Thermodynamics of n-Alkane+Dimethylsiloxane Mixtures Part 3.-Excess Volumes B Y ERIC DICKINSON AND IAN A. MCLURE* Department of Chemistry, The University, Sheffield S3 7HF Received 24th April, 1974 Excess volumes have been measured dilatometrically for binary mixtures of six n-alkanes (n- pentane, n-hexane, n-heptane, n-octane, n-decane and n-tetradecane) with four linear dimethyl- siloxanes (dimer, trimer, tetramer and pentamer) at 303.2 K, and for four other mixtures. The sign and magnitude of the excess volumes depend intimately upon the chain lengths of the oligomers. The phenomenological corresponding states theory of Patterson is shown to reproduce qualitatively the experimental chain length dependence. Suggestions for the success of the model are discussed briefly in terms of the properties of the pure components, and the disadvantages of two other ap- proaches-the van der Waals one-fluid model and the Prigogine cell theory-are identified.An application of Brsnsted’s principle of congruence to ternary mixtures is discussed. In previous papers measurements of vapour pressures and gas-liquid critical temperatures for mixtures of the type n-alkane + linear dimethylsiloxane were analysed in terms of the relative strengths of the unlike pairwise interaction energies. We now present the results of measurements of excess volumes of mixing for a large number of these systems and attempt to show that their chain length dependence may be interpreted in terms of a simple corresponding states approach. Linear dimethylsiloxanes are completely miscible with n-alkanes over a broad range of temperature and pressure.In binary mixtures where the components are of very different molecular size, phase separation is observed 2 * 3 above a lower critical solution point (near to the gas-liquid critical temperature of the more volatile component). At low reduced temperatures, however, away from the gas-liquid critical region, there is complete miscibility. Hence, since each dimethylsiloxane oligomer (up to the high polymer) is liquid at ambient temperatures and pressures, the limiting factor in an isothermal study of this type is the shorter liquid range of the n-alkanes. At 303.2 K and 1 atm pressure, we are restricted to paraffins having between 4 and 19 carbon atoms in the chain. Mixtures involving the following constituents are considered here : n-pentane, n-hexane, n-heptane, n-octane, n-decane, n-tetradecane and n-octadecane ; hexamethyldisiloxane (dimer), octamethyltrisiloxane (trimer), decamethyltetrasiloxane (tetramer) and dodecamethylpentasiloxane (pen- tamer).EXPERIMENTAL The excess volumes of mixing were measured using the continuous dilution dilatometer described p r e v i ~ u s l y . ~ Isothermal compressibilities, required to correct the observed volume changes for the effects of variations in the hydrostatic pressure, were taken from the work of Orwoll and Flory and Sadler.6 The dimethylsiloxanes were obtained by fractionally distilling the appropriate Hopkin and Williams MS 200 silicone fluids. All final samples had a purity of better than 99.0 mol % as indicated by gas-liquid chromatography with a 2 m silicone rubber (SE 30) stationary phase, The samples of n-pentane (99.5 mol %), n-hexane (99.0 mol %), n-heptane (99.5 rnol 2328E .DICKINSON A N D I . A . MCLURE 2329 %) and n-decane (99.0 mol %) were obtained from British Drug Houses and were used as received. The Newton Maine n-octane and n-octadecane, and the Phillips n-tetradecane, contained less than 1.0 mol % impurity. RESULTS Excess volumes were determined at 303.2 K over the whole composition range for the mixtures of six n-alkanes (n-pentane, n-hexane, n-heptane, n-octane, n-decane and n-tetradecane) with four linear dimethylsiloxanes (dimer, trimer, tetramer and pentamer). Also investigated were the systems n-heptane + pentamer at 323.2 K, TABLE 1 .