COMPARISONS OF D/R MODELS OF EQUILIBRIUM ADSORPTION OF BINARY MIXTURES OF ORGANIC VAPORS ON ACTIVATED CARBONS Gerry O. Wd Ls Alams Natinal Labratry, Mail St K-486, Ls Alams, NM 87545 E-mail: gerry@lanl.gv; Tel: (505) 667-984; Fax: (505) 665-19 Intrductin A variety f equatins and sets f equatins ( mdels ) have been rsed fr calculating ttal and cmnent equilibrium adsrtin caacities f mixtures f vars n activated carbns. The bases fr such multicmnent adsrtin mdels are always the adsrtin istherm equatins and arameters f individual cmnents. Many istherm equatins have been rsed fr describing adsrtin istherm data (Freundlich, Langmuir, Langmuir/Freundlich, Dubinin/Radushkevich, Planyi, Kisarv, Vacancy Slutin Mdel, and Jhns). The Dubinin/Radushkevich adsrtin istherm equatin [1] is the mst versatile, rven, and useful mdel fr redicting, as well as describing, equilibrium adsrtin caacities f rganic vars n rdinary cmmercial activated carbns []. Fr secialized carbns the mre general Dubinin/Astakhv equatin [3] with an additinal arameter can be used. These Dubinin equatins have the advantages f including: a) carbn rerty arameters, b) var rerty arameters, and c) temerature. Therefre, nly mixture mdels based n the Dubinin/Radushkevich (D/R) adsrtin istherm equatin were included in this study. The single var D/R equatin fr n mles adsrbed (e.g., ml/g) in equilibrium with its var ressure (r cncentratin in any cnsistent units) can be exressed as: W = R T { } n ex - ln ( sat/) (1) Vm ß E where W (cm 3 /g) is the micrre vlume f the adsrbent and E (kj/ml) is its reference adsrtin energy; V m (cm 3 /ml) is the liquid mlar vlume f the adsrbate, β is its affinity cefficient (relative t the reference), and sat is the var ressure f its unadsrbed bulk frm at temerature T. One f the best features f the D/R equatin is the inclusin f the affinity cefficient, which allws the alicatin f the arameters f micrre vlume and reference adsrtin energy measured with ne var t redict adsrtin caacities f ther vars. A thrugh review with cmilatins and crrelatins f affinity cefficients has been ublished [4], which makes single var istherms easily redictable. Benzene is usually chsen as the reference var (β = 1.0). In searching and reviewing the diverse and scattered literature we recgnized the need t summarize rsed multicmnent mdels and ublish them tgether in ne lace. Since they have been validated based n different and limited data sets (ften fr light gases, nt vars f cndensable chemicals), we als saw a need t cmare mdels with a cmmn set f multivar data. Mdels Tested Mixed D/R Istherm Equatins: The simlest extensins f the D/R equatin t mixtures f miscible cmnents invlve mle fractin (x i ) weighting f affinity cefficient (β i ), artial liquid mlar vlume (V mi ), and adsrtin tential (ε i = RT ln[ si / i ]) arameters f the cmnents i t calculate ttal mlar caacity, n T, fr the mixture. This was first rsed by Bering et al. [5]: W = e n T T ex - () VmT ß T E where β T = Σ x i β i V mt = Σ x i V mi ε T = RT Σ x i ln( si / i ) This equatin will be called the Bering1 mdel. Bering et al. [5] als rsed an alternative in which the relative artial ressures f the cmnents f the mixture are set equal t the rati f the sum f the ressures f the cmnents t the sum f the standard state ressures. Fr a binary mixture this is: + s1 s1 s = (3) 1 1 + The crresnding D/R equatin fr a multicmnent mixture is: W RT Σ = si n T ex - ln (4) V Σ mt ß T E i This equatin will be called the Bering mdel.
