J.C.S. Faraday I, 1980,76,2496-2506Thermodynamics of Liquid Mixtures of Nitrous Oxide and XenonBY JosB R. S. MACHADO AND KEITH E. GUBBINSSchool of Chemical Engineering, Cornell University,Ithaca, New York 14853, U.S.A.AND LELIO Q. LOBO AND LIONEL A. K. STAVELEY*Inorganic Chemistry Laboratory, University of Oxford, Oxford OX1 3QRReceived 1st November, 1979The total vapour pressure, excess volume and excess enthalpy of the system nitrous oxide + xenonhave been measured as a function of composition ; the vapour pressure and excess volume measure-ments are at 182.32 K (the triple-point of N20) and those for the excess enthalpy are at 184.05 K.The vapour pressure results have been used to estimate the excess Gibbs energy. The mixture exhibitsa positive azeotrope at a N,O mole fraction of = 0.08.The experimental results are compared withvalues calculated from perturbation theory for non-spherical molecules, using an intermolecular poten-tial model that includes dipolar and quadrupolar electrostatic terms. Agreement is good for the pro-perties of both the pure fluids and the mixture and is a substantial improvement over theories thatneglect the acentric nature of the intermolecular forces.This paper is one of a series devoted to an examination of the effects of electro-static forces and molecular shape on the thermodynamic properties of binary liquidmixtures of which one constituent is polar. The experimental values of the primaryexcess thermodynamic functions are compared with those calculated from the per-turbation theory developed by one of us (K.E. G.) and his collaborator^.^'^ Twoprevious papers dealt with the systems xenon + hydrogen chloride, xenon + hydrogenbromide and hydrogen chloride + hydrogen bromide 4 * and hence were largelyconcerned with the influence of a dipole in the polar species. Here we present theresults of an experimental and theoretical study of the system nitrous oxide(dinitrogen oxide, N,O)+xenon, selected to serve as a model for mixtures of thetype quadrupolar molecules + non-polar molecules. It was clearly desirable tochoose as the polar component a compound with a relatively large quadrupolemoment. A suitable choice might seem to be carbon dioxide, but the relatively hightriple-point temperature and pressure of this substance rule it out on practical grounds,at least as far as studies of a mixture with a rare gas are concerned.The moleculeof nitrous oxide has a quadrupole moment ( Q = -3.65 x e.s.u. cm2 =- 12.2 x( Q = -4.3 x C m2) and though it has a dipolemoment this is so small ( p = 0.166 D = 0.55 x C m) that one would expectthat its influence on the properties of the N,O + Xe system would be negligible.C m2) which is not much smaller than that of carbon dioxidee.s.u. cm2 = - 14.3 xEXPERIMENTALThe experimental techniques used to measure the excess Gibbs energy of mixing GE, thevolume of mixing VE and the enthalpy of mixing HE have already been described.8-10 Thevapour pressures were measured with a Texas Instruments quartz spiral gauge, which hadbeen calibrated against a wide bore mercury manometer and a dead-weight piston gauge.249MACHADO, GUBBINS, LOB0 A N D STAVELEY 2497The pyknometer used for the volume determinations had been calibrated using a 99.96 molpercent sample of ethane and the results of Haynes and Hiza." In the heat of mixingexperiments, temperature was measured with a copper resistance thermometer which hadbeen calibrated by measuring the vapour pressure of ethane, using the vapour pressureequation of Goodwin et u1.l' The total volume of the upper compartment of the calori-meter, where N20 was condensed, was 3.16 cm3 ; the lower chamber, into which the xenonwas condensed, had a capacity of 8.66 cm3.GE and VE were determined at the triple-point of