wll gve the same close packng densty at nfnte pressure, e η C, but ths would be a very arbtrary defnton and t would agan ncur severe complca- η C tons f mxtures of dfferently szed molecules are consdered. Ths smple analyss shows that whle there s always only one value of the absolute adsorbed amount correspondng to a pressure or fugacty, both net and excess adsorbed amounts may have two correspondng pressure or fugacty values. Fgure shows ths qualtatve behavour, wth excess adsorpton omtted as t wll be qualtatvely smlar to net adsorpton, but wll le somewhere between the two curves shown. For completeness the parameters used to generate Fgs., 2, 3 and 4 are gven n the Supplementary Informaton to ths paper. The fact that there s always one fugacty whch corresponds to a gven adsorbed amount already ponts to the 57 fact that the natural thermodynamc varable to choose s the absolute adsorbed amount, also n vew of the fact that numercal algorthms for the soluton of multcomponent adsorpton equlbrum are robust for the monotoncally ncreasng absolute adsorpton (Mangano et al. 24). 3 The deal adsorbed soluton theory (IST) equatons The basc equatons for the IST were derved orgnally by Myers and rausntz (965). The set of equatons to be solved n terms of fugactes, whch can be obtaned from the expressons n Santor et al. (25) assumng an deal adsorbed soluton, can be summarsed as follows: φ y f (Ψ)x, 2 Nc (5) These are the equlbrum relatonshps for each adsorbed component, where f s the fugacty at whch (a) mount adsorbed (b) mount adsorbed Concentra on bsolute Net Fugacty bsolute Fg. Comparson of absolute and net adsorpton versus a concentraton and b fugacty Net 3

4 572 dsorpton (27) 23: (a) 4 bsolute.7 Net adsorp on No solu on Concentra on More than one solu on Fugacty One solu on -3 Net (b) bsolute Fg. 4 Net adsorpton reduced grand potentals for two components wth a selectvty of each pure component s at the same reduced grand potental, Ψ, and temperature of the mxture. The reduced grand potental s defned by the Gbbs adsorpton sotherm ΨΨ f q (f )dlnf, 2 Nc Fugacty Fg. 2 Comparson of reduced grand potentals defned n terms of absolute and net adsorpton versus a concentraton and b fugacty bsolute adsorp on sngle feasble solu on always exsts Net Fugacty Fg. 3 bsolute adsorpton reduced grand potentals for two components wth a selectvty of 5 (6) To close the problem, the total number of adsorbed moles can be found assumng zero mass or volume change upon adsorpton q t Nc x ) (7) ( q f Myers and Monson (24) arrve at the commonly adopted formulaton where q q abs, whle Talu (23) arrves at the same equatons but q q net. Recently Furmanak et al. (25) ponted out the fact that Talu s approach leads to a reduced grand potental that has a maxmum above a certan pressure. Ths can be understood consderng the fact that once the net adsorbed amount becomes negatve the ntegral equaton, Eq. 6, wll reach a maxmum. The fact that there s not a one-toone mappng of fugacty and reduced grand potental led them to abandon further testng of the IST based on net adsorpton (Talu 23), snce t would not be clear whch root of Eq. 6 one should use. Fgure 2 shows the qualtatve shape of the reduced grand potental calculated usng absolute and net adsorbed amounts. Ths shows that the lmts at nfnte pressure are opposte, the absolute adsorpton value s +, whle the net adsorpton value s. gan qualtatvely a formulaton based on the excess adsorbed amounts n Eqs. 5, 6, 7 would yeld results smlar to those of net adsorpton. The ssues assocated wth the soluton of Eq. 6 are made clearer f we consder Fgs. 3 and 4 whch show the reduced grand potentals calculated usng a Langmur adsorpton sotherm for two components wth a selectvty of fve havng the same saturaton capacty for thermodynamc consstency (Ruthven 984). In the absolute adsorpton framework, Fg. 3, one can always fnd a soluton, whch s represented by a horzontal lne parallel to the fugacty axs,.e. the lne for whch the reduced 3

