J. Chem. SOC.,Faraday Trans. 2,1983,79, 41-55 Phase Equilibria in Binary Mixtures Part 1.-Miscibility Gap with Upper or Lower Critical Point BY FRANTISEKVNUK School of Metallurgy, The South Australian Institute of Technology, P.O. Box 1, Ingle Farm, South Australia 5098, Australia Received 14th April, 1982 The general applicability to binary systems of a recently presented, and here extended, new model of liquid-liquid phase equilibria is demonstrated for n-heptane +acetic anhydride, methanol + cyclohexane, mercury +gallium, sodium +ammonia and polystyrene +cyclohexane. New relationships are derived which accurately describe the two-phase equilibria in binary mixtures not only in the critical region but over the whole range of coexistence. By the use of new composition coordinates and order parameters (which are introduced and discussed) the model reveals some novel interrelationships between the coexisting phases.It is shown that the composition and properties of the coexisting phases are mutually linked by the inherent symmetry features of these systems. 1. INTRODUCTION It has been established experimentally that the physical and thermodynamic properties of coexisting phases in a binary system, made up of components A and B, obey the following relations:' as the critical temperature, T,, is approached. In these expressions X1and X2 are the compositions of the coexisting phases (in terms of the component A), a and p are the so-called critical exponents, E: is the reduced temperature I( T,-T)/T,I,Klo and K20 are proportionality constants and X, is the critical composition to which both X1and X2 converge as T -+T,.Relations (1) and (2) hold only at temperatures within the so-called critical range, i.e. a few degrees below T,. For temperatures further away from the critical point it was postulated that additional terms should be added to eqn (1)and (2), which now take the forms and In a new model of phase behaviour in binary mixtures proposed recently3 it was shown that the compositions of the coexisting phases can be conveniently described not only in the critical region but over the whole range of coexistence 41 PHASE EQUILIBRIA IN BINARY MIXTURES by the relations where (7) and The constants Q and M in eqn (7)and (8)are related to Kloand K20in eqn (1)and (2), while n =p and p = 1-a.They can all be obtained directly from the experimental data, as shown in section 5. However, to achieve this extended range of validity, one must express the composition coordinates in a more appropriate form. 2. NEW COMPOSITION PARAMETERS In common practice the composition parameters of the coexisting phases, X, are expressed in molar fractions (x), volume fractions (4) or mass fractions (w) defined by and where A and B are the compositions of A and B, respectively, expressed in molar, volume or mass quantities and AB(xf +xH) = (4: +4B)= (w, +w, ) = 1. One can convert mole fractions (x) into volume (4)or mass fractions (w)via the well known relation where for mole fraction and for mass fraction, pA, pB and MA,MBbeing the densities and molecular weights of the components A and B, respectively.As the result of this non-linear correlation expressed in eqn (11) the shape of the binary phase diagram is unevenly altered when replotted from the (T-x) to (T-4)or (T-w)coordinates (see fig. 1). Once it was thought that the coexistence F. VNUK 43 I I I 1-0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09 1.0 x, atomic fraction (full line) or w, mass fraction (broken line) of Hg FIG.1.-Temperature-composition phase diagram of mercury and gallium. The shape of the miscibility gap in the liquid state is more symmetrical when the compositions are expressed as atomic fractions than when expressed as mass fractions.The points show the experimental data of Predel,I3 the full line represents the calculated values given by eqn (34) and (35). curve was more symmetric in volume fraction than in mole fraction, but experience has shown that there is no simple rule about this. Since (x 1-xz) # (41 -&) # (w -w2), except at T = T,, the constants xc, Klo, K20,/3 and a in eqn (1) and (2) take different values with different composition parameters. These shortcomings can be removed if one expresses the composition parameters as and where A and B are expressed, as before, in molar, volume and mass quantities, respectively. With this new set of composition parameters the conversion from molar to volume