J . Chern. SOC., Faraday Trans. 1, 1982, 78, 2265-2274 Temperature and Density Dependence of the Self-diffusion Coefficient of n-Hexane from 223 to 333 K and up to 400 MPa BY KENNETH R. HARRIS? Diffusion Research Unit, Research School of Physical Sciences, Australian National University, Canberra, ACT 260 1, Australia Received 20th October, 198 I Self-diffusion coefficients for liquid n-hexane measured by the n.m.r. spin-echo technique are reported at 223, 248, 273, 298 and 333 K at pressures up to 400 MPa. Owing to a marked non-linear volume dependence, the results could not be directly fitted to the rough hard-sphere model of diffusion, but reduced diffusion coefficients were fitted as a function of a reduced volume, expressed as the ratio of the molar volume to a ' hard-core volume '.The temperature dependence of an equivalent hard-sphere diameter calculated from this hard-core volume was found to be adequately represented by an equation origmally proposed by Protopapas, Andersen and Parlee for liquid metals: CT = a,[l -B(T/T,):]. The constant B, 0.072, is of similar magnitude to that found to fit CH, and C,H, self-diffusion data, 0.069, and is consistent with diameters derived from the viscosity measurements of Brazier and Freeman. Between 273 and 333 K where the diffusion and viscosity results overlap, the group Dvt is constant, at a given temperature, with t = 0.97: consequently the Stokes-Einstein equation is not obeyed. As part of a continuing study of the effect of temperature and pressure on diffusion in dense liquids and gases, measurements of the self-diffusion coefficient of n-hexane, made between 223 and 333 K at pressures up to 400 MPa, are reported.and other laboratorie~~*~ has been concerned with 'simple' fluids of small rigid-core molecules, or water.5 The flexible nature of the n-alkane chain gives rise to an equilibrium mixture of conformers of different shapes which is liquid over a wide range of pressure. This equilibrium between the conformers is known to be pressure dependent,6 the percentage of gauche configurations increasing with pressure, whereas in the solid only trans configurations are found. It is therefore of interest to see whether these factors have any influence on the transport properties of chain molecules, and what deviations from the predictions of hard-sphere theories generally applied to the rigid-core molecules can be observed in this case.Most recent experimental work from this1* EXPERIMENTAL The n-hexane sample used was Merck (Darmstadt, Germany) Uvasol spectroscopic grade, distilled and dried over molecular sieves. A gas chromatographic analysis using a Supelco (Bellefonte, Penn., USA) column (5% SP-1200+ 1.75% Bentone 3) showed no evidence of unsaturated or non-homologous impurities. The self-diffusion coefficients were measured by the n.m.r. spinxcho method : the apparatus used and technique followed have been described in earlier papers.', The sample was confined in a glass and 316 stainless-steel bellows cell. In some of the experiments reported here, a phase-lock loop detector, linear to better than 0.2%, was used in place of the diode detector t Present address : Department of Physical and Inorganic Chemistry, University of Adelaide, Adelaide, South Australia 5001. 22652266 SELF-DIFFUSION OF LIQUID n-HEXANE of earlier work.? Calibration of the quadrupole coil used for generating the magnetic field gradient was carried out by measuring the positions of the first minima of the n.m.r.echo. Excellent agreement (1 %) was found between these results and results obtained earlier using tracer diffusion coefficients for water and benzene as standards.’ Temperatures were measured to f0.05 K with calibrated Pt resistors and pressures to f0.02 MPa using calibrated Heise Bourdon gauges (Dresser Industries, Newtown, Conn., USA.).