Anil Saran

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POTENTIAL PARAMETERS FOR LIKE AND UNLIKE INTERACTIONS ON MORSE POTENTIAL MODELANIL SATLANL n d i.'ln" As i^wclation Foil Tiin OiTLTi\'ATiUiN 01' Soifnck , Ca lc utta -32.{Rcccivvd, Mruj 15, I9(i:i)A B ST R A C T , Viscosity clatii )\avo boon utibbcvl tn dolni'miiio tlio Ibrco constants for tlio ]\I ir.iO iioLoiihal nijiln] foi' Ar, N j, l ie ,  COi iini] O j. Those paramotoin can roprmluco Iho o\porimental viscosity clal.i m oio salisfar'iniAy ill,in llio piuamotors ilotonniuoj from Iho isoconil viijal ;mrl crystal prniioi'Ly cj liii A sot of f ninlmiatioii rules loi unlilio inioractions iiiis boon ])ro]io‘;o J  and lo s lu l in rolatiun to (ho m lci ddlnsion cocnicioni^s, Tho results thus obt.im oJ L.I10W a good iigrnoiiioni, willi tlio ox.jjoiimontal v alue, uvailiiblo in iho literatnroI  N T  R  O D U  (; 0  NOf ilm vanoiLS fomu of llu' ml.iT-mol(‘niliir potential, the liCmiarrl-Jonea (J2 : 0) model and Buckingbani Exp-,«i\ model arc qiiik realistic, and are, tluTefore, invariably used for < alcidat.ing tlio transpojd, and oquililiriiim propiM’ties of gases. EeronUy, Tvononaloiv, Ta^dor and Hii’selifiddcr (100la) Iumr' ealcnlaU'd the second virial coc [Iickmt fui tbo Morse potimtial and tlicse liavo been utilised by Konowalow and Hirsclifeldco’ (lOOJb) for dtdermining the potential paraiiK'ters for some non­polar molocnli's from the vii'ial eoefficinits and crystal data. Ovit a vide range of temperatm\), tlio tlu^oretical second virial (‘odlicienfs determined from tho Mouse potential agree very w'oll v'itb the expernnental data and are quite compar­able with tlioso obtained by using Lemiard-Jones (12 0) and BuekingliamExp-,six potential.Mur.se potential i.s written in the form,9i(r) =  c{cxp[-2(c/o-)(r-c„,)]-2cxp[-(c/o-)(»--r„,)]] ... (1)where (j>[r) is the potential energy of tlici two molcoule.s .separated liy a distance r, E is tho raaxiinum attraction energy at r — a  is the finite value of r at wdiich (j)[r) — 0, and tlm parameter c is related to the v idth of the potential well. Tho quantity c i.s given by tho relation56I =  (t (1  +  1 / c  In 2 ) (2)V^ ery recently, Lovell and Ilir.schfcldcr (1062) have evaluated the collision integrals for tho ga.se,s obeying tlic Mor.sc' potential and lienee tlie evaluation of tho parameters from transport data i,s also iiossiblc. Usually viscosity data arc used for this purpose. Very accurate data lor the viscosity of Ar, Ng, He, CO^  and Og4914D2 Anil Saranarc availiiblc for a ■wide range of temperature. We liavc licrc utilised those data to ohtain the potential parameler.s for these STd s^tances.A set of c!oml)jnation rules has h(?en propost^d to determine the unlike niole- eular inUn-acitions from a knowledge of the like interaetions. Tliesc ^^ T^ c so chosen as to fit in with tli(‘ expeiiimmtal data on the intci-diffusion ooeflicieiits of the systems He-Ar, He-N2, CO^ -No and C/O^ -Oj given by Walker and Westenberg (1958, 1959 and 1900) and by workers in this laboratory.C A J . C U  L  A T  I  O N S O F  ] ’ 0  T  E  N T  I A L  P  A R  A M E  T  E  Tw S F R O M  V I S C J O S I T YVarious methods can lie employed to determine the potential parametets from the experimental viscosity data. In the present paper, we have ii.sed the ratio method. The \'iscosity data for Ar, lie and O.