5. GASES LIQUIDS AND LIQUID MIXTURES By N. G. Parsonage (Imperial College of Science and Technology London S . W.7) THE most recent occasions on which these topics have been reviewed in the Annual Reports were liquid mixtures by McGlashan' in 1962 simple fluids by Rowlinson' in 1959 and gases liquids and liquid mixtures by Rowlinson in 1955.3 Gases.-Studies of intermolecular forces usually conclude by bemoaning the lack of data of sufficient quality and over sufficient range of temperature, for the second virial coefficient of gases (B) composed of simple molecules. An important recent improvement in this situation was the publication of accurate low-temperature data for argon and krypton by Weir et d4 The Burnett method was used and one noteworthy feature was the manometry.The latter involved a mercury manometer of square cross-section (to facilitate the observation of the meniscus) and led to an accuracy of 2 x atmos. Values of B were given for argon from 8@-190"~ (triple-point = 83.8"~) and for krypton from 110-225"~ (triple-point = 116.0"~) the absolute accuracy ranging from 10 ~ r n . ~ mole-' at the lowest temperature to 1 ~ m . ~ mole- at the highest. Substantial disagreement below 120"~ was found with the only previous set5 of low-temperature virial coefficients for these gases, this being attributed to the greater attention paid to adsorption corrections in the present work. The new data requires a deeper and narrower potential-well than is given by a Lennard-Jones 6-12 potential. Best agreement with experiment was found for a Kihara potential in which all three parameters were treated as adjustable.The potential recommended is with r = R b and y o as the diameter of the hard core such that u = co if R < "p and with the values ~ / k = 163-7 & 0-6",o = 3.15 & O*OOA y = 0.164 (argon) and ~ / k = 213-9 & 1.3" (T = 3.42 0-01 A y = 0.126 (krypton). For comparison the well-depths ~ / k for the Lennard-Jones 6-12 potential as given by Hirschfelder et d6 are 119-8 or 122" for argon and 171 or 158" for krypton. M. L. McGlashan Ann. Reports 162,59 73. J. S. Rowlinson Ann. Reports 1959 56 22. J. S. Rowlinson Ann. Reports 1955 52 56. R. D. Weir I. W. Jones J. S. Rowlinson and G. Saville Trans. Faraday SOC. 1967 63 1320. B. E. Fender and G. D. Halsey J . Chem. Phys.1962,36 1881. J. 0. Hirschfelder C. F. Curtiss and R. B. Byrd The Molecular Theory of Gases and Liquids, Wiley New York 1954 p. 11 10 58 N. G. Parsonage The Burnett method has also been employed by Suh and Storvick' in measurements on methyl chloride at higher temperatures (200-350") and pressures (up to 500 lb. in.-2) although the data above 300" are probably unreliable because of decomposition. A completely different approach is being adopted by Bottomley. In a recent paper with Spurling' he determined dB/dT by a differential method and combined these values with data for B at one temperature to give B for CS,, CH,COCH, CH,Cl and CH30H in the range 50-150". Analysis of B was made in terms of two types of potential for each substance. For CS2, these were the Lennard-Jones 6-12 potential with and without a quadrupole-quadrupole interaction term the latter leading to a quadrupole moment of k9.23 x esu.cm.2 which seems too large. For CH,COCH3 and CH,Cl Stockmayer potentials with and without dipole-induced dipole terms were used the values of the dipole moments and the polarisability being taken from the literature. Each mixture was considered in two ways (a) using the simplest and (b) using the most sophisticated potential for each component. Taking the geometric and arithmetic combining rules for E and 0 respectively, predictions based on (b) were in general in much better agreement with experiment than those using (a). Binary systems involving CF, molecules of which have a large octopole moment are beginning to be studied although the temperatures employed are probably too high for contributions to the virial coefficients from this source to be important.Douslin et al.' have reported a very comprehensive set of PVT measurements by the Burnett method on CF4 + CH4 from 0-350" and with pressures ranging from 1 6 - 4 0 atmos. Values are given for the cross-virial coefficients B,, Cl12 C122 Dll12 Dl12, and Ill,,, which are related to the overall virial coefficients by the equations B = Bllx + 2B,,x1x2 + 4Ol1 12x:x2 + 6Dl122~:~; + 4D1222x1x; + D2222x~. A corresponding-states treatment of the B, values was attempted using for reduction the Boyle temperature (7'') and the Boyle volume (V = TdB/dT,=,,). When the Boyle parameters for the cross interaction were evaluated from those for the pure substances using the usual combining rules TI, = (TIBT2J+ and V = ('" + '$,)' the B12 data failed to fit the corresponding-states plot but better results were obtained when T 1 2 B and '12 were taken directly from the experimental data.The size of the discrepancy was quite striking e.g. TI,, derived from experiment was 467" almost 50" lower than the value obtained from the above combination rule. Kalfoglou and Miller" used the Burnett method for B of CF4 + He at 30" and at 100" intervals from 100-500" with pressures ranging from 3-50 atmos. They were unable to fit their values for B with any simple potential. They also studied He + Ar. Since the data B& c = c11,x + 3c,12x:x + 3c,,,x,x; + c,,,x; D = D,,,,Xf + 8 ' K. W. Suh and T. S. Storvick Amer.Inst. Chem. Engineers J. 1967 13 231. G. A. Bottomley and T. H. Spurling Austral. J . Chem. 1967,20 1789. D. R. Douslin R. H. Harrison and R T. Moore J . Phys. Chem. 1967,71 3477. l o N. K. Kalfoglou and J. G. Miller J . Phys. Chem. 1967 71 1256 Gases Liquids and Liquid Mixtures 59 for pure He and pure Ar were fitted best by 6-exp and Lennard-Jones 6-12 potentials respectively both of these were considered for the fitting of the B12 data but none of the five sets of combination rules tried gave reasonable agreement with experiment. Sass et a2." have also used the Burnett method to obtain data with which to test proposed combining rules. They also found that the rules this time for the Benedict-Webb-Rubin equation were in