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Phase rule

The phase rule, formally known as the Gibbs phase rule, is a cornerstone of thermodynamics that quantifies the degrees of freedom in a multiphase equilibrium system by relating the number of independent components and phases present.^[1] It is mathematically expressed as F = C - P + 2, where F represents the degrees of freedom (the number of intensive variables, such as temperature, pressure, or composition, that can be independently varied without altering the number of phases), C is the number of components (the minimum number of independent chemical species required to define the system's composition), and P is the number of phases (distinct, homogeneous physical states like solid, liquid, or gas).^[2] This rule applies to closed systems at thermodynamic equilibrium where components can freely distribute between phases, assuming only pressure-volume work and no external fields or reactions that impose additional constraints.^[3] Formulated by American physicist and mathematician J. Willard Gibbs in his seminal 1876–1878 work On the Equilibrium of Heterogeneous Substances, the phase rule emerges from the conditions for phase stability derived from the Gibbs-Duhem equation, which links chemical potentials, temperature, and pressure across phases.^[4] Gibbs' derivation underscores that equilibrium requires equality of chemical potentials for each component in every phase, leading to the rule's predictive power for phase diagrams and transitions.^[5] For instance, in a single-component system like pure water (C = 1), the rule explains the invariant triple point where solid, liquid, and vapor coexist (P = 3, F = 0), univariant melting curve (P = 2, F = 1), and bivariant single-phase regions (P = 1, F = 2).^[1] The phase rule's significance extends beyond pure thermodynamics, enabling the analysis of complex multicomponent systems in practical applications. In materials science, it guides the interpretation of binary alloy phase diagrams, such as the eutectic in silver-copper systems, where C = 2 yields regions of varying phase assemblages and degrees of freedom.^[2] In geology and geochemistry, it informs petrogenesis by estimating pressure-temperature conditions from mineral assemblages, as seen in the invariant equilibrium of andalusite, kyanite, and sillimanite in the Al₂SiO₅ system at specific pressures around 3.8 kbar and 500°C.^[3] Variations of the rule account for constraints like fixed pressure (reducing the +2 to +1) or chemical reactions, enhancing its utility in fields from chemical engineering to planetary science.^[5]

Definition and Fundamentals

Gibbs Phase Rule Equation

The Gibbs phase rule provides a fundamental relation for the equilibrium of heterogeneous systems and is mathematically expressed as

F = C - P + 2

where F denotes the number of degrees of freedom or variance of the system, C is the number of independent chemical components, and P is the number of coexisting phases.^[3]^[6] This equation quantifies the constraints imposed by phase equilibrium on the system's intensive state variables.^[3] Formulated by American physicist Josiah Willard Gibbs in his landmark 1876–1878 publication On the Equilibrium of Heterogeneous Substances,^[4] the rule emerged from his analysis of thermodynamic potentials in multiphase systems. The +2 term specifically arises because temperature and pressure serve as the two primary intensive variables that can be independently varied in such systems, assuming no additional external fields or special constraints like fixed volume or chemical reactions.^[6]^[3] In this context, F indicates the maximum number of intensive thermodynamic variables—such as temperature, pressure, or composition—that can be freely adjusted while maintaining the specified number of phases at equilibrium, without altering the phase assemblage.^[3] The rule thus encapsulates the dimensionality of the equilibrium manifold for heterogeneous substances under standard thermodynamic conditions.^[6]

Interpretation of Variables

In the Gibbs phase rule, the number of components, denoted as C, represents the smallest number of independent chemical constituents required to describe the composition of all phases in a system at equilibrium. These constituents are chemically independent substances or mixtures of fixed composition from which the phases can be prepared, accounting for any chemical equilibria or stoichiometric constraints that may reduce the effective independence. For instance, in a system consisting of water and sodium chloride (salt), C = 2, as the salt is treated as a single molecular species despite its dissociation into ions in solution; the ions are not counted as additional components because their concentrations are linked by the dissociation equilibrium.^[7]^[2] The number of phases, P, refers to the count of distinct, homogeneous regions within the system that have uniform physical and chemical properties throughout, separated by interfaces or boundaries. Each phase can be a solid, liquid, gas, or other state, provided it maintains internal uniformity; for example, in a simple system like pure water, the solid ice, liquid water, and water vapor constitute three separate phases when coexisting. In multicomponent systems, such as the aforementioned water-salt mixture, phases might include solid salt crystals, the saturated aqueous solution, and water vapor, each with potentially different compositions but homogeneity within.^[7]^[2] Degrees of freedom, F, indicate the number of intensive thermodynamic variables—such as temperature (T), pressure (P), or phase compositions (e.g., mole fractions)—that can be independently varied without disrupting the equilibrium among the phases, as governed by the relation F = C - P + 2. A system with F = 0 is invariant, meaning no variables can be changed (e.g., a triple point); F = 1 is univariant, allowing one variable to vary (e.g., along a coexistence curve); and higher F values permit more flexibility. Importantly, F only considers intensive properties, which are independent of the system's total size or extent, excluding extensive variables like total volume or mass that scale with the amount of material; this distinction ensures the rule applies universally to systems of varying scale, particularly in closed systems where no matter is exchanged with the surroundings.^[7]^[2]

