Research: Science & Education
The Behavior of a Pair of Partially Miscible Liquids S. R. Logan Department of Chemistry, University of Ulster, Coleraine BT52 1SA, N. Ireland
The phenomenon of combining two liquids to obtain two liquid phases is a well-known one. Such a system may be used constructively in the laboratory and is widely used to separate solutes by liquid–liquid extraction. Quite often, the rationale for the existence of these two phases is expressed in terms of the differing characteristics of the two molecules involved, coupled to the adage that like mixes freely with like. Two-component systems that may exhibit two liquid phases are often discussed in relation to the Gibbs phase rule. In the context of the undergraduate laboratory, systems with an upper critical solution temperature that may be achieved at atmospheric pressure, such as aniline/hexane (1) or butanol/water (2), are of particular interest. However, the phase rule, while it enables us to evaluate the maximum number of phases a system may consist of and the number of degrees of freedom for any permissible number, does not help us in regard to what number will actually be present. The Thermodynamic Approach If the question is pressed, why do the two liquid components not form a single liquid phase, then there are other thermodynamic approaches that offer more illumination. At constant temperature and pressure, the only spontaneous changes available to a system are those for which the Gibbs energy, G, shows a decrease. So, mixing should occur if the corresponding value of ∆G is negative. Where ∆G is positive, the mixing process will be purely hypothetical. The aim of this paper is to show the extent to which the behavior of two liquid components can be rationalized on this simple basis. The Gibbs energy is defined as G =H–TS
(1)
so that, at constant temperature, T, we have: ∆G = ∆ H – T ∆S
(2)
The entropy change, ∆S, for a mixing process at constant volume is almost invariably positive, so that the Gibbs energy change will be negative unless the enthalpy change, ∆H, is sufficiently positive that it exceeds T ∆S. In that eventuality, mixing would not be expected to occur, so that all the difference terms in eq 2 would be hypothetical quantities, relating to a process that does not take place. This explanation, however, is a little bit misleading, in that it seems to imply that mixing is an all-or-nothing phenomenon. If one considers a pair of dissimilar liquids such as water and chloroform, each component has a small though finite solubility in the other: the top layer is water saturated with chloroform and the lower layer is chloroform saturated with water. To understand this, it is necessary to examine eq 2 in rather greater detail. In mixing two gases, at the same temperature and pressure, the entropy may be shown (3) to increase by the amount ∆S = { nR {x ln x +(1 – x ) ln (1 – x)}
(3)
where x denotes the mole fraction of one component and (1 – x) that of the other, n is the total number of moles of the two substances, and R is the gas constant. This expression may be derived using either classical or statistical thermodynamics. For two liquids that mix freely, the same equation is applicable, since the process of mixing is essentially the same as in the gas phase. As both x and (1 – x) must be less than one, the logarithms in eq 3 are both negative, so that (except at x = 0 and x = 1) ∆S is everywhere positive, with a maximum value of nR ln 2 at x = 0.5. If the total amount of substance in the mixture is taken as one mole, this gives the greatest value of ∆S as R ln 2. Enthalpy Changes in Real Systems Experimental measurements of the enthalpy of mixing have been made for several pairs of liquids that mix freely to form one phase. For very similar substances, a parabolic curve is found, which may be represented by the equation ∆H = A x(1 – x )
(4)