N20 (182.32 K) and H E at a slightlyhigher temperature (184.05 K), since attempts to measure H E at the triple-point of one ofthe two components might present experimental difficulties owing to the partial solidificationof that component.The xenon used was research grade quality, of purity > 99.995 mol percent.Thenitrous oxide was taken from a cylinder (medical grade ; at least 99.0 mol percent) andfractionated in the laboratory low-temperature c o l ~ m n . ~ The purity of the sample usedwas checked by the constancy of its triple-point pressure, for which the value of (87.866f0.001) kPa was obtained (mean value of six determinations, as measured directly with amercury manometer ; cf. 87.864 l 3 and 87.853 kPa 1 4 9 15). Our measured value of thevapour pressure of liquid xenon at the triple-point of NzO was 247.742 kPa.The valuesinterpolated from the results of Michels and Wassenaar l 6 and Theeuwes and Bearman l 7are 247.434 and 247.306 kPa, respectively. The differences between these figures and oursare covered by a difference of w 0.03 K in the temperature scales.The average of six determinations of the molar volume of liquid N20 at its triple-pointwas (35.487k0.002) cm3 mol-1 (cf. 35.80 l 5 and 35.46 cm3 mol-l 18). Three determina-tions of the molar volume of xenon at the same temperature gave (46.453+0.010) cm3 mol-1(cf. 46.409 l7 and 46.534 cm3 mol-1 19).In the evaluation of C", VE and H E from the actual experimental measurements, valuesfor certain physical properties must be adopted and it will be convenient to summarize theseand to indicate their source.The value of 182.26 Kreported by Blue and Giauque l4 was estimated ' ' 9 'l to be equivalent to 182.32 K onIPTS-68. B forNzO at room temperature, which is required in assessing the quantity introduced into thepyknometer (or calorimeter), was taken from the work of Couch et dZ2 and Schampet ~ 1 .~ ~ The value so obtained for 298.15 K was -133.2 cm3 mol-I. B for N20 at182.32 K was calculated to be -400 c1n3 mol-1 from the equation of Pitzer andThe acentric factor co was estimated as 0.160 from the vapour pressure data of Couch et al."and the critical constants listed by mat hew^.'^ The necessary B values for xenon weretaken from Brewer's report.26 For mixtures of the two gases it was assumed that the crossvirial coefficient BIZ is the arithmetic mean of those of the two pure components.For HE (for the use made of the following quantities, see Lewis et ~ 1 .' ~ ) : (1) We usedour own values of the molar volumes of the pure liquids, of the volume change on mixingand of the vapour pressures of the pure components at the temperature of the H E determina-tion. The molar volume VG of gaseous NzO at this temperature was estimated by correctingfor gas imperfection as far as the second virial coefficient. VG for xenon was interpolatedfrom the results of Streett et ~ 1 . ~ ~ The composition of the vapour phase in the calorimeterafter mixing was calculated from our Redlich-Kister equation for GE (vide infva).(2) The molar enthalpy of vaporization AHv of NzO at 184.05 K was estimated as 16.59 kJmol-1 from the value of 16.56 kJ mol-1 found by Blue and Giauque l4 at the normal boilingpoint.AHv for xenon was interpolated from table 6 of the paper of Streett et ~ 1 . , ~ * thevalue so obtained being 11.93 kJ mol-l. (3) The coefficient of expansion a, of NzO at itssaturation vapour pressure was derived from the results of Leadbetter et u1.,l5 while thatfor xenon was interpolated from table 7, ref. (28). The values obtained were a,(NzO) =2.1 x K-l and a,(Xe) = 2.5 x lW3 K-l at 184.05 K. (4) The isothermal compressibilityKT of liquid NzO was calculated to be 0 . 9 ~ MPa-' from the approximate relation 29KT w aTV/(AH,-RRT). For Xe, a value of 2.3 x lop3 MPa-l was interpolated fromtable 7, ref.