Thermodynamic Derivation

Intensive Variables and Constraints

In multiphase thermodynamic systems, the key intensive variables that describe the state at equilibrium are temperature (T), pressure (P), and the chemical potential (\mu_i) of each component i. These variables are intensive because they do not depend on the size or extent of the system, allowing the equilibrium conditions to be specified independently of the amounts of each phase.^[8]^[2] For a system to be in thermodynamic equilibrium across multiple phases, certain conditions must hold uniformly. Specifically, the temperature and pressure must be the same in every phase, ensuring thermal and mechanical equilibrium, respectively. Additionally, the chemical potential of each component must be equal in all phases where that component is present, reflecting diffusive equilibrium and the minimization of the Gibbs free energy. These equality conditions impose fundamental constraints on the system's state.^[9]^[8] In a system with P phases and C components, the equilibrium constraints arise from the equality of chemical potentials across phases, yielding C(P-1) independent equalities (since the equalities are transitive, there are P-1 per component). These constraints limit the number of independent intensive variables that can be freely specified while maintaining equilibrium. The total number of possible intensive variables includes the system-wide T and P, and the independent composition variables in each phase—specifically, C-1 independent mole fractions per phase (since the fractions in each phase sum to unity), yielding a total of $2 + P(C-1).^[2]^[9]^[8] The Gibbs phase rule applies under the assumption of no chemical reactions occurring within the system, as reactions would introduce additional equilibrium conditions that effectively reduce the number of independent components C by the number of independent stoichiometric constraints (typically r-1 for r reactions). External fields, such as gravitational or electric fields, are also excluded, as they could add further constraints on the intensive variables.^[8]^[9]

Step-by-Step Derivation

The derivation of the Gibbs phase rule proceeds by systematically counting the intensive variables required to specify the equilibrium state of a system with C components distributed among P phases, then subtracting the number of independent constraints imposed by thermodynamic equilibrium conditions. This approach, originally developed by J. Willard Gibbs, relies on the fundamental requirements that temperature T and pressure P are uniform throughout the system, and that the chemical potential \mu_i of each component i is equal across all phases.^[4]^[2] The intensive variables consist of T and P, which are shared across all phases, contributing 2 variables. For compositions, each phase is described by the mole fractions x_i^\alpha of the C components in phase \alpha, but the normalization condition \sum_{i=1}^C x_i^\alpha = 1 reduces the independent compositional variables per phase to C - 1. With P phases, the total compositional variables are thus P(C - 1). The overall number of intensive variables is therefore

$2 + P(C - 1).

^[2]/12:_Phase_Equilibrium/12.02:_Gibbs_Phase_Rule) Equilibrium imposes constraints through the equality of chemical potentials: for each of the C components, the chemical potential must be the same in every phase, yielding C independent equalities per pair of phases. Since these equalities are transitive, there are P - 1 independent equations per component, for a total of C(P - 1) constraints. Notably, no additional independent equations arise from the uniformity of T and P, as these are already treated as single system-wide variables rather than phase-specific ones.^[4]^[2] The number of degrees of freedom F, or the number of intensive variables that can be independently varied without altering the number of phases, is the total number of variables minus the number of independent constraints:

F = [2 + P(C - 1)] - C(P - 1).

Simplifying the expression gives

F = 2 + PC - P - CP + C = C - P + 2.

