Communication Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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The Gibbs Phase Rule: What Happens When Some Phases Lack Some Components? Deepika Janakiraman*
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Department of Chemistry, Indian Institute of Technology, Madras, Chennai, India ABSTRACT: The derivation for the Gibbs phase rule, provided in physical chemistry text books, often assumes that all the components are present in all the phases coexisting at equilibrium. However, very often we have situations where all the phases at equilibrium do not have all the components, the binary eutectic system being a classic example. The melt (miscible solution of A and B) coexists with pure solid A and pure solid B at the eutectic point. The solid phases are one-component phases, whereas the system itself is binary. How does the phase rule, derived under the above-mentioned assumption, apply to the eutectic point? Using a simple example, we demonstrate that all the components need not be present in all the phases to arrive at the phase rule. KEYWORDS: Second-Year Undergraduate, Misconceptions/Discrepant Events, Physical Chemistry, Phases/Phase Transitions/Diagrams
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RECAPITULATING THE TEXTBOOK DERIVATION OF THE GIBBS PHASE RULE The Gibbs phase rule has been an essential part of the physical chemistry curriculum and taught to generations of students. The rule states that if a system consists of C components and P phases existing in equilibrium, the number of degrees of freedom or the variance (F) is given by1−3 (1) F = C − P + 2. At the heart of the Gibbs phase rule lies the Gibbs−Duhem equation, which is given by2,4,5
superscript denotes the phase in which it is present). This results in a total of (P − 1) equations or constraints for a single component. When there are C components, one obtains the number of constraints (or equations) = C(P − 1).
(4)
The constraints reduce the number of variables that can be independently varied. Therefore, the degrees of freedom (independent variables), F, for a system at equilibrium with C components in all P phases, coexisting at equilibrium, is F = variables − constraints = 2 + P(C − 1) − C(P − 1)
C
∑ Ndi μi + SdT − Vdp = 0. i=1
= C − P + 2,
(2)
which is the classic Gibbs phase rule.
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For the benefit of a student, we shall revisit the implications of the above equation and how it leads to the conventional derivation for the Gibbs phase rule. The Gibbs−Duhem eq (eq 2) suggests the following: If we consider a single phase with C components, the infinitesimal changes in their chemical potentials (μi’s), temperature, and pressure have to obey the above relation and cannot be varied arbitrarily. Fixing the temperature, pressure, and chemical potential for C − 1 components automatically fixes the chemical potential for the Cth component. Therefore, for a single phase with C components, there are 2 + (C − 1) variables that one could vary. When there are P phases with each phase consisting of C components, P(C − 1) chemical potentials, along with temperature and pressure, can be varied independently according to the Gibbs−Duhem equation. Therefore, the number of variables = 2 + P(C − 1).
DO THE ASSUMPTIONS IN THE ABOVE DERIVATION ALLOW US TO EMPLOY THE PHASE RULE TO THE BINARY EUTECTIC SYSTEM? The standard derivation for the Gibbs phase rule has made an assumption that all the components in the system are present in all the phases. Let us consider a textbook example of the binary eutectic system and see if this assumption holds.2,5−7 The simple eutectic system is a binary mixture where the two components are completely miscible in the liquid (melt) phase but immiscible in the solid phase. One has to bear in mind that the reduced phase rule, F = C−P + 1, is used for most binary systems. The pressure is kept constant for the ease of representing the phase diagram. The phase diagram for the eutectic system, provided in Figure 1, has five distinct regions, which are as follows 1. Liquid (melt): Both components are present in this phase (C = 2, P = 1, F = 2; all values of F for the eutectic system are
(3)
This expression does not assume equilibrium among the phases. At equilibrium, each component has the same chemical potential across phases, i.e., the chemical potentials for, say, component 1 in the coexisting phases will obey μ11 = μ21 = ··· μP−1 = μP1 (subscript on μ denotes the component and the 1 © XXXX American Chemical Society and Division of Chemical Education, Inc.