poor agreement with their PVT data which was for CO + C2H4 at 40 50 75, 100 and 125" and at pressures from 5-500 atmos.Zandbergen and Beenakker' have calculated B values from the volume change on mixing the pure gases. In this way they avoided the considerable dependence on accurate values for the pure components which is a weakness of the usual way of determining B from values of the overall B of the mixture. Systems studied were N + H, Ar + H, and Ar + N and pressures were from 3-100 atmos. The largest volume change measured was 25 ~ m . ~ mole- '. The results suggested that there were important deviations from the arithmetic-mean combining rule for o, and also that &' < ( E E ) ~ for N + H2 and Ar + H,. They have discussed13 these results in terms of the cell-type cor-responding states theories of Prig~gine'~ and Scott15 and have found that the behaviour of the systems lie between those of the 2- and 3-liquid approxima-tions of Scott.Workers in the same laboratory16 have measursd the enthalpy of mixing (AH,) for the same three systems at temperatures from 150-293"~ and at pressures from 30-1 30 atmos. using an isothermal flow-calorimeter. They ~bserved'~ that near the critical temperature of one component it is the H behaviour of that component which mainly determines AH i.e. AH,(x P T) zz x(Ho(xP T ) - Ro(P T)) where the superscript 0 refers to the pure substance and R indicates a molar quantity. They again found that the behaviour lay between those corresponding to the 2- and 3-liquid rhodels. Measurements of the solubilities of benzene naphthalene and anthracene in compressed argon and oxygen have been used by Bradley and King18 to determine B12 for each of the six binary systems at one temperature only.The assumption was made that since the intermolecular-force parameters for Ar and 0 are nearly equal then any difference in the B12 values for Ar and 0 with any organic substance must arise from specific interactions between the 0 and the aromatic molecules. The differences observed were in all cases less than the experimental uncertainty in agreement with the conclusion of Tsubomura and Mulliken' that the charge-transfer absorption of these oxygen systems does not arise from the formation of stable complexes. A. Sass B. F. Dodge and R. F. Bretton J . Chem. and Eng. Data 1967 12 168.l 2 P. Zandbergen and J. J. M. Beenakker Physica 1967,33 343. P. Zandbergen and J. J. M. Beenakker Physica 1967,33,366. I4 I. Prigogine The Molecular Theory of Solutions. North Holland Publishing Co. Amsterdam, Is R. L. Scott J . Chem. Phys. 1956 25 193. l6 M. Knoester K. W. Taconis and J. J. M. Beenakker Physica 1967 33 389. 1957. M. Knoester and J. J. M. Beenakker Physica 1967,33,410. H. Bradley jun. and A. D. King jun. J. Chem. Phys. 1967,47 1189. l9 H Tsubomura and R. S. Mulliken J . Amer. Chem. Soc. 1960,82 5966. 60 N . G. Parsonage Storvick st al. have published several papers in which B is calculated for systems in which the intermolecular potential includes terms for polar inter-actions as well as the usual repulsive and attractive terms. They have handled the configurational partition function by expanding the exponential term before integration in the manner of Pople.20 Thus Suh and Storvick21 have treated the case of non-spherical polar gases by taking the dipole-dipole term as a perturbation on the Kihara core potential with the cores used being selected from a consideration of the geometry of the hydrocarbon homomorph.Examples of the cores used were a triangular prism for CHC1 and a pentagon for CH,COCH,. They found little improvement over the Stockmayer potential for the hydrogen-bonded substances (NH, H,O and CH,OH) but the re-maining compounds (CC12F, CHCl,F CHCl, C2H5Cl CH,Cl CH,COCH,, and CH,F) showed better accord between theory and experiment. De Rocco, Spurling and Storvick22 treated the case of molecules considered as having their centres of force distributed uniformly over a spherical shell and having a central dipole or axially-symmetric quadrupole.There was a considerable im-provement over the Stockmayer potential for CH,CI and reasonable agreement for C6H6 (for which they chose @ = -& 15 x e.s.u. cm.2). Following on from this they2 have calculated C for spherical-shell molecules with an embedded axially-symmetric quadrupole but ignoring non-additivity. The results were generally inadequate to represent the experimental C values. Spurling and Mason24 have questioned the need for the consideration of off-centre dipoles proposed by Dymond and Smith.,’ They point out that the off-centre dipole may be replaced by an infinite number of multipoles centred on the origin and they maintain that values for Band 93 (the dielectric second virial co-efficient) can be calculated to within experimental accuracy by retaining only the dipole and quadrupole terms.However they only give values for CHF,. Since the papers of Stogryn and Hirschfelder,26 on the contribution of dimers to physical properties interest in this topic has continued. Buluggiu and Foglia’ ’ have calculated the concentrations of diatomic ‘molecules’ in Ne Ar Kr and Xe assuming Morse potential interactions and using the Wentzel-Kramers-Brillouin method to determine the vibrational energy levels. Their results covered a range of reduced temperatures (= ~ T E ) from 1-5. Barua and co-workers2* have extended their previous calculation of the contri-bution to B of bound dimers in polar gases to cover metastably-bound dimers.The co-existence of two gas phases (actually with T > T for either pure component) has been found to occur in two types of system mixtures of one polar and one non-polar gas e.g. NH + N, and mixtures of He + a heavy 2o J. A. Pople Proc. Roy. SOC. 1954 A 221,498,508. 21 K. W. Suh and T. S. Storvick J. Phys. Chem. 1967 71 1450. 2 2 A. G. de Rocco T. H. Spurling and T. S. Storvick J . Chem. Phys. 1967,46 599. 23 T. S. Storvick T. H. Spurling and A. G. de Rocco J. Chem. Phys. 1967,46 1498. 24 T. H. Spurling and E. A. Mason J. Chem. Phys. 1967,46,404. 2 5 J. H. Dymond and E. B. Smith Trans. Faruduy SOC. 1964,60 1378. 