-EXCESS VOLUME PARAMETERS A i (IN EQUATION 1) FOR n-ALKANE+ DIMETHYL- SILOXANE MIXTURES AT TEMPERATURE T.bv IS THE STANDARD DEVIATION ; np IS THE NUMBER OF EXPERIMENTAL POINTS. n-alkanes (1) + hexamethyldisiloxane (2) component 1 n-pentane n-hexane n-heptane n-octane n-decane n-tetradecane n-oct adecane component 1 n-pentane n-hexane n- hep tane n- oc tane n-decane n-te tradecane n-tetradecane TI K 303.2 303.2 303.2 303.2 303.2 303.2 303.2 A0 0.128 0.112 - 0.107 - 0.26 I -0.915 - 1.459 - 1.903 A1 - 0.043 0.001 0.005 - 0.009 -0.189 - 0.299 - 0.540 A2 - 0.010 0.049 - 0.025 - 0.023 - 0.020 - 0,033 - 0.224 n-alkanes (1)+ octamethyltrisiloxane (2) T/K 303.2 303.2 303.2 303.2 303.2 303.2 323.2 A0 0.050 0.177 0.03 1 - 0.059 - 0.407 -0.871 - 1.234 A1 - 0.01 6 0.000 - 0.030 - 0.057 - 0.057 -0.204 - 0.103 A2 - 0.020 0.064 - 0.014 0.019 0.027 0.008 0.096 n-alkanes (1)+ decamethyltetrasiloxane (2) component 1 TI K A0 A1 A2 n-pentane 303.2 -0.111 0.020 0.013 n-hept ane 303.2 0.129 0.014 -0.044 n-decane 303.2 -0.150 0.028 -0.044 n-hexane 303.2 0.149 0.049 0.040 n-oct ane 303.2 0.051 0.047 0.031 n-tetradecane 303.2 - 0.535 -0.102 0.002 n-pentane/ n-decane 303.2 0.167 0.018 0.032 =V 0.001 0.002 0.002 0.004 0.003 0.003 0.003 =V 0.002 0.002 0.001 0.002 0.002 0.001 0.004 OV 0.001 0.001 0.002 0.001 0.002 0.001 0.001 n-alkanes (1) + dodecamethylpentasiloxane (2) component 1 TIK A0 A1 A2 0, n-pen t ane 303.2 -0.194 0.026 0.044 0.002 n-hexane 303.2 0.146 0.034 -0.002 0.003 n-hep tane 303.2 0.187 0.031 0.007 0.001 n-hept ane 323.2 0.188 0.056 0.015 0.002 n-octane 303.2 0.136 0.O00 0.082 0.002 n-tetradecane 303.2 - 0.341 - 0.028 0.005 0.002 n-decane 303.2 -0.023 -0.002 0.007 0.001 UP 11 11 10 12 7 13 14 nP 9 11 9 11 8 11 7 nP 11 10 11 10 10 11 9 nP 10 11 6 9 9 10 102330 THERMODYNAMICS OF TI-ALKANE -k DIMETHYLSILOXANE MIXTURES n-tetradecane + triiner at 323.2 K, n-octadecane + diiner at 303.2 K, and n-pentane/ n-decane + tetramer at 303.2 K.The fractional volume changes A V/ V for each mixture were fitted by least squares ’ to an equation of the form -0.4 I I I I I I I I 1 I I I where #2 is the volume fraction of component 2.* Table 1 lists the coefficients Ai, the standard deviation gV and the number of experimental points np. (Molar excess volumes YE, if required, may be calculated using the densities of Orwoll and Flory and Pretty.*) The values of oV fluctuate within the range 0.001 to 0.004%, but it should be noted that since standard deviations derived from dilution dilatometry, especially those from a single run, tend to overestimate the intrinsic precision of the data, the actual accuracy may be slightly less than that implied by table 1.Fig. 1 shows a plot of the fractional volume change AV/V at volume fraction 0.5 against the chain length of the n-alkane component of the mixture. Solid curves are drawn through the sets of points corresponding to a given siloxane. 0.1 1 DISCUSSION Any theory of chain molecule liquid mixtures could be reasonably expected to reproduce the following features of the experimental data : (a) the increasingly negative excess volumes as the relative size differences between the two components become (i) Fractional volume changes and volume fractions are more closely related to the actual experimental quantities than excess volumes per mole and mole fractions.(ii) A plot of the last two variables produces a very skewed curve for many of these mixtures. (iii) Excess volumes and compositions computed on a molar basis are entirely inappropriate for comparing the present results with those for mixtures containing polymeric alkanes or siloxanes, * We express the results in this form for the following reasons.E . DICKINSON AND I . A . MCLURE 233 1 larger ; (6) the small but positive excess volumes with hydrocarbons of intermediate chain length; (c) a distinct "cross-over point" in the individual siloxane curves at around hydrocarbon number n = 6.With these criteria