or mass units (and vice versa) is now a simple and straightforward operation, i.e.ai,Wi =KiCi i = 1,2;j = 4,w. (15) The symmetry of phase diagrams then remains unchanged irrespective of the choice of C, @ or W as the composition coordinates. In fact, we have 16) PHASE EQUILIBRIA IN BINARY MIXTURES and so that the adjustable constants M, n, Q and p in eqn (5) and (6) remain the same in all three systems of units, while @o, Wo=KjCo j=@, W. (18) The conventional fractional units X, # and w are related to the new compositional units C, @ and W by a simple relation 3. ORDER PARAMETER The present model assumes that in a state of complete miscibility the interactions (or contacts) between the like and unlike molecules of A and B are energetically favoured to the same extent.Below T, (or above T, in systems with a lower critical temperature) the like interactions, A-A and B-B, are energetically more favoured than the unlike interactions, A-B, with the result that the molecu!es -4 and B tend to be surrounded more by like than by unlike molecules and the once homogenous mixture separates into an A-rich and a B-rich phase. As the tem- perature interval IT,-TI increases, the interactions between similar molecules become more and more likely, leading eventually to complete unmixing. Under such conditions the order parameter P can now be defined as where rzl is the fraction of like interactions, A-A and B-B, nu is the fraction of unlike interactions, A-B, in the mixture. For a miscibility gap with an upper critical point we have: P=O atT>T, O<P<1 atO<T<T, and generally P r= 1for T << T,.As pointed out by Bragg and Williams4 and Borelius' the order parameter specified in this manner has a temperature dependence where F(nl,nu) is the energy required to replace two unlike groupings A-B with two like ones, A-A and B-B. If F(nl,nu)/k is replaced by M(T,-T)"T('-")and substituted in eqn (8) one obtains f(T)= 1-P(T) (22) where 1/2 P(T)=tanhM($-1)" =1-(5)c2 F. VNUK 45 is the newly defined order parameter. Since P(T) has the same numerical value irrespective of the choice of units for the composition coordinate. 4. SYMMETRY Symmetry features are one of the most appealing aspects of the relationships pertaining to coexisting phases in the critical region.Both the classical and the lattice gas version of the Ising model postulate perfect symmetry in properties of the coexisting phases so that, e.g. in gas-liquid systems, there should be, at all temperatures, the following density relationship: Pc-Pg =PI -pc. (24) In practice this is not so and these symmetry features, if present at all, are confined to a very narrow temperature region in the vicinity of the critical point.6 With binary liquids the situation is even less satisfactory7 and the symmetry requirements are absent even in the critical region, i.e. for mole fraction x 1x1-xcl# Ixc-x2I. (25) Several methods have been suggested to convert the asymmetric shape of the coexistence curve into a symmetric one.? It was claimed that improved symmetry can be obtained if the composition variable is expressed in volume fraction instead of mole fraction.’ While this was the case with many binary mixtures, opposite behaviour has been observed in others.” The present model does possess very distinct symmetry features but on a different level.As implicit in eqn (10) and (11)the two branches of the coexistence curve are completely symmetric with respect to the newly defined “rectilinear diameter” dT).This new kind of symmetry becomes apparent when the phase diagram is replotted with its composition coordinate in 1nC as shown in fig. 2. One can further improve this representation and make both branches of the coexistence curve completely symmetric with respect to C, or W = 1.This is achieved by expressing the composition variable as where x1, 41, w1 and x2, 42, w2 are the molar, volume and mass fractions of the coexisting phases at equilibrium. t Malesinska’ succeeded in converting asymmetric coexistence curves into symmetric ones by expressing the compositions in modified units where q2/ql is a slightly temperature dependent “asymmetry function”. PHASE EQUILIBRIA IN BINARY MIXTURES 340 * 330. In C FIG. 2.