Research: Science & Education
The Gibbs Energy Basis and Construction of Melting Point Diagrams in Binary Systems Norman O. Smith Department of Chemistry, Fordham University, Bronx, NY 10458 The importance of melting point diagrams needs no emphasis. Although the various types of melting behavior in binary systems are often shown in textbooks, there is usually no description of the extent to which deviations from ideality on the part of the solutions plays a role. Why may one system that forms a complete series of solid solutions show a maximum melting point while another shows a minimum? Why is the type with a maximum rarely found? This article describes how deviations from ideality, expressed in terms of activity coefficients or excess Gibbs energies, along with the thermodynamic properties of the components themselves, determine the melting behavior. Implementation of the methods described here provides an enlightening exercise in thermodynamics. Bakhuis Roozeboom (1) presented a qualitative classification of various kinds of behavior, but the quantitative description had to await the formulation of the concept of activity by G. N. Lewis (2). The presence of intercomponent solid compounds and the polymorphic behavior of solids are excluded from this treatment. Deviations from ideality shown by liquid solutions have been described quantitatively for many years, but such data for solid solutions are comparatively recent. Solid state activities have been found through studies of sublimation pressure (3), emf (4), melting behavior (5), and still more recently for isomorphous salts, distribution between solid solutions and their saturated aqueous solutions (6, 7). A number of examples have been found of salt pairs that show ideal behavior, and negative and positive deviations (7). An excellent review of alkali halide pairs (5) is also available and is a useful source of information on their melting behavior and related quantities. Seltz developed a graphical method for determining melting point diagrams for both ideal (8) and non-ideal (9) systems, but his procedure is less convenient than that described below. In this article it will be assumed that the deviations are known quantitatively, regardless of how they were found, and, from these, the melting point diagram will be determined. As a preliminary, let us consider only the broad categories of miscible, partially miscible, and immiscible liquids and solids, corresponding to increasing “antisocial” behavior. On the basis of these categories the types of melting behavior shown in Figure 1 may be distinguished, although combinations of these types are also possible. Types I, II, and III are for pairs of components that are miscible in both the liquid and solid states. Types IV and V are possible behaviors with partially miscible solid components and are described as the peritectic and eutectic types, respectively. (The first five type numbers correspond to those assigned by B. Roozeboom (1) even though, in the light of what follows, it would have been more logical to reverse the numbering of Types I and II.) In Type VI there is miscibility in the liquid but no measurable miscibility in the solid; in Type VII there is partial miscibility in the liquid superimposed on Type VI. Finally, if both liquid and solid phases show immiscibility, there is no binary behavior, for the components behave independently. It should be noted that the combinations of partial miscibility or of immiscibility in the