For a less similar pair of substances, a skewed parabola may be found, and this may be accommodated (4) by the modified equation ∆H = x (1 – x)[A + B(1 – 2x ) + C (1 – 2x)2 + D (1 – 2x )3] (5) We may, for the present, proceed on the reasonable assumption that an expression of this type would also describe the hypothetical enthalpy of mixing of a pair of immiscible liquids. When the parameters are adjusted for optimum fit to the experimental data, the values of C and D are usually quite small, so these terms will be neglected. This will tend to simplify the discussion, with only the slightest effect on the results. So, to obtain an expression for ∆G as a function of the molar composition, for a total of one mole of the two components, we substitute eqs 3 and 5 into eq 2 to obtain ∆G = x(1 – x)[A + B(1 – 2x)] + RT [x ln x + (1 – x) ln(1 – x)](6) It may be helpful to refer to two terms used in regard to binary liquid mixtures. Ideal solutions are those that obey Raoult’s law and of course they mix freely in all proportions. For such solutions, the entropy change is as given in eq 3 and the enthalpy change is zero. More generally, the enthalpy change is found to be nonzero and the term “regular solution” was introduced by Hildebrand in 1929, on the basis of a definition (5) that means the entropy change is as given in eq 3. So eq 6 is applicable to mixing processes that form regular solutions. It will also be used here in regard to solutions whose behavior is far from ideal, thus demonstrating how the regular solution concept can provide a rational basis for the behavior of partially miscible liquids. A word of explanation is perhaps advisable. It is not being claimed that, where a binary liquid mixture forms two
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Research: Science & Education
∆H 1500
T ∆S a
1000
500
0
∆G 0.5 x
200
∆G/kJ mol-1
0 0.5
x
-200
b
-400
c
0
-600
d
-200 Figure 1. Plots of ∆H and T ∆S (scale on left) and ∆G (scale on right) vs x, using eq 6, with A = 7500 J mol{1, B = 1000 J mol{1, and T = 298 K.
layers, the Gibbs energy change is given by eq 6. Rather, this equation is being used to evaluate the hypothetical Gibbs energy change that would take place if a single phase were to be formed. The curve of ∆G against the mole fraction is then used to deduce the disposition of the two components, which minimizes the Gibbs energy of the system. In some instances this is achieved by having two phases rather than complete mixing. The close similarity between these predictions of regular solution theory and the phase behavior observed may be taken as a posteriori justification for the application of eq 6 to such systems. The shape of the curve obtained from eq 6 clearly depends on the magnitude of A in relation to R T. If A is much less than 4RT ln 2 and B is small, then ∆G is everywhere negative over the range, 0 < x < 1. But if A exceeds 4RT ln 2, ∆G will be positive at x = 0.5, so that a mixture of this molar composition will not form. However, close to x = 0 and close to x = 1, ∆G will be negative, so that the two components are not in fact totally immiscible. To establish the existence of these regions where ∆G is negative, it is sufficient to point out that, on the basis of eq 3, d∆S/dx = ∞ at x = 0 and { ∞ at x = 1, whereas, from eq 5, d∆H/dx has the corresponding values of (A + B) and ({ A + B). This means that, close to the extremities, the T∆S term must predominate over the ∆H term, no matter how large A is. The variations of ∆H, T∆S and ∆G with composition, for arbitrary values of A and B, are shown in Figure 1. This demonstrates the important distinction between the shapes of the curves of ∆H and T∆S, particularly at very high and very low mole fractions. The parameter A has been allocated a value in excess of 4RT ln 2, so that ∆G is positive at x = 0.5. However, close to x = 0 and x = 1, ∆G is negative, with minima that are not equidistant from these limits, in view of the skewed nature of the parabolic curve for ∆H (consistent with the finite value allocated to B ). The interpretation of Figure 1 is thus that if one combines the components to give a mixture with 0 < x ≤ 0.04, or with 0.90 ≤ x < 1.0, then a single phase of this exact composition will result. If, however, the calculated mole fraction lies between the values of 0.04 and 0.90, then no such phase can be formed. Instead, at equilibrium there will be two phases, with mole fractions close to x = 0.04 and 0.90, the compositions of the two minima in ∆G shown in Figure 1. 340
Figure 2. Plots of ∆G vs x , using eq 6, with T = 298 K and the other parameters as follows: (a) A = 7500 J mol{1, B = 1000 J mol{1; (b) A = 6450 J mol{1, B = 860 J mol{1; (c) A = 5400 J mol{1, B = 720 J mol {1; (d) A = 4350 J mol{1 , B = 580 J mol {1.