Anther arach has been taken in weighting adsrtin tentials. Xie et al. [6] used: ε Τ = Σ x i ln (γ i x i sati / i ) (5) where γ i are activity cefficients, which in their alicatins fr three binary mixtures (benzene-hexane, benzene-entane, and hexane-entane) n tw carbns were aarently taken as unity. They did nt use the D/R equatin, but a similar ne with the same weighted terms and an additinal emirical arameter fr micrre size distributin hmgeneity. Using this weighting arach with the Bering1 Equatin () gives: Σ W RT x = i ln (xi sati / i) n T ex - (6) V mt E ßT Taking the standard reference cmnent ressures t be x i sati (suerscrit indicates ure cmnent) is the same as assuming Rault s Law and an ideal adsrbed slutin (see later discussin). Therefre, Equatin (6) will be referred t as the Bering1-IAS mdel and tin. Mixture D/R istherm equatins such as Equatins (), (4), and (6) give nly the ttal mles f mixture adsrbed. The cmnent distributins (e.g., as mles n i r mle fractins x i ) must be knwn r determined indeendently by anther assumtin and equatin. Lewis Equatin: The equatin mst ften used t btain mles f mixture cmnents when ttal mlar caacity is knwn r calculated is: n Σ i = 1 (7) n i where n i are the reference adsrbed mlar caacities f the ure cmnents and n i are the mlar caacities f the cmnents in the adsrbed mixture. It is based n an emirical crrelatin btained fr adsrtin frm cnstant ttal ressure mixtures f hydrcarbn gases by Lewis et al. [7]. The Lewis Equatin fr a binary mixture in terms f mle fractins and ttal mlar caacity (ml/g) is: 1 nt x1 n1 x + n = (8) Xie et al. [6] rsed a vlumetric frm f the Lewis Equatin fr nn-ideal mixtures: Vmi ni = 1 Vmi ni Σ (9) Partial mlar vlumes V mi fr each mixture cmsitin must be btained frm indeendent mixing data. The Lewis Equatin can als be used t calculate ttal mlar caacity when the mle fractins are determined by anther methd (see belw). Since the riginal Lewis crrelatin was btained frm mixtures at cnstant ttal ressure, the reference adsrbed mlar caacities n i shuld be calculated at i = Σ j, the ttal ressure f the cmnents, rather than at i. Hwever, Lavanchy et al. [8] and Sundaram [9] bth successfully alied the Lewis Equatin assuming Rault s Law fr an ideal slutin fr which i = i /x i. Fr these cases the rer D/R standard reference ressure is si = sati, the saturatin var ressure f the ure cmnent i. Planyi Adsrtin Ptential Thery: One f the mst ular theries relating adsrtin f vars f single ure chemicals is the Planyi Thery [10]. Lewis et al. [7] and Grant and Manes [11] have develed it fr mixtures. The latter assumed: a) a liquid-like adsrbate mixture in which the adsrtin tential f each ure adsrbed cmnent is determined by the ttal adsrbate vlume f the mixture, b) Rault s Law as the relatinshi between the artial ressure f each cmnent and its adsrbate mle fractin, and c) the adsrbate vlumes are additive. Accrding t the Planyi Thery all characteristic curves (adsrtin caacities vs. adsrtin tentials) n a given adsrbent are suerimsable t frm a single curve by using crrelating divisrs fr the adsrtin tentials. This crrelating divisr can be a) mlar vlume calculated at the biling int crresnding t adsrtin ressure [7], b) nrmal biling int mlar vlume [11], r, mre generally, c) the affinity cefficient β f Dubinin [1,3]. This thery fr mixtures states that: (RT / β 1 ) ln (x 1 f 1s / f 1) = (RT / β ) ln (x f s / f ) = etc. (10) Fugacities f i (and f is fr saturated vars) used by Grant and Manes fr high-ressure gases can be relaced with artial (and saturated var) ressures i r cncentratins C i at nrmal atmsheric cnditins. Grant and Manes [11] used the additivity f mlar vlumes: V T = n T Σ x i V mi (11) and the sum f mle fractins x i in the adsrbate equal t unity t calculate the numbers f mles f each cmnent adsrbed. Hwever, this assumtin f additivity is nt a necessary art f the Planyi mixture thery. Mixture istherm equatins, the Lewis Equatin, Mlar Prrtinality, r any ther way f calculating r measuring ttal adsrbed mlar caacity can be cmbined with the mle fractins btained frm Equatin (10) t get mlar caacities f mixture cmnents. Ideal Adsrbed Slutin Thery (IAST): Myers and Prausnitz [1] are credited with the thermdynamically