$ RESULTS Density data from the literature*-l1 were fitted to an equation of state of the form 4 K = z aipi i=o ai = aoi + a I i / T where K is the linear secant modulus K = v,(P-Pa)/(v, - V ) (2) and Va is the molar volume at atmospheric pressure, pa.The temperature depend- ence of ai is based on that of the constant B in the Tait equation of state.12 This polynomial expression was found to give a better fit to the data than that equation which has only two parameters. The values of Vo were fitted to the equation 3 i - 0 Vo = C pi( T - 273.1 5)i (3) To reduce systematic errors in the amalgamation of several sets of data, values of K were calculated using the reported values of Yo for the different sets, but these were normalised relative to a datum at 298.15 K taken as a reference point (131 .612 cm3 m01-l)~~ before being fitted to eqn (3).The molar mass of n-hexane was taken as 86.178 g mol-l. The accuracy of the densities is ca. 0.2%. The coefficients aij and pi are given in table 1. This equation of state was used to obtain extrapolated densities above 200 MPa at 248 K and 150 MPa at 223 K. The experimental results are given in table 2 and, reduced by the factor 1 /d T, are shown plotted in fig. 1. The precision is estimated to be k 1.5% and, taking into account the calibration, the accuracy _+ 2.5%. The spin-echo measurements of Douglass et a/. obtained at atmospheric pressure between 182 and 333 K1* and at room temperature to 50 MPa15 are consistent with the data reported here. The average value of D at 0.1 MPa and 298.15 K, 4.18 x m2 s-l, lies between the two tracer values in the literature,l67 l 7 4.26, and 4.09, x lop9 m2 s-l.Three measurements were also made on n-heptane at atmospheric pressure using the sample described in ref.(16), theresultsbeing0.397,1.46and3.04 x lop9 m2 s-lat 195.5,250.0and299.7 K, respectively. These values are consistent with those of Douglass and McCalll* and the tracer value at 298.15 K, 3.016 x loy9 m2 sP1.l6 The parallel nature of the isotherms in fig. 1 suggests that the data could be fitted to an equation of the form D / d T = E l + % K/(l+E3/K) (4) It is a pleasure to thank Professor N. J. Trappeniers, Dr K. 0. Prins and Mr T. Jongeneelen of the Van der Waals Laboratorium, University of Amsterdam, for making available the design of this detector, and Mr P. Smith of the School Electronics Unit for its construction. $ The calibrations were made by the Division of Applied Physics, CSIRO, Sydney, N.S.W.It was found that the calibrations of Heise pressure gauges may differ by as much as 1 % from the factory calibration on delivery, and consequently these were checked periodically.K . R. HARRIS 2267 where with 298.15 K as the reference temperature, T,. Eqn (4) was more sxcessful than a four-term polynomial. The standard deviation of the fit, obtained by a non-linear least-squares method using weighting factors corresponding to the experimental errors for D, T and V, was 1.2%. That the fit was satisfactory is shown by the deviation plot, fig. 2. The coefficients E~ and 8i are given in table 3. V , = V - el( T - T,) - 0,( T - T,)2 ( 5 ) TABLE COEFFICIENTS FOR EQN (1) AND (2) 1 a0i Q l i Pi 0 -0.100 475 x 104 0.482 533 x lo6 0.127 292 x lo3 1 0.498 727 x 10' 0.605 314x 10' 0.162 834 3 0.403 108 x lop4 -0.866 104 x lop2 0.157 563 x 4 -0.241 280 x lo-' 0.493 288 x - 2 -0.208 482 x lo-' 0.428 990 x 10' 0.335 495 x 10-3 standard deviation 1 1 MPa 0.04 cm3 mol-' DISCUSSION COMPARISON WITH MOLE CU L A R-D Y N A M I CS CALCULATIONS FOR THE HA R D-S PHE R E F L U ID Some of the results reported here have been used in a discussion by Dymond and WoolP* of the diffusion of various solutes at infinite dilution in n-hexane solutions.The dependence of the ratio of the solute intra-diffusion coefficient to the solvent self-diffusion coefficient on the density and on the solute-solvent mass and size ratios was interpreted in terms of a rough hard-sphere model based on a comparison of experimental data with molecular-dynamics computer simulation results for tracer diffusion in the hard-sphere fluid It proved possible to interpret much of the self-diffusion data for rigid-core molecules using similar rough hard-sphere