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liquid with miscibility in the solid are not included. This is because greater molecular dissimilarity of the components is needed for partial miscibility in the liquid than is needed for partial miscibility in the solid. If, therefore, deviations in the liquid are sufficiently positive to produce even partial liquid miscibility, the deviations in the solid will be expected to be too large for any solid miscibility. The quantitative treatment is based on the concept of Gibbs (10), according to which, at true equilibrium, the chemical potential of the ith component, µi, is the same in all phases. For a liquid solution in equilibrium with a solid solution, both containing i, µ i, = µi°(,) + RT ln ai, must equal µis = µ i°(s) + RT ln ais, where a is activity. The standard state for i in the liquid is chosen throughout to be the pure i(,) at the same temperature, whether that state be stable or metastable, and analogously for the standard state for i in the solid. It follows that µi°(s) – µ i°(,) = RT ln ai,/ais. The left side is the negative of the Gibbs energy of fusion, ∆fus Gi, at temperature T. Furthermore, ai = γixi, where γi is the activity coefficient and xi the mole fraction. We have, therefore, since RT ln γi, is the partial molar excess Gibbs energy of i in the liquid, G iE(,), and analogously for the solid, {∆fusG i = RT ln (xi,/xis) + GiE(,) – G iE(s). Finally, applying this to the components A and B, and replacing xA, by x, and xB, by 1 – x, xAs by y, and xBs by 1 – y, we have RT ln (x/y) + GAE(,) – GAE(s) + ∆ fusGA = 0
(1)
RT ln [(1 – x)/(1 – y)] + GBE(,) – GBE(s) + ∆fusGB = 0 (2) GAE(,) and G BE(,) are related through GE(,) = x G AE(,) + (1 – x) GBE(,) and analogously for GE(s), where GE(,) is the molar excess Gibbs energy of the liquid. Furthermore, GE(,) and GAE(,) are related through GAE(,) = G E(,) + (1 – x) (dGE(,)/dx), and similarly for B. Analogous relations can be written for the solid solutions. To solve eqs 1 and 2, the excess quantities must be expressed in terms of concentration, bearing in mind that the Gibbs-Duhem relation Σxi dGiE = 0 (p, T constant) must be followed for both liquid and solid solutions. For this reason these quantities may be expressed in the form G E = xAxB (g 0 + g1xB + g2 xB2 + …) , where g0, g1 , g2 , … are constants referring to the liquid or solid phase as appropriate. This yields GAE = (g0 – g1) xB2 + 2 (g1 – g2) xB3 + 3g2xB4
(3)
GBE = (g0 + 2 g1 + 3g2) xA2 – (2g1 + 6g2) xA3 + 3g2xA4 (4) to three terms. Frequently, only one parameter, g0, is needed for a sufficiently accurate description (g1 = g2 = 0), and the solutions are termed regular (5). If two parameters, g0 and g1, are needed they are termed subregular. For regular solutions, then, G E = g0xAxB, or G AE = g0xB2, GBE = g0 xA 2, g 0 being zero for ideal, > 0 for positively deviating, < 0 for negatively deviating solutions, whether liquid or solid. In what follows, emphasis will be on regular solutions where possible, even at the sacrifice of some accuracy, because of their simplicity. Extension of the treatment to subregular solutions should be self-evident.
Journal of Chemical Education • Vol. 74 No. 9 September 1997
Research: Science & Education
(a)
For regular liquid and solid solutions, eqs 1 and 2 become
(b)
(c)
(d)
(e)
(f)
RT ln (x / y) + g0,(1 – x)2 – g0s (1 – y)2 + ∆fusGA = 0
(5)
RT ln [(1 – x) / (1 – y)] + g0,x2 – g0s y2 + ∆fusGB = 0
(6)
which can be solved as shown later. One may often assume that ∆fusH and ∆fusS are independent of T, in which case ∆ fusG is given by ∆fusH – T (∆fus H/T°), T° being the melting point of the pure component. A further widely used assumption is that SE, the excess molar entropy of both the liquid and solid solutions, is zero at all temperatures under consideration, which implies that GE is temperature independent. Before illustrating how eqs 5 and 6 or similar equations in two unknowns may be solved, the following may be noted concerning Types I, II, and III. Table 1 gives the behavior for a series of fictitious systems all comprising the same pair of components with properties given in the table footnote, but with various arbitrary g 0, and g0s values. It is clear that the difference between the magnitudes of the deviations in the liquid and solid phases determines the type: as g0, – g 0s decreases from positive through zero to negative, one passes from Type II through Type I to Type III. Furthermore, Type I is more likely the greater the difference between ∆ fusGA and ∆fus GB at any one temperature, although Table I