in mind, we discuss in detail the phenomenological corresponding states treatment developed by Patterson, and compare the conclusions with those resulting from two other approaches, the simple van der Waals model and the Prigogine cell theory. CORRESPONDING STATES THEORY OF CHAIN MOLECULE LIQUID MIXTURES According to Prig~gine,~ the configurational thermodynamic properties of an oligomeric liquid are related to the corresponding reduced quantities by reduction parameters which depend only upon the chain length n. With the exclusion of any combinatorial term, the molar Gibbs free energy G,, the molar volume V, and the molar entropy S, are given at temperature T and pressure p by the equations G,(p, T, n) = G(p, T)U*(n), U*(II) = N , g ( n ) ~ ; (2) V,(p, T, i z ) = V(p, T)V*(n), (3) S,,,(p, T, n) = S(p, T)S*(n), S*(n) = IVAc(n)k.(4) V*(n) = NAr(n)a3 : The dimensionless reduced quantities G, Pand are functions of the reduced tem- perature Tand the reduced pressure p . The reduction parameters U*, Y*, and S* are related to an arbitrary spherical reference molecule,' with characteristic energy c and size a, by the effective numbers of segments q(n), r(n) and c(n), which are propor- tional respectively to the molecular surface area, the molecular volume and the number of external degrees of freedom of the chain molecule. The temperature and pressure reduction factors, T* = T/T and p* = p / p , are given by T*(n) = U*(n)/S*(n) = q(n)&/c(n)k, p'k(n) = uyn>/ V*(17) = q(n)&/r(n)a3.( 5 ) (6) In terms of the phenomenological formulation of the corresponding states the molar principle, as introduced by Hijmans volume of the mixture is expressed in the form where (p") and (T) are the reduced pressure and temperature of the mixture, and (V*) is the average volume reduction parameter. For a binary mixture a t negligible pressure, the excess volume is and developed by Patterson,". Vrn(p, T ) = V(<jj>, < T>)< v*>, (7) V: = ( x V: + x v;> P( ( T) j - x VT P( T, ) - x2 V ; V( T2). (8) In eqn (8), ( V * ) has been replaced by a mole fraction average of V$ and V:. (T) is related to TI and 4; by the surface fraction average where XI is the surface fraction of component 1 : (T) = T/(T*) = XITl + XzT*, (9) X I = x,pTV"f(X,pT~T+x,pqV;).(10) Expansion of the reduced volume of each pure component about that of the mixture yields the equation VE = - (~1x2 VT - x ~ X , Vq)AT(d P/d( T)) - (+)(XI X i V t - N ~ X : V;)(AQ2(d2 Pjd( T ) 2 ) - ( & ) ( x ~ X ~ V T - X.~X:VZ)(AT)~(~~ P/d( Q3),2332 THERMODYNAMICS OF n-ALKANE f DIMETHYLSILOXANE MIXTURES where terms of order (AT)" = (TI - T2)" or higher are ignored. gous to eqn (1 1) may be developed for the other excess functions. Expressions analo- REDUCTION PARAMETERS A N D REDUCED EQUATION OF STATE Before employing an explicit expression such as eqn (1 1) to calculate the excess mixing properties of a group of conformal substances, it is necessary first to define a self-consistent set of reduction parameters and a reduced equation of state.In testing the corresponding states principle from methane to polymethylene, Patterson and Bardin l 3 evaluated reduction factors for the n-alkanes. To facilitate comparison with the n-alkane analysis, we use the same procedure here for the linear dimethyl- siloxanes (the details are discussed elsewhere 14). TABLE 2.-RELATIVE VOLUME AND TEMPERATURE REDUCTION PARAMETERS OF THE LINEAR DIMETHYLSILOXANES dimethylsiloxane oligoiner V*/ V* T*/T; dimer 1.259 0.H2 trimer 1.746 0.943 tetramer 2.228 0.992 pentamer 2.723 1.025 The volume and temperature reduction parameters listed in table 2 were derived from the molar volumes and isobaric expansivities of the dimethylsiloxanes.8 Follow- ing Patterson et aZ.,13 n-octane is taken as the arbitrary reference substance (with reduction parameters Vo* and To*).The n-alkane pressure reduction factors are essentially independent of chain length, whereas an analysis of the thermal pressure coefficient data of the dimethylsiloxanes suggests that their relative pressure reduc- tion factors p*/p$ decrease with increasing chain length. l4 However, until this effect is substantiated by measurements of dimethylsiloxane p , V, T properties over a much larger temperature range, it seems reasonable to stay with an average value of p*/p$ = 0.804.