cnsistent Ideal Adsrtin Slutin Thery, smetimes called the Myers-Prausnitz thery. They assumed Rault s Law and the cncet f equality f sreading ressures Π i fr each cmnent: i RT n Π i i = d i A (1) 0 i where n i is the number f mles f ure cmnent i in the adsrbed hase btained frm a ure cmnent istherm fr a var ressure i. The value f i is that crresnding t the sreading ressure. A is the secific area f the srbent. Grant and Manes [11] stated that their adsrtin thery fr mixtures and the IAST are ractically equivalent if the crrelating divisr is mlar vlume. A majr difficulty with the IAST mdel is the requirement that the adsrtin istherms (actually, the n i / i ratis as functins f i ) be accurately defined t zer ressure and caacity, s that they can be integrated. Sme have used Freundlich and ther istherm equatins with this rerty r have fit the lwer cverage rtin f exerimental r theretical istherms with emirical equatins that can be integrated analytically. Sundaram [9] truncated a lgarithmic exansin f the inverted D/R Equatin (1) t get the Henry s Law limit and aly the IAST. Alternately, Grant and Manes [11] inted ut that the integratin difficulties fr the IAST culd be vercme by using any Planyi-tye crrelatin. Subsequently, Lavanchy et al. [8] derived analytical slutins fr the integratins f the Dubinin/Radushkevich and Dubinin/Astakhv equatins t calculate sreading ressures. Their D/R-Ideal Adsrbed Slutin equatin fr sreading ressure is: Πi W ß i E RT x i sati = 1 erf ln Vmi R T ß i E i (13) where erf is the classical errr functin, which can be arximated [16] by a series: 3 5 7 9 x x x x (14) erf (x) = x - + - + -... 3 10 4 16 In alying this mdel, sreading ressures fr the cmnents are balanced by adjusting the mle fractins, which must add u t unity. A vlumetric frm f the IAST can be called the Vlumetric Adsrbed Slutin Thery (VAST). Since mlecules f mixture cmnents ccuy different vlumes, their evaratin rates and crresnding ressures shuld be rrtinal t vlume fractins, nt mle fractins (all intermlecular interactins being equal). Als, since activated carbn is a vlume filling srbent, we shuld have a filling ressure, rather than a sreading ressure. Equatin (13), then, can be used t equate filling ressures and calculate vlume fractins, which can be cnverted t mle fractins by knwing artial mlar vlumes in the crresnding mixture. As with the Planyi Adsrbed Ptential Thery, which als gives adsrbed mixture cmnent distributins, the IAST and VAST require a secnd equatin t determine ttal and cmnent adsrbed caacities. Exclusin Theries: Mdels based n exclusin assume that each adsrbate in a mixture reduces the srbent available fr adsrbing the ther(s). In Mlar Exclusin the adsrbates reduce the number f surface sites r area; in Vlume Exclusin they reduce adsrtin vlume. The adsrbates are still cnsidered indeendent and existing as if they are in the ure state; nly the area r vlume t be filled is less fr each because f the resence f the ther. Dng and Yang [14] rsed a Vlume Exclusin Mdel fr the D/R equatin, such that fr cmnent 1 f a binary mixture: ( ) R T V = s1 1 W - V ex ln (15) ß 1 E 1 where V 1 and V are the micrre vlumes ccuied by adsrbed mixture cmnents 1 and frm the ttal micrre vlume W. If the D/R istherms fr the ure cmnents shw significantly different micrre vlumes, W 1 and W, these can be used. Benefits f the Dng/Yang simle vlume exclusin mdel include: a) slvable by simle matrix slutins withut the need fr iteratin, even fr multile cmnents, and b) yielding bth distributins and quantities f adsrbed mixture cmnents withut a secnd equatin. Prrtinality Theries: The simlest mdel fr redicting adsrtin caacities f mixtures frm knwn distributins (r distributins frm ttal binary mixture caacity) is the Mlar Prrtinality Mdel (r Methd). It incrrates the assumtin that the amunts adsrbed frm a var mixture are rrtinal by adsrbate mle fractins t the amunts n i that wuld have been adsrbed frm a ure var at the same artial var ressure (r cncentratin). In ther wrds, the different cmnents d nt interact excet t deny adsrtin t ne anther. This assumes a limited number f mles (adsrtin sites r surface area) can be cvered (the Langmuir istherm assumtin). Fr a binary var mixture (tw vars excluding air cmnents) the ttal n T and individual amunts (e.g., ml/g) n i adsrbed accrding t Mlar Prrtinality is: n T = x 1 n 1 + x n (16).