(28).For GE and VE : (1) The triple-point temperature of NzO.(2) Second virial coefficients (B), (also needed in the H E calculations)2498 THERMODYNAMICS OF N20+XeRESULTSThroughout this paper, nitrous oxide is designated 1 and xenon 2. Table 1 givesour results for the total vapour pressure P as a function of xl, the liquid mole fractionof N20. The system is markedly non-ideal, forming a positive azeotrope which isnot far from being a tangent azeotrope. GE was evaluated by Barker's method,30minimizing the pressure residuals RP = Pexp -Pcalc. The values in table 1 of yl, themole fraction of N20 in the vapour in equilibrium with the liquid mixture, are calcu-lated. A three-term Redlich-Kister equation was found to be adequate for GE,namelywith A = 1.1829 (aA = 0.0015) ; B = 0.0532 (a, = 0.0028) ; C = 0.0425 (ac =0.0056), the CJ being the standard deviations of the parameters.G: = (448.2k0.6)J mol-I.GE/RT = x,(l- xl)[A + B(2~1- 1) + C(2~1- 1)2] (1)TABLE ~.-VAPOUR PRESSURE AND EXCESS MOLAR GIBBS ENERGY OF THE SYSTEM NITROUSOXIDE XENON (2) AT 182.32 K. Rp = PeXp-~,,1,.X1 Y1 P/kPa Rp/Pa GE/J mol-'0.000 000.092 840.191 380.309 940.438 390.566 210.751 560.855 720.922 731.000 000.000 000.090 630.154 570.207.970.252 1 30.295 030.381 320.480 450.607 101 .ooo 00247.742249.539246.745240.267231.168218.731191.942163.939136.37987.866-841- 56172223- 284- 92- 17-0149.6274.3378.7436.3433.2335.9226.4113.70The molar volumes of mixtures of known composition and the derived values ofVE are recorded in table 2.These values of VE refer to mixing at the saturationvapour pressure. (The difference from the values at zero pressure is negligible.)They fit the equationwith D, E and F, respectively, equal to 2.617, -0.878 and 0,100 em3 mol-l, thestandard deviation ofthis fitting beinga = 0.006 cm3 mol-l. V t = 0.654 cm3 mol-I.The results for HE are represented in table 3 in the form adopted in a recentpublication on the methane + ethylene system.31 The HE values have been calculatedboth for mixing at the saturation vapour pressure, HE(Ps), and at zero pressure,HE(0). As the vapour pressures under the prevailing experimental conditions werecomparatively low, the differences between HE(P,) and HE(0) are very small.TheHE(0) values fit the equationwith G = 2.3574 ; H = -0.0273 ; J = 0.5827, the standard deviation beinga = 5.5 J mol-l. Hi(0) = 901.8 J mol-'.VE = x,(l -x,)[D+E(2x, - 1)+F(2x1 - 1)2] (2)HE(0)/RT = XI( 1 - x,)[G + H(2~1- 1) + J ( 2 ~ 1 - 1)2] (3MACHADO, GUBBINS, LOB0 A N D STAVELEY 2499TABLE 2.-MOLAR VOLUMES AND EXCESS MOLAR VOLUMES OF LIQUID MIXTURES OF NITROUSOXIDE XEN XENON (2) AT 182.32 K AND AT THE SATURATION VAPOUR PRESSURE^0.000 000.152 780.284 770.392 230.503 150.639 440.772 500.892 710.910 071 .ooo 0046.45345.20643.94242.82141.58939.98838.37136.85336.62435.48700.4280.61 10.6690.6530.5470.3900.1900.1510-0.004- 0.003- 0.0010.000- 0.0020.009- 0.000- 0.010IRV is the volume residual, = VE- VFalc, where VFalc is the excess molar volumecalculated from eqn (2).TABLE 3.-EXCESS MOLAR ENTHALPY OF THE SYSTEM NITROUS OXIDE (l)+XENON (2) AT(1 84.05 0.01) KaHE(Ps) HEW) RHn /mol n2 /mol X1 Q/J /J mol-1 /J mo1-l /J mol-10.020 52 0.053 47 0.2750 58.533 754.0 753.9 - 5.60.023 47 0.044 39 0.3442 59.847 842.8 842.7 6.00.040 12 0.048 36 0.4513 81.273 899.0 898.8 2.40.047 53 0.039 09 0.5455 78.297 890.8 890.7 - 4.60.043 01 0.022 87 0.6469 55.889 838.2 838.1 - 0.70.047 34 0.014 51 0.7562 43.114 706.2 706.1 1.8a Q is the energy supplied to the calorimeter to maintain it at the initial temperature.RH is the enthalpy residual, = H E - HFalc, where HFalc is the excess molar enthalpy calcu-lated from eqn (3).Finally, the values of SE at 182.32 K, derived on the assumption that HE at thistemperature