^  in the. temperature range 100°K-]()0()'dv and for CO^  in the temperature range i'00'’K-lOOG'^ 'K were iitilised\ for this i^mrpose. 'On the Ohapman-Enskog theory, the coefficient of viscosity i] is given to the first apjiroxiiuation as (Hirst.hfelder H ah, 1954).|„l , , j y , _ 2 0 C  9 . - } x V T x i l 7  L '/ JiX iU  - (3)where T  is the absolute temperature, M  is the molecular weight of the .substance, Q(a,2)>i«p. /p+j collision integrals depending on c and as tabu­lated by Lovell and Hirsehfelder (196-).Let and ?/.j be the viscosity coefficients at temperatures T^ and To re.spec- tively. TheniVi j L J (4)By knowing experimentally the quantities and 9;.,, values of c and cjlc can be so adjii,steel tliat the right hand side of cqii. (4) becomes ccpial to the experimental [ijjt/i]. A parLicular value of [972/ViJ cl)o.scn and for various values of c, values of cjk were found satisfying the eqn. (4). Thus for one particular value of wo found corresponding values of c and ejk. The values of c were plotted against cjk for various values of [yjr/j] and from the intcrsecition of the eurvc.s c and cjk were found. The values of cr v^ as then calculated using eqn. (3) at two or three temperatures and moan taken Eqn, (2) nou^  detei’iilines the value of The values of the force eon.stant.s thus obtained arc shown in the Table T. Eor the .‘■akc ol comparison the values obtained by Konowalow and Hirsehfelder (lOGlb) from \drial and crystal data are also included in the table.It is .seen from the table, that tiie potential parameter.  ^ dctcrinined from viscosity and those from the virial and crystal data arc mutually not quite con-Potential Parameters for Like and Unlike, etc. 493sifttent. Tlxis tyfxe of discrejiiincy l)oi\\Toii tlic poU'niial ])iiranu’tprs dcLormined from tho non-iM]iiilibmiiii and thi‘ rqmhbrmm j)roj)i'rtK’S of fluids lias l)Oi'n ob­served 111 the case of other potential models as xvell.TABLE TValues ol the force constanls on Morse potential modelSubsiancoFrom vihcosiWe/L7i frAt ’Kun Viri.il iiml c'r5^ >^ Ull d-ita 0 f/r7v trA i , n .ArN ,JToCOjG.7 6 1 0 .0  5 1 f) 01201)2Uli)0OR3 101 3 007 2 . G 2 2  3 90S 3 G ti5 .05 53 SS24 1172 925 -4 .5 0 7  —3 057 —141.8  134 43 380 3 .5 7 93 855 -J 030C O M P  A U I  H O N \V I  T  IT E  X  1^  E  14 I  ]\I E  X  T  In order to test the rehabihly ol the foicc eon,slants thus deteimined, it is necessary to see how far they K'pi'odnco the experimental data over a nude range of temperature.TABLE IIExperimental and calculated viscosity coefficient,s for Ar^  N^ , lie, CO^ j and O , at different temperaturesSabalancoA rr) A 10' till-111 Hw. cijlculalud with lorcc' °KV X 107 g/i-m 800 oxptcoiiiUuits fromViscosity Virml and ciystnl property data100 839» 841 709120 993“ 1003 921140 