CORRELATION OF DIFFUSION COEFFICIENTS IN DILUTE SOLUTIONS


 Herbert Porter
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1 Applctons of holup t to expln the effect of ffusvty on the vporzton of lqus n pckngs n to estmte effectve nterfcl res for mss trnsfer hve been outlne. ACKNOWLEDGMENT The uthors wsh to cknowlege support of ths work (Prts I, 11, n 111) uner Grnt G200 of the Ntonl Scence Founton n Contrct No. AT(301)1463 of the Atomc Energy Commsson. NOTATION = effectve nterfcl re, sq. ft.i cu.ft. c = constnt n Equton (5) D = ffusvty of solute n gs, sq. ft./hr. D, = meter of sphere possessng the sme surfce re s pece of pckng, ft. G = superfcl gs rte, 1b.I (hr.) (sq.ft.) h, = opertng holup, cu.ft./cu.ft. l, = sttc holup, cu.ft.1cu.ft. ht = totl holup, cu.ft./cu.ft. k, = gsphse mss trnsfer coeffcent, lb. moles/ (hr.) (sq. ft.) (tm.) k, = gsphse mss trnsfer coeffcent, lb. moles/ (hr.) (cu. ft.) (trn.) L = superfcl lqu rte, 1b.l (hr.) (sq.ft.) m = constnt n Equton (5) MM = men moleculr weght of gs, 1b.llb. mole n = constnt n Equton (5) P,, = men prtl pressure of nert gs n the gs phse, tm. Greek Letters p = lqu vscosty, centposes = gs vscosty, 1b.l (hr.) (ft.) p = lqu ensty, g./ml. = gs ensty, 1b.lcu.ft. o = surfce tenson, yneslcm. o = vo frcton, cu. ft./cu.ft. Subscrpts w = for wter systems bs = for bsorpton work wthout chemcl recton vp = for vporzton work LITERATURE CITED 1. Jesser, B. W., n J. C. Elgn, Trns. Am. Inst. Chem. Engrs., 39, 277 (1943). 2. Lev, M., Tower Pckngs n Pcke Tower Desgn, 2n e., The Unte Sttes Stonewre Compny, Akron, Oho (1953). 3. Meht, J. J., n R. H. Prekh, S.M. thess, Mss. Inst. Technol. (1939). 4. Surosky, A. E., n B. F. Doge, Zn. Eng. Chem., 42, 1112 (1950). Prts I n II presente t the A. I. Ch. E. Sfrngfel mcetny, Prt III t the New Yorn?nPetng. CORRELATION OF DIFFUSION COEFFICIENTS IN DILUTE SOLUTIONS C. R. WILKE n PIN CHANG Unversty of Clforn, Berkeley, Clforn The ffuson coeffcent s normlly efne n ssume n ths stuy to be the proportonlty constnt n the rte equton wrtten for unrectonl mss trnsfer s follows : Equton 1) s strctly pplcble n el lute solutons n whch convectve trnsport ue to volume chnges on mxng s neglgble, n n whch other possble moes of mss trnsfer re not opertve. Ths pper represents n ttemp to generlze the relton of P to convenently vlble proper tes of lute solutons so s to permt estmton of ffuson coeffcents for engneerng purposes. PREVIOUS COlRRELATION In the erler pper by Wlke (10) metho of correltng ffuson coeffcents ws propose on the bss of qulttve conclusons of the Eyrng theory(3) n the StokesEnsten relton. It ws shown tht the group TIDq, esgnte s the ffuson fctor F, ws essentlly nepenent of temperture for vlble systems. Furthermore F coul be represente s smooth functon of moll volume for ffuson of vrous solutes n gven solvent. In generl t ws ssume tht ths functon extrpolte nto the StokesEnsten equton t very lrge solute moll volumes. DEVELOPMENT OF NEW CORRELATION Sources of Dt. At the tme of the prevous work so few t were vlble for ffuson of sngle solutes n vrety of solvents tht the effect of solvent propertes, coul not be brought nto generl correlton. In specl effort to obtn sutble t of ths kn compnon expermentl stuy (2) ws conucte nvolvng the ffuson of one n toluene n we vrety of hyrocrbon Pge 264 A.1.Ch.E. Journl June, 1955