5 dsorpton (27) 23: grand potentals are equal, because for both components the reduced grand potentals wll go to nfnty at nfnte pressure. In the net adsorpton framework, Fg. 4, there are regons where one can fnd a soluton, whch s not unque as ponted out by Furmanak et al. (25). The net adsorpton framework wll yeld one feasble soluton n a mxture for postve reduced grand potentals up to the maxmum correspondng to the more weakly adsorbed component. bove ths pont, gven that at a fxed fugacty the mxture grand potental wll be an nterpolaton between the pure component grand potentals (Mangano et al. 24), there s also a regon between the two maxma where there s no soluton. For negatve values of the reduced grand potental there s only one soluton, but ths s physcally mpossble as the correspondng fugacty of the more weakly adsorbed component s smaller than that of the more strongly adsorbed speces. Clearly also the solutons at a fugacty hgher than the one correspondng to the maxmum for the more strongly adsorbed component wll be physcally mpossble snce the soluton wll gve a lower reference fugacty for the less strongly adsorbed components. The regon of no soluton s not a purely hypothetcal case. s a smple practcal example one could consder the breakthrough curve for oxygen on 5 zeolte usng helum as the carrer gas. Whle one can argue that n ths case a good approxmaton s obtaned assumng that helum s not adsorbed, as demonstrated by Brandan et al. (26) the error of makng ths assumpton s approxmately 3% and not neglgble because oxygen s not strongly adsorbed. In addton to ths, a general and rgorous thermodynamc model should be applcable to any bnary mxture. In ths case at low pressures the net adsorpton of oxygen s postve gven that Ruthven and Xu (993) report a dmensonless Henry law constant of 4.6 at 33 K, whle helum adsorpton s not zero, but the net adsorbed amount s negatve gven that at room temperature helum pycnometry s used to measure the skeletal densty of mcroporous materals,.e. the dmensonless Henry law constant s close to the porosty and clearly less than one. Ths s a system for whch at low pressures one would expect the IST to provde accurate predctons, but the formulaton based on the net adsorpton framework s not applcable because at all condtons the reduced grand potental of helum s negatve and there are no solutons n the regon bar as can be seen n Fg. 5. If one s stll not convnced by these smple arguments, there s further clear evdence that the IST formulated n the net adsorpton framework s not correct. From a smple nspecton of Eq. 7, one can see that the total amount adsorbed cannot be defned f the net amount adsorbed of one of the components n the mxture at the reference state s zero. The frst occurrence would correspond to the maxmum Net adsorp on of the reduced grand potental for the weakest component n the mxture. Whle these smple arguments are suffcent to understand that net adsorpton cannot be used n a thermodynamc framework to arrve at an alternatve formulaton of the IST, we stll need to understand what caused the dfference and demonstrate that even startng from the vewpont of net adsorpton one should arrve at the same equatons as Myers and Monson (24). 4 Dervaton of the Gbbs adsorpton sotherm and the IST The key thermodynamc quantty n adsorpton equlbra s the reduced grand potental and the correspondng Gbbs adsorpton sotherm (Talu 23; Myers and Monson 24). Myers and Monson (24) use the absolute adsorpton and arrve at the classcal result (see for example Ruthven 984) Ψ μ S μ S q dlnf (8) where μ s the chemcal potental of the sold on a volume S bass n the absence of adsorbate. Talu (23) constructs a thermodynamc framework based on net adsorpton and arrves at what s apparently a dfferent defnton Ψ net μ S μ S( ) (9) where μ S ( ) s the chemcal potental of the sold at zero pressure. Clearly μ S ( ) μ ( ) S q net dlnf Oxygen ressure, bar Helum Fg. 5 Net adsorpton reduced grand potentals for oxygen and helum on 5 zeolte at 296 K. arameters used to calculate the curves are gven n Table (2) 3