1140“ 1157 1077160 1293“ 1303 1220180 1447“ 1457 1377200 1594“ 1602 1527220 1739“ 1743 1071240 1878“ 1878 1813260 2014“ 2000 1952280 2145“ 2125 2083494 A n i l  S a r a nT A B L E  H - ~ { c . r m t d )fSubalanco T  ‘ /vV y  107 g/fin . Koc oxpLV X 10'^o a lcu la to d  wiUi I'orco fo n s ta n ts  fro mVjauosiby V ir ia l aacl r r y s ta l  p ro p e r ty  d a l a;joo 2 2 7 0 « 2 2 S 4 220 9A rso o 4021'^ -1017 4 7 0 1100 0 5302^ S3 80 5 5 9 3100 0 ‘JS^ 70S 02 0120 8 2 0 '’ 8 3 2 737110 01 S'* D.j 5 SOSlo o lOOS*^ 1 072 0 7 2N .180 1183'* 1180 1 0 8 42 0 0 1295f* 1 2 9 0 1 1 9 42 2 0 1 403'* 1 3 ') l 1 3 0 121 0 ISOS'* 1493 1 4 1 02 0 0 lo o s '* IS S S 1 5 1 02 8 0 IGOC'5 1081 16003 0 0 1 7 8 0 '’ 1 709 J0 9 R80 0 3403^^ 3S04 3 5 0 31 000 4011'^ 4004 4 1 5 8100 047 '* 941 —120 lo o s '* 1002 —MO 118 2 « 1 1 7 8100 1290'* 1 2 8 8 —180 139S« 139 32 0 0 1490 '* 14 9 S —n o 2 2 0 ISOS'* 1 S 9 4 —2 4 0 1 0 9 2 “ 1 0 8 8 —2G0 ITSO** 1 7 8 3 —2 SO 1 8 8 8 “ 1 8 7 5 —3 0 0 1087 '* 1 9 0 3 —8 0 0 3840*^ 3861 -1 0 0 0 4455*^ 4S 2 1Poteniial Parameters for Like and Unlike, etc.TABLE il  (goM .)4951? X lOTg/t 1 1 1  socCNpl.’1 N 10" g/um  hoc. c nluuliilr)fl w 1 ( h fuioocoiihlm ita IromVlKCOHlfcy V i rial and  ci-ystnl p rop erly  d a ta1015 '’ 10152 20 1112 '’ 1111 —2-iO 1209 '’ 1212 —2 0 0 1303 '’ 1307280 1 4 0 0 '’ 1405 —30 0 1495'^ 1109 —800 3391^ 3383100 0 3 9 3 5 '' 39SG —100 TCSf 794 -120 9 1 0 —140 lOGl*^ 1080IGO 1 2 0 2 '’ 121 6ISO 1 3 4 K 134 8 —2 00 14 76*^ 1473 —2 200 :-1G04<^ 1.587 ~240 1728'' 17072G0 1 8 45 '’ 18172 8 0 1958 '’ 1913 —3 00 2 07  K 2031800 4115« 4 0 5 9 -1000 4720'^ 4 7 1 7 ~ab=  Jo lin B to n , II . L  and Clrilly, E .  11 — V asilosco , V (1 945), (1 9 4 2 )cd,7ol)iiston , H . 1j ., an d  Mc'Closkoy, =  TrBTitz, M  , an d  Zink, l l .,  (1930)K . E  (1 9 4 0 )e. --V a lu o s  oblainod Jii fio m  tlio inLorpolation of availab le  high  to m p e ra tu ro  visonssit.y d a ta .ViscosityExpciimcntal and calculated viseo.sitv codlieicnts in the wide range of temperature for Ar, N., He. CO^  and Oo are given in the Table II. The496 Anil Sarancalculuic'd valufs of bho viscosity coofficionts for Ar and using the force cons­tants fi‘(jiii vjjial and crystal data arc also giv^ en in tlie table for comparison. It can be setui fimn the taldo tliat th(i cxpei’inicntal viscosity data are much better reproduiiblc vith the force constants determined m this paper than Avith tliose from virial and crystal jnojierty data,rj O M n  1 N A  T  I  O N lb U L 10 S F  O R U N L I K E  M O L E C U L A R  I  N T  E  R A C T  1 O NThe i>ropiTties of gas mixtures are dependent on the forces betveen unlilfo iiiolecu](‘s. To jiiinciple, thi'se forces can be detei'mmed from the accurate ex- peninental data on the transport and equilihrium jiroperties of mixtures In ih(" abs(ui(;e of accurate expi'rmiental data it voidd lie v i'iy  heljifiil to be able to predict the interaction for unlik(‘ molecules lioni the kno\ni mteraclion foi likc\ nu)lec,nles. In  tins w ay the properties ol gas mixtures could lu' calculated from the data on the pure components, (bm bim ng rules lor the Mouse potential modi‘1 can b(i olitaincd liy Aviiting the eqn. (1) in the form^  ... (5)FolloAVing Zener ( 1 9 3 1 ) ,  we get the parameter b for unlike interactionu as. . .  ( 6 )which, Avhen exjiressed in terms of Ihc constants of the Morse jjoteiitial, gives1  r c,  ^ 3  ] “ L O-j (T.. J.. (7)Folloving the siiggesUon of Srwastava and Siivastava (195S), the combina­tion rule for the parameter c is taken as(8)The usual combinatioii rule foi- tr, udiich is exactly valid for the rigid splierc, is given by<rn =  il< i^+crt] (9)Tlic Analue of is fixed by ei^ s. (7) and (9). The fourth parameter /*,„ of tlio Morse potential for the unlike interaction is then(>■«); ! =  < i^a [ l +  2 ]L Ci2 J . . .  (10)However, it was found tliat tlie agreement with the calculated values of Iho inter- diffusion eoetiieient from the prescrihcd combination rules [Eqs. (7)-(l0)] and theexperimental values was not satisfactory. Consequently. v:& have tried the geometric mean rule for cTjj, i.o.Potential Parameters for Like and Unlihef etc, 497<^ 12 =  k lX O -g ]* (1 1 )The agreement between tlie calculated and the experimental values of the inter- diffusion coefficients improve by using tlu' geometric mean rule for cr^ g. Conse­quently, we suggest, cqs. (7), (8), (10) and (Jl) as tlio combination rules for unlike interactions on the Morse potejitial Tlic unlike interaction parameters for the gas pairs He-Ai-. He-T^ g COg-Ng and COg~Og determined by using the like inter­action parameters of Table I and the comliinatiun rules are given in Table III, The combination ruh's for the gas jiairs sbowm in Table IIT have been tested by comparing with the expi'rimcnt'il values of the interdiffusion coefficients. To the first a])proxiination, the inter-difiiisioii coefficient i« given as (Hirsch- felder et al , 1951).W  1 -  0.002028'■ '  p o - - ; ,(1 2 )where p  is the pressure in atmosphere, ilf^  and Jfg are molecular weights of species L and 2, T  is the absolute ieinperaiurt', tr^ g is the collision diameter,[Cia, Tjg*] is the collision integral and is the reduced temperature equalto AT/eja.TABLE IIIUnlike force constants on Morse potential model.Gas pairs Cl 2 rv ’^ J'K ffi -A (»-»ih''AH e— Ar 5 . D 3(>. 3 3 012 3 366H e— N j G 1 3 1 .8 3 .113 3.467CO.— N . 6 .6 134.3 3 .630 4 304C O j— Oi! 5 .5 138 6 3 .7 5 0 4 .2 2 3The experimental and the calculated values of over extensive range of temperatures arc slioAni in Table IV. It may be seen tliat except for the measure­ments of iSrivastava (1950) on the lle-Ar system, the agreement between the experimental and the calculated values of D^ ., is only lair. A large part of the dis­crepancy iiuiy he attributed to the uncm-tainties in the cxiienmental method of Walker and Westenberg (J95S, 1959 and 19G0). Consequently, till better data are available to provide a iiioKi definite answoi’, the combination rules for the Morse potential as suggested in this paper may be used for calculating unlike interactions. . . _b498 Anil SaranT A B LE IVExperimental and calculated values of inter-diffusion coefficient for the systems Ho-Ar, fle-Ng, COn-Ng and COo—Og at 1 atm.Systom scm-/sec exp t.D l l  ('m"/aoo calcnlatocluainp th eco m b m alio n  nile^ proposed273 0 . G40« 0 G402H8 0 701« 0 .7 0 2300 0 720*’ 0  752303 0 .7 0 0 a 0 765318 0 825« 0 .8 3 0400 1 247*' ] 224H e — A r f)00 1 R31*' 1 780600 2 557*' 2 432700 3 . 