2 u;v 8 6 I U xv 4 o m polvents rngng from hexne through tetrecne. Dt were lso obtne for ffuson of orgnc cs n severl solvents. These new t were supplemente by certn other t from the lterture, nclung ll the t reporte n the prevous pper (lo), to prove bss for the present evelopment. All t whch supplement those presente n Tbles 2 through 5 of referencel0) re presente n Tble 1. FIG. 1. DIFFUSION IN WATEP.. G D GO SOLUTE MOL AL VOLUME, CU. Cm /qm msl 6  Effect of Solute Moll Volume. Fg 1 ure 1 shows the ffuson s 4  functon of moll volume for vr 03 I ous solutes n wter bse on t x 0 BENZENE BROMOBE NZ E NE v CARBON TETRACHLORIDE] Q IODINE OTULUE NE  from Tble 2 of reference 10. Moll volumes use throughout ths work re vlues t the norml bo!?g pont estmte for complex molecules by the tomc contrbutons of LeBs (I, 6) s summrze n Tble 2. As ncte n Fgure 1, F s I I, smooth functon of moll volume hvng loglog slope of bout oo t low moll volumes n pprent V, c.c /gm mol ly mergng smoothly wth the I I c I. 2 V I gv vs 5: 1 \,SLOPE = ACETIC A BENZOIC 0 ClNNAMlC '0 FORMIC v v, c.c./gm. mol V, c.c./qm. mol FIG. 3. DIFFUSION OF ORGAPI'IC ACIDS I N TOLUENE. FIG. 4. DIFFUSION OF ORGANIC ACIDS IN CARBONTETRACHLORIDE. Vol. 1, No. 2 A.1.Ch.E. Journl Pge 263
3 TABLE NAL DIFFUSION DATA FOR VARIOUS SYSTEMS MOLECULAR WEIGHT OF SOLVENTS FIG. 5. EFFECT OF SOLVENT MOLECULAR WEIGHT. StokesEnsten equton whch requres slope of 113 t hgh moll volumes. On the ssumpton tht molecules re sphercl wth rus equl to (3V/4nN)1I3 the StokesEnsten equton my be wrtten s follows:  = X 10' 3 (2) DS Equton (2) s shown s otte lne on Fgure 1. The generl behvor of the curve of Fgure 1 reltve to the StokesEnsten equton consttutes resonble evence n fvor of the propose metho of correlton. Over the mle rnge of moll volumes the curve of Fgure 1 my be stsfctorly represente by lne of slope 0.6. Berng n mn the theoretcl lmttons of the ssumpton t s convenent to ssume tht the ffuson fctor s proportonl to P6 over the mle rnge. The proportonlty of Dy to V0.6 ws use by Thkr n Othmer (8) n ther representton of the correlton for ffuson of substnces n wter. To explore the mol volume effect n nonqueous systems severl solvents were stue s shown n Fgures 2 through 4. The group DqlT my be represente stsfctorly s proportonl to VO.8. It s therefore fr generlzton tht DylT s proportonl to Vo.6 n the meum moll volume (Supplementry to Tbles 2 to 5 of reference 10) DXlW DX106 Temper sq. cm./sec. sq. cm./sec. Solute Solvent ture, "C. (obs.) (clc.) Acetc c Acetone Crbon tetrchlore Amy1 lcohol Ethyl lcohol Anle Ethyl lcohol Bromobenzene Chloroform nhexne Benzoc c Acetone Crbon tetrchlore Benzotrchlore Bromobenzene Cyclohexne mcymene Ethyl benzene Ethl ether nhexne Mestylene Trnsecln mx ylene Bromform Acetone Ethyl lcohol Ref f Bromonphthlene Ethyl lcohol b Bromonphthlene ~ rnbromotoluene Crbon tetrbrome Crbn tetrchlore Cyclohexne Decln Dbenzyl ether nhexne Methyl nphthlene Tetrln Crbon tetrchlore C yclohexne Decln Doxne nheptne nhextne Isooctne Kerosene Tetrln g j g g 1.01 g 3.25 g 3.11 g 2.86 g 1.03 g.735 g 2.22 g Cnnmc c Acetone U Crbon tetrchlore U Ethyl benzote Acetone h Benzo trchlore h Ethyl cette h Ntrobenzene h Pge 266 A.1.Ch.E. Journl June, 1955