-Coexistence curve of n-heptane +acetic anhydride. The compositions are represented in new composition parameter C =A/B instead of the conventional mole fraction x = A/(A+B),where the components A and B are expressed in their respective molar quantities. The two branches of the coexistence curve are completely symmetrical with respect to their "rectilinear diameter".The points are the experimental data of Nagarajan et a1." and the curve represents the calculated values as given by eqn (29) and (30). 304 302 30C T/K 2 98 2 96 294 I -8 -6 -4 -2 0 2 4 6 8 In u FIG.3.-(a) Conventional representation of the miscibility gap in the polystyrene +cyclohexane system. The coexistence curve shows a large degree of asymmetry. 4 is the volume fraction of polystyrene. (b)The same diagram displaying perfect symmetry when the compositions are expressed in modified composition parameters u, specified by eqn (26). (Experimental data of Nakata er d"). By this transformation one can convert even the most asymmetric phase diagram into a fully symmetric one as is shown in fi 3(a) and (b),where the miscibility I!?gap in the polystyrene + cyclohexane system is shown in conventional representa- tion, T plotted against 4, and in its new composition units, T plotted against lnu.F. VNUK 47 5. APPLICATION OF THE NEW MODEL (a) n-HEPTANEtACETIC ANHYDRIDE These new concepts are now applied to the binary system, n-heptane + acetic anhydride, which was experimentally investigated by Nagarajan et a/.’’ between 275.7 K and the critical temperature. The separation temperature was determined on 76 samples ranging in composition from 0.04 to 0.88 mole fraction of n-heptane. In their analysis the authors disregarded the experimental data for those samples where the separation temperature was ca.0.85 IS below the critical point, since here the gravity effects would be expected to have caused “significant distortion of experimental measurements3~.1” In the present reanalysis of these experimental data no prior assumption is made as to the values of T,and x,. These quantities are determined by extrapolation from those experimental data which would not have been affected by the gravity effects. The selection of the experimental data for the analysis (i.e. data with negligible gravity effects) is assessed by the model prediction. According to the model g(T)= (c~c~)”~= c0exp Q(T,-TIP (7) i.e. (C1C2)1’2is an exponential function which can be either monotonically increasing or monotonically decreasing over the whole coexistence range.The experimental data of Nagarajan et a1.l’ show that (C1C2)1/2is monotonically increasing between 275.65 and 341.47 K and decreasing between 341.47 and 341.67 K. This change in behaviour is attributed to the gravity effects, and it is assumed that the gravity- affected data are located in the temperature interval of ca. 2(Tc-341.67), i.e. ca. 0.4 K. Consequently the samples having separation temperature between 34 1.27 K and the critical temperature are not included in the present analysis. The remaining 41 sets of data points were then processed as follows: (i) Molar fractions (xl,x2) were converted into the new composition parameters (Cl,C2) according to eqn (19). (ii) The constants M, n and T, were obtained from eqn (8), which for this purpose is written as: The solution consists of a search for a linear plot of the left-hand side of eqn (27) against In [(TJT)-11 with a value of T, giving the minimum sum of squared residuals.Such a plot with T,= 341.648 K is shown in fig. 4(a). On this linearized plot In (2M) and n are the intercept and slope, respectively. (iii) With this value of T, the constants Co, Q and p were then determined from eqn (7), which is rewritten as Here again, a value of p is sought to give an optimal linear plot of the left-hand side of eqn (7a) against (Tc-T)’. Such a plot, with p = 0.842,is shown in fig. 4(6). The constants obtained by this method are: M = 2,4001 n = 0.2847 T,= 341.648 Q = -0.01326 p = 0.842 Co= 0.9463 PHASE EQUILIBRIA IN BINARY MIXTURES 341.648In (7-1) FIG.4.