As we shall see, the compositions are not precisely those of the minima. In allocating numerical values to the parameters in eq 6 to generate Figure 1, ∆H was put only slightly greater than T∆S at x = 0.5. The minima in ∆G persist even if ∆H is made much greater than this, but the greater the disparity between ∆H and T∆S at x = 0.5, the farther the minima are pushed toward the outer limits. For example, if A is increased to 15,000 J mol{1, the minima come at x = 0.0016 and 0.9964. The Criterion for the Existence of Two Phases We want now to derive the conditions under which a mixture of the two components will exist as two phases of different compositions, rather than as a single phase. Also, where two phases exist, we want to determine their respective compositions. In seeking these, we will apply the simple criterion that the Gibbs energy of the total system should be minimized. Suppose we have one mole of a mixture in which the mole fraction of component A is x 2. If this were to separate into two phases of mole fractions x1 and x 3, there would be (x 3 – x 2)/(x 3 – x 1) moles of the former and (x 2 – x 1)/(x 3 – x 1) moles of the latter. It follows that, on a plot of the Gibbs energy change against the mole fraction, x, if the respective Gibbs energy changes of the three points are ∆G1, ∆G2, and ∆G3 and these three points are collinear, then:
∆G 2 – ∆G 1 ∆G 3 – ∆G 2 ∆G 3 – ∆G 1 = = x2 – x1 x3 – x2 x3 – x1
(7)
The total Gibbs energy change, ∆Gf , of the final two phases will be given by the simple sum
∆G f =
x3 – x2 x –x ∆G 1 + 2 1 ∆G 3 x3 – x1 x3 – x1 ∆G 3 – ∆G 2 ∆G 2 – ∆G 1 = ∆G 1 + ∆G 3 = ∆G 2 ∆G 3 – ∆G 1 ∆G 3 – ∆G 1
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Research: Science & Education
0
0.5
x
1
400
3 -400
4 5
T /K
∆G/kJ mol-1
2
350
6 7 -800
8
Figure 3. Plots of ∆G vs x, using eq 6, with A = 6450 J mol{1 and B = 860 J mol{1, for the following values of T: (1) 278 K; (2) 298 K; (3) 318 K; (4) 338 K; (5) 358 K; (6) 378 K; (7) 398 K; (8) 418 K.
So in this instance, where the three points are collinear, the Gibbs energy change is the same for one mole of composition x 2 as for the two phases of compositions x 1 and x 3. However, if the points are not collinear, then we must have an inequality. Where the smooth curve through the three points is concave upwards, the lowest value of ∆G will be achieved by the single phase, of composition x 2. If the curve is concave downwards, then the system of two phases will have a ∆Gf value that is lower than ∆G2. This same argument is of course applicable to the phases of compositions x 1 and x 3. In the end analysis, the compositions of the two phases that will then be formed will be such that, on the plot of ∆G versus x, when we draw a line through these two points there is no composition for which the ∆G value lies below this line. This means that the requisite line is the common tangent to the two portions of the curve, on either side of the section that is concave downwards. This line is illustrated on Figure 1. The points of contact of this line differ only very slightly from the positions of the two minima referred to earlier. In Figure 2, plots are shown of ∆G as a function of the mole fraction, for four sets of assumptions regarding ∆H. In curve a, ∆G is positive over a central range of values x, between two minima; in curve b, ∆G shows a maximum between two minima, but is nowhere positive; in curve c, there is no maximum and therefore only one minimum, but this curve has a region that is concave downwards, between two points of inflection; curve d has no points of inflection and therefore will exhibit only one phase, regardless of the nominal composition. For each of the curves a, b, and c, the common tangent has been drawn and the contact points mark the respective compositions of the coexistent phases, present when the nominal composition of the sample lies between these two values of x . In Figure 2, by assigning a finite value to the parameter B in eq 6, we have generated curves that are asymmetrical, so that the common tangent does not touch at the two minima. Also, as the mixing process is assumed less endothermic, one of the minima in the curve disappears by merging with the intervening maximum. However, complete miscibility has not been attained at that point. Where no asymmetry is assumed (6), all three features merge into one minimum at x = 0.5, at which stage the two components become fully mis-
300
0
0.5 x
1
Figure 4. Plot of the values of x at which the common tangent touches, vs T , using the data in Fig. 3.