26 D. E. Stogryn and J. 0. Hirschfelder J. Chem. Phys. 1959,31 1531 1545.27 E. Buluggiu and C. Foglia Chem. Phys. Letters 1967 1 82. 28 A. Saran Y. Singh and A. K. Barua J . Phys. SOC. Japan 1967,22,77 Gases Liquids and Liquid Mixtures 61 compound (polar or non-polar). Jones and Kay2' have investigated this phenomenon in He + n-C4Hlo. A perpetual problem is the reconciliation of the interaction potentials derived from gas data with those from solid-state data. Klein and Munn3' have taken the potentials found by Munn and Smith3' to give the best pre-dictions of B and transport data for Ne Ar Kr and Xe. These potentials were of the form where r = R/o. They calculated the sublimation energy lattice parameter, and bulk modulus for the solids both assuming additivity and allowing for non-additivity by means of the Axilrod-Teller term in the two sets of computa-tions.Inclusion of the latter term considerably improved the agreement with experiment but it was still not good e.g. for Ar the experimental heat of sub-limation is 1846 cal. mole-' as against calculated values of 2112 and 1932 cal. mole- ' respectively. The theory of Monchick and Mason,32 which is based on the assumption that the period of a collision is too short for there to be any appreciable re-orientation during it has been frequently used in the interpretation of the transport data of non-spherical molecules. Thus Pal and B a r ~ a ~ ~ measured the viscosities @l) of H2S SO2 and NH up to 200" by the oscillating-disc method but were unable to fit their data to the above theory. Burch and Raw,34 on the other hand successfully used the theory to fit their data on theq of NH and CH3NH2 to Stockmayer potentials.For mixtures quite good agreement with experiment was obtained with a geometric-mean combination rule for E~~ and the polar coefficient &12 and an arithmetic-mean rule for c12. A very extensive series of determinations of theq of He N, and their mixtures have been reported by Kao and K~bayashi.~' Pressures ranged from 10-500 atmos. and temperatures from -90-50" with at least five different compositions being studied under each set of conditions. At inter-mediate pressures (ca. 120 atmos.) they found both a maximum and a minimum in the graph ofq against mole-fraction. At low pressures only the maximum was found and at high pressures the graph was monotonic. Saxena and M a t h ~ r ~ have pointed out that the potential parameters which fit the high temperature diffusion data of Westenberg et ~ 1 .~ ~ do not reproduce other properties satisfactorily. New self-diffusion (D measurements on argon gas 29 A. E. Jones and W. B. Kay Amer. Inst. Chem. Engineers J . 1967,13,717,720. 30 M. L. Klein and R. J. Munn J . Chem. Phys. 1967,47 1035. 'I R. J. Munn and F. J. Smith J . &em. Phys. 1965,43,3998. 32 L. Monchick and E. A. Mason J . Chem. Phys. 1961,35 1676. 3 3 A. K. Pal and A. K. Barua Trans. Faraday SOC. 1967,63,341. 34 L. G. Burch and C. J. G. Raw J . Chem. Phys. 1967,47,2798. 3 5 J. T. F. Kao and R. Kobayashi J . Chem. Phys. 1967,47,2836. 36 S . C. Saxena and B. P. Mathur Chem. Phys. Letters 1967,1,224. 37 R. E. Walker and A.A. Westenberg J . Chem. Phys. 1958,29,1139,1147; 1959,31,519; 196432, 436; A. A. Westenberg and G. Frazier ibid. 1962 36 3499 62 N . G. Parsonage in the range 77.5-121"~ (also at 294"~) have been presented by de Paz et u Z . ~ * There is good agreement with the results of Winn39 and consistency with both q and B data. Barua et aL4' have reported an improvement in design for the trennschaukel, an apparatus due to Clusius and Huber4' for the measurement of the thermal-diffusion factor (a) by the cascade amplification of the normal separation obtained in a 2-bulb experiment. Until now trennschaukels had always given somewhat low values for a but good agreement was found in this case with 2-bulb results for Ar + N2 and He + Ne. Ghosh Batabyal and Barua4, used the improved apparatus to examine the composition dependence of a for H + He as previous measurements of the thermal conductivity @) had suggested that inelastic collisions involving rotational relaxation were im-portant in this system.Although a graph of 01 versus mole-fraction showed a pronounced minimum the results were in poor agreement with the treatment of Monchick et a1.43 for inelastic processes. A note by H ~ m p h r e y s ~ ~ suggests that 01 in HD + D2 is only slightly composition-dependent in contrast to predictions based on the likelihood of inelastic collisions. Thermal-diffusion factors of mixtures containing small concentrations ( < 5 %) of all possible isotopically-substituted hydrogen molecules in He3 and He4 have been measured by van de Ree et ~ 1 .~ ~ over the range 100-500". Mixtures containing H, D,, and T conformed to the Chapman-Enskog theory with a modified Bucking-ham 6-exp potential and taking the values of the parameters from B andq data. For the HD HT and DT systems it was necessary to include a term in a proportional to the shift of the centre of mass. Systems containing H have also been studied by Mason et aE.46 They have measured the diffusion thermo-effect the inverse of thermal diffusion for H + Ar H + COz and H2 + CH4 at pressures of 1-20 atmos. and room temperature. The results give no -surprises but this appears to be the first measurement of this effect at pressures greater than 1 atmos. An unusual determination of'transport properties nas oeen maae DY Lame-vale et ~ 1 . ~ ~ They have measured the velocity (W) and absorption (a) of ultra-sound in He and Ar up to 1 3 0 0 " ~ and in Ar plasma at 8000"~.They use the expression where f = frequency and y = C,/C,. For a monatomic gas without ionisation 38 M. de Paz B. Turi and M. L. Klein Physica 1967 36 127. 39 -E. B. Winn Phys. Rev. 1950,80 1024. 40 A. K. Batabyal A. K. Ghosh and A. K. Barua J . Chem. Phys. 1967,47,448. 41 K. Clusius and M. Huber Z . Naturforsch. 1955,10a 230. 42 A. K. Ghosh A. K. Batabyal and A. K. Barua J . Chem. Phys. 1967,47,452. 43 L. Monchick R. J. Munn and E. A. Mason J . Chem. Phys. 1966,45 3051. 44 A. E. Humphreys J . Chem. Phys. 1967,47,874. *' J. van de Ree J. Los and A. E. de Vries Physica 1967,34,66. 