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Received: May 21, 2018 Revised: August 24, 2018
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DOI: 10.1021/acs.jchemed.8b00377 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Communication
as well as the mole fractions of pure solids A and B are always equal to unity. Similar arguments hold for the region with hypereutectic composition.5 It is evident that in 4 out of 5 regions (barring the melt phase), all of the coexisting phases do not contain all of the components. Though the existence of all components in all phases was an asusmption in deriving the phase rule, textbooks have always applied the phase rule, without hesitation, to find the variance in all regions of the eutectic system! Phase Rule with Phases That Lack Some Components
A diligent student would immediately question the validity of using the Gibbs phase rule for the binary eutectic system. The assumption of all components (C) in all phases (P) is clearly violated for the simple eutectic system. However, applying the Gibbs phase rule does give results that are physically reasonable and verified to be correct for generations. How does the phase rule scale, and scale correctly, when some of the coexisting phases are deficient in some components? The aim of this article is to address this question and demonstrate using a simple example that the phase rule holds independent of the composition of the coexisting phases. Let us assume a situation where there are P phases in equilibrium. All C components are present in P − 1 phases, resulting in (P − 1)(C − 1) variables. The Pth phase is deficient in one of the components and has only C − 1 components, leading to C − 2 quantities that can be varied. Therefore, according to the Gibbs−Duhem equation, the
Figure 1. Phase diagram for a simple binary eutectic system
evaluated according to the reduced phase rule). One can vary both temperature and composition of the melt, giving rise to F = 2 in this region. 2. Liquid + Solid A (region enclosed by DGE in Figure 1): The liquid phase consists of both components and the solid consists of only A (C = 2, P = 2, F = 1). In this region, the melt composition at a given temperature is always given by the tieline to the phase boundary DE. Upon reducing temperature for a given % composition (x-axis of the phase diagram), more of solid A precipitates out, leaving the melt richer in B. The single degree of freedom in this region is temperature. Fixing the temperature fixes the melt composition, which is given by the DE curve. The solid, which is pure A, has a mole fraction equal to 1, leaving temperature as the only independent parameter. It must be noted that both the phase boundary DE and the region DGE have F = 1! 3. Liquid + Solid B (Region enclosed by FHE in Figure 1): In this region, solid B precipitates out of the melt and is otherwise similar to the region DGE. 4. The eutectic point (point E in Figure 1): The three phases, liquid, pure solid A, and pure solid B, coexist (C = 2, P = 3, F = 0). At point E in Figure 1, the melt solidifies completely without preferentially precipitating solid A or B. Though the melting behavior of the eutectic solid resembles that of a pure substance, microscopic observation reveals that it is a mixture with alternate lamellaes of A and B and not a homogenous solid solution with an isomorphous structure (arising from the immiscibility of A and B in the solid phase). As a result of the distinct phase boundaries between A and B layers and the absence of a uniform crystal structure, the eutectic solid is biphasic.8,9 5. Solid: For a solid with hypoeutectic composition (GECA region in Figure 1), the microstructure shows primary crystals of A (which started forming in the DGE region) and smaller crystals with the eutectic composition (the melt composition will be the eutectic composition at the eutectic temperature below which it solidifies completely). This is effectively a twophase region (of pure solid A and pure solid B) with C = 2, P = 2, and F = 1. The single degree of freedom is temperature here
number of variables = 2 + (P − 1)(C − 1) + 1(C − 2), (6)
where “2” stands for the varying temperature and pressure. As for the constraints, C − 1 components exist in P phases at equilibrium. The condition of equal chemical potentials across phases will result in (C − 1)(P − 1) equations or constraints. The last component, namely the Cth component that exists only in P − 1 phases, gives rise to P − 2 constraints. As a result, the number of constraints = (C − 1)(P − 1) + 1(P − 2). (7)
Therefore, the variance or the degrees of freedom (F) for this system is F = 2 + (P − 1)(C − 1) + 1(C − 2) − (C − 1)(P − 1) − (P − 2) = C − P + 2.
(8)
This is a very reassuring result! Even though a phase is deficient in one of the components, the Gibbs phase rule remains unchanged. In other words, when we remove one component from a phase, we reduce the number of variables we can specify by one, but the number of constraints also reduces by one!4,5 This keeps the phase rule unchanged in form. If we have several phases deficient in several components, we can arrive at the desired composition by sequentially removing one component from a phase at a time. Thereby, it becomes evident that the number of variables and constraints would change by the same number.