6 574 dsorpton (27) 23: Table arameters used to calculate curves n Fg. 5 arameter q q S b f +b f Source q S 4.3 (mol kg ) Mathas et al. (996). ssumed sngle ste Langmur as heat of adsorpton of O 2 s ndependent of loadng b Oxy.48 (bar ) Ft of data at 296 K from Talu et al. (996) b He.3 (bar ) ssumed dmensonless Henry law constant of.42 (approxmate porosty) and a value of 5 for O 2 from Ruthven and Xu (993). The deal selectvty s 36: for O 2 /He. For helum the fugacty coeffcent s always approxmately,.e. f ρ S 42 (kg m 3 ) Estmated from crystal densty as reported by Frst et al. (2) Both Talu (23) and Myers and Monson (24) start ther respectve defntons from the defnton of the nternal energy, but then Myers and Monson (24) opt to use the Helmholtz energy as the potental n the sold phase and the Gbbs energy for the flud phase, correctly ndcatng that what s beng neglected s not mportant at low pressures. Gven that the equlbrum between two phases s establshed when the chemcal potental s the same for each component n both phases, t should be natural to formulate the equlbrum problem usng the Gbbs energy for both phases. Talu (23) nvokes a seres of thermodynamc relatonshps and derves the Gbbs Duhem equaton and from ths the Gbbs adsorpton sotherm. n unambguous dervaton of the Gbbs adsorpton sotherm for both net adsorpton and absolute adsorpton frameworks s needed n order to resolve ths fundamental ssue and understand the orgn of the apparent dscrepancy. We start wth the classcal soluton thermodynamcs approach (Ruthven 984) and smply wrte the total Gbbs energy of the system and use for the sold varables on a volume bass. The equvalent relatonshps on a mass bass are obtaned usng M S when multplyng the chemcal potental of the sold and result smply n a change n the unts of the varables and are nterchangeable f the sold densty whch ncludes the mcropores, ρ S, s known (Brandan et al. 26). G n μ + V S μ S (2) where n q V S. If we are consderng that the adsorbed phase s at equlbrum wth a flud phase we can also wrte μ μ The dfferental of the Gbbs energy s gven by (22) dg SdT + Vd + μ dn + μ S dv S (23) From the total dfferental of the Gbbs energy we obtan the Gbbs Duhem equaton To defne absolute adsorpton we need to defne the state of the sold wthout adsorbate G V S μ S and dg S dt + V d + μ S dv S and the correspondng Gbbs Duhem equaton S dt + V d V S dμ S (27) If we assume constant temperature and subtract the two Gbbs Duhem relatonshps we obtan V S d ( μ S μ ) ( S V V ) d n dμ (28) We now ntroduce the addtonal assumpton that n the system the quantty of sold does not change and hence we are consderng a system of constant volume, e V V V S. If we now recall that for a flud at constant temperature dμ dlnf we fnally can arrve at μ S μ S whch s the result obtaned by Myers and Monson (24) wth a slghtly dfferent dervaton. We now proceed n the net adsorpton framework. For ths now we have to replace the sold wth an dentcal volume flled wth flud whch would be at equlbrum wth the adsorbed phase. In ths case we have G Flud n G μ n V S dlnf dg Flud S Flud dt + V S d + μ dn G and the correspondng Gbbs Duhem equaton S Flud dt + V S d n G dμ (25) (26) (29) (3) (3) (32) (33) SdT + Vd n dμ V S dμ S (24) t constant temperature, f we subtract the two Gbbs Duhem relatonshps we now obtan 3