46 E. A. Mason L. Miller and T. H. Spurling J . Chem.Phys. 1967,47 1669. 47 E. H. Carnevale L. C. Lynnworth and G. S. Larson J . Chem. Phys. 1967,46 3040 Gases Liquids and Liquid Mixtures 63 this may be written asq = const. aW/f2. Theq results agree within a few per cent with independent measurements by normal methods up to 1300"~. At 8000"~, there is reasonable agreement with the calculations of Amdur and Mason.48 Smith et ~ 1 . ~ ~ have extended the previously mentioned treatment of polar gases by Monchick and Mason3' to the case of quadrupolar gases. They found that the quadrupolar contributions to q and DI1 although small must be taken into account if sufficiently accurate parameters for the spherically-symmetric part of the potential are needed for them to be used in conjunction with B values to give the quadrupole moment (0).They also found that the thermal-diffusion factor is the most sensitive to 0 of the transport coefficients. The results of these calculations have been used by Spurling and Mason" to obtain 0 for 9 gases. The values are all reasonable. Loaded spheres that is spherical molecules in which the centre of mass lies at a distance 6 from the centre of the sphere (of diameter cr) have received further attention. Mueller and Curtiss5' have considered collisions between such molecules quantum mechanically and have derived expressions for q 3t and D up to second order in 6/0. Sandler and Dahler52 have shown that a thermal-diffusion factor of the correct order of magnitude can be calculated for the system D + HT by considering it as a system of loaded spheres.Alternative treatments in terms of elastic collisions or rough spheres give values which are too small by factors of ca. 100 and ca. 10 respectively. The Knudsen effusion method for the determination of vapour pressures has been critically examined by Ward et They used a-Pu and gold rendered radioactive by neutron irradiation to give them high detection sensitivity so as to be able to examine the distribution of directions in the escaping molecules. They found that at their pressures ( < 1 x 10- torr) most of the effusing molecules came from the cell wall rather than from a collision in the gas phase and that they could indeed obtain an image of the interior of the cell. Computer with various values for the loss of molecules at the interior surface agree with the experiment and point to the importance of avoiding wall losses.Liquids.-Liquid Ar continues to attract a large amount of attention both experimentally and theoretically. McCain and Ziegler' have reported some new measurements on the vapour-pressure curve from 114.40"~ to the critical temperature which they found to be 150.65 & 0.02"K. There is good agreement with both Clark et ~ 1 . ~ ~ and van Itterbeek et aL5' 48 I. Amdur and E. A. Mason Phys. Fluids 1958,5 370. 49 F. J. Smith R. J. Munn and E. A. Mason J . Chem. Phys. 1967 46. 317 'O T. H. Spurling and E. A. Mason J . Chem. Phys. 1967,46,322. 5 1 J. J. Mueller and C. F. Curtiss J . Chem. Phys. 1967,46 283 1252. " S. I. Sandler and J. S. Dahler J . Chem. Phys. 1967,47 2621. 53 J. W. Ward R.N. R. Mulford and M. Kahn J . Chem. Phys. 1967,47 1710. '4 J. A. Ward R. N. R. Mulford and M. Kahn J . Chem. Phys. 1967,47 1718. " W. D. McCain jun. and W. T. Ziegler J . Chem. and Eng. Data 1967 12 199. A. M. Clark F. Din J. Robb A. Michels,T. Wassenaar and T. Zwietering Physica 1951,17,876. " A. van Itterbeek. J. de Boelpaep 0. Verbeke F. Theewes and K. Staes Physica 1964,30,2119 64 N. G. Parsonage Thomaes et aL5* have continued their measurements ofq of liquefied gases by the capillary-flow method with a paper on CO and N2. Fitting their results to the Theorem of Corresponding States they found good agreement for the reducing parameters with those from critical data for the monatomic gases, CO and N2 but large discrepancies appeared for O2 and CH, e.g. for O2 Elk = 126.41"~ (crit.) 102.27"~ b) and CJ = 3-34 8 (crit.) 2.72 8 (q).De Bock et ~ 1 . ~ ~ have examined the pressure dependence ofq of liquid Ar and O2 up to 150 atmos. by observing the change in electrical resistance of a quartz crystal when at resonance in the fluid due to the viscous damping. Interpretation of the effects of isotopic substitution on the vapour pressures of liquids is usually done in terms of the frequency shifts in going from the liquid to the vapour phase for both the parent and the isotopically-substituted compound.60 Van Hook6' has made such an analysis of his results for four methylacetylenes from 167-255"~. When the acetylenic hydrogen atom was substituted there was an important contribution from the effect on dimerisation. Measurements near the critical point must be carefully designed so as to reduce density differences arising from the gravitational field.This is underlined by the work of Schmidt et aE.62 on C for Xe near its critical point. They found that the singularity occurred 0.32" below the recognised value for T,. V ~ r o n e l ~ ~ had made similar observations for Ar and 02 the discrepancy then being ca. 0.2". The results were discussed with reference to the height of the calori-meter (ca. 10 cm. in Schmidt's work) and the occurrence of peaks in c at different temperatures for material at different heights in the vessel. Determinations of adiabatic compressibility by measuring the velocity of sound and the density have been frequent. Thus Boelh~uwer~~ has examined six n-paraffins from - 20-200" and at pressures up to 1400 atmos.by observ-ing the time for the return of an echo signal to the nearest 0.1 psec. from an oscilloscope trace. Aziz et a1.,65 on the other hand have employed a resonance technique for Ar Kr and Xe up to the neighbourhood of their critical points. The data conformed well to a corresponding-states plot. However a similar study on CF466 showed that this compound obeyed a similar corresponding-states plot to CCl, but markedly different from the simpler molecules. It is suggested that this may be due to a steeper repulsive potential for CF and CCl, which is in accord with the use of an effective 7-28 potential for such compound^.