-n-Heptane + acetic anhydride miscibility gap. (Experimental data of Nagarajan et a/.") (a) Linearized plot of eqn (8) and (27) to obtain constants M, n and T,. Slope = 0.2847. (6) Linearized plot of eqn (7) and (28) to obtain numerical values of Q, Coand p. Slope = -0.0133. giving g(T)= 0.9463 exp [ -0.01326(341.648-T)0.842] (34 lT648 0.2847f(T)= 1-tanh 2.4001 A-from which the values C1and C2are calculated c1,2 = g(T)[f(T)l*l. When these calculated values are converted to the conventional composition para- meters x1 and xz and compared with the experimental data of ref. (lo), they show very good agreement over the whole temperature range under investigation as seen from the deviation plot in fig.5. Note that the calculated value of Tc(= 341.648 K) comes very close to the experimental value as reported by Nagarajan et al. (= 341.672 K), while their value of xc (0.4707) is significantly lower than the calculated value (0.4862). The phase diagram, redrawn within the new coordinates (T against In C) to underline the significance of g(T)and f(T),is shown in fig. 2. (6) METHANOL+CYCLOHEXANE The data on n-heptane + acetic anhydride," analysed in the preceding section, are extensive but do not have the same degree of experimental precision as the F. VNUK 49 I I I I I Id* 11 2 90 300 310 320 330 34 0 T/K FIG.5.-Deviation plot for the compositions of the coexisting phases in the n-heptane +acetic anhydride system [ref. (lo)].Full circles (@) represent the experimentally determined and interpolated composi- tions of the acetic-anhydride-rich phase, empty circles (0)those of the n-heptane-rich phase, in terms of the mole fraction of n-heptane. They are compared with the calculated values obtained from eqn (29) and (30) and converted to mole fractions according to eqn (19). 1 I 1 I-0.2 L I 1 2 3 4 5 6 (318.565 -T)0-708 -6 -5 -4 -3 318.565In (-f--1) FIG.6.-Liquid-liquid phase equilibria in the methanol +cyclohexane system. (Experimental data of Becker et al.'*) (a) Linearized plot of eqn (8) to obtain M, n and T,. ih) Linearized plot of eqn (7) to obtain Co,Q and p. data of Becker et a1.,12 who remeasured the miscibility gap in the system methanol + cyclohexane within 15 K of its critical point.They used the same point-by-point determination method as Nagarajan et al. but employed a more accurate experi- mental technique (turbidity measurements with electro-optical indication). Their experimental data (excluding those within 0.5 K of the critical point) were processsed as outlined in the preceding section. The linearized plots according to eqn (7) and (8) are given in fig. 6(a) and (b),and they show only a very small scatter. The calculated critical constants (Tc= 318.565 K and xc = 0.516) come very 50 PHASE EQUILIBRIA IN BINARY MIXTURES 0.4 0.2 - em T,= 318.565K Y 0- 0 0 0 0- 0 0-0-0 -o-o-oe-.-. 0 . 0-q 0 -0.2 b 0 0 .L ' I I I 1 close to the quoted experimental values, i.e. T,= 318.50f0.05 K and xc= 0.513 *0.002. Complete formulae for the compositions of the two branches are c1,2 = g(mf(n1" where g(T) = 1.068 exp [ -2.924 x lW'(318.565 -T)0.708] and (3 18; 65 1)0.2976f(T)= 1-tanh 2.534 ~-(33) The experimental accuracies claimed by the authors are: kO.05 K in tem- perature and f0.001 in mole fraction. Within these limits there is perfect agreement between the experimental data and those calculated by eqn (32) and (33), as can be seen from the deviation plot (fig. 7). (c) ME R CU R Y + G A L L I U M This is an example of a miscibility gap in a metallic system. The cornpositions of the coexisting liquid phases were investigated by Predel13 and their res ective resistivities near the critical point were measured by Schurmann and Parks!4 The diagram is shown in fig.I. It can be seen that the range of coexistence extends over 170K and at the lowest temperature of liquid coexistence the phases are almost completely unmixed. When the compositions are expressed as mole frac- tions, the diagram is nearly symmetric. This feature is lost when the compositions are expressed in mass fractions. The compositions were read from the diagram and processed as outlined earlier. Again, we have where g(T)= I.072expi-1.114~ 10-3(475.644-T)1~083] (34) F. VNUK 51 0 I FIG. 8.-Deviation plot for the compositions of coexisting phases in the binary system Hg+Ga. (Experimental data of Predel.13) Calculated values obtained from eqn (34)and (35).Ax = 100 (xeXpt-Xcalc). and f(T)=l-tanh2.475(--1)475.644 0.3458. .T (35) The comparison between the experimental and calculated values is displayed on the deviation plot in fig. 8. (d) SODIUMtAMMONIA This binary mixture of two very dissimilar substances has attracted considerable interest, since it represents one of the better known cases of the metal-to-non-metal transition.” Furthermore, it has been claimed that in this system one can observe a clear and well defined transition from the Ising-type critical behaviour near T, to the mean-field behaviour at temperatures ca. 2 K below TC.l6,l7The transition is detected by an abrupt change in (n in the present model) from