has the same value as at 184.05 K, conform to the equationwith K = 1.1747; L = -0.0805; M = 0.5402.TS; = 453.6 J mol-l.In fig. 1, GE, HE and TSE are plotted against xl, the mole fraction of N20. Thedependence of VE on x1 is shown graphically in fig. 2. All four curves are fairlysymmetrical, the most skewed being that for VE. It will be noted that TSE and VEare both positive and relatively large. The total vapour pressure data are shown inSE/R = ~ l ( l -xl)[K+L(2~1- l)+M(2~1- 1)2] (4)fig. 3.COMPARISON WITH THEORYThe theoretical approach has been fully described in previous papers,4.3 2 * 33 sothat only a brief outline of the method is given here. The Helmholtz free energy Afor the mixture is expanded in powers of the anisotropic part of the intermolecula2500 THERMODYNAMICS OF N,O+Xepotential energy for a pair of molecules of species c1 and p, about the free energyA . for a reference mixture of spherical molecules. The reference potential z& isdefined to be an unweighted average over the orientations of the full potential uaB.800LIII 222 600aw" 400 uw"%2 00I I I I0.2 0 . 4 0.6 0.8x1FIG. 1.-Excess molar Gibbs free energy at 182.32 K and excess molar enthalpy at 184.05 K forN20+Xe, plotted against xl, the mole fraction of nitrous oxide. Points are experimental data,lines are from eqn (5).The dashed line is the experimental excess entropy, obtained from TSE =HE- GE.0.6r(.-. I 8 0.4EL.m20.20 0.2 0.4 0.6 0.8x1FIG. 2.-Excess molar volume YE for N20 + Xe at 182.32 K from experiment (points) and eqn ( 5 ) (line)MACHADO, GUBBINS, LOB0 AND STAVELEY300200 2 24 \ a,100250 1----I 1 I Ix, Y ( N 2 0 )FIG. 3.-Vapour-liquid equilibrium for N20+Xe at 182.32 K from experiment (points) and eqn (5)(solid line). The dashed line is the result calculated using van der Waals one-fluid theory withisotropic n,6 potentials for each of the pair interactions.With this choice of reference the first-order term A l vanishes and the series to third-order is used as the basis for a simple Pad6 approximantThis expression is in good agreement with computer simulation results for dipole-dipole and quadrupole-quadrupole potentials and for anisotropic overlap potentialsof the type used here.Comparisons of theory and experiment therefore provide atest of the intermolecular potential models used. The procedure for calculatingproperties of the reference system was as described by Gubbins and Twu 32 andinvolved an expansion of the n,6 fluid properties about those for a 12,6 fluid, togetherwith the use of van der Waals one-fluid theory to relate the properties of the 12,6mixture to those of a pure 12,6 fluid. The Gosman et al.34 equation of state wasused for the free energy of the pure fluid and the equations of Gubbins and Twu 32were used for the integrals J and K that arise in the A , and A 3 terms in eqn (5).Forthe J' integrals the equation of Nicolas et al.35 was used, since it is more accuratethan that previously given by Gubbins and Twu.For the xenon/xenon interaction the Lennard-Jones 12,6 potential model wasusedUXe/Xe = uowith the parameters given previ~usly.~ This gives an excellent fit to the data for thepure coexisting gas and l i q ~ i d . ~ For the N,O/N,O interaction we initially used thepotential modelwhere ugs6) is the (isotropic) n,6 p ~ t e n t i a l , ~ up, . . . uQQ are the dipole-dipole. . . quadrupole-quadrupole potentials and uov( 101 + 01 1) and Udis( I01 + 01 1) are theleading terms in a spherical harmonic expansion of the anisotropic overlap anddispersion terms ; here 101 and 01 1 are the values in the expansion.Detailed( 6 ) (12,6)UN~OIN~O = Ub",6) + u , p + upQ+UQp+UQQ+Uo,(lol+o1 I)+ UdiS(101+011) (72502 THERMODYNAMICS OF N,O+Xeexpressions for these potentials are given in ref. (4) and (32). Multipole