Viscosity and Diffusion Coefficients of Polar Gas Mixtures Jae Ho Bae and Thomas M. Reed, 111 Department of Chemical Engineering, University of Florida, Gainesville,Fla. 32601
The canonically angle-averaged pair potential energy function with temperature-dependent parameters i s applied to polar-polar and polar-nonpolar gas mixtures. The viscosity of five binary mixtures and one ternary mixture of polar gases and the diffusion coefficients of four polar-polar mixtures are reproduced within experimental error b y this potential. The parameters for mixtures are approximated from those of pure components, determined from viscosity data, by the use of the combining rules. The calculation was extended to 22 polar-nonpolar gas mixtures for viscosity and diffusion coefficients. The canonically angleaveraged potential i s successful in correlating transport properties of gases and gas mixtures containing polar, nonassociated molecules.
T h e transport properties of nonpolar gases and gas mixtures caii be estimated from molecular pair potential energy fuiictioiis. However, the properties of polar gases and mistures are more difficult to predict, mainly because the interactions betweeii polar molecules are so comples t'hat a suitable potential fuiictioii is difficult to obtain. Furt'hermore, collision integrals containing orientation depeiideiice are difficult to compute. The most exteiisively used potential for polar molecules is the Stockmayer potential (Stockniayer, 1941), in which the angle-dependent dipole-dipole interaction is added to the Leniiard-Jones (1 2-6) potential. 31oiichick and 3IIasoii (1961) used this model for correlation of traiisport, properties of 1)ol:ir gase.3. In evaluating the collisioii integrals for this potential, they assumed t'hat the relative orientation of tivo molecules did not change during the biliary collisioii. Thus, the collision integral is calculated ai: a fuiiction of orieiitation, giving equal weight to each orieiitatioii. The resulting table of collisioii integrals has three ernl~iricallyadjurtable parameters. I n a different approach taken by I3ae and Reed (1967), it \vas assumed that the relative orieiitatioii of two molecules during collisioii changes so railidly that the collisioii can be ibcd by n caiioiiically aiigle-averaged potential. I n this model, the dipole-dipole and dipole-induced dipole iiiteraction ternis are added to the Leiiiiard-Jones (12-6) model. The angle-depeiident parts of the potential are canonically averaged over the coordiiiates for relative molecular orientat,ioiis. The resulting potential becomes teniperature-dependent.
Thih potential can be arranged into the familiar LeniiardJoiiei (12-6) form with temperature-dependent poteiitial paranieteri.
The usual tables of 12-6 collisioii integrals may be used 36
Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
(Hirschfelder et al., 1954) with reduced temperature T* = k T / e l j * and collision diameter u i j = 2-1'6rij*, to calculate properties, once values of e i j o , T i l o , dipole moments, and polarizabilities are assigned. The poteiit,ial proved to be as successful as t'he Stockmayer potential in correlat,ing the transport properties of polar gases ( h e and Reed, 1967; Mason and Moilchick, 1962). These potentials, however, have not' previously been tested 011 the traiisport properties of polar-polar gas mistures. Such data are available aiid the viscosity and diffusion coefficients of binary and ternary mistures are calculated from the parameters of like-pair interaction by the use of conibiiiiiig rules for unlike-pair parameters. The results are compared with those for the Stockmayer potential aiid experiment. The calculation is also extended to polar-nonpolar mistures. The parameters for unlike-pair interaction are also calculated directly from biliary diffusion coefficients and compared with those obtained by the combiiiiiig rules. Combining Rules for Unlike-Pair Interaction
I n the absence of parameters for unlike-pair interaction deterniiiied from experimental data, these parameters have to be approximated from those for like pairs by the use of t,he combining rules. It is apparent from Equation 3 that e i j * caiiiiot be a geometric ineaii of e l , * aiid ell*. Furthermore, ril* caiiiiot be an arithmetic mean of rii* and rj,*. 011the other hand, the dispersion force parameters are expected to follow the usual combining rules. eijo =
(€*iO€jj0)1'2
(5)
With these rules and the dipole moments and polarizabilities, til* and rij* can be calculated by Equations 3 aiid 4, respectively. When one member of the pair is nonpolar, t i l * and rtj* are no longer temperature-dependent, since the term in p i 2 p j 2 is theii zero. The potential of Equation 1, however, is not trivial e ~ e nin treating polar-lionpolar mixtures, because the dipole-dipole term is not negligible iii evaluating the likepair parameters e i l o and r i i o (required for e i l o and r t j o , Equations 5 and 6) from data on the single-component polar systems.
~ _ _ _ _ _ _
~
Table I. Potential Parameters and Molecular Constants Molecules
KH3 H20 SO, H c'1 CH3C1 CH2C12 CH30H
CzHbOH (CBd20 (CrHdzO (CH3)zCO
P/k,
O K
168 4 200 1 257 1 256 9 213 2 352 6 247 2 283 3 304 5 362 8 154 9
ro, A
3 3 4 3 5 5 4 5 5 6 6
842 439 702 910 015 464 528 276 077 180 376
H , Debyer
1 47 1 85 1 63 1 08 1 87 1 57 1 70 1 69 1 30 1 15
2 88
cy,
A3
2 1 3 2 4 6 3 5 5 8 6
26 49 72 63 56 48 23 62 16 73 33
~
~~~
Table II. Viscosities of Mixtures of Two Polar Molecules Vircositv X 1 O7 G Cm-' Sec-1 Mole Fract. 1
Exptl.