This can easily be ext ended t any number f var cmnents. Similarly, Vlume Prrtinality states: V T = z 1 V 1 + z V (17) Any single-var istherm, including the D/R, can be used t calculate the ure var adsrtin caacities n i r vlumes V i. Standard State Otins: The Lewis, Prrtinality, and Exclusin Mdels are all interlatins between caacities f the ure cmnents. The ressures i and/r reference standard ressures si at which these ure cmnent caacities n i are t be calculated by Equatin (1) is u fr discussin. The first tin is the Single Var Istherm (SVI), where i = i, the artial var ressure f cmnent i in equilibrium with the mixture, and si = sati, the saturatin var ressure f ure i. The secnd is the Ideal Adsrbed Slutin (IAS) assumtin, where i = x i / i r si = x i sati (nt bth). Similarly, the third is the Ideal Vlumetric Slutin (IVS), where i = z i / i fr vlume fractins z i. Frm the Lewis crrelatin we als have the ssibility that the reference caacities n i shuld be calculated at the ttal ressure, s that i = Σ i. In this aer we will exlre these tins with the abvementined mdels. Cmarisns Database Selectin: Criteria fr selecting mixture equilibrium adsrtin data fr testing f redictive mdels required listings f: a) D/R arameters f the ure cmnents (r data frm which they culd be derived); b) var hase ressures f cmnents in adsrbed mixtures; c) adsrbed hase caacities and distributins; d) infrmatin n cnditins, such as temerature; and e) an activated carbnaceus srbent. Fur surces f data meeting these criteria were selected: Lavanchy et al. [8] ublished such data fr 0 binary mixtures f chlrbenzene and carbn tetrachlride n an activated carbn at 98 K. Steckli et al. [15] tabulated data fr 18 benzene and 1,-dichlrethane mixtures n the same carbn at 93 K. Xie et al. [6] ublished 4 data fr three binary mixtures (benzene-hexane, benzeneentane, and hexane-entane) n an activated carbn and a carbn mlecular sieve. He and Wrch [16] ublished gas hase ressures and adsrbed hase caacities fr 13 mixtures f benzene and isranl at 303 K. Althugh the latter did nt give the D/R arameters fr the ure cmnents, they did list calculated ure cmnent sreading ressures/rt fr var ressures 400 4800 Pa. Frm the benzene sreading ressures and Equatin (13) we calculated a best-fit reference adsrtin tential f E = 10.9 kj/ml and micrre vlume f W = 0.517 cm 3 /g. This gave a ttal f 93 binary mixtures t study. Table 1 lists the mixtures and D/R arameters used fr mdel cmarisns. Affinity cefficients were calculated frm mlecular arachrs.[4] Cmnent Mlar Vlumes: In alying the mdels discussed abve a questin is what t use fr cmnent mlar vlumes V mi in adsrbed mixtures. Data n vlume changes un mixing vs. mixture cmsitin are necessary fr exact values; hwever, they are seldm available. On the ther hand, mlar vlumes f ure liquids are readily calculated frm liquid densities d L and mlecular weights M w as V mi = M w /d L. Dng and Yang reviewed ther equatins fr estimating mlar vlumes abve nrmal biling ints [14]. Since ne gal f ur wrk is t redict adsrtin caacities f cmnents f a wide variety f liquid mixtures using a minimum amunt f inut data, which must be readily available, in this aer we chse