7 dsorpton (27) 23: V S dμ S ( ) V V S d n dμ + and by ntegraton V S μ S μ S ( ) (35) Ths s the result obtaned by Talu (23), but t s not for the same thermodynamc functon. Ths ssue s easly resolved f we defne the reduced grand potental consstently as the dfference between the chemcal potental of the sold and the chemcal potental of the sold wthout adsorbate, but at the same temperature and pressure of the system, snce from Eq. 33 at constant temperature n G dμ V S d then V S μ S μ S ( n n G ) dlnf n G dμ ( n n G ) dlnf V S (34) (37) whch s the classcal result and shows that absolute adsorpton s the fundamental thermodynamc varable n adsorpton thermodynamcs. Snce equlbrum requres that the two phases should be at the same pressure f surface effects are neglected (rausntz et al. 999), e the classcal soluton thermodynamcs approach whch s at the bass of the IST, the grand potental should not be defned relatve to zero pressure. rgorous defnton of the reference state would appear to requre the use of a oyntng correcton (rausntz et al. 999) n Eq. 5 n order to take nto account the dfference between the pressure of the reference state for the pure adsorbate and the pressure of the mxture. Ths s n fact neglgble because for a sngle adsorbate the change n chemcal potental due to a change n pressure at constant temperature and constant amount adsorbed combnng Eqs. 24 and 27 s Δμ V S ( ) V Δ ΔμS S (38) n n Δ( μ S μ ) S If one assumes that the change n chemcal potental of the sold due to a change n the external pressure s not affected by the presence of the adsorbate, whch s reasonable for a rgd crystal, then the RHS of Eq. 38 s zero. Even f ths s not exactly true, one can see that the resultng oyntng correcton factor would be very small gven that t s calculated from the dfference of Δ and Δμ S and not Δ alone as for a normal lqud phase (rausntz et al. 999). d n net dlnf (36) n dlnf 575 Ths s the reason why one can use the Helmholtz energy for the adsorbed phase n the dervaton of the IST (Myers and Monson 24) and the resultng formulaton, Eqs. 4 and 5, s not lmted to low pressures f fugactes are used to correct for the gas phase non-dealty. The fnal step for the closure of the equatons of the IST s the dervaton of the total amount adsorbed n an deal adsorbed mxture. The dfferental of the grand potental at constant temperature for the mxture s gven by V S d ( μ S μ ) S n dμ (39) whch can also be wrtten for the pure component V S d ( μ S μs) n dμ V S n dμ d ( μ S μ S Snce the reference state s chosen as that of the pure component at the same temperature and grand potental of the mxture and gven that the adsorbed phase s deal dμ d ( ) μ S μ d ( ) μ S S μ S and substtutng nto Eq. 39 we have n n or (Myers and Monson 24) x n (44) Tot n Talu (23) usng the ncorrect reduced grand potental arrves at a smlar expresson n terms of net adsorpton n net Tot x net n net but ths fnal expresson s clearly ncorrect snce t s undefned at hgher pressures whch correspond to the net adsorpton of the pure components beng zero. 5 Conclusons ) dμ (4) (4) (42) (43) (45) In the formulaton of a thermodynamc framework for mxed gas adsorpton we have shown that only the absolute adsorbed amount has a one to one correspondence between fugacty and adsorbed amounts. Both net and excess adsorpton wll ntally ncrease lnearly and then go through a maxmum and fnally become negatve f pressure (or fugacty) s suffcently hgh. Ths leads to the 3