-f- The temperature derivatives of the reduced volume are derived from the experi- mental data of the pure components 5 9 6- * through the relations : dV/dT = apVT*, (12) (1 3) d2 V/d T2 = VT*2(du,/dT) + ( I /V)(dV/dT)2, d3V/dT3 = 2T*2(dV/dT)(d~,/dT) +(I /V)(dV/dT)(d2V/dT2)+ VT*3(d2ap/dT2), (14) where ap is the isobaric expansivity.The dimensionless quantity (a,T)-l appropriate to the mixture is taken as the arithmetic mean of the values of (ct,T)-l which each of the pure components would have at a reduced temperature (T} equal to that of the mixture. [The pure component values of (a,T)-l should, of course, be identical for substances obeying the same principle of corresponding states.] The values of CI,, (dcr,/dT) and (d2m,/dT2) are taken from the work of Orwoll and Flory' and Pretty.' COMPARISON OF EXPERIMENT A N D THEORY The theoretical excess volumes, as calculated from eqn (1 1) at 303.2 K, are Fractional volume changes AV/V for the equimolar mixtures illustrated in fig.2. I- The general conclusions are unaffected by small changes in the parameter p * . Since the aim is not to obtain perfect agreement with experiment, the exact values are not very important,E . DICKINSON AND I . A . MCLURE 2333 are plotted against the chain length of the hydrocarbon. As in fig. 1, each curve represents the separate mixing of an individual siloxane with a series of n-alkanes. A comparison of experiment and theory reveals several notable features. The sign and magnitude of the experimental and calculated excess volumes depend intimately upon the relative chain lengths of the components. With regard to the 0.2 0. I 0 - 0.1 -0.2 - 0.3 kh a 2 - 0.4 - 0.5 - 0.6 3 4 5 6 7 8 9 10 I I 12 13 1 4 1 n FIG. 2.-Volume changes of n-alkane + dimethylsiloxane mixtures as predicted by the Patterson phenomenological treatment.The fractional volume change A V/ V at mole fraction 0.5 is plotted against the chain length n of the alkane component of the mixture. Solid curves are drawn through the sets of points corresponding to a given siloxane : 0, dimer ; A, trimer ; 0, tetramer ; V , pentamer. calculated volume changes, the form of the curves arises solely from the relative sizes of the reduction parameters ; it is not the product of some judiciously chosen adjustable parameter. The array of curves in fig. 2 shows good qualitative agreement with the gross features of the experimental data (fig. l), but the calculated excess volumes are generally larger than those observed experimentally.Especially noteworthy is the prediction of a “cross-over region” around IZ = 5 : the experimental lines appear to intersect at n z 6 . Again, however, the magnitude of AV/Vis overestimated, at this point by a factor of about 2+. Clearly, the phenomenological treatment outlined above offers neither a good fit nor a rigorous interpretation of the excess volumes of n-alkanes + linear dimethyl- siloxanes. It does demonstrate, however, that the apparently complex volumetric behaviour of these mixtures is usefully described by a simple application of the principle of corresponding states. Other models, such as the van der Waals approxi- mation,l tend to reproduce badly the excess volumes of these systerns,1’14 especially for those mixtures containing oligomers whose molecular sizes are very different.Although the curves calculated from the van der Waals equation exhibit l4 an apparent periodic dependence upon the chain length, it is one with turning points corresponding to the mixtures in which the sizes of the