does not show this. The curvatures of the liquidi and solidi are not readily predictable, but at temperatures in the vicinity of the onset of partial miscibility the lines will show a nearly horizontal portion. In a Type I diagram, the greater the concavity of the curves from below the greater the proximity to a Type II diagram; the greater the concavity from above, the greater the proximity to a Type III. The details of calculating the phase boundaries for equilibrium between liquid and solid solutions, applicable to Types I, II, and III and parts of IV and V, will be illustrated by the system KBr/KI, the parameters for which are given in Table 2. As g0, y and x < y, respectively, must always be expected in Types II and III over a limited temperature range. It is a help in estimating the original x, y values to know whether there is a maximum or minimum in the melting point. At such points x = y = xm, and if this restriction be imposed on eqs 5 and 6, it follows readily that
xm =
{b ±
2
b – 4ac 2a
where a = (g0, – g 0s) (∆fusSB – ∆fusS A), b = 2 (g 0s – g0,) ∆fusS B, and c = (∆fusSB) (∆fusHA ) – (∆fusS A) (∆fusHB) + (g0, – g0s) (∆fusSB ), valid only for regular solution behavior in both phases and with ∆fusH and ∆fusS independent of T. If xm so calculated is < 0, > 1, or imaginary, there is neither maximum nor minimum. The temperature Tm, corresponding to xm, is given by [(g 0, – g0s) xm2 + ∆ fusH B]/∆fusS B, from which it is obvious whether there is a maximum or a minimum, although the magnitude of g0, – g0s may suggest this (see discussion of Table 1 above). The dependence of xm and Tm on g 0, – g 0s was pointed out in 1935 (11). Application of the above expressions to the KBr/KI system is, strictly, incorrect, because the liquid is not regular. However, taking T as an average of
Table 2. Themodynamic Data for Construction of Melting Point Diagrams System A/ B
Type
MgO/ NiO
I
p -C6H4Br2 /p -C6H4BrCla
I
T A° 3075 360.5
KCl/RbCl
I
1044.0
Pd/Ag
I
1823
T B° 2230 337.7 993.0 1234
∆fusH A (J/mol)
∆fusH B (J/mol)
g 0, (J/mol)
g 0s (J/mol)
Ref
77,400
50,625
0
0
13
20,540
18,760
0
457
3, 14
26,285
18,410
84
1500
5
15,990
11,000
1607
{1012
11
Pt /Au
I
2047
1336
21,970
13,100
22120
22,300
11
Si/ Ge
I
1683
1232
11,100
8,300
7100 b
5000 b
15, 16
d -carvoxime/ l -carvoxime
II
345
345
16,200
16,200
2890 b
{740 b
17
,
KBr/ KI
III
1007.0
954.0
25,520
24,015
g 0 = 439 g 1, = 13
6805 – 2.627T
5, 7
NaBr/ NaI
III
1020.0
933.0
26,107
23,598
g 0, = 653 g 1, = { 21
g 0s = 6730 g 1s = {1600
5
Cu/Au
III
1356
13,010
12,680
1000 b
7250 b
18
A/Bc
IV
580
485
16,100
12,580
7100
d
see text
NaCl/ NaI
V
1073.8
933.0
28,158
23,598
g 0, = 2259 – 0.2T g 1, = { 640 – 4.6T
e
5
NaF/ NaCl
VI
1269.0
1073.8
33,135
28,158
g 0, = 1416 g 1, = 1283
—
5
NaF/ RbF
VI
1269.0
1066.0
33,135
22,930
g 0, = 375 + 2.658T g 1, = {4.1040T g 2, = 4.702T
—
5
A/ Bc
VII
425
375
21,250
16,875
7200
—
see text
1336
aThe
data in ref 3 have been reworked and the solids found to fit the regular solution model satisfactorily. bEstimated by the present author. c This is a fictitious system, chosen as being more suitable than an actual system for the present purpose. dG E(α) = 0; G E(α) = 12435; G E(β) = 5700; G E(β) = 0 J/mol, where α, β refer to the two conjugate solid phases—see text. A B A B eG E(α) = 0; G E(α) = 21066; G E(β) = 9649; G E(β) = 0 J/mol—see previous footnote. A B A B
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Journal of Chemical Education • Vol. 74 No. 9 September 1997