4 Ethylene brome Formc c nheptyl brome nhexyl brome Ione Ione Ioobenzene Methyl oe Ntrobenzene noctyl brome P yrne 1, 2, 4, 5Tetrchlorobenzene 1, 2, 4Trchlorotoluene Ethylene chlore Acetone Crbon tetrchlore Heptne Hexne Crbon tetrchlore C yclohexne Doxne Ethyl lcohol Heptne Hexne nhexne Methyl cyclohexne noctne ntetrecne Methylene chlore Acetone Ethyl benzote Ethyl cette Octne Ethyl lcohol ndecne ndoecne nheptne nhexne ntetrecne h h h b ppere to correlte the t most successfully. Fgure 5 shows the group DqlT s functon of moleculr weght for ffuson of gven solutes n number of solvents. Although there s conserble sctter of the ponts lne of slope 112 on the loglog plots correltes ech system moertely well. As n the cse of the mollvolume effect there re obvously other fctors nvolve so tht use of solvent moleculr weght s stsfctory only s frst pproxmton. The t of Trevoy n Drckmer(9) re for 0.50 mole frcton of phenol n vrous hyrocrbons so tht the solvent moleculr weght s use prmrly to show the tren. Generl Correlton for Unssocte Lqus. From the results of the preceng secton t ws conclue tht n equton of the frllowng form woul express the effects of solute n solvent: TM'/ D = const. (3) r] VoP6 Fgure t shows loglog plot of DIT vs. the group TVO.~/M~/~ for we vrety of unssocte solvents embrcng the t of Tble 2 of ths pper n Tble 5 of reference 10. The metho of plottng ws selecte to spre the t n best llustrte the scope of the correlton. The lne through the t hs slope 1 s requre n the ssumptons of the correlton n my be expresse by the Pn, n C. R. Wlke, "Some Mesurements of Dffuson n Lqus," J. Phys. Chem. (n press). bznterntonl Crtcl Tbles 5, (1929). CMuchn, G. E., n G. P. Fermnn, 2. physk. Chem., 121, 180 (1926). Herzog, R. O., et l., 2. physk. Chem., (A) 167, 329 n 343 (1933). C'e Mone, H., J. phys. rum, 7, (1936). foholm, L. W., Me. Nobelnst., 2, 23 (1913). ghmmof, B. R., n R. H. Stokes, personl communcton to J. H. Hlebrn (Sept. 21, 1954). ILDurnmer, E., 2. korg. u. llgem Chem., 109, 49 (1919). %tokes, R. H., P. J. Dunlop, n J. R. Hll., Trns. Fry Soc., 49, 886 (1953). jwtts, H., B. J. Aler, n J. H. Hlebrn, J. Phys. Chem., 23, 659 (1955). rnge for both queous n nonqueous solvents. However, t must be recognze tht specl structurl fetures of molecules n other moleculr nterctons my be mportnt n certn cses n tht therefore the propose reltonshp s t best n oversmplfcton utlze to obtn prctcl result. Effect of Solvent Propertes. Stuy of the effect of solvent propertes n ton to vscosty centere on the behvor of the group DqlT for ffuson of sngle solutes n vrety of solvents. A we vrety of vrbles such s moll volume, het of vporzton, moleculr weght, etc., were exmne. Of these the solvent moleculr weght Dt for 155 ponts mong 123 fferent solutesolvent systems re expresse by the correlton wth n verge evton of 12% between clculte n observe results. Correlton of Assocte Lqus. ASsocte lqus such s wter n other hyrogenbonng solvents mght be expecte to show evton from the correlton of Fgure 6. Fgure 7 shows the plot of DIT vs. qvo.6/m1/2 for ffuson n wter. The best lne through the t flls clerly bove the otte lne representng Fgure t. Ths evton s n the recton corresponng to ssocton of the solvent. By ssgnng moleculr weghtt to the solvent equl to 2.6 tmes the nomnl moleculr weght of wter one cn brng the t of Vol. 1, No. 2 A.1.Ch.E. Journl Pge 267