two different chain molecules approach equality, e.g. the mixtures dimer + n-tetradecane, etc. No positive volume changes are predicted. Neither is there a tendency for the theoretical curves to intersect at a given hydrocarbon number as is observed experimentally. Whilst positive volume changes can be produced from the van der Waals equation by invoking appropriate2334 THERMODYNAMICS OF n - A L K A N E -I- DIMETHYLSILOXANE MIXTURES deviations from the Berthelot combining rule, the artifice does nothing to satisfy any of the other objections.The qualitative success of the Patterson approach may be interpreted graphically. A plot of reduced volume against reduced temperature is invariably concave upwards (see fig. 3), thereby leading to negative volume changes for binary mixtures where y2 (T)! (T)” T I T FIG. 3.-The dependence of the reduced volume of a mixture ?(<?)) ypon the reduced temperature { f>. For the case (f> = <i”>’, A V/ V is negative ; alternatively, if (2‘) = ( T ) ” , A V/V is positive. p:=pZ. Sets of mixtures falling into this category are the binary systems formed from two n-alkanes,I6 two dimethylsiloxanes or two perfluoro-n-alkanes. For these systems the surface fraction XI is approximately equal to the volume or segment fraction $1, and, in fig.3, the reduced volume of the mixture V((F)’) lies almost di- rectly below the reduced volume of the unmixed pure components, r( .fl) + +2 v(.f2), since <T>Z41Tl+$21;. (15) Conversely, for mixtures of conformal substances with different pressure reduction factors, positive volume changes are possible if The inequality (16) is most likely to be satisfied if I TI - 7’1 is small. In comparison with their hydrocarbon analogues, the high reduced temperatures of the dimethyl- siloxanes decrease only slowly with increasing chain length-a feature which has been attributed l 8 to the greater siloxane chain flexibility. Positive excess volumes occur with components having similar reduced temperatures, but in mixtures of dimethylsiloxanes with n-alkanes of extreme chain length (large or small) negative volume changes are expected, and found.If we suppose further, that the individual n-alkane and siloxane molecules are subdivisible into arbitrary identical segments with characteristic parameters E and 0, then, from eqn (6), a siloxane molecule has associated with it a valuz of ( q / r ) larger than that for a n-alkane. This might be interpreted on a molecular basis by postu- lating that, due to shielding by the peripheral methyl groups, a -SiO - siloxane seg- ment makes fewer “extcrnal contacts” than does a --CH2CH2- alkane segment. V( ( T ) ”) = V( x1 TI + x2 T2) > (b 1 V( TJ + 4 2 V( T2). (16)E . DICKINSON A N D I . A . MCLURE 2335 The parameter c has no direct influence on the sign of V i ,but it does, through the ratio (c/q), determine the values of the pure component reduced temperatures. As the chain length dependence of (c/q) is much smaller for the dimethylsiloxanes than for the n-alkanes, much smaller volume changes are expected for binary mixtures of the former, a prediction which is borne out by experiment.6 THE PRIGOGINE AVERAGE POTENTIAL CELL MODEL Prigogine’s average potential approach for chain molecule mixtures involves a model of the liquid state composed of a lattice of cells whose volume varies as a function of the temperature, pressure and composition.Each oligomer interacts with neighbouring “segments” of adjacent oligomers via qz contacts, where z is the co-ordination number of the quasi-lattice. If the two different chain molecules are considered to be composed of equal sized segments interacting identically with segments of adjacent molecules according to a well-defined pair potential (that is, if, in the Prigogine nomenclature, p = 6 = 6 = 0), then the species will differ only in their relative values of the parameters q(n), r(n) and c(n).In testing the applicability