models.20 The density dependence of the self-diffusion coefficient of the hard-sphere fluid is reproduced by using a temperature dependent core diameter, 0, as an adjustable parameter.The diffusion coefficient, D,,,, calculated at a given state point using this hard-sphere diameter, is found to differ from the experimental value by a factor which is often independent of both temperature and density. This second disposable parameter is taken to represent the slowing of translational diffusion due to the inelastic collisions of real polyatomic molecules and it is known as the translational-rotational coupling constant, A .That it should be independent of density and very nearly independent of temperature for essentially spherical molecules has been argued by Chandler. 2o Thus the self-diffusion coefficient is given by D = ADsHs A d I where n is the number density and C is a 'correction' factor relating the self-diffusion coefficient, DSHS, obtained directly from molecular-dynamics calculations for the2268 SELF-DIFFUSION OF LIQUID II-HEXANE TABLE 2.-sELF-DIFFUSION COEFFICIENTS OF Il-HEXANE 248.15 273.15 298.15 223.15 0.1 48.7 51.4 100.7 150.4 200.5 252.9 292.3 0.1 0.1 25.8 55.7 100.8 154.4 197.5 250.3 292.9 305.0 308.0 350.6 391.2 0.1 25.0 25.2 99.8 152.3 198.7 248.6 249.5 302.8 344.0 393.8 0.1 0.1 0.1 0.1 27.6 50.0 101.4 105.3 153.0 200.1 249.5 290.8 295.8 299.1 353.5 8.347 8.649 8.664 8.89 1 9.075 9.229b 9.367b 9.460b 8.103 8.103 8.303 8.494 8.725 8.943 9.089 9.244b 9.356b 9.386b 9.395b 9.493b 9.584b 7.856 8.084 8.085 8.557 8.792 8.964 9.125 9.128 9.278 9.384 9.503 7.598 7.598 7.598 7.598 7.889 8.073 8.397 8.418 8.645 8.834 9.005 9.133 9.148 9.157 9.306 1.32" 0.88," 0.86," 0.6 1 0.433a 0.31," 0.1 7," 2.08 2.08 1.68 1.37 1.02 0.76, 0.59," 0.443" 0.358" 0. 342" 0.339" 0.263" 0.2 1 6" 2.96 2.47" 2.49 1.51" 1. 14" 0.9 1," 0.7 1 , 0.70ga 0.56," 0.47," 0.38," 4.15 4.21 4.21 4.16 3.32 2.85 2.08 2.05 1.66 1.32 1.07 0.89," 0.89," 0.86," 0.70," 0.22,"K. R.HARRIS 2269 TABLE 2.-continued 333.15 0.1 21.9 42.0 69.8 99.1 99.3 148.3 203.9 249.0 296.2 347.6 392.4 7.215 7.512 7.719 7.944 8.136 8.137 8.399 8.643 8.814 8.972 9.124 9.242 5.97 4.85 4.19 3.56 3.09 3.02 2.43 1.91 1.62 1.35 1.14" 0.962a a Experiments in which diode detection was used. The phase-lock loop detector was used Extrapolated densities lying outside region of data fitted by eqn (1) and for the remainder. 333K 3 110 120 130 V/cm3 mol-' 1c0 FIG. 1 .-Self-diffusion isotherms (D/Th) as a function of molar volume for n-hexane. hard-sphere fluid to that given by simple Enskog theory.21 This latter quantity is a function of the reduced density, n* = no3. The series of experimental isotherms is therefore fitted to what is really a family of hard-sphere potentials given by a(T).The density dependence of D,,, at reduced densities of the order found in liquids is such that DsHs is essentially linear in the molar volume, V , and DymondZ2 has given2270 6 4 2 h -k L o Q -2 -4 -6 SELF-DIFFUSION OF LIQUID n-HEXANE 100 110 120 130 1 40 150 I.',/cm3 mol-' FIG. 2.-Deviation plot for self-diffusion data fitted to eqn (4) and (5). 0, 223; @, 248; a, 273; ., 298; A, 333 K. TABLE 3.-cOEFFICIENTS FOR EQN (4) AND (5), EQN (9) AND (lo) AND EQN (9) AND (13) lo9 D:eqn (4) & (5) D*:eqn (9) & (10) (q*)-l:eqn (9) & (13) (q*)-':eqn (9) & (13) E, -0.287 010 x 10' ll -0.269 799 x 10' -0.390 101 -0.355 281 E , -0.477 509 x 10, c3 -0.469 259 x 10, -0.482 531 x 10, -0.478 343 x 10, c2 0.150 482 x 10-1 [, 0.143 694 x 10-l 0.202 714 x lo-, 0.186 267 x 8, -0.495 595 x 10-l 5, -0.447 498 x -0.511 774 x lop3 -0.546 286 x lop3 8, 0.224 741 x C2 0.200 519 x lop5 0.273 280 x 0.142 285 x standard deviation (%) 1.2 1.6 1.1 2.8 T range/K 223-333 223-333 298-37332 27 3-3 3 3" a simple equation expressing this.The diffusion isotherms of simple liquids, such as ethylene,23 often show the same behaviour. However, the non-linear dependence of D for n-hexane (see fig. 1) contrasts strongly with that predicted by the model. That a similar effect is observed for