347*» 3 158800 4 202*' 3 .9 5 6000 6 .1 4 9 *' 4 .8 3 71000 0 .3 1 0 *' 5 7822 4 3 ,2 0 . 477c 0 .6 1 027 5  0 0 5960 0 .0 2 93 0 0 .0 0 .7 3 9 ^ 0 .7 2 8303 55 0 . 719c 0 743332 6 O .S I K 0 866400 1.218«* 1 .1 8 3H e — Ns 6 0 0 1 . 782<i 1 .7 2 4GOO 2 434^' 2 .3 4 4700 3 .1 4 4 '* 3 .0 3 8800 3.040<* 3 .8 4 9900 4 .7 5 2 '* 4 .6 5 01000 5 6.52<* 5 500300 0 .1 7 1 '* 0 .1 5 7400 0 .3 0 0 '* 0 .2 6 4500 0 .445** 0 .3 9 2600 O-GIO** 0 .5 3 9C O j— Ns 700 0 .793** 0 .7 0 4800 0.998** 0 ,8 3 79 0 0 1 .2 1 5 '* 1 .0 8 51000 1.408** 1 .2 9 9Potential Parameters for Like and Unlike  ^ etc,TABLE IV {covJiL)Systems499COj—OjT “KD ll cm-/sec oxptE jn  cm ’/scc culcuhitod using the combiutttiou rules proposed300 0 160^ 0 150400 0 270* 0 203500 0 400« 0 391600 0 505" 0 538700 0 740« 0 .7 0 5800 0 92 S'; 0 .887900 1.142*^ 1 0801000 1 380" 1 302a — Si'iv'aatava, K . 1’ (1950). b =Wallc(>r, R . R . urul We.sl-enberg, A. A., (1959). c —i ’mil, R-, and jSiivastava, 1 H., (1901). d =W alUor, R . R . nnd WoHtouberg, A. A., (195S). e =\Vidkor, R . E . im<] Wostenbcrg, A. A,, (1960).A C  K  F O W L E D  C{ M E N TThe author is gratoful to ]Vof. R. N. vSrivastava, D.So., F.N I., for suggesting the problem and for liis valuable guidamit" tliroiighout the progress of the work. Thanks are also duo to Mr. P. K. CUiakraborti, for sonic lidpfid discussions.R E F E R E N C E SHirechfolder, Curtiss and Bird, 1954, Molorulur Theory of Gases and Liquids, (John W iley and Sons, Iiu*., N, Y .)Johnston, H. L. and McCloHkoy, K . E  , 1940, J .  Phya. Clmm , 44, 1038.Johnston, H . L ., and Grilly, E . R ,  1942, J .  Phys. Chem., 46, 948,Konowalow, D D , Taylor, M. H ., and lIiiRchfolder, J .  O , 1961a, Phya. Fluids, 4, 0215, Koiiowalow, E .  E . ,  and Hirscbfoldor, J  O., 1961b, Phya, Fhticla, 4, 629.Lovell, S. E . and Hirschfeldoi-, J .  0 ,1 9 6 2 ,  Vniv. of Wisoonsin, W 1S-A F.21.Paul, B .,  and SWvu.stavn, I .  B ., 1961, Ind. Jour Phya., 85» 523.Srivastavu, B . N. and Snvastava, K  1’., 1958, J .  Chem. Phya., 28, 643.Srivastavu, K . P., 1959, Phyaicn, 25, 571.Trautz, M., and Zink, R ., 1930, A7in. Phyaik, 7, 427.Vasilosco, V ., 1945, Ann Physique, 20, 292,W alker. R . E . and Wesienborg, A. A., 1958. J .  Chem Phya., 29, 1139.Wttlkor, K , E . tmd Westenborg, A A., 1959, J .  Chem. Phya., 81. 619.W alker, R . E .,  and Westenborg, A. A , 1900, J .  Chem. Phya., 82, 436.Zoner, C., 1931, Phya. Rev., 37, 650,

Potential Parameters for Like and Unlike Interactions on Morse Potential Model

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Potential Parameters for Like and Unlike Interactions on Morse Potential Model

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