5 TABLE 2ATOhllC VOLUMES FOR COMPLEX MOLECULES, MOLECULAR VOLUMES FOR SIMPLE SUBSTANCES Atomc Volumes Bromne 27.0 Ntrogen, n seconry mnes 12.0 Crbon 14.8 Oxygen (except s note below) 7.4 Chlorne 24.6 Oxygen, n methyl esters 9.1 Hyrogen 3.7 Oxygen, n methyl ethers 9.9 Ione 37.0 Oxygen, n hgher ethers n esters 11.0 Ntrogen, oub!e bone Oxygen, n cs 12.0 Ntrogen, n prmry mnes 10.5 Sulfur For For For threemembere rng, s n ethylene oxe, euct fourmembere rng, s n cyclobutne, euct fvemembere rng, s n furn, thophene, euct For pyrne, euct 15 For benzene rng, euct 15 For nphthlene rng, euct 30 For nthrcme rng, euct 47.5 H, 0, N, Ar co coz so2 NO D,O * *Estmte vlue. Moleculr Volumes 14.3 N NH, 31.2 H,O 29.9 H& 3c1.7 cos 34.0 Clz 44.8 Br TABLE 3COMPARISON OF ASSOCIA TION PARAMETERS WITH ASSOClAl ION NUMBERS OF JACOBSEN Solvent 4ssocton prmeter, x Wter 2.6 Methyl lcohol 1.9 Ethyl lcohol o Ether 1.o Heptne 1.0 *At 20 C. Assocton number * hol, llustrte n Fgure 9, x s foun to be 1.5. It s of nterest to compre the vlues of x wth the ssocton numbers recommene by Jcobsen (4) from ntermoleculr freelength reltonshps s gven n Tble 3. Although Jcobsen s ssocton numbers re lrger thn the present ssocton prmeters the greement n orer of the solvents suggests tht the generl concept of the ssocton effect my be vl. The results further suggest tht the methos of Jcobsen mght be use to select n ssocton prmeter whch normlly woul le between the vlues of 2.6 for wter n 1.0 for unssocte solvents. By use of the gven ssocton prmeters the t or ffuson n wter re correlte by Equkon (5) wth n verge evton of bout 6%. Dt for methyl lcohol re precte wthn 11%. It shoul be note tht the expermentl t for methyl lcohol systems re known to be of rther low precson n the orgnl source. 7v0.6 M 12 FIG. 6. DIFFUSION IN UNASSOCIATED LIQUIDS. Fgure 7 squrely onto the curve of Fgure 6. Thus the correlton for ffuson n wter n lso n nonssocte solvents my be expresse by the generl equton The ssocton prmeter x s ntrouce to efne the effectve moleculr weght of the solvent wth respect to the ffuson process. For nonssocte solvents z = 1 n for wter x = 2.6. Dffuson n methyl lcohol s shown smlrly n Fgure 8, nctng n ssocton prmeter of 1.9, n for ffuson n ethyl lco DISCUSSION Generl Comment. The correlton represente by Equton (5) s stsfctory for estmton of ffuson coeffcents n lute solutons wth suffcent precson or most engneerng purposes,.e., bout 10% verge error. It must be emphsze tht the ffuson process s extremely complex n tht ny rgorous tretment must conser solutesolvent ntercton n more etle mnner thn the present relton coul possbly mply.? Although the present functonl reltonshp of ffuson coeffcent to solute moll volume rests upon some qulttve theo *For 285 ponts mong 251 solutesolvent systems of ths stuy.?dffuson of one n romtc hyrocrbons, for exmple, bs been exclue from the present correlton becuse of known comples forrnt on. Pge 268 A.1.Ch.E. Journl June, 1955
6 retcl founton, the relton tonshp or some mprove corre e here. Stuy of the correlton shp to solvent moleculr weght lton woul be hghly esrble, n evtons from constncy of s strctly emprcl. Some theo Only tenfol rnge of vscosty the group DTIT over more extenretcl bss for the ltter rel s covere by the solvents present sve temperture n vscosty rnges s especlly neee. Although evtons n constny of DTIT hve been observe n mght well be expecte for strongly nterctng solutesolvent systems such s one n romtc solvents n cetc c n ethylene glycol, use of the group seems justfe for precton of the effect of temperture on D n bsence of expermentl t I I V 0.6 M "2 FIG. 7. DIFFUSION IN WATER. Comprson wth Other Correltons. Olson n Wlton (5) hve propose generl form of correlton of ffuson coeffcents bse on surfcetenson lowerng of the solvent by hhe solute. In vew of the specl t