substances such as benzene, tetramethylsilaneZ4 and c h l o r ~ f o r m ~ ~ suggests that it is quite general at high densities. Because of this non-linearity, it is not possible to obtain a good estimate of the coupling constant, A , in this particular case, though this can be done in favourable circumstances if the isotherms are sufficiently linear at low density.24 It is convenient to represent the diffusion results in the form of the reduced diffusion coefficient introduced by Dymond26 where the superscript co denotes the dilute gas value of the product (nD) for the hard-sphere fluid, i.e.3/8 ( R T / ~ M ) ~ o - ~ , and y0 is the volume of random close packing, La3/2/2. After substitution of these values in eqn (7), D* can be expressed in terms of experimental quantities as D* = aD DV-t(M/RT)i (8)K. R. HARRIS 227 1 where an = 5.029 x los molf. If the hard-sphere model applied exactly, the product AD* would have the same functional dependence on n* as the molecular-dynamics data, wirh the appropriate choice of 0.Though this is not the case for n-hexane, the D* isotherms do seem to form a family of curves. Over the range of state points covered by these experiments, D* can be fitted as a function of molar volume by condensing the data onto a single 28 through the transformation Q cg -2 -1 - 6 - where T, is again a chosen reference isotherm (298.15 K). It was found that D* could then be satisfactorily fitted to an equation similar to eqn (4): O O e 5 0 O A - A 0.. 0 A a 0 A - I I I I I The coefficients Ci and ti are given in table 3 and the residuals are plotted in fig. 3. 4 The fact that the hexane results conform to eqn (lo), and that the D* isotherms depend only on the reduced density, i.e. on the ratio of the molar volume to a hard-core volume, albeit temperature dependent, assigned to the molecules themselves, could be interpreted as support for a free-volume model of diffusion.This is what one expects and finds for substances which can be modelled as hard spheres, rough or smooth. However, the quite empirical eqn (4) fits equally well and data of quite high precision would be needed to decide between eqn (4) and (10). The form of eqn (9) for the close-packed volume is quite arbitrary. However, it is possible to make an estimate of the dependence of 0, and hence V,, on temperature. In one of the earliest comparisons of the diffusion coefficients of the hard-sphere fluid with that of real liquids, Protopapas et al.29 attempted to relate the hard-sphere diameter 0 to the intermolecular potential function.With the assumption that the bottom of the well was parabolic, they derived a simple formula for 02272 SELF-DIFFUSION OF LIQUID n-HEXANE where a, is the separation at the well minimum and T, the melting temperature. B is a constant which can be related to the ratio of the vibrational amplitude of a molecule to the intermolecular separation at the melting temperature. For the large number of metals examined by Andersen and his coworkers, B was equal to 0.1 12. In earlier work,2s using comparisons between hard-sphere and experimental values for the quantity D*, estimates of a have been obtained for CH, and C2H,, for both of which A is ca. 1. These are shown plotted against Ti in fig. 4. A good correlation is obtained with eqn (1 1) in both cases, with B = 0.069.The line for CH, is slightly concave: this reflects the wide range of temperature for which data are available (up to 1.7 TJ, i.e. well beyond the range for which the approximation of a parabolic well shape might be expected to hold. Values for n-hexane are also shown (table 4). These were calculated from eqn (10) assuming 0 at 298.15 K to be 0.566 nm, a value obtained by making an approximate fit of the low-density data to the hard-sphere equations.'? 26 0.55 I X 15 20 10 15 FIG. 4.-Temperature dependence of the (equivalent) hard-sphere diameters of CH, (go = 0.430, nm), (0); C,H, (go = 0.447, nm), (A); n-C,H,, (oo = 0.624, nm) (El, D ; x , ~ 7 ; ~ ~ W, v ' O ) .K. R. HARRIS 2273 TABLE 4.