Testing Intermolecular Potential Functions usingTransport Property DataPart 3.-Binary Diffusion Coefficient of Methane + PerfluoromethaneBY ANTHONY A. CLIFFORD AND ERIC DICK INS ON*^Department of Physical Chemistry, University of Leeds, Leeds LS2 9JTANDG. PETER MATTHEWS AND E. BRIAN SMITHPhysical Chemistry Laboratory, South Parks Road, Oxford OX1 342Received 26th April, 1976The binary diffusion coefficient DI2 for methane+ perfluoromethane has been measured by atwo-bulb method at 302.8, 323.2 and 342.9 K, and the application of a simple I' inversion " techniquegives three points on an effective spherically averaged intermolecular pair potential. Comparisonwith the spherical CHs-CF4 potential obtained from a full inversion of interaction viscosities indi-cates differences in the separation at a given energy of the order of 2 %.Of the three main transport coefficients of a binary mixture-viscosity, diffusionand thermal conductivity-only the mutual diffusion coefficient D1 is determinedprimarily by the cross term interactions.On this basis, therefore, D12 should be oneof the most useful non-equilibrium sources of information about intermolecularforces between unlike species. Unfortunately, however, it is difficult to measureaccurately and has been determined for most systems with a precision of only 1-2 %at the very best1In this work, DI2 for methane+perfluoromethane is reported at three tempera-tures using N,+H, as a standard system to calibrate the apparatus.2 CH4+CF4is the simplest example of a mixture of an aliphatic hydrocarbon with an aliphaticperfluorocarbon, a type of binary system that has been the subject of many thcrmo-dynamic investigations due to an interest in the cross interaction energy, which isanomalously weaker than that predicted from the normal combining rules.Sincethe viscosities qmix of CH,+CF4 mixtures have recently been measured over a widetemperature range,4* we are now in a position to compare the extent to which theDI2 and qmiX data can be represented by the same spherically averaged intermolecularpair potential.DETERMINATION OF Dl2The two-bulb apparatus is based upon the design of van Heijningen et al; itwill be described in detail elsewhere.'In brief, a composition difference Ax is set up at constant temperature betweentwo chambers connected by a capillary tube of known length and diameter, and arelaxation time is determined by monitoring the changing difference between thet Present address : Physical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ.291 291 8 CH4-CF4 POTENTIAL FUNCTIONthermal conductivities of the gases in the two chambers.The conductivity changesare followed in situ by two self heating thermistors, one in each chamber, which formpart of a sensitive bridge circuit.For a given pressure p and mean composition x, Oi2 is determined from a plotof ln(Ax(t)-Ax(t+$T)} against the time t , where T is the total time over whichmeasurements are taken. This procedure avoids the difficulty of needing to knowAx@) or ( h x ( t ) - A x ( ~ ~ ) } , each of which is difficult to measure because of long termelectrical drift and the necessity for establishing a quasi-stationary state.Correctionsare applied for the finite volume of the connecting tube and the difference betweenits effective and geometrical length^.^r3'(a) 0 . 8 2 2 0.802 4 6 8p-llatrn-'FIG. 1.-The pressure dependence of D12 for N2+Hz. D12p is plotted against l/p (tube length30.00mm, i.d. 2.06mm): (a) 302.8K; (b) 323.2K; (c) 342.9K.Much of the total error in the two-bulb experiment is associated with uncertaintiesin the geometry of the apparatus, in particular the diameter of the connecting tube.It is useful therefore to calibrate the apparatus with a standard system for which veryaccurate data have already been obtained. We choose N2 + H2 since Carson et aZB2have recently measured Di2 for this system near room temperature to a stated pre-cision of 0.2 %.Fig. 1 shows plots of our values of Di2p against l/p for the equi-molar N2 +H2 mixture at 302.8, 323.2 and 342.9 K (tube length 30.00 mm, i.d.2.06* mm). Within the experimental scatter, D I 2 p is independent of pressure at eachtcrnperature, thus indicating the absence of Knudsen effects.6TABLE 1 .-BINARY DIFFUSION COEFFICIENT DI2 OF CH4+ CF4D12(1 atm, x = 0.5)/cm2 s-1literature-this work aT/ K302.8 0.148 0.126"0.145120.1435323.2 0.167342.9 0.186Based on Ol2 for N2+Hz at 300 K estimated from ref. (2) ; the uncalibrated values are higherby 0.7 %. b Extrapolated to 302.8 K.C Calculated from mixture viscositiesA . A . CLIFFORD, E . DICKINSON, G . P . MATTHEWS AND E. B . SMITH 2919In fig. 2, uncorrected values of D12p for CH,+CF, (x = 0.5) at 302.8, 323.2 and342.9K are plotted against I/'. DI2p is independent of pressure in the range0.13-0.35 atm, but above -0.4 atm the data become spurious, probably dueto the presence of convection.1o The increased scatter at 342.9 K is caused by0.1851 Ip-I latm-'FIG. 2.