of this simplified Prigogine model to the n-alkane+ dimethylsiloxane mixtures, we consider three different but related procedures for generating the chain length dependent parameters. In the first method, r(n) is calculated by arbitrarily designating a segment to be one of the units -CH,, -CH2CH2-- or -SiO-, each of which is thereafter taken to be of the same size. The parameter q(n) is derived from the relation (17) with z = 10, and the values of c(n) are taken from the corresponding states analysis of Simha and H a ~ 1 i k .l ~ The pure n-alkane component of the mixture is used as reference substance. A comparison of the experimental volume changes with those obtained by this procedure reveals that, with the exception of mixtures involving hexamethyldisiloxane and certain of the lower n-alkanes, the excess volumes predicted from structural effects alone are an order of magnitude smaller than those observed experimentally. The relative magnitudes of the predicted volume changes are, however, in fair agreement with experiment. The correct order is obtained at the extremes of hydrocarbon chain length, and a reversal is implied (with the exclusion of the dimer) in the region n-octane to n-decane.The form of the results is insensitive to the exact value of the co-ordination number z. We use, in the second method, values of r(n) and c(n) which are in compliance with the thermodynamic reduction parameters V*(n) and S*(n) defined in eqn (3) and (4). Expressed as linear dependences upon the chain length, these are given by 13.14 q ( 4 = KZ - 2)/zIr(n) + (2/4 rI = (nl+0.84)/2.0, (1 8) r2 = (n, + 0.16)/0.929, (19) ~1 = ( ~ 1 + 6.0)/2.006, (20) c2 = (n, + 2.02y0.873, (21) where the subscripts 1 and 2 refer to the n-alkane and dimethylsiloxane components respectively. The same reference substances as before are employed, and q(n) is again defined by eqn (17) with z = 10. The excess volume for each mixture is pre- dicted to be always small and negative irrespective of the relative chain lengths of the components.This feature is a direct consequence of the similarity in the values taken2336 THERMODYNAMICS OF n-ALKANE + DIMETHYLSILOXANE MIXTURES by the ratios (ql / r , ) and (q2/r2) : that is, it is due to the fact that the parameter p*(n) is nearly the same for both o1igomers.T As stated previously, it might be expected on simple molecular grounds that - --SO- .- siloxane segments would make significantly fewer external contacts (due to shielding by peripheral methyl groups) than -CH2CH2- n-alkane segments. To take some account of this effect in the third method, we restrict each middle siloxane segment to one less contact than its hydrocarbon counterpart by use of the relation and keep all other conditions identical to those of the previous procedure. The results of this manoeuvre are shown in fig.4 : small positive and negative chain length q2 = MZ - 3)/Zi+ (414, (22) 6 a 10 12 14 16 n FIG. 4.-Volume changes calculated from the simplified Prigogine cell theory. The fractional volume change A V/ Y at volume fraction 0.5 is plotted against the chain length n of the alkane com- ponent of the mixture. Solid curves are drawn through the sets of points corresponding to a given siloxane : 0, dimer ; A, trimer ; 0, tetramer ; V, pentamer. dependent volume changes, together with a “cross-over point” (AV/V = 4 x ; nl = 6) in excellent, but almost certainly fortuitous, agreement with experiment. p ; is now significantly greater than p ; ; or, in molecular terms, (q/r), >(q/r)2.The number of external degrees of freedom, 3c, enters the mixing expression through and large differences in molecular flexibility tend to increase ITl - T21 thereby pro- ducing a larger negative VE. Thus it is seen that, if the appropriate values of q(n), ~ ( n ) and c(n) are inserted into the Prigogine cell m0de1,~ predictions similar to those of fig. 2 can be generated. However, the