cible. The conditions for the existence of two phases are derived in this book (6 ) by an alternative argument that employs partial molar quantities. Effects of Temperature In eq 6, the temperature appears explicitly in the second term, representing T∆S. This identifies the source of the major effect of a temperature change on the curve of ∆G against composition. Additionally, the enthalpy of mixing is likely to show some variation with temperature. Indeed, it was shown that in mixing n-hexane with n-hexadecane, the enthalpy change decreases by more than 70% between 293 and 323 K (7); therefore A and B cannot be regarded as strictly temperature-independent parameters. However, because the values involved here were quite small, such effects, while real, are likely to be minor. Thus we can realistically hope to replicate the influence of temperature on the miscibility of two liquids by looking at the developments in the curves produced by eq 6 as the temperature is changed. As a starting point, let us take the situation represented in Figure 2b, where calculations at 298 K imply the existence of two phases, with compositions given by x = 0.10 and 0.85. The effects of variations in temperature, with the parameters A and B held constant, are shown in Figure 3. As the temperature is increased, the values of ∆G become more negative, in such a way that the minima in ∆G move slightly closer and the maximum becomes less prominent and moves to lower values of x (Fig. 3). In effect, increasing temperature brings about changes that closely resemble those from decreasing the enthalpy of mixing (Fig. 2). By using the common tangents to the two portions of each curve, the compositions of the two phases can be evaluated at each temperature, up to 398 K. At 418 K, the curve shows no points of inflection and we deduce that there will be one phase only. In Figure 4, these x values are plotted as a function of temperature, leading to a curve broadly comparable to those found experimentally. Conclusions The above calculations show that a simple assessment of the Gibbs energy change on mixing two liquids can explain, at least qualitatively, some of the phenomena observed
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in relation to the miscibility of two liquids. It is appropriate to add that significant simplifications are implicit in eq 6, easily sufficient to account for any disparities with experimental findings. Principally, there is the question of the correct expression for the entropy of mixing, bearing in mind that the systems giving rise to two liquid phases are those in which the molecules would not be expected to undergo strictly random movements. Since these systems almost invariably involve quite dissimilar molecules, it is perhaps unrealistic to assume that the molecules of the two components can be disposed randomly within a single phase. Also, there is the problem of the correct expression for ∆H for the mixing of components that do not mix freely, and on its variation with temperature. In view of the hypothetical nature of this process, experimental work can offer little guidance. Some binary systems exhibit a lower critical solution temperature, an occurrence not predicted on the basis of eq 6. In these systems, very strong and specific attractive interactions occur between molecules of the two components, so that a third species is effectively formed. The assumptions made here
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in regard to ∆S have not involved the possible presence of a third component, so that the results should not be expected to accommodate the consequences of such an occurrence. Acknowledgments I wish to acknowledge helpful discussions with G. P. Shannon of Mathematics and constructive comments from a reviewer. Literature Cited 1. Glasstone, S. Textbook of Physical Chemistry, 2nd ed.; Macmillan: London, 1953; p 724. 2. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1995; p 331. 3. Glasstone, S. Textbook of Physical Chemistry, 2nd ed.; Macmillan: London, 1953; p 226. Atkins, P. W. Physical Chemistry, 4th ed.; Oxford UP: London, 1990; p 160. 4. Larkin, J. A.; McGlashan, M. L. J. Chem. Soc. 1961, 3425. 5. Hildebrand, J. L.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions; Van Nostrand/Reinhold: New York, 1970; p 3. 6. Hildebrand, J. L.; Prausnitz, J. M.; Scott, R. L. Ibid., p 20. 7. McGlashan, M. L.; Morcom, K. W. Trans. Faraday Soc. 1961, 57, 581.
Journal of Chemical Education • Vol. 75 No. 3 March 1998 • JChemEd.chem.wisc.edu