8 576 dsorpton (27) 23: mportant concluson that t s not possble to develop a rgorous verson of the IST based on ether the net or the excess adsorbed amounts. Clearly f net and excess adsorbed amounts are used to approxmate absolute adsorpton, then values obtaned usng excess adsorpton wll be closer to the true soluton, but the correct approach to follow s to defne the sold volumes as dscussed n Brandan et al. (26) and carry out the predctons drectly usng absolute adsorbed amounts. Havng establshed on qualtattve grounds that the defnton of the IST equatons obtaned by Talu (23) are nconsstent, we have proceeded to prove that the key ssue n Talu s dervaton s the fact that the equlbrum between the phases s not defned at the same pressure, whch s the startng pont of classcal flud phase equlbra formulatons. Ths small nconsstency leads to a set of equatons whch s ncorrect, does not have unque solutons and s undefned when the lghter components reach the condtons where net adsorpton s zero. When the correct reference state s used, the orgnal IST formulaton s recovered whether one starts from absolute or net adsorbed amounts, arrvng at the concluson that there s only one defnton of deal adsorbed soluton. The analyss presented s a further ndcaton that absolute adsorpton s the thermodynamc varable to use n descrbng adsorpton. Whle net and excess adsorpton can be used to report expermental results, t s stll necessary to determne the densty of the mcroporous sold, whch ncludes the mcropores (Brandan et al. 26), n order to be able to use the data n adsorpton process smulatons and consstent thermodynamc frameworks for mxed gas adsorpton. cknowledgements Fnancal support from the ESRC through grant E/J277X/ s gratefully acknowledged. Open ccess Ths artcle s dstrbuted under the terms of the Creatve Commons ttrbuton 4. Internatonal Lcense ( creatvecommons.org/lcenses/by/4./), whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made. References Brandan, S., Brandan, V.: On the propertes of equatons of state at nfnte pressure. IChE J. 53, (27) Brandan, S., Mangano, E., Sarksov, L.: Net, excess and absolute adsorpton and adsorpton of helum. dsorpton, 22, (26) Frst, E.L., Gounars, C.E., We, J., Floudas, C..: Computatonal characterzaton of zeolte porous networks: an automated approach. hys. Chem. Chem. hys. 3, (2) Furmanak, S., Koter, S., Tezyk,.., Gauden,.., Kowalczyk,., Rychlck, G.: New nsghts nto the deal adsorbed soluton theory. hys. Chem. Chem. hys. 7, (25) Gumma, S., Talu, O., Net dsorpton: thermodynamc framework for supercrtcal gas adsorpton and storage n porous solds. Langmur 26, (2) Malbrunot,., Vdal, D., Vermesse, J., Chahne, R., Bose, T.K.: dsorbent helum densty measurement and ts effect on adsorpton sotherms at hgh pressure. Langmur 3, (997) Mangano, E., Fredrch, D., Brandan, S.: Robust algorthms for the soluton of the deal adsorbed soluton theory equatons. IChE J. 6, (24) Mathas,.M., Kumar, R., Moyer, J.D., Schork, J.M., Srnvasan, S.R., uvl, S.R., Talu, O.: Correlaton of multcomponent gas adsorpton by the dual-ste Langmur model: pplcaton to ntrogen/oxygen adsorpton on 5-zeolte. Ind. Eng. Chem. Res. 35, (996) Myers,.L., Monson,.: hyscal adsorpton of gases: the case for absolute adsorpton as the bass for thermodynamc analyss. dsorpton, 2, (24) Myers,.L., rausntz, J.M.: Thermodynamcs of mxed-gas adsorpton. IChE J., 2 27 (965) rausntz, J.M., Lchtenthaler, R.N., de zevedo, E.G., Molecular Thermodynamcs of Flud-hase Equlbra. 3rd Ed. rentce Hall TR, Upper Saddle Rver, (999) Ruthven, D.M.: rncples of dsorpton and dsorpton rocesses. Wley, New York (984) Ruthven, D.M., Xu, Z.: Dffuson of oxygen and ntrogen n 5 zeolte crystals and commercal 5 pellets. Chem. Eng. Sc. 48, (993) Santor, G., Lubert, M., Brandan, S.: Common tangent plane n mxed-gas adsorpton. Flud hase Equlb. 392, (25) Srcar, S.: Gbbsan surface excess for gas adsorpton revsted. Ind. Eng. Chem. Res. 38, (999) Talu, O.: Net adsorpton of gas/vapor mxtures n mcroporous solds. J. hys. Chem. C. 7, (23) Talu, O., L, J., Kumar, R., Mathas,.M., Moyer, J.D., Schork, J.M.: Measurement and analyss of oxygen/ntrogen/5-zeolte adsorpton equlbra for ar separaton. Gas Sep. urf.,, (996) 3