t use ure liquid mlar vlumes (0 5 C), even fr cm nents f adsrbed mixtures. Table 1. Mixtures and D/R arameters used fr calculatins Binary Mixture Cmnents Carbn Designatin Dubinin/Radushkevich Istherm Parameters (Benzene Reference) Data Surce Micrre Adsrtin Affinity Ref Vlume (cm3/g) Ptential (kj/ml) Cefficients Chlrbenzene Carbn Activated Carbn U-0 0.448 17.00 1.17, 1.06 8 Tetrachlride Benzene 1,-Dichlrethane Activated Carbn U-0 0.448 17.00 1.00, 0.91 15 Benzene Hexane Carbn Ml Sieve J-1 0.469 18.70 1.00, 1.8 6 Benzene Pentane Carbn Ml Sieve J-1 0.469 18.70 1.00, 1.11 6 Hexane Pentane Carbn Ml Sieve J-1 0.469 18.70 1.8, 1.11 6 Benzene Hexane Activated Carbn GH-8 0.60 15.0 1.00, 1.8 6 Benzene Pentane Activated Carbn GH-8 0.60 15.0 1.00, 1.11 6 Hexane Pentane Activated Carbn GH-8 0.60 15.0 1.8, 1.11 6 Benzene Isranl Activated Carbn B-4 0.517 10.9 1.00, 0.8 16
Results and Discussin Ttal Caacity Calculatin Cmarisns: The first test f the mdels discussed abve with the tins and data discussed abve was hw well their calculated ttal adsrbed mixture caacities cmared with rerted ttal caacities. Calculatins were dne fr each f the 1 mdels and tins listed in Table and each f the 93 mixtures. Average and Standard Deviatins frm the exerimental values are listed in Table ; these reresent measures f accuracy and recisin, resectively. Table. Ttal Caacity f Mixtures Calculatins: Mdels Average Deviatins (Accuracy) and Standard Deviatins (Precisin) frm Exerimental Values fr 93 Binary Mixtures. Mdel and Otin Ttal Caacity (mml/g) Average Std Dev Vlume Prrtinality-SVI -0.318 0.47 Vlume Prrtinality-IAS 0.008 0.38 Vlume Prrtinality-IVS -0.003 0.38 Mlar Prrtinality-SVI -0.93 0.46 Mlar Prrtinality-IAS 0.04 0.38 Mlar Prrtinality-IVS 0.03 0.38 Lewis -SVI -0.36 0.50 Lewis -IAS -0.008 0.38 Lewis -IVS -0.008 0.38 Bering1-SVI -0.304 0.47 Bering1-IAS -0.003 0.38 Bering1-IVS -0.001 0.38 Bering-SVI -0.464 0.63 Bering-IAS 0.051 0.4 Bering-IVS 0.043 0.4 Mlar Exclusin-SVI 0.168 0.46 Mlar Exclusin-IAS 0.498 0.79 Mlar Exclusin-IVS 0.540 0.81 Vlume Exclusin-SVI 0.168 0.46 Vlume Exclusin-IAS 0.498 0.79 Vlume Exclusin-IVS 0.540 0.81 SVI = Single Var Istherm IAS = Ideal Adsrbed Slutin IVS = Ideal Vlumetric Slutin Table shws that the best (and equivalent) recisins f the mdel redictins were btained fr the Vlume Prrtinality, Mlar Prrtinality, Lewis, and Bering1 mdels with the Ideal Adsrbed Slutin and Ideal Vlumetric Slutin tins. Of these, the Bering1 mixture istherm Equatin () with the Ideal Vlumetric Slutin tin [ε T = RT Σ z i ln(z i sati / i )] had the best average accuracy. The Single Var Istherm tin [(ε T = RT Σ z i ln( sati / i )] rduced wrse recisins and accuracies in these fur mdels. The Vlume and Mlar Exclusin mdels redictins were significantly wrse than thse f the fur best, yielding results equivalent t ne anther due t the assumtin that mlar vlumes were the same in mixtures as in the ure states. Bering mdel redictins were significantly wrse that thse fr the riginal Bering1 mdel. Mle Fractin (Distributin) Mdels : The secnd test was f thse mdels that can calculate distributins f adsrbed mixture cmnents, in sme cases starting with calculated r exerimentally knwn