cell theory equations are based upon a quasi-lattice partition fun~tion,~ and many assumptions of questionable validity have to be made in the specification of relative segment sizes and the lattice co-ordination number. This apparently large degree of arbitrariness, as applied to mixtures of dissimilar chain molecules, limits the usefulness of the cell theory in interpreting the thermodynamic behaviour f If c is set equal to unity throughout, the system simplifies to a set of binary mixtures of rigid rods of different lengths.The associated volume change, negative for all values of n1 and n2, arises out of the extra contracts at the end of a chain molecule over those in the middle. So, even for a mixture in which 6 = 0 = 0, the structural factor conceals inherent “ energetic effects ” in the form of the parameter q.E. DICKINSON AND I . A . MCLURE 2337 of this set of systems. However, only very unreasonable choices of q(n), r(n), c(n) and z alter the basic qualitative picture. THE TERNARY MIXTURE n-PENTANE + n-DECANE + DECAMETHYLTETRASILOXANE Brransted’s principle of congruence 2o states that the thermodynamic properties of a mixture of homologues are identical, with the exception of the Gibbs paradoxical term, to those of a molecule of average chain length ii = C xini, 1 where x i is the mole fraction of the species with chain length ni. The principle implies, for instance, that an equimolar mixture of n-pentane + n-decane has the same molar volume as a pseudo-n-alkane of chain length 6 = 7.5.Hence, although n-pentane and n-decane each give a negative equimolar excess volume with deca- methyltetrasiloxane (the values at 303.2 K are -0.042 and -0.099 cm3 mol-l respectively), a mixture with the pseudo-n-alkane of chain length 7.5 should at the same temperature exhibit a positive volume change, in line with any realistic interpola- tion of the n-heptane and n-octane points of fig. 1. The prediction is fulfilled qualitatively with V:(xs = 0.5) = Vnl(x, = 0.5, x5 = 0.25, xlo = 0.25) -x,Vm(xs = 1) -(1 -x,)Vm(x5 = 0.5, xlo = 0.5) equal to (0.098 & 0.007) cm3 mol-l, where x,, xs and xi0 are respectively the mole fractions of decamethyltetrasiloxane, n-pentane and n-decane.The interpolated value from fig. 1 has a positive value of (0.053 & 0.005) cm3 rnol-l, although the excess volume relative to the three unmixed pure components is still, of course, negative. We thank Miss C. W. Green for her preliminary experimental work on this prob- lem. E. D. acknowledges receipt of a S.R.C. Studentship. E. Dickinson, I. A. McLure and B. H. Powell, J.C.S. Faraday I, 1974, 70, 2321. P. I. Freeman and J. S. Rowlinson, Polymer, 1959, 1, 20. D. Patterson, G. Delmas and T. Somcynsky, Polymer, 1967, 8, 503. E. Dickinson, D. C. Hunt and I. A. McLure, J. Chem. Thermodynamics, submitted for publica- tion. R. A. Orwoll and P. J. Flory, J. Amer. Chem. Soc., 1967, 89, 6814. P. A. Sadler, Ph.D. Thesis (Sheffield, 1971). A. J. Pretty and I. A. McLure, unpublished results. I. Prigogine, The Molecular Theory of Solutions (North-Holland Amsterdam, 1957), chap. 16. ’ D. B. Myers and R. L. Scott, Ind. Eng. Chem., 1967, 55, 43. lo J. Hijmans, Physica, 1961, 27, 433. l1 D. Patterson, Rubber Chem. Technol., 1967, 40, 1. l2 D. Patterson and G. Delmas, Disc. Faraday SOC., 1970, 49, 98. l3 D. Patterson and J. M. Bardin, Trans. Faraday SOC., 1970, 66, 321. l4 E. Dickinson, Ph.D. Thesis (Sheffield, 1972), chap. 9. l 5 T. W. Leland, J. S. Rowlinson and G. A. Sather, Trans. Faraday SOC., 1968, 64, 1447. l6 J. S. Rowlinson, Liquids and Liquid Mixtures (Butterworth, London, 2nd edn., 1969), chap. 4. l7 E. E. Erickson, M.S. Dim. (UCLA, 1969). lB D. Patterson, S. N. Bhattacharyya and P. Picker, Trans. Faraday SOC., 1968, 64, 648, l9 R. Simha and A. J. Havlik, J. Amer. Chem. SOC., 1964, 86, 197. 2o J. N. Brsnsted and J. Koefoed, Kgl. Danske Vid. Selsk. (Mat.-Fys. Medd.), 1946, 22, 17. 1- -74