^' The ultrasonic-absorption coefficient (a) in liquid Ar has been determined 5 8 J. P.Boon J. C. Legros and G. Thomaes Physica 1967,33,547. 59 A. de Bock W. Grevendonk and H. Awouters Physica 1967,34,49. 'O J. Bigeleisen and M. G. Mayer J . Chem. Phys. 1947,15,261; J. Bigeieisen ibid. 1961,34,1485. '' W. A. van Hook J . Chern. Phys. 1967,46 1907. 62 H. H. Schmidt J. Opdycke and C. F. Gay Phys. Reo. Letters 1967,19,887 '' A. V. Voronel' and P. G. Strelkov Pribory i Tekhn. Eksper. 1960 No. 6 111 (translation: 64 J. W. M. Boelhouwer Physica 1967,34,484. " R. A. Aziz D. H. Bowman and C. C. Lim Canad. J . Chem. 1967,45,2079. " R. A. Aziz C. C. Lim and D. H. Bowman Canad. J . Chem. 1967,451037. 67 S. D. Hamann and J. A. Lambert Austral. J . Chem. 1954,7,1; S. D. Hamann ibid. 1960,13,325. Instr. Exptl. Tech. (U.S.S.R.) 1960 No. 6 970) Gases Liquids and Liquid Mixtures 65 by Swyt et ~ 1 .~ ~ from 90-145"~ and at pressures up to 100 atmos. The bulk viscosity h,) is then given by the equation : 01 PW3 (Y - 1)h + f q r l o = - - - - ~ f 2 2 X 2 { c 3 } where f is the frequency h the thermal conductivity andq the shear viscosity. The ratioq,h varies somewhat with density but is in fair agreement with the predictions of the Ri~e-Allnatt~~ and hard sphere7* theories of transport processes. A similar examination of liquid N2 has been made by Singer and Lunsford7 with similar conclusions. Determination of the structure of fluids by direct methods have made considerable progress. The X-ray work of Pings' group will be discussed in the section concerned with the theory of fluids. Br has been studied by X-ray diffraction by Gruebel and Clayton.72 They found that the radial distribution function had two peaks of approximately equal intensity at'4.06 and 5.66 rather than the single peak at 4.6 A which they had expected.They interpreted this as being due to the molecules undergoing hindered rather than free rotation. Weinberg73 has studied the second-harmonic scattering when CCl, is illuminated by laser light. He discussed the polarisation induced in terms of the local field and was thereby able to interpret his results as showing that between 5 and 55" there was a progressive break-up of clusters of orienta-tionally-related molecules. This conclusion is supported by the study of the depolarisation of the laser Raman scattering by Murphy et al.74 They found a significant non-zero depolarisation ratio due to the lower symmetry of the scattering CC14 molecules in the pure liquid and that in mixtures this ratio varies with the nature and concentration of the second constituent (CS2, and its mixtures with C6H6 have been shown by light-scattering and n.m.r.relaxation methods to have considerable order but CC14 + C6H,N02 was di~ordered.~ A completely new technique for studying molecular motion which has particular application to liquids is the scattering of cold neutrons.76 Two types of process elastic and inelastic occur. The former leads to a strong line broadened by the Doppler effect the width of the line giving a measure of the speed of the scattering particles. This information can therefore be related to the self-diffusion coefficient.The inelastic effect arises from the neutron taking up a phonon and can therefore give information on frequency C6H6 P - C ~ H ~ B ~ Z CyClO-C6H12 n-C7H16 CH,CN and CHSOH). C6HsN02 '* D. S. Swyt J. F. Havlice and E. F. Carome. J . Chem. Phys. 1967.47 1199. 69 S. A. Rice and A. R. Allnatt J . Chem. Phys. 1961,34 2144; A. R. Allnatt and S. A. Rice ibid., " H. C. Longuet-Higgins and J. P. Valleau Mol. Phys. 1958 1 284. 71 J. R. Singer and J. H. Lunsford J . Chem. Phys. 1967,47,811. 72 R. W. Gruebel and G. T. Clayton J . Chem. Phys. 1967,47 175. 73 D. L. Weinberg J . Chem. Phys. 1967,47 1307. 74 W. F. Murphy M. V. Evans and P. Bender J . Chem. Phys. 1967,47 1836. '' A. Szoke E. Courtens and A. Ben Reuven Chem. Phys. Letters 1967,1,87. 76 P. EgelstaE Discuss. Faraday SOC.1967 No. 43 149; B. K. Aldred R. C. Eden and J. W. White, 1961,34 2156; P. Gray and S. A. Rice ibid 1964,41 3689. ibid. 1967 No. 43 169 66 N . G. Parsonage distribution within the phase (there are no selection rules to be obeyed). Since isotopic substitution can lead to very large changes of the scattering cross-section it is sometimes possible to assign motion to specific regions of the molecule. The major advances in the theory of fluids have centred on the Percus-Yevick (PY) and the (Convoluted) Hypernetted Chain (HNC) approximation^.'^ They both lead to integral equations from the solution of which the radial distribution function (g(r)) and the total (h(r)) and direct (c(r)) correlation functions may be evaluated. h(r) and c(r) are defined by the equations h(r) = g(r) - 1 and h(r12) = c(r12) + pjc(r13)h(r23)dT3 where p is the number density.The approximation differ in which of the cluster-type integrals to omit from the expressions for h(r) and c(r) but R o ~ l i n s o n ~ ~ ' has shown that the PY approximation can also be deduced from the requirement that c(r) should be short-ranged. Two ways are commonly used for deriving the equation of state from the expressions for h(r) and c(r). These are via the compressibility equation of Zernike and Prins kT(ap/dP) = 1 + pJh(rl,)dz2 and via the virial equation of Clausius. Although both of these equations are exact they lead to different expressions for the equation of state thereby demon-strating an inconsistency which is introduced by the PY and HNC approxima-tions.The approach by way of the compressibility equation is rather more frequently used. Comparisons of these theories have mostly been with molecular dynamics and Monte Carlo experiments for hard spheres at high densities, and with the virial coefficients for hard spheres at low densities. From these the PY approximation seems to be the better. However for molecules inter-acting according to the Lennard-Jones 6-12 or the 6-exp potentials there is evidence that the HNC might be preferable at low temperatures but not at high.77c Mikolaj and Pings7* have discussed the respective