requre no ttempt wll be me to compre ther metho quntttvely wth the present correlton. Schebel(7) hs ftte the correlton of Wlke to generl equton nvolvng the moll volumes of solute n solvent bse on the curves for wter, methyl lcohol, n benzene. In vew of the specl stncton evelope bove between wter n methyl lcohol s ssocte solvents n benzene s n unssocte solvent the bsc ssumptons use by Schebel re beleve to be n error. Thkr n Othmer (8) hve propose the followng generl equton evelope through the reference substnce metho : l 0  X ll I vo.6 M '4 FIG. 8. DIFFUSION IN METHYL ALCOHOL. For ffuson n wter only t ws ssume tht the group Dql.1 ftte the temperture behvor better thn the StokesEnsten group DqlT. However snce the rto [ql.ll / [q/ TI chnges only 10% for wter between 0" n 80 C. t s ffcult to justfy choce between the two wys of representng the temperture epenence on the bss of the reltvely lmte t presently vlble. Contrry to conclusons of the present stuy, for ffuson of sngle solute n vrous solvents t 20 C. Equton (6) oes not permt vrton of Dq,O wth solvent moleculr weght. Applcton of Equton (6) to thrtysx representtve systems nvolvng ffuson of vrous solutes mong twentyone unssocte solvents gve rther unstsfctory results, wth n verge evton of over 30 % between clculte n expermentl ffuson coeffcents. Vol. 1, No. 2 A.1.Ch.E. Journl Pge 269
7 CONCLUSION It s beleve tht Equton (5) represents n mprovement over prevous correltons of ffuson coeffcents n lute solutons. Due cuton of course shoul be observe n extenng the metho fr beyon the rnge of vrbles n types of systems nclue n the present evelopment. To fcltte use of the metho revse ffusonfotor chrt s gven n Fgure 10 to nclue the ssocton prmeter n moleculr weght of the solvent. ACKNOWLEDGMENT Assstnce of Reserch Corporton through grntn s grtefully cknowlege. Q) 0  X I VO.6 M /2 FIG. 9. DIFFUSION IN ETHYL ALCOHOL. N WATI ON C = concentrton, g. moles/cc. D = ffuson coeffcent, sq.cm./ see. F = ffuson fctor, TJDq, (OK.) (see.) (sqxm.) (centpose) L, = ltent het of vporzton of solvent L, = ltent het of vporzton of wter M = moleculr weght of solvent N = Avogro s number, molecules per mole NA = ffuson rte of component A, g. moles/ (see.) (sqxm.) V = moll volume of solute t norml bolng pont, cc.lg. mole t =temperture, C. T = temperture, OK. x = ssocton prmeter, multple of nomnl moleculr weght of solvent to gve effectve vlue Z= stnce n recton of ffuson q = vscosty of soluton, centpose qw = vscosty of wter, centpose qso = vscosty of solvent t 20 C., centpose 0) v) I Y u u s: Pge 270 SOLUTE MOLAL VOLUME, FIG. 10. GENERALIZED DIFFUSIONFACTOR CHART. CU. cm./gm. mol. A.1.Ch.E. Journl LITERATURE CITED 1. Arnol, J. H., In. Eng. Chem., 22, 1091 (1930). 2. Chng, Pn, n C. R. Wlke, J. Phys. Chem. (n press). 3. Eyrng, H., J. Chem. Phys., 4, (1936). 4. Jcobsen, Bertl, Assocton Numbers n Lqu Systems from Intermoleculr Free Length Reltonshps, Krolnsk Insttute, Stockholm (n press). 5. Olson, R. L., n J. S. Wlton, In. Eng. Chem., 43, 701 (1953). 6. Perry, J. H., Chemcl Engneers Hnbook, McGrw Hll Book Compny, Inc., New York (1950). 7. Schebel, E. G., In. Eng. Chem., 46, 2007 (1954). 8. Thkr, N. S., n D. F. Othmer, In. Eng. Chem., 45, 589 (1953). 9. Trevoy, D. J., n H. G. Drckmer, J. Chem. Phys., 17, 1117 (1949). 10. Wlke, C. R., Chem. Eng. Progr., 45, 219 (1949). Presente t A. I. Ch. E. brew York meetnp. June, 1955
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