momentsand the anisotropic polarizability value were taken from independent experimentalmeasurements and the remaining parameters (e, a and n in the n,6 potential and theparameter 6 that occurs in uOv) were obtained by fitting the theory to saturated liquiddata in the usual way.4 Eqn (7) was found to give an excellent fit to the data for puregaseous and liquid N20. However, an equally good fit was obtained by omittingthe anisotropic overlap and dispersion terms- U ( n , 6 )and this potential model was the one finally used.Values of the potential parametersfor this model are shown in table 4. The value of the quadrupole moment of- 3.65 x e.s.u. cm2 was obtained by the direct method of magnetic susceptibilityanisotropy and is estimated to be accurate to kO.25 x(8) UN20/N20 - 0 + + upQ + uQp + uQQe.s.u. cm2.TABLE 4.-POTENTIAL PARAMETERS ---P QXe+Xe 231.5 3.961 12 0 0N20+ N2O 261.9 3.771 15 0.166' - 3.656Xe+N20 243.0b 3.881b 13.4 -pair (&/k)/K" a/Aa nu e.s.u. cm e.s.u. cm2--a Like-pair parameters from orthobaric liquid density and pressure. NzO+ N20 para-metersfromeqn(8). Xe+N20parameters&andafrom GF and Vg, II fromeqn(l1). Thesevalues correspond to cXe/N20 = 0.987 and vXe/N20 = 1.004, where 5Xe/N20 a ~ ~ d l y x ~ l ~ ~ o arethe usual parameters in the modified Lorentz-Berthelot rules, &ab = <ab(&aa&bb)' and Gab =iqab(oaa+ ebb).TIKFIG.4.-Orthobaric liquid density of NzO from experiment (points, Couch et aE.22 and this work)and theory (lines). The solid line is based on the Pad6 approximant of eqn (5) with theanisotropic potential model, eqn (8) ; the dashed line is for the isotropic potential model, eqn (9)MACHADO, GUBBINS, L O B 0 AND STAVELEY 25037000500030001000200 240 280T/KFIG. S.-Vapour pressure of N20 from experiment (points, Couch et aLZ2 and this work) and theory(lines). Key as in fig. 4.Comparison of theory (solid line) and experiment for the orthobaric liquid densityand vapour pressure for N20 are shown in fig. 4 and 5. The average deviationbetween theory [eqn (5)] and experiment for the temperature range 182.32 K (triple-point) to 295 K was 1 % for pressure, 0.3 % for liquid density and 2 % for gasdensity.At the critical point itself (309.58 K) the errors in the predicted gas andliquid densities are larger ( z 10 %), since the predicted critical point lies x 0.5 Kabove the experimental value. The predictions of a simple isotropic potential, then,6 model,are also included in fig. 4 and 5 (dashed lines). These curves represent the bestpossible fit to the data using this simpler model. It should be stressed that the samethree adjustable parameters ( E , Q, n) are involved in both the isotropic model of eqn(9) and in the anisotropic model of eqn (8) and the procedure for obtaining theseparameters is identical in the two cases.The anisotropic model is seen to be insubstantially better agreement with experiment and eqn (8) is thus superior to eqn (9)as an effective potential for the pure fluid. Similar comparisons have been made forcarbon dioxide, ethane and ethylene by M a ~ h a d o , ~ ~ with the same conclusion in eachcase. A comparison of theory and experiment 2 2 for pressures of the compressedgas is made in table 5.(9) - ug'6)'N20IN20 -Agreement is within 1 % or better for most points.For the xenon/nitrous oxide interaction a simple n,6 model was usedUXe/N20 = (10)The addition of an anisotropic overlap term uOv( 101) and a quadrupole-induceddipole-quadrupole term uQindQ(O0O) to this potential model resulted in no improve-ment to the fit and these terms were therefore omitted. The value of nXe/N20 wasestimated from the geometric mean rule 37IZXe/N20 = (nXe/XenN20/Nz0)'* (1 1)The values of E ~ ~ / ~ ~ ~ and oXeINZ0 were obtained by requiring agreement betweentheory and experiment for GZ and V: and are included in table 42504 THERMODYNAMICS OF N,O+XeTABLE 5.