System (CHJ20 942 983 816 1031 706 1070 609 1106 508 1145 1179 409 308 1220 218 1254 1279 0 156
Av. potential
Stockmayer potential
+ SO,,Temp. 308.16OK 987 1033 1074 1109 1146 1181 1218 1250 1272
0 0 0 0 0 0 0 0
984 1029 1067 1102 1139 1175 1210 1246 1269
Temp. 353.16"K Polar-Polar Mixtures
The 1)oteiitial parameters for polar molecule^ are listed in Ta1;lc 1: together with tlil)ole moinriits and polarizabilities. Tlic paraiiieter.* w r e 1)revioudy (letermined from gas vis, (CH3)20aiid c o d y tlntn ( h e and Reed, 1967). H o w ~ e rfor (C,Hs)uO, they were deterriiiiied from the Titaiii (1930, 1933))followiiig the method Iirevioiisly reported (13ae and Reed, 1967). The two extreme teniperatures of the datn were i i w l in the evduatioii of the 1)otential parameters. This teriil)eixtiire clifferelice in each case wis at least 200°K. T h e v d w s of dipole monieiits and polarizabilities are taken from Laiidolt-norii.teiii (1951). Tlie parameters for 11011polar nioleciile.: were those used by I\Iasoii and 1Ionchick (1962) aiitl ni'c not reprocluced liere. Viscosity. hccortliiig t80 t h e rigorous kiiietic theory of 1iiulticoii~l)oiieiitgas mistures! t h e coefficients of viscosity of 11- c o 1111)01 i en t nii st II r~s a re give 11 h y I*:(I11a t i o lis 8.225, 26, aiid 27 of Hirschfelder et nl. (1954). T h e hypothpticnl viscosity of unlike-pair iiiteractioii appearing iii t,hc f o r i n d a s is obtained by
'fhe viscosities of p l a r gas niixtures are calculated using the combiiiing rules of Equations 5 and 6. In Table I1 the calculated viscositips of binary syxtems cIf3c1 SOr, (CI13),0 SO2, alitl (CII&O CH3C1 are compared with esperiniental data :\nd tlioac of the Stockniayer poteiit'ial (Cliakmhorti ant1 Gray, 1966). The nbmlute deliation of calculated values from esperimeiit for these systems is le,+ t h a n 1%. However, the 'teliih CIl3OH H g 0 a11d (12FIjOTI €I@ exhibit h g e r deviation than the other three. Tlie calculated result? are compared in Figures 1 and 2 with experimciital datii (Silgardo and Storrow) 1950) and with those olitailied froin the Stockmayer poteiitinl. The combining rules and the parameters of 11abon ant1 RIoiichick (1962) were used in the calculatioii of viscosities froiu the Stockmayer potential. Because of the differelice in ordinate scale hetween Figures 1 and 2, the deviatioii of calculated values from experiment is somewhat exaggerated in Figure 1, compared with Figure 2. In f:ict, t h e absolute average deviations are 1.1 and 2.4% in Figure 1, and 3.6 and 2.8y0 in Figure 2, for the Stockmayer and averaged potential models, respec tively . The visco.sities of ternary niistures of (CHJzO CH3C1 SO,are calculated and compared with experinient in Table 111. The values for the Stockniayer poteiitial are taken from Chakraborti and Gray (1966), The deviatioii of calculated
+
+
+
+
+
+
+
0 0 0 0 0 0 0 0 0
1114 1169 1204 1253 1305 1333 1377 1.110 1464
1120 1183 1222 1270 1319 1347 1380 1421 1469
Sy-ten, C"&1
+ SO1 Temp
961 810 721 611 496 430 352 252 134
0 0 0 0 0 0 0 0 0
955 833 714 631 508 396 310 232 153
1130 1156 1183 1206 1231 1256 1273 1292 1310
1134 1156 1178 1194 1218 1241 1259 1276 1293
1118 1168 1206 1254 1300 1334 1368 1421 1464
308 1 6 O K 1135 1160 1184 1201 1225 1248 1266 1282 1297
Temp 353.16"K
0 0 0 0 0 0 0 0
947 817 715 606 517 411 314 207
1286 1319 1343 1377 1400 1428 1456 1487
1291 1325 1351 1378 1400 1426 1449 1475
1289 1325 1351 1377 1400 1426 1450 1474
+
Sj\ten1 (CH3)20 CH3C1, T e m p 308.16"K 0 0 0 0 0 0 0 0 0
954 778 TO1 599 492 396 301 198 123
975 999 1009 1024 1041 1054 1070 1086 1099
973 1002 1014 1030 1047 1063 1078 1094 1106
972 1002 1009 1025 1042 1058 1074 1091 1105
Temp 353.16'K
0 0 0 0
937 809 719 600
1109 1129 1142 1166
1111 1137 1155 1178
1111 1134 1150 1172
0 0 0 0
526 412 331 239
1176 1197 1212 1232
1191 1212 1225 1241
1185 1205 1220 1236
Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
37
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