-EQUIVALENT HARD-SPHERE DIAMETERS FOR n-HEXANE [a (298.15 K) = 0.566 nm] 223.1 5 0.5747 (298.15) 248.15 0.5713 323.15 273.15 0.5683 348.38 (298.15) (0.5660) 373.36 333.15 0.5635 273.15 (298.15) 303.15 333.15 (0.5660) 0.5639" 0.5625a 0.56 17" 0.56Bb (0.5660) 0.5655b 0.5628b Calculated from falling-slug-viscometer re~ults.~2 Calculated from rolling-ball-visco- meter results.lo Again a straight line is obtained, with the parameter B taking the value of 0.072.It appears therefore that eqn (1 1) is suitable for correlating hard-sphere diameters for liquids conforming to the model and equivalent hard-sphere diameters for those for which D* is a function of reduced density alone, and that this can be done over a fairly wide range of temperature. COMPARISON WITH THE VISCOSITY It is of interest to compare the density dependence of the diffusion coefficient with that of the viscosity.At a given temperature the product (Dq) for the intra-diffusion coefficient of a particular solute in a range of solvent of differing viscosity almost always increases as the viscosity, q, increases3O9 31 and the following relationship, recurrent in the literature, has been proposed Dqt =f(q t < 1. (12) This equation, which is more general than the Stokes-Einstein equation, may also be applied to self-diffusion. Measurements of the viscosity of n-hexane have been reported for temperatures between 298 and 373 K by Dymond et aZ.32 and between 273 and 333 K by Brazier and Freernan.'O The former were obtained with a falling-slug viscometer, the latter with a rolling-ball viscometer. The two sets of data are, however, not concordant, differing by as much as 10% at 333 K and 400 MPa, whereas the experimental error is of the order of 2%.Both sets are in reasonable agreement with the capillary- viscometer results of Kuss and P01lmann~~ at 313 K and 0.1 to 150 MPa, and as it is not clear which set is to be preferred both are used in this analysis. The viscosities have been fitted to an equation analogous to eqn (10) where V,' is again given by eqn (9). The reduced viscosity q* is given byz6 where q7"0 is the dilute gas value for the hard-sphere fluid and eqn (14) may be reduced, in a similar way to that for D*, to q* = a,, ~ v % ( R M T ) - ~ (1 5 )2274 SELF-DIFFUSION OF LIQUID n-HEXANE with a, = 6.0348 mol-4. The coefficients of eqn (13) are given for the two different sets of viscosities in table 3.The fit of the falling-slug results is better than that of the rolling-ball data. Approximate diameters calculated as described above are also plotted in fig. 4. Those obtained from the data of Brazier and Freemanlo are in very good agreement with the diameters calculated from the diffusion coefficients (cf. CH,28), but those calculated from the viscosities of Dymond et af.32 are not, giving the much lower value of 0.046 for B. The closeness of the value of B obtained from the combined self-diffusion and former set of viscosity data for n-hexane to that found for methane and ethylene suggests that any change in conformation with pressure has little effect on the apparent hard-sphere diameter. Both sets of viscosities were also fitted to eqn (1 2), the falling-slug and rolling-ball results yielding values of 0.93 0.01 and 0.97 f 0.01, respectively, for the exponent t.Again the former gave the better fit. With either set, it is apparent that the Stokes-Einstein relationship is not obeyed. K. R. Harris, Physica, 1978, 94A, 448. L. A. Woolf, J. Chem. SOC., Faraday Trans. I , 1982, 78, 583. J. Jonas, D. Hasha and S. G. Huang, J . Phys. 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Solution Chem., 1979, 8,461 ; D. F. Evans, T. Tominaga 31 H. J. V. Tyrrell, Sci. Prog., 1981, 67, 271. 32 J. H. Dymond, K. J. Young and J. D. Isdale, Znt. J . Thermophys., 1981, in press; J. D. Isdale, 33 E. Kuss and P. Pollmann, 2. Phys. Chem. N . F., 1969, 68, 205. York, 1977), p. 660. and H. T. Davis, J. Chem. Phys., 1981, 74, 1298. J. H. Dymond and T. A. Brawn, High Temp. High Press., 1979, 11, 571. (PAPER 1 / 163 1)