Research: Science & Education TA° and TB°, namely, 980 K, and g 0, – g0s as 439 – [6805 – 2.627 (980)] = {3792 J/mol, gives xm = 0.32, Tm = 938 K, a surprisingly accurate result. It should be observed that the reference above to the unlikelihood of miscibility in the solid being combined with partial miscibility in the liquid translates into the unlikelihood of g 0, – g 0s being both large and positive. This would appear to account for the rarity of Type II behavior. Readers wishing to construct Type I, II, or III diagrams may use any of the systems given in Table 2 for these types. It is to be assumed that g0, g1 , ... are temperature independent unless otherwise stated, as are ∆ fusH and ∆fusS. Comparison with experimental results may be made by consulting the references cited. Agreement with experiment better than within a few degrees for a given composition should not be expected because of the approximations made and because the experimental location of the solidi, in particular, is usually open to some uncertainty. The presence of partial miscibility in either the liquid or solid arises for regular solutions when g0 > 2RT (12), and is treated as follows. If an A-rich (α) phase is in equilibrium with a B-rich (β) phase, aA must be the same in both (provided the standard states are the same), and so must aB. Hence
yβ = 0.09. For the α–β equilibria eqs 7 and 8 become at, say, 500 K,
aA = xAα exp (GAE(α)/RT) = x Aβ exp(GAE(β)/RT)
(7)
994R ln x + 133(1 – x)2 + 2566 (1 – x) 3 + 7181 = 0 = fA
(8)
which may be solved by the Newton–Raphson method of successive approximation. Accordingly, if x is an estimated root, a closer approximation is found by solving (dfA/dx)∆x = fA for ∆x, and the root determined iteratively as x′ = x – ∆x. In the present example the exact root is found to be x = 0.39, the solubility of A(s) in B at 994 K. Analogously, eq 6 becomes
aB = (1 – xAα) exp (GBE(α)/RT) = (1 – xAβ ) exp (GBE(β)/RT)
After expressing the excess Gibbs energies in terms of mole fractions the equations are solved for xAα and xAβ. Two simplifications of these equations are commonly encountered. First, if the liquid or solid phases are regular the miscibility gap is symmetric in mole fraction, and eqs 7 and 8 reduce to xAα exp [g0 (1 – xAα)2/RT] = (1 – xAα) exp [g0(xAα)2/RT] (9) since xAα = xBβ and xBα = xAβ. An upper critical solution temperature is predicted to be at Tc = g0 / 2R, where xA = xB = 0.5. Second, if the gap is a wide one, but asymmetric, the assumption that Raoult’s law is valid for the major component and Henry’s law for the minor component in both the α and β phases may be used. Thus GAE(α) = 0; GBE(α) = k
(10)
GAE(β)
(11)
= k′;
GBE(β)
=0
where k and k′ are constants. A Type IV diagram may be thought of as a superposition of a solid miscibility gap on Type I. Its calculation will be illustrated by using the data for the (fictitious) Type IV system given in Table 2. Three sets of calculations at various appropriate temperatures are made, one set for the A-rich solids (α)–liquid equilibria (eqs 1 and 2), another set for the B-rich solids (β)–liquid equilibria (eqs 1 and 2), and a third set for the α–β equilibria (eqs 7 and 8), all in conjunction with eqs 10 and 11, as stated in the table. For the α(s)–liquid region at 535 K, for example, eqs 1 and 2 become, respectively, 535R ln (x/yα) + 7100 (1 – x)2 – 0 + 1249 = 0 = fA 535R ln [(1 – x) / (1 – yα)] + 7100x2 – 12435 – 1297 = 0 = fB solved as shown earlier. The solution is x = 0.45, yα = 0.96 (points on the liquidus and solidus for the α(s)–liquid region). Similarly, at 494 K, for example, eqs 1 and 2 become 494R ln (x / yβ) + 7100(1 – x)2 – 5700 + 2387 = 0 = fA 494R ln[(1 – x) / (1 – yβ)] + 7100x2 – 0 – 233 = 0 = fB for the β(s)–liquid region, the solution for which is x = 0.04,
yAα exp (0) = yAβ exp(5700 / 500R) (1 – yAα) exp (12435 / 500R) = (1 – yAβ)exp (0) the roots of which are yAα = 0.96, yAβ = 0.24. In this way the boundaries of the three 2-phase areas are outlined. The calculations are extended to where these boundaries intersect, and the intersections locate the peritectic at 506 K, with xA, = 0.20, xAα = 0.96, and xAβ = 0.24. The construction of the Type V diagram, a superposition of a solid miscibility gap on Type III, will be illustrated by the system NaCl/NaI (Table 2). The procedure is the same as for the Type IV system above, except that g0 and g1 are temperature dependent and the invariant temperature is below both TA° and TB°. The eutectic is found to be at 850 K, where xA, = 0.40, xAα = 0.96, and xAβ = 0.25. In Type VI the limited solid miscibility has become immeasurably small, so the solid phases are A(s) and B(s). In eq 5, therefore, y = 1, and in eq 6, y = 0. Only eq 5 is needed to locate the A(s) solubility curve, and only eq 6 the B(s). Taking the system NaF/NaCl as an illustration (Table 2), eq 5 becomes, for a representative temperature of 994 K,