merits of the two theories in the light of their data on the X-ray scattering from fluid argon near the critical point. Expressing the assumptions in the forms c(r) = g(r) (1 -exp[u(r)/kT]} for PY and c(r) = h(r) - lng(r) - u(r)/kT for HNC they deduced from their experimentally determined correlation functions the function u(r) which should be independent of T and p.They found that u(r) was insensitive to T but varied considerably with p e.g. the well-depth ~ / k changed from 120-95" over the range p = 0.784.28. It is suggested that these variations with p may be due to many-body interactions. Rushbrooke and Silbert79 and Rowlinson" have shown how the presence of three-body forces may be incorporated into the PY and HNC theories and the former have shown that this modified HNC theory leads to the approximately linear decrease of the well-depth of the effective two-body potential with increase in 77 (') J. S. Rowlinson Reports Progr. Physics 1965 28 169.(*) G. S. Rushbrooke Discuss. Fataday SOC. 1967 No. 43,7; (') J. S. Rowlinson ibid. 1967 No. 43 243. 78 P. G. Mikolaj and C. J. Pings J . Chem. Phys. 1967,46 1401 1412. l9 G. S. Rushbrooke and M. Silbert Mol. Phys. 1967,12 505. J. S. Rowlinson Mol. Phys. 1967 12 513 Gases Liquids and Liquid Mixtures 67 p. Verlet’ has shown that considering only two-body forces the PY and HNC theories may be considered as the first members of two hierarchies of ap-proximations. Machine evaluations for the distribution and correlation functions for the second approximations (PY2 and HNC2) are becoming available and it does appear that they represent considerable improvements.B2 Harris and ClaytonB3 have re-examined the intensity of X-ray scattering from Ar and Xe near their triple points and found much better agreement especially for the outer peaks with the predictions of the HNC theory for a liquid of Lennard-Jones molecules than was obtained from the much earlier measure-ments of Campbell and Hildebrand.B4 One of the surprising things about the Percus-Yevick approximation is that an exact solution of the integral equation is known for hard spheres.77 This analytic solution breaks down at high densities (po3/2$ 3 0-8) and it has been suggested that this point corresponds to a fluid-solid transition.Temperley” has indeed found alternative solutions for these higher densities but HutchinsonB6 has pointed out that these solutions are unphysical in that they would lead to negative values for the intensity of radiation scattered at some angles.Hutchinson then proceeded to show that there can be no acceptable solutions other than the original ones. Considerable interest is being shown in the model of a liquid as a system of random-packed spheres. Thus the ratios of the peak positions in the radial distribution function derived by Scott et from a mechanical model of random-packed spheres have been shown to be in good agreement with the corresponding ratios derived from the X-ray study of Ar and Xe near their triple points.B3 Bernal’s group continue to be very active in this field. TheyB8 have examined the polyhedra which are formed when the planes are drawn bisecting and perpendicular to the lines joining the molecules. They found a high incidence of pentagonal faces and of polyhedra with 13-15 faces results which are in general agreement with their previous work.With the wealth of results for hard-sphere fluids it is not surprising that attempts to use perturbation treatments to extend these to more realistic potentials have been made.B Recently Barker and Hender~on,~’ using this approach have found that for a square-well potential the results are in good agreement with Monte Carlo and molecular-dynamics calculations for this potential. They concluded that failure of the perturbation treatment to con-verge for still more realistic potentials is due to the “softness” of the repulsive potential rather than the presence of the attractive well. L. Verlet Physica 1965 31,959. D. Henderson S. Kim and L. Oden Discuss. Faraday SOC. 1967 No. 43,26.83 R. W. Harris and G. J. Clayton Phys. Rev. 1967,153,229. 84 J . A. Campbell and J. H. Hildebrand J . Chem. Phys. 1943,11,334. 85 H. N. V. Temperley Proc. Phys. SOC. 1964,84,339. 86 P. Hutchinson Mol. Phys. 1967,13,495. G. D. Scott J. D. Bernal J. Mason and K. R. Knight Nature 1962,194,956. J. D. Bernal and J. L. Finney Discuss. Faraday Soc. 1967 No. 43.62 R. W. Zwanzig J . Chem. Phys. 1954,22 1420; J. S. Rowlinson Mol. Phys. 1964,7 349; ibid., J. A. Barker and D. Henderson J . Chem. Phys. 1967,47 2856. 196t48 107. C 68 N . G. Parsonage Phenomena in the neighbourhood of the critical point has been well covered in the report of the conference of that name.9' The report shows clearly and comprehensively the analogies which can be drawn with magnetic phenomena near the Curie point which is a consequence of the similarity between the Grand Partition Function for a lattice-gas and the Constant Field Partition Function for the Ising model.Particular attention was given to the indices with which various thermodynamic quantities approached their critical values e.g. Lt (InlP - P,I/lnIp - p,l} = 6. P=P, P = P, R o ~ l i n s o n ~ ~ " has discussed the experimentally observed values for these indices together with certain restrictions upon the possible values which they can adopt. More recently Green et from observations on Xe COz SF,, and Ar have concluded that 6 is 5 rather than 4.2 a value which was accepted at the conference by Rowlinson but disputed by Fisher.91b Liquid Mixtures.