^[2]/12:_Phase_Equilibrium/12.02:_Gibbs_Phase_Rule) This is the Gibbs phase rule, where the +2 accounts for the freedoms in choosing T and P. If external conditions fix one variable, such as pressure in a constant-pressure process, the degrees of freedom reduce by 1, yielding F = C - P + 1. This adjustment reflects the loss of one independent variable while the constraints remain unchanged.^[2]

Applications to Simple Systems

Single-Component Systems

In single-component systems, consisting of a pure substance (C=1), the Gibbs phase rule simplifies to F = 3 - P, where F is the number of degrees of freedom, and P is the number of phases. This formulation, derived from the general rule F = C - P + 2, dictates the variability of intensive variables like temperature and pressure.^[2]^[3] For a single phase (P = 1), the system is bivariant (F = 2), allowing independent variation of temperature and pressure within broad regions of stability, such as the solid, liquid, or vapor domains. When two phases coexist (P = 2), the system becomes univariant (F = 1), restricting equilibrium to specific curves, for example, the vapor pressure curve where liquid and vapor phases are in balance at a fixed temperature-dependent pressure, or the melting curve separating solid and liquid. At the triple point (P = 3), the system is invariant (F = 0), fixing both temperature and pressure uniquely for the coexistence of three phases. A representative case is the triple point of water, at T = 0.01^\circC (273.16 K) and P = 611.657 Pa, where ice, liquid water, and vapor equilibrate.^[3]^[10]^[11] The pressure-temperature (P-T) phase diagram for a pure substance illustrates these features: extensive areas denote bivariant single-phase regions, monovariant lines trace two-phase boundaries (e.g., sublimation, vaporization, and fusion curves), and invariant points mark triple points. The liquid-vapor coexistence line concludes at the critical point, where the meniscus vanishes and the phases become indistinguishable, transitioning to a single supercritical fluid phase with F = 2. These monovariant lines and invariant points highlight the phase rule's role in defining precise conditions for phase transitions in pure substances.^[12]^[6]

Binary Systems

In binary systems, where the number of components C = 2, the Gibbs phase rule simplifies to F = 4 - P, where F is the degrees of freedom and P is the number of phases. For a single phase (P = 1), F = 3, allowing independent variation of temperature T, pressure P, and the composition variable x (the mole fraction of one component). With two phases (P = 2), F = 2. At fixed pressure, the effective F' = 1, such as varying temperature T, which determines the compositions of the coexisting phases. For three phases (P = 3), the system has F = 1.^[2] Binary phase diagrams, typically constructed at constant pressure (isobaric sections), provide a visual representation of phase equilibria in these systems, plotting temperature against composition. In two-phase regions, tie-lines connect the compositions of the coexisting phases, and the lever rule determines the relative amounts of each phase: the fraction of one phase is the length of the tie-line segment opposite to it divided by the total tie-line length. These regions often appear as lens-shaped areas bounded by liquidus and solidus curves, indicating partial miscibility. At constant pressure, the effective degrees of freedom reduce to F' = C - P + 1 = 3 - P, reflecting the constraint on pressure and emphasizing T and x as variables.^[13] A key feature in many binary systems is the eutectic point, where three phases—typically two solids and a liquid—coexist in equilibrium, making the system invariant (F' = 0) at fixed pressure, with temperature and phase compositions fixed at the eutectic values. Below this point, the two solid phases form without an intervening liquid. In contrast, isomorphous systems exhibit complete solid solubility, lacking a eutectic; the Cu-Ni alloy system is a classic example, where a single solid phase (α) spans all compositions at lower temperatures, with phase boundaries shifting upon varying temperature.^[14] The Pb-Sn system, however, displays a eutectic at approximately 61.9 wt% Sn and 183°C, where liquid decomposes into α (Pb-rich solid) and β (Sn-rich solid) phases, illustrating limited solid solubility and a pronounced three-phase invariance. In metallic alloys, varying temperature along an isobar alters phase boundaries, enabling control of microstructure through heat treatment, as the degrees of freedom dictate the stability of phases like solid solutions or intermetallics.

Advanced Examples and Cases

Multicomponent Electrolyte Solutions

In multicomponent electrolyte solutions, such as those involving multiple salts dissolved in water, the Gibbs phase rule must account for the dissociation of salts into ions, which influences the effective number of independent components. Consider a system comprising four uni-univalent salts—NaCl, KCl, NaNO₃, and KNO₃—along with water; nominally, this constitutes five components (the four salts plus water). However, upon complete dissociation, the system yields five chemical species: Na