ttal adsrbed mlar caacities. Table 3 lists these mdels with tins and resulting measures f accuracy and recisin in alying them t data. It als lists the numbers f the 93 binary mixtures fr which we were able t calculate mle fractins between 0 and 1. Only results fr ne cmnent f each binary mixture were used fr these measures, since the same results wuld be btained fr the ther f each air. T avid the effect f which cmnent was chsen frm each binary mixture, we averaged the abslute values f mdel residuals (calculated minus exerimental mle fractin fr ne f the cmnents). Table 3 shws that nly fur f these mdels were able t calculate mlar distributins fr all 93 mixtures: IAST, VAST, Planyi, and Vlume Exclusin-SVI. Of these, the best accuracy and recisin were fund fr the IAST and the wrst fr Vlume Exclusin-SVI; fr VAST and Planyi they were intermediate and very similar. The Lewis and Mlar Prrtinality mdels calculated mle fractins are very sensitive t the values f ttal and reference mles inut, which exlains their r erfrmances. One ther tin that was tried with the IAST and Planyi mdels was mle fractin weighting f the affinity cefficient: β i = Σ x j β j (j includes i). This rduced much wrse accuracy and recisin measures than using individual ure cmnent β i. Cmbined Equatin Mdels : The third test was t calculate bth exerimental distributins and exerimental ttal caacities by cmbining tw equatin mdels: a) IAST, VAST, and Planyi mdels were used t calculate the mle fractins f binary cmnents; b) then the equatins and tins listed in the first clumn f Table 4 were used t calculate ttal and cmnent adsrbed caacities. The latter were then cmared with rerted exerimental values. Table 4 shws measures f accuracy (average deviatins frm exerimental values) and recisin (standard deviatins).
Table 3. Mlar Distributin Calculatins: Mdels Average Abslute Deviatins (Accuracy) and Standard Deviatins (Precisin) f Calculated Cmnent Mle Fractins frm Exerimental Values fr One Selected Cmnent f Each Binary Mixture. Average Abslute Value f Residuals Standard Deviatin f Residuals Number f Mixtures that Culd be Calculated Lewis -SVI 0.659 0.67 31 Lewis -IAS 0.664 0.338 43 Lewis -Ttal Pressure 0.731 0.34 63 Mlar Prrtinality-SVI 0.66 0.80 31 Mlar Prrtinality-IAS 0.671 0.376 43 Mlar Prrtinality-Ttal Pressure 0.803 0.383 63 Ideal Adsrbed Slutin Thery 0.054 0.071 93 Vlumetric Adsrbed Slutin Thery 0.07 0.090 93 Planyi Adsrtin Ptential Thery 0.07 0.088 93 Mlar Exclusin-SVI 0.86 0.40 88 Mlar Exclusin-IAS 0.19 0.6 53 Mlar Exclusin-IVS 0.195 0.9 53 Vlume Exclusin-SVI 0.09 0.111 93 Vlume Exclusin-IAS 0.135 0.157 88 Vlume Exclusin-IVS 0.137 0.160 88 SVI = Single Var Istherm IAS = Ideal Adsrbed Slutin IVS = Ideal Vlumetric Slutin Table 4. Cmbined Mdels Average Deviatins (Accuracies) and Standard Deviatins (Precisins) f Calculated Cmnent Caacities frm Exerimental Values fr Bth Cmnents f 93 Binary Mixtures. Mdel and Otin IAST VAST Planyi Average Deviatin (mml/g) Standard Deviatin (mml/g) Average Deviatin (mml/g) Standard Deviatin (mml/g) Average Deviatin (mml/g) Standard Deviatin (mml/g) Vlume Prrtinality -IAS 0.0144 0.318 0.0408 0.497-0.0018 0.395 Vlume Prrtinality-IVS 0.0150 