994R ln (1 – x) + 3982x2 – 2566x3 + 2093 = 0 = fB the root of which is x = 0.24, the solubility of B(s) in A at the same temperature. The intersection of the solubility curves determined in this way locates the eutectic point at 962 K and x = 0.33, which is close to the experimental value (5). Increasing the deviations in the liquid algebraically decreases the solubility of both components, raising the eutectic temperature. Type VII superimposes partial miscibility in the liquid on Type VI. Using the data in Table 2 for a (fictitious) Type VII system, the boundaries of the miscibility gap are first determined. As the liquid is regular eq 9 is used. At 425 K, for example, xAα exp [7200 (1 – xAα)2/425R] = (1 – xAα) exp [7200(xAα)2/425R] which can be quickly solved by trial and error to give xA = 0.61. The gap thus reaches from xA = 0.39 to 0.61 (symmetrical) at 425 K. The upper critical solution point is at 7200/2R or 433 K, with xAα = xAβ = 0.5. The solubility curves are then determined as shown above for Type VI. Their intersection gives the eutectic temperature (372 K) and composition (xAβ = 0.05), and the intersection of the A(s) curve with the miscibility gap gives the other invariant point (413 K, xAα = 0.68, xAβ = 0.32). The completed diagram shows that the upper portion of the A(s) curve is concave from above, but that the lower portion has the opposite curvature. They are, in fact, one and the same calculated curve with an experimentally unrealizable S-shaped middle portion. Of the 15 systems given in Table 2, the data for five have been utilized above to show how diagrams for Types I to VII can be calculated. The data for the remaining 10 systems should provide ample additional material for further exercise. Several of the systems have been treated as regular although they are actually subregular, and the tempera-
Vol. 74 No. 9 September 1997 • Journal of Chemical Education
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Research: Science & Education ture dependence of ∆fusH and ∆fus S has been ignored. Although this sacrifices some accuracy, it was done for simplicity, and gives results acceptable for the present purpose.
7. 8. 9. 10.
Literature Cited
11. 12.
1. Bakhuis Roozeboom, H. W. Z. phys. Chem. 1899, 30, 385–412. 2. Lewis, G. N.; Randall, M. Thermodynamics, 2nd ed.; Rev. by K. S. Pitzer and L. Brewer; McGraw-Hill: New York, 1961; Chapter 20. 3. Walsh, P. N.; Smith, N. O. J. Phys. Chem. 1961, 65, 718–721. 4. Wachter, A. J. Am. Chem. Soc. 1932, 54, 2271–2278. 5. Sangster, J.; Pelton, A. D. J. Phys. Chem. Ref. Data 1987, 16, 509–561. 6. See, for example, McCoy, W. H.; Wallace, W. E. J. Am. Chem. Soc. 1956, 78, 5995–5998; Oikova, T.; Balarev, Chr.; Makarov. L. I. Russ. J. Phys. Chem. 1976, 50, 205–208.
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13. 14. 15. 16. 17. 18.
Smith, N. O. J. Solution Chem. 1992, 21, 1051–1068. Seltz, H. J. Am. Chem. Soc. 1934, 56, 307–311. Seltz, H. J. Am. Chem. Soc. 1935, 57, 391–395. Gibbs, J. W. The Collected Works of J. Willard Gibbs; Longmans: New York, 1928; Vol. 1, Chapter 3. Scatchard, G.; Hamer, W. J. J. Am. Chem. Soc. 1935, 57, 1809–1811. See, for example, Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969; p 236. v. Wartenberg, H.; Prophet, E. Z. anorg. allgem. Chem. 1932, 208, 369–379. Campbell, A. N.; Prodan, L. A. J. Am. Chem. Soc. 1932, 54, 553–561. Stöhr, H.; Klemm, W. Z. anorg. allgem. Chem. 1939, 241, 305–323. Thurmond, C. D. J. Phys. Chem. 1953, 57, 827–830. Adriani, J. H. Z. phys. Chem. 1900, 33, 453–476. Bennett, H. E. J. Inst. Metals 1962, 91, 158 as quoted in Hansen, M. Constitution of Binary Alloys, 2nd ed.; McGraw-Hill: New York, 1958; pp 198–203.
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