-Tne Average Potential model originally proposed by Prigogine et and S ~ o t t ' ~ has been reviewed at length by Bellemans et In particular they separate the fundamental propositions from certain additional assumptions which were made in the original papers e.g.the use of a Taylor series expansion to represent the properties of the mixture in terms of those of a single reference substance. In effect they94 use a number of reference substances to construct empirical relations between the reduced thermodynamic quantities and the reduced values of T and P. In this way, they were able to overcome the former drawback that in order for example, to predict the excess volume it was necessary to know the temperature deriva-tives of the volume of the reference substance with an accuracy greater than the data could justify.For the reduction parameters for the interaction between dissimilar molecules they found that the use of the usual combination rules (geometric mean for E ~ ~ arithmetic mean for oI2) in general only failed to give the sign of the excess quantity correctly when this quantity was small. Further they found that often quite small changes in s12 and o12 from these values were sufficient to achieve good agreement although the amount of skew of the graph of excess quantity versus mole fraction was often in the wrong direction. Streett and Staveley'' have also discussed the possibility of explaining the excess volumes ((VE) of 8 binary mixtures of liquefied gases by means of the Average Potential model without using the Taylor expansion. They took the data for Ar to define a reduced volume versus reduced tempera-ture curve for P = 0 and considered the average potentials represented by 9 1 Conference on Phenomena in the Neighbourhood of Critical Points.Ed. M. S. Green and J. V. Sengers. National Bureau of Standards Miscellaneous Publication 233 Washington D.C. 1966. (a) J. S. Rowlinson p. 9; (b) M. E. Fisher p. 25. 92 M. S. Green M. Vicentini-Missoni and J. M. H. L. Sengers Phys. Rev. Letters 1967 18 1113. 93 I. Prigogine A. Bellemans and A. Englert-Chwoles J . Chem. Phys. 1956,24 518. 94 A. Bellemans V. Mathot and M. Simon Adv. Chem. Phys. 1967 11 117. 9 5 W. B. Streett and L. A. K. Staveley J . Chem. Phys. 1967,47,2449 Gases Liquids and Liquid Mixtures 69 Scott's 1- 2- and 3-liquid models. They took a range of values for the energy parameter ( E ) for the pure substances and used the usual combination rules, but they were unable to obtain good agreement with experiment.The 2- and 3-liquid models were about equally good but both gave bad agreement for the simplest systems Ar + Kr and Ar + Ar + CH,. Mastinug6 has also used the Average Potential model to discuss his results for V E for solutions of small quantities (0.5-2.0%) of H in N2. The technique adopted was to measure with a differential device the pressure developed when similar volumes of solvent and solution were vaporised into similar volumes. The partial molar volume of Hz was found to be 39.784 ~ m . ~ mole- I whereas a 2-liquid treatment gave the surprising prediction of 166-35 ~ m . ~ mole-'.Of course the wide disparity in the molecular sizes would render this theoretical treatment inappropriate here and Mastinu observed that the same approach applied to mixtures of molecules with similar parameters led to agreement to better than 5 %. Fuks and Bellemansg7 have given some values admittedly of lower accuracy than usual for this field for GE and VE for CH + Kr and N2 + CH,. Findenegg and Kohler9* have also employed the model in their discussion of CUE of binary mixtures of CH,Br.CH,Br and CHzCl.CH2Cl with C6H6. The system is complicated by the change of the ratio of trans- to gauche-forms during the mixing process but allowance for this was made using the dielectric constant data99 for these solutions. C of the CHzCl.CHzCl mixture was slightly positive and was attributed to a simple change in the trans:gauche ratio; for the CH2Br.CH2Br solutions on the other hand it was large and negative and the predicted change in cell size from the model was insufficient to lead to a change of the rotational heat capacity of the required size.Vilcu and Bellemansloo have considered the extension of the model to moderate pressures. They express the reduced volume_ in a power series in the reduced pressure (P) the coefficient of the term in P o being taken from the work of Simon and Mathot' on several substances whilst the coefficients of the terms in f" and P2 were chosen to fit the PV data of liquid Ar up to 300 atmos. Applying these equations to CO + CH and Ar + CH they found that there should be extrema in V E at ca.200 and ca. 100 atmos. respectively although the former result is less certain. Fluid mixtures at high pressure have been examined by Throop and Bear-man''' using the PY equation assuming a Lennard-Jones 6-12 potential. Calculations were given for Ar + Kr Ar + Xe Ne + Kr and Ne + Xe at supercritical temperatures and at densities up to twice the critical density. For constant pressure processes UE V E and GE were all positive at low p, increasing with p to a maximum and then decreasing to small positive or negative values. Constant volume processes gave much smaller excess quan-96 G. Mastinu J . Chem. Phys. 1967,47,338; Rev. Sci. Znstr. 1967,38 1114. '' S. Fuks and A. Bellemans Bull. SOC. chim. belges 1967,76 290. 98 G. H. Findenegg and F. Kohler Trans. Faruduy SOC.1967.63,870. A. Neckel and H. Volk Z . Elektrochem. 1958,62 1104. loo R. Vilcu and A. Bellemans Bull. SOC. chim. belges 1967,76 325. G. J. Throop and R. J. Bearman J . Chem. Phys. 1967,47,3036. 9 70 N. G. Parsonage tities but were more complicated. Snider and Herringtonlo2 have treated a mixture for which the intermolecular potential is represented by a hard-sphere repulsive potential and an otherwise uniform attractive potential. For a pure system of this type Longuet-Higgins and Widornlo3 had found the equation of state : where 5 =7r03p/6 and 0 is the hard-sphere diameter. The first term is that found for hard spheres by the scaled-particle theorylo4 or by the PY theory using the compressibility equation. Lebowitz' O5 had previously extended the scaled-particle model to mixtures.Snider and Herrington found good agree-ment with experiment for the HE SE and VE of Ar + CH, Ar + Kr Ar + N2, O2 + Ar Ar + CO CO + CH4 O2 + N, and N + CO but much poorer agreement for CCl + C(CH,) and CCl + cyclo-C6Hl,. The method also failed for AH of Ne in Ar this being attributed to the less uniform potential experience by the Ne molecules as a consequence of their greater freedom of movement. Scaled-particle theory which as mentioned above gives the same