0.318 0.054 0.40-0.0176 0.449 Mlar Prrtinality-IAS 0.040 0.318 0.036 0.407 0.0108 0.391 Mlar Prrtinality-IVS 0.088 0.30 0.0351 0.404 0.0143 0.391 Lewis -IAS 0.0139 0.318 0.036 0.40-0.006 0.395 Lewis -IVS 0.0146 0.318 0.05 0.40-0.006 0.396 Bering1-IAS 0.0139 0.318 0.044 0.40-0.006 0.395 Bering1-IVS 0.0147 0.318 0.05 0.486-0.0018 0.395 Bering-IAS 0.03 0.34 0.0464 0.408 0.0056 0.401 Bering-IVS 0.086 0.35 0.068 0.455 0.0018 0.403 IAS = Ideal Adsrbed Slutin IVS = Ideal Vlumetric Slutin Table 4 shws that the tw mdels with the best (and same) cmbinatin f accuracy and recisin measures were the IAST-Lewis -IAS and IAST-Bering1-IAS cmbinatins. The Planyi mdel usually had better average accuracy, but wrse recisin. The Ideal Vlumetric Slutin assumtin gave n better (ften wrse) results than the Ideal (Mlar) Adsrbed Slutin assumtin. Likewise, VAST was n imrvement ver IAST.
6 Calculated Cmnent Caacities (mml/g) 5 4 3 1 y = 1.0018x R = 0.9435 0 0 1 3 4 5 6 Exerimental Cmnent Caacities (mml/g) Figure. 1. Cmarisn f calculated and exerimental caacities f 186 cmnents f binary adsrbed mixtures using the IAST-D/R-Bering1-IAS cmbinatin f equatins and tins. The linear least squares sle (frced zer intercet) and squared crrelatin cefficient quantify average accuracy and recisin, resectively. Figure 1 shws a cmarisn f cmnent caacities fr bth cmnents f the 93 binary mixtures calculated by the IAST-Bering1-IAS cmbinatin mdel with exerimental values. The largest sitive deviatins were btained with carbn tetrachlride and isranl. Deviatins include bth exerimental and mdel errrs. Cnclusins We cnclude frm this study that the best mdel fr calculating equilibrium mlar distributins f cmnents f adsrbed binary mixtures f rganic cmunds using knwn single-cmnent Dubinin/Radushkevich istherm arameters is the Ideal Adsrbed Slutin Thery. An analytical slutin (Equatin 13) fr the necessary istherm integratins (Equatin 1) avids the rblem f n Henry s Law limit. The errr functin (erf) in this slutin can be calculated by a series exansin. This IAST-D/R mdel can be extended t multile cmnents; hwever, an iterative slutin (easily dne by cmuter) is required. A secnd equatin is needed with the IAST-D/R t calculate ttal and cmnent adsrbed caacities. The best nes were fund t be the Lewis (Equatin 7) and the Bering1 (Equatin ) with the Ideal Adsrbed Slutin assumtin, si = x i sati, fr single-var istherm cntributins and fr mixed adsrtin tential cntributins, resectively. The IAST-D/R-Bering1-IAS cmbinatin seems t be easier t aly than the IAST-D/R-Lewis -IAS, requiring fewer D/R equatin calculatins. After determining mle fractins x i f adsrbed mixture cmnents by equating sreading ressures (Equatin 13) f cmnents and requiring that Σ x i = 1, ne can calculate cmnent i mlar caacity (e.g., ml/g) by the Bering1-IAS cmbinatin: xiw ni = Σxj Vmj R T ex - E Σxj ln (xjsatj /j) Σxjß j (18)