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CONCLUSIONS The assumption that all C components are present in all P phases in the standard textbook derivation is superfluous. It makes it appear as if the rule has a caveat, which is not true. This has seldom been brought to the notice of students, B
DOI: 10.1021/acs.jchemed.8b00377 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Communication
(16) Castellan, G. W. Phase equilibria and the chemical potential. J. Chem. Educ. 1955, 32, 424. (17) Hollingsworth, C. A. An Interpretation to the Phase Rule. J. Chem. Educ. 1952, 29, 464. (18) Battino, R. The critical point and the number of degrees of freedom. J. Chem. Educ. 1991, 68, 276. (19) Alberty, R. A. Components in Chemical Thermodynamics. J. Chem. Educ. 1995, 72, 820. (20) Franzen, H. F. The true meaning of component in the Gibbs phase rule. J. Chem. Educ. 1986, 63, 948.
though several articles have been published on the phase rule over the years.10−17 There could be situations that lay constraints in addition to the equilibrium condition (i.e., equal chemical potential for a component across phases) (i) peritectic system,4 where the two components react for a product under a specific composition and temperature,2 (ii) an azeotropic mixture where the composition of the liquid and vapor is the same at all temperatures,1,2 and (iii) liquid−vapor critical point, where the density of the liquid becomes equal to that of the vapor.18 These conditions must be treated as additional constraints, which will further reduce the degrees of freedom. Although the issue of additional constraints has been addressed in the literature,12,19,20 the invariant nature of the phase rule irrespective of the composition of the phases is rarely appreciated. It is necessary to understand this aspect to apply the phase rule to binary and multicomponent systems.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Deepika Janakiraman: 0000-0003-1309-5603 Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS The author thanks Prof. K. S. Viswanathan, IISER, Mohali for his valuable comments and Prof. E. Prasad, IIT, Madras for bringing a discussion on the eutectic system in Physical Chemistry by Silbey and Alberty to the author’s notice. This work was supported by the Department of Science and Technology, India under the INSPIRE Faculty fellowship programme.
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REFERENCES
(1) Atkins, P. W.; de Paula, J. Atkins’ Physical Chemistry, 8th ed.; W. H. Freeman and Company: New York, 2006; Chapter 6. (2) Prutton, C. F.; Maron, S. H. Principles of Physical Chemistry, 4th ed.; The Macmillan Company: New York, 1965; Chapter 14. (3) Deming, H. G. An Introduction to the Phase Rule. Part I. J. Chem. Educ. 1939, 16, 215. (4) Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: Boston, MA, 2002; Chapters 7 and 12. (5) Silbey, R. J.; Alberty, R. A.; Bawendi, M. G. Physical Chemistry, 4th ed.; John Wiley and Sons, Inc.: Hoboken, NJ, 2005; Chapter 6. (6) Toshev, B. V. On Gibbs’ Phase Rule. Langmuir 1991, 7, 569. (7) Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry, 2nd ed.; Oxford University Press: New York, 2000; pp 659−663. (8) Loxham, J. G.; Hellawell, A. Constitution and Microstructure of Some Binary Alkali Halide Mixtures. J. Am. Ceram. Soc. 1964, 47, 184. (9) Berg, R. W.; Kerridge, D. H. Raman mapping in the elucidation of solid salt eutectic and near eutectic structures. J. Raman Spectrosc. 2002, 33, 165. (10) Redlich, O. On the phase rule. J. Chem. Educ. 1945, 22, 265. (11) Alper, J. S. The Gibbs Phase Rule Revisited: Interrelationships between Components and Phases. J. Chem. Educ. 1999, 76, 1567. (12) Jensen, W. B. Generalizing the Phase Rule. J. Chem. Educ. 2001, 78, 1369. (13) Mindel, J. Gibbs’ Phase Rule and Euler’s Formula. J. Chem. Educ. 1962, 39, 512. (14) Hillert, M. Gibbs’ phase rule applied to phase diagrams and transformations. J. Phase Equilib. 1993, 14, 418. (15) Daub, E. E. Gibbs phase rule: A centenary retrospect. J. Chem. Educ. 1976, 53, 747. C
DOI: 10.1021/acs.jchemed.8b00377 J. Chem. Educ. XXXX, XXX, XXX−XXX