equation of state as the PY approximation (using the compressibility equation) had been applied to non-planar gases in non-polar solvents and in water by Pierotti,lo6 but this work has been criticised as far as water is concerned by Ben-naim and Friedmanlo7 on the ground that it leads to an incorrect tem-perature-dependence for the surface tension.Dymond'" has presented solubility data for 11 gases in (CH,),O at 25" and for 9 gases in cyc1o-C6H,, at various temperatures. All the results refer to P = 1 atmos. Byrns and Mazolog have observed that AE for isotopic mixtures can be closely predicted from the quantum-mechanical version of the theorem of corresponding states if it is assumed that the system is composed uniformly of particles of mass rImT-a where x is the mole-fraction of the particles of mass ma. No complete justification but only a plausible argument is given for this. Harrison and Winnick'" have measured VE at three temperatures for 5 binary mixtures composed from the even n-alkanes Cl0 to (216. VE was always negative and ranged up to -0-14 ~ m .~ mole-'. Deviations from the principle of congruence' l1 averaged only 0-002 ~ m . ~ mole-Since the discovery by Rowlinson et ~ 1 . l ' ~ of the partial miscibility of N. S. Snider and T. Herrington J . Chem. Phys. 1967,47 2248. H. C. Longuet-Higgins and B. Widom Mol. Phys. 1964,8 549. J. L. Lebowitz Phys. Rev. Series A 1964,133,895 ; J. L. Lebowitz E. Helfand and E. Praestgaard, R. A. Pierotti J . Phys. Chem. 1963 67 1840; [bid. 1965,69 281 A. Ben-naim and H. L. Friedman J . Phys. Chem. 1967,71,448. J. H. Dymond J . Phys. Chem. 1967,71 1829. F. L. Byrns and R. M. Mazo J . Chem. Phys. 1967,41 2007. C. Harrison and J. Winnick J . Chem. and Eng. Data 1967 12 176. J. Hijmans and T. Holleman Mol. Phys. 1961,4,91. ''' A. J. Davenport and J.S. Rowlinson Trans. Faraday SOC. 1963,59,78. lo4 H. Reiss H. L. Frisch and J. L. Lebowitz J . Chem. Phys. 1959 31 369. J . Chem. Phys. 1965,43 774 Gases Liquids and Liquid Mixtures 71 systems composed of one short and one long n-parafin the phase diagrams of a number of other such systems have been studied 9ver a'wide range of pressures with similar re~ults."~ Recent systems studied include C3H8 with CH, C02,andN,,114 He + CH,,"' CO withn-C8H,,,n-C ,H,,,n-C,3H,8, H20."* In addition Larkin et ~ 1 . " ~ have examined the phase equilibria of solutions of liquid sulphur in 10 solvents. Analogies between the plait point in a 3-component system at constant temperature and pressure and the situa-tion in a 2-component liquid system at constant temperature (but variable pressure) have been pointed out by Widom.'20 The non-ideality of fluorocarbon-hydrocarbon mixtures which is much larger than would be predicted by solubility-parameter theory has continued to attract attention.Gilmour et have reported on the vapour-pressures of three such alkane-perfluoroalkane mixtures and on the liquid-liquid phase diagrams of these and 8 similar systems. The deviations from solubility para-meter theory in contrast to the composition at the critical solution temperature, were not very sensitive to differences in the molar volumes of the components. This suggested that a formulation of the theory in terms of volume- or surface-fractions was required and subsequent analysis led to a preference for the latter. Fenby and Scott122 have determined HE calorimetrically for 34 of the 78 possible binary systems of the form C6H,F6- + C6H,F6-, and also C6F6.They found a wide variety of types of behaviour. Qualitative interpreta-tion was given in terms of physical (positive) and chemical (negative) contribu-tions but in the absence of auxiliary data on the position of the chemical equilibrium no quantitative discussion was possible. Much of the discussion'23 of transport processes in dense fluids has been concerned with the Ri~e-Allnatt~~ theory. R i c ~ i ' ~ has found that D for H, T, Ne Ar and CH in liquid N is almost independent of mass. Only a theory due to BearmanI2' conforms with this. Measurements of D of N2 and 0 in H20126 show a temperature-dependence which is ca. 10 times and n-C16H34,116 C6H6 -k H20,117 CyClO-C6H, f H20 and n-C,H,,+ -b for C3H7.CsHs C6F6 n-C,H9.C6HS CsF'H and n-C4H9.C6H5 + A.B. Rodrigues and J. P. Kohn J . Chem. and Eng. data 1967,12191 ; J. M. Beaudoin and J. P. Kohn ibid. 1967 12 189. 'I4 J. G. Roof and J. D. Baron J. Chem. and Eng. Data 1967,12,292. C. K. Heck and M. J. Hiza Amer. Inst. Chem. Engineers J. 1967,13 593. G. Schneider Z. Alwani W. Heim E. Horvath and U. Frank Chem.-1ng.-Tech. 1967,39,649. Z. Alwani and G. Schneider Ber. Bunsengesellschaft Phys. Chem. 1967,71,633 C. H. Rebert and K. E. Hayworth Amer. Inst. Chem. Engineers J. 1967,13 118. B. Widom J. Chem. Phys. 1967,46,3324. J. B. Gilmour J. 0. Zwicker J. Katz and R. L. Scott J . Phys. Chem. 1967,71 3259. 'I9 J. A. Larkin J. Katz and R. L. Scott J. Phys. Chem. 1967,71 352.lZ2 D. V. Fenby and R. L. Scott J . Phys. Chem. 1967,71,4103. lZ3 J. A. Palyvos and H. T. Davis J . Phys. Chem. 1967,71,439; A. F. Collings and C. J. Pings ibid., 1967,71 3710; J. Palyvos K. D. Luks I. L. McLaughlin and H. T. Davis J . Chem. Phys. 1967 47, 2082; Ching Cheng Wei and H. T. Davis ibid. 1967 46 3456. F. P. Ricci Phys. Rev. 1967 156 184. 125 R. J. Bearman J . Chem. Phys. 1960 32 1308; F. P. Ricci ibid. 1966 45 3897. R. T. Ferrell and D. M. Himmelblau. J . Chem. find En(!. Durcr. 1967. 12. 1 1 1 72 N. G. Parsonage greater than predicted by Longuet-Higgins and Pople.' 27 A laminar-flow technique has been used by Turner et d.'z8 for the study of the thermal dif-CHCI, CH,COCH + C6H6 and CH3COCH + H 2 0 at 25" have been determined by the spin-echo method.lZ9 A diaphragm cell has been used for the measurement of D12 in CH,OH + CH3.C6H5 at 250.13* fusion Of cc1 + CyClO-C6H1,. D12 for C6H6 + CyClO-C6H12 CH3COCH3 + H. C. Longuet-Higgins and J. A. Pople J. Chern. Phys 1956 25 884. 127 lZ8 J. C. R. Turner B. D. Butler and M. J. Story Trans. Faraday Soc. 1967,63,1906. lt9 D. W. McCall and D. C. Douglas J . Phys. Chern. 1967,71,987. lJo L. W. Shemilt and R. Nagarajan Canad. J. Chern. 1967,45,1143