where the summatin index j includes i, j is a cmnent j ressure (r cncentratin) in the var hase in equilibrium with the adsrbed mixture, satj is a var ressure (r cncentratin) that wuld be in equilibrium with the bulk ure single cmnent j at the same temerature T, V mj is a ure cmnent liquid mlar vlume, β j is a ure cmnent affinity cefficient, E is the reference (benzene) adsrtin energy f the activated carbn, and W is its micrre vlume. The crresnding Lewis -IAS equatin is: n i [5] Bering BP, Serinskii VV, Surinva SI. Adsrtin f a gas mixture. Cmmunicatin 7. Jint adsrtin f a binary mixture f vars n activated charcal. English translatin frm Izv Akad Nauk SSSR Ser Khim 1965;5:769-776 [6] Xie Z, Gu D, Wu J, Yuan C. Adsrtin equilibrium f var mixture n activated carbn. In Fundamentals f adsrtin 6, Meunier F, ed. 1998;19-4 Elsevier, New Yrk [7] Lewis WK, Gilliland ER, Chertw B, Cadgan WP. x j = x i n T = x i (19) j W x R T j satj ex - ln V ß mj je j 1 where j again includes i. The results in Table 3 indicate that the Lewis and Mlar Prrtinality equatins shuld nt be used t calculate mlar distributins frm ttal mlar adsrbed caacities. They are, hwever, useful fr calculating ttal and cmnent mlar adsrbed caacities frm mlar distributins btained exerimentally r frm searate calculatins (Tables and 4). The IAST is successful fr mixtures f cmunds with similar adsrtin istherms, e.g., Tye I fr rganic cmunds n activated carbn. Mixed Langmuir r Langmuir-Freundlich mdels may als give gd results, but lack the redictive caability f the D/R equatin. When cmnent istherms are dissimilar, The IAST may be less successful. References [1] Dubinin MM, Zaverina ED, Radushkevich LV. Srtin and structure f active carbns. I. Adsrtin f rganic vars. Zh Fiz Khim 1947;1:1351-136 [] Steckli F. Dubinin s thery fr the vlume filling f micrres: An histrical arach. Adsr Sci Technl 1983:10:3-16 [3] Dubinin MM, Astakhv VA. Develment f the cncets f vlume filling f micrres in the adsrtin f gases and vars by micrrus adsrbents. Izv Akad Nauk SSSR Ser Khim 1971:5-11 [4] Wd GO. Affinity cefficients f the Planyi/Dubinin adsrtin istherm equatins: A review with cmilatins and crrelatins. Carbn 001;39:343-356 Adsrtin equilibria: Hydrcarbn gas mixtures. Ind Eng Chem 1950;4:1319-136 [8] Lavanchy A, Stckli M, Wirz C, Steckli F. Binary adsrtin f vars in active carbns described by the Dubinin equatin. Adsr Sci Technl 1996;13:537-545 [9] Sundaram N. Equatin fr adsrtin frm gas mixtures. Langmuir 1995;11:33-334 [10] Planyi M. Causes f frces f adsrtin. Z Elektrchem 190;6:370-374. Theries f adsrtin f gases. General survey and sme additinal remarks. Trans Farad Sc 193;8:316-333 [11] Grant RJ, Manes M. Adsrtin f binary hydrcarbn gas mixtures n activated carbn. I & EC Fundamentals 1966;5:490-498 [1] Myers AL, Prausnitz JM. Thermdynamics f mixedgas adsrtin, A I Ch E J 1965;11:11-17 [13] Lide DR, ed., CRC Handbk f Chemistry and Physics, 75 th Editin 1994;A-101, CRC Press, Inc., Bca Ratn, FL [14] Dng SJ, Yang RT. A simle tential-thery mdel fr redicting mixed-gas adsrtin. Ind Eng Chem Res 1988;7:630-635 [15] Steckli F, Wintgens D, Lavanchy A, Stckli M. Binary adsrtin f vaurs in active carbns described by the cmbined theries f Myers- Prausnitz and Dubinin (II). Adsr Sci Technl 1997;15:677-683 [16] He VH, Wrch E. Zur berechnung des nichtidealen gemischadsrtins-gleichgewichtes benzen-isranl-activkhle B 4 unter verwendung exerimentell bestimmter aktivitatskeffizienten. Z hys Chemie Leizig 198;6:1169-1177