its Ising value (p= 0.325) to the mean-field value (p= 0.50) at E ~0.01.The experimental data for this ~ysteml~’~~were reanalysed by Das and Greer.l9 They fitted the data with a two-term Wegner expansion as modified for liquid mixtures by Ley-Koo and Green2 [eqn (3)], and claimed “an excellent fit”.19 They achieved this by assigning fixed values to @(0.325),A(0.50) and T, (232.55 K) and determining Kloand Kll by a standard fitting procedure. When the experimental data of ref. (20) supplemented by the interpolated values of Das and Greer” are treated by the present model, a very good agreement is obtained between the experimental/interpolated and calculated data over the whole range of coexistence. The calculated critical parameters (T,= 231.52 K and xc= 0.0404) are virtually identical with the experimental ones (T,= 231.55 K x, = 0.04111, while n =0.389 (as compared to two values of Das and Greer, i.e. 0.5 and 0.325).The compositions of the coexisting phases are given by C1,J= -x1,2 - g(T)[f(T)]’l1-x1,2 (19931) where g(T)=4.232 x exp [-3.313 x 1OP3(231.52-T)1.128] (34) PHASE EQUILIBRIA IN BINARY MIXTURES 0.02 2 % 0.04 -- --0 -y+.ok, c .-Y wE 0.06- 0.94 - e, 2< 0.08- // ;O 231 23: T,= 231.52 K 0.101 0.2 c I 210 215 220 225 230 TI K FIG. 9.-(a) Coexistence curve in Na+NH3 system. Experimental data of Kraus and Lucasse" and interpolated values of Das and Greer'' shown as circles (0).Experimental data of Chieux and Sienko16 are shown as crosses (+) but these were not used in the evaluation of the constants in eqn (36) and (37).(b)Deviation plot showing the extent of scatter between the experimental and calculated values. Ax = 100 (Xexpt -Xcalc). and 0.389 f(T)= 1-tanh 1.685(23;52 1) , (37) The phase diagram, together with the deviation plot, is shown in fig. 9. (e) P OLY STY R EN E +cY CLO HEX ANE" The respective molecular weights of the components in this system differ by more than 4 orders of magnitude and consequently the coexistence curve is one of the most asymmetric, having the critical composition ca. 0.032 volume fraction of polystyrene. Lowering the temperature by 3 K results in a four-fold increase in the concentration of polystyrene in the polystyrene-rich phase, but an eighteen-fold decrease in the cyclohexane-rich phase.At still lower temperatures the solubility of polystyrene in cyclohexane becomes so small as to be of the same order as the width of the experimental error. As a result, the solubilities of polystyrene in cyclohexane can be obtained only with a low relative accuracy [see fig. 3(a)]. However, the new model makes it possible to determine the composition of the cyclohexane-rich liquids with greatly improved accuracy on the basis of the symmetry properties expressed in eqn (26). F. VNUK 53 0 0 -0.10 294 296 298 300 II . "302.0 302.5 303.0 303.5 T/K FIG. lO.-Deviation plot for the compositions (in terms of mole fraction of polystyrene) of coexisting phases in polystyrene+cyclohexane system [ref.(ll)]. Note the change in temperature scale at T> 301 K. Full circles (0)represent the cyclohexane-rich liquid. Calculated values obtained from eqn (38) and (39). A4 = 100 (4expr-Llc). First the function f(T)is derived from those values of 41 and q52 where the experimental results are most reliable, i.e. within the temperature range 0.029 < (T,-T)<2.983 K. This gives (303;53 1)0.3313f(T)= 1-tanh 6.752 ______-The function f(T)is then extrapolated to lower temperatures and the values of 41are calculated from By this method the calculated values are: 0.000 89 at 299.174 K, 0.000 41 at 296.65 K and 0.000 16 at 292.61 K, as compared with the corresponding values of Nakata et a/.," viz.0.0012, 0.0009 and 0.0009. With these values one can now calculate g(T)and obtain a full expression for the composition of both branches of the coexistence curve: where g(T)=3.3993 X exp [-0.4288(303.653 -T)0.516] (39) and (303T653 1)0.3313f(T)= 1-tanh 6.752 A-The comparison between the calucated and experimental values is shown in fig. 10. 