-cOMPARISON OF THEORY AND EXPERIMENT FOR GASEOUS N20, SHOWN AS THEPRESSURE P AT WHICH THE GAS HAS THE DENSITY pP/kPaT K plrnol dm-3 expt 22 calc.243.15243.15258258.15273.15273.15288.15288.15288.15303.15303.15303.15303.15348.15348.15398.15398.15398.15423.15423.15423.150.32180.70290.29931.1940.48271.8520.451 11.2782.9780.42431.4633.0435.17415.0717.653.8959.4823.4477.94113.5211.7860812166082 02710143 04110142 5344 46010143 0415 0686 28420 27130 40710 13520 27130 40710 13520 27130 407612123761 12 06510193 09010172 5544 50410163 0625 0906 23020 68731 48110 08920 16030 70110 10020 11930 508A comparison of theory and experiment for the mixture data is shown in fig.1-3.Since the experimental values of GF and V: are used in fitting parameters, the com-parison tests the ability of the theory to correctly predict the HE curve and the shapesof GE and VE. Agreement between theory and experiment is excellent for GE andVE; for HE the theory predicts values that are z 4% too high for the equimolarmixture. Good agreement is obtained for the pressure values (fig.3) and the azeo-trope is correctly predicted. The pressure values predicted assuming simple n,6isotropic potentials for all three interactions are also included in fig. 3. In thesecalculations the properties of the n,6 mixture are related to those for a 12,6 mixturein the usual way 3 2 and the properties of the latter are calculated from the van derWaals one-fluid theory ;3 values of n, E and c for each pair interaction were obtainedby the same procedure as for the anisotropic potential models described above. Asseen from fig. 3 the anisotropic potential model gives considerably better agreementwith experiment than the isotropic n,6 model alone. The excess properties were alsocalculated using these simple isotropic potential models.Good results were obtainedfor VE and for H: (both G: and V: were again used to fit parameters, so that theoryand experiment must always agree for these properties). However, the predictedHE and GE curves were not of the correct shape.CONCLUSIONThe thermodynamic data reported here for N,O+Xe mixtures, as well as theexisting data for pure N 2 0 and Xe ,are in good agreement with theoretical predictionsusing simple intermolecular potential models. The calculations indicate that thMACHADO, GUBBINS, LOB0 AND STAVELEY 2505effect of electrostatic forces is significant, but that these may be successfully approxi-mated by a multipole series terminated at the quadrupole-quadrupole term.Sincemultibody potential terms are omitted in these calculations the potential modelsshould be regarded as effective pair potentials suitable for the liquid phase. More-over, potential terms for anisotropic overlap (shape), anisotropic dispersion andinduction forces, found to have a negligible effect in the calculations given here,may be significant in calculations over a wider range of temperature and pressure.Despite these reservations, the anisotropic potential models developed here are asubstantial improvement over isotropic potential models with the same number ofadjustable parameters.Xe + N 2 0 mixtures offer the simplifying features that one of the components (Xe)is spherical, while the other (N20) is linear and has only a weak dipole moment.Weare in the process of studying the mixtures N20+C2H, and N,O+HCI and shallreport results for these shortly. Ethylene is non-polar, but possesses a non-axialquadrupole moment (Le., the quadrupole moment has two independent components),while hydrogen chloride has a relatively large dipole moment.This work was supported by grant ENG 7682101 from the National ScienceFoundation. 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