-The pressure dependence of D12 for CH4+CF4. D12p is plotted against 1/p (tube length30.00 mm, i.d. 2.06 mm) : (a) 302.8 ; (b) 323.2 K ; (c) 342.9 K.a lower thermistor sensitivity and poorer temperature control. The values of D12(1 atm, x = 0.5) listed in table 1 are based upon the calibration system N2 +H2, forwhich Ol2 (1 atm, x = 0.5) = 0.803, cm2 s-l at 300 K.The result at 302.8 K iscompared with previous literature values. 99INVERSION OF VISCOSITIESViscosities of gaseous CH,+CF, mixtures have been reported as a function ofcomposition by Maitland and Smith (295-1022 K) and Gough et aL5 (150-320 K).The two sets of experimental ymix data agree well in the overlap region near 300 K,but the interaction viscosities y12 derived from them are inconsistent due to differentvalues being taken for the viscosity of pure CF4. Because of high slip-flow effects,Maitland's value is the more likely to be in error, so in order to remove artefacts inthe final potential which would arise from such a discontinuity in y12, the Maitlandvalue for pure CF, is reduced by 1.2 % to coincide with that of G o ~ g h .~ Values ofthe parameterATz =where Sz!'; ')* and 2)* are respectively the cross term collision integrals appro-priate to diffusion and viscosity, are taken from the Barker-Bobetic-Maitland-Smith(BBMS) potentia1,l and these are then used to recalculate interaction viscositiesfrom the adjusted qmix data. (The function chosen here is not crucial, since a 3 %change in AT2, the maximum range observed for all the potentials considered in thiswork, alters q12 by no more than 0.5 %. In fact, values of AT2 were generated fromthe final viscosity potential listed in table 2. The difference in q12 is 0.2 % at the lowesttemperature rising to 0.5 % at the highest, well within the experimental uncertainty of+2 %.2920 CH4-CF4 POTENTIAL FUNCTIONThe interaction viscosities are " inverted " as described previously l4 for purecomponent viscosities.In essence, the method seeks to relate each value of thecollision integral obtained from the experimental data to the coordinate of a pointon the potential energy function at an energy defined, initially, by an approximatepotential energy function. The derived function can be iteratively refined if reason-TABLE 2.-CH4-CF4 POTENTIAL OBTAINED FROM INVERSION OF INTERACTION VISCOSITIESr/nm0.35410.35810.36210.36610.37000.37400.37800.38200.38600.38990.39380.39570.39880.40040.40240.4047{UWkIlK31 30.02547.02052.01633.01280.0984.0736.2529.9359.2218.7114.247.3- 10.3-31.8- 52.5- 72.8rinm0.40740.41060.41430.41860.42350.42910.43520.43680.43840.44010.441 70.44330.44480.44630.4477{U(r)lkl/K- 92.4-111.5- 130.0- 147.7- 164.3- 179.4- 191.9- 194.4- 196.6- 198.3- 199.5- 200.0- 199.4- 197.6- 195.7r/nm {UW/kllK0.4489 -193.30.4508 - 190.50.4533 - 187.20.4561 -0.4591 -0.4623 -0.4657 -0.4695 -0.4736 -0.4780 -0.4827 -0.4879 -0.4933 -0.4992 -0.5054 -83.579.374.669.664.058.151.745.037.830.122.013.4rlnm0.51 190.51860.53850.55840.57830.59820.61810.63800.65790.67780.69770.7 1760.73750.99521.1942{UWlk 1 /K- 105.4- 98.0- 79.0- 63.9- 52.0- 42.5- 35.0- 29.0- 24.1- 20.2- 16.9- 14.3- 12.2- 2.0- 0.7r /nmFIG.3.-The CH4-CF4 intermolecular potential energy U(r) as a function of the separation r : -,two-iteration inversion of q12 ; 0, simple inversion of DI2.able assumptions are made about its behaviour beyond the range defined by experi-ment. Preliminarily, several long and short range extrapolations were tried, and thenumber of iterations was varied. The data were inverted assuming a range of valuesof the well depth E , and in each case the fit to the interaction second virial coefficientA. A . CLIFFORD, E . DICKINSON, G . P . MATTHEWS AND E . B. SMITH 2921of Douslin et aZ.15 was examined. Depending on the choice of extrapolation andnumber of iterations, the optimum value of ~ / k lay somewhere in the range 180-220 K :this represents the limit of precision that could be obtained from the macroscopicinformation alone.To obtain a more precise estimate of E it is necessary to make an assumption aboutthe behaviour of the intermolecular energy function at long range.There are notheoretical estimates of the r6 or r-8 terms in the multipole expansion for CH4-CF4,and so we have assumed that, since the CH4-CH4 interaction is closely representedby a (20 : 6) function,16 a moderately conformal CH4-CF4 potential will also be reason-ably represented by a suitably scaled (20: 6) function in the long range attractiveregion. The final potential is insensitive to the short range extrapolation, but a one-term Barker function l 3 is chosen as this was found to be the most satisfactory forThe final potential is listed in table 2 and illustrated in fig.3. It is obtained aftertwo iterations from a BBMS starting potential as further iterations produced noimprovement in the fit to the viscosities. The form is independent of the startingpotential within -1 % in r, the viscosities and virial coefficients being fitted withr.rn.s. deviations of 1.1 x lo-' kg m-l s-I and 1.9 cm3 mol-1 respectively. Thecharacteristic parameters are ~ / k = 200 K and CT = 0.3975 nm, and the ratio of thewell depth to that of CH4-CH4 (0.922) is in good agreement with the ratio of the Boyletemperatures (0.91 7).15CH4-CH4.INVERSION OF DIFFUSION COEFFICIENTSSince the D I 2 values cover a relatively narrow temperature range, a full inversionanalogous to that of y12 is not feasible.We employ instead the simplified inversionprocedure described by Clancy et al. ;17 this is effectively a one-iteration inversionfrom the BBMS starting p0tentia1.l~ While it would be preferable to estimate Eindependently of the vl2 analysis, we can do no better than assume Elk = 200 K.The result of this inversion is shown in fig. 3. In terms of the separation r, the pointsobtained from inverting D12 differ from the viscosity potential by 1.8-2.2 %, with theformer having the smaller intermolecular separation.The difference between the two potentials is explicable in terms of experimentaluncertainties in D1 and q (- & 2 % each) and ignoring errors inherent in the analy-sis itself.However, these latter errors may be significant here since the inversion ofdata for CH4+ CF4 mixtures differs in two important respects from most of thosepreviously considered : (i) Each molecule contains 12 internal degrees of freedom,whilst the kinetic theory upon which the inversion method is based is properly validonly for monatomic species. (ii) The actual anisotropic pair interaction may perhapsbe represented only rather crudely by a potential of spherical symmetry. Addition-ally, there is the fact that the quantity y12 is not defined uniquely by the mixtureviscosities ; the pure component values and some approximate knowledge of thepotential form (through AT2) are also required.This work represents the first direct attempt to describe the form of the intermole-cular potential between hydrocarbon and perfluorocarbon molecules.Despite itsobviously approximate nature, we believe that this CH4-CF4 potential may usefullyaugment previous information obtained from the study of liquid mixtures. Thus,on the basis of deviations from the Berthelot combining rule suggested by liquid statethermodynamic measurement^,^ we can estimate, assuming conformality, the welldepth of the spherically averaged CF4-CF4 interactionE(CF~-CF~) = (E(CH~-CF~)/O.~O~)~/E(CH~-CH~) = (223 & 10) K2922 CH4-CF4 POTENTIAL FUNCTIONWe note, finally, that the refinement of potential energy functions obtained in thismanner, and the extension of the methods to other systems, is limited in the main bydeficiencies in the quality and range of the available thermophysical data.T.R. Marrero and E. A. Mason, J. Phys. Chem. Rex Data, 1972, 1, 1.P. J. Carson, P. J. Dunlop and T. N. Bell, J. Chem. Phys., 1972, 56, 531.J. S. Rowlinson, Liquids and Liquid Mixtures (Butterworth, London, 2nd edn., 1969).G. C. Maitland and E. B. Smith, J.C.S. Faraday I, 1974, 70,1191.D. W. Gough, G. P. Matthews and E. B. Smith, J.C.S. Faraday I, 1976,72,645.R. J. J. van Heijningen, A. Feberwee, A. van Oosten and J. J. M. Beenakker, Physica, 1966,32, 1649. ’ A. A. Clifford, E. Dickinson and R. S. Mason, to be published.* E. P. Ney and F. C. Armistead, Phys. Rev., 1947,71, 14.Lord Rayleigh, Theory of Sound (Macmillan, London, 1878), vol. 11, p. 295.lo A. A. Clifford, E. Dickinson and P. Gray, J.C.S. Faraday I, 1976,72, 1997.C. R. Mueller and R. W. Cahill, J. Chem. Phys., 1964, 40, 651.l 2 A. T. Hu and R. Kobayashi, J. Chem. Eng. Data, 1970, 15, 328.l3 G. C. Maitland and E. B. Smith, Mol. Phys., 1971, 22, 861 and references therein.l4 D. W. Gough, G. C. Maitland and E. B. Smith, MoI. Phys., 1973,25,1433.D. R. Douslin, R. H. Harrison and R. T. Moore, J. Phys. Chem., 1967, 71, 3477.l6 G. P. Matthews and E. B. Smith, to be published.l7 P. Clancy, D. W. Gough, G. C. Maitland, G. P. Matthews and E. B. Smith, Mol. Phys., 1975,30, 1397.(PAPER 6/801