6. SUMMARY AND CONCLUSIONS (i) A new model for phase equilibria in binary mixtures has been proposed. In its versatility and applicability to a diversity of binary mixtures and over an extended temperature range it is superior to existing models. It achieves these advantages PHASE EQUILIBRIA IN BINARY MIXTURES with an economy of adjustable constants which deserves repeated emphasis in this summary.Comparing the performance of the present model with that of Wegner’s ex ansion2 as applied to the n-heptane+acetic anhydride system by Nagarajan et al!’ one observes: (a) The difference in “order parameter” (x2-xl) for the coexisting phases in the n-heptane +acetic anhydride system was satisfactorily fitted to eqn (3) by Nagarajan et al. with four constants (ISlo,Kll, p and A) when compositions are expressed in mole fractions and with five constants (Klo,KI1, K12,p and A) when compositions are expressed in volume fractions. In the new model the “order parameter” expressed as ratio (C1/C2)1’2(where C1< C2)can be satisfactorily fitted to eqn (8) with two constants M and n (= p) which have the same value irrespective of whether the composition coordinates are expressed in mole, volume or mass units.(b)The “rectilinear diameter”, $(x1+x2), for the coexisting phases in the n-heptane +acetic anhydride system was satisfactorily fitted to eqn (4) by Nagarajan et al.” with five constants (xc, K20,K21,a, A). In the new model the “geometrical rectilinear diameter”, (C1C2I1”,can be satisfactorily fitted to eqn (7)with three constants Q, p and Co,where Cois the critical composition in the appropriate composition parameters. (ii) The expressions for the “order parameter” [eqn (S)] and the “rectilinear diameter” [eqn (7)]can be readily used for formulating simple relationships between the compositions of the coexisting phases and T, i.e. C2,@2, ui= g(T)/f(T) (41) where C1,a1,W1and C2,Q2, Wzare the new composition parameters related to the fractional composition parameters x1, 41,w~1and x2, 42, w2 by (iii) The above formulations are not only simple but they also reveal a new and intrinsically appealing type of symmetry, so much sought in all analyses of critical phenomena.One can see that the two branches of the coexistence curve are perfectly symmetrical with respect to the new “rectilinear diameter”, g (T)= (C1C2)li2.This symmetry can be displayed in a striking manner if the temperature- composition diagram is plotted with the abscissa in In C, as shown in fig.2. (iv) The new mathematical formulations hold over the whole range of coexistence, including T = T,. This unique feature endows the model with an extrapolative power which enables one to determine the critical. parameters T, and xc by extrapolation of the experimental data obtained at temperatures further away from T, where the disturbances due to gravity effects are less severe or negligible.(v) The model suggests a new and more appropriate expression for the order parameter which is one of the most important but rather elusive quantities in the study of coexisting phases. By relating the order parameter P to the probability of like and unlike contacts in the mixture an expression for the temperature dependence of P is obtained: P = tanhivl($-I)‘*. This expression defines P uniquely, irrespective of the choice of composition coordinate. F. VNUK 55 (vi) Exponent n in eqn (23)is identical with the critical exponent p in eqn (1).This p is considered in current theories to be a universal constant with a fixed value for all binary systems (p = 0.312 from the lattice-model expansion technique and p = 0.325 from the normalization group calculations*’). The present model suggests that p is not a universal but an adjustable constant, and (in the same sense as Tc)has a characteristic value which varies from system to system. It appears, however, that in a given system n (=p)remains unchanged even when the coexisting phases are specified by their respective physical properties (e.g. refractive indices, densities, electrical resistivities, etc.) instead of compositions. -i-(vii) The principles applied to the coexisting phases in miscibility gaps are applicable also to other types of phase equilibria, as described in the following paper, and to other cooperative critical phenomena, as we hope to discuss in subsequent papers.+‘E.g. in the system mercury i-gallium, analysed in section S(c 1, the order parameter ,,75;44 1,)o.345aP = tanh 2.475 ~-when calculated from the compositions of the coexisting phases [ref. (13)], and 0.34’ 6;4 7 34P = tanh 0.630 ____ -when calculated from the electrical resistivities of the coexisting phases [ref. (1411 in the temperature range 469.31 < T/K<476.53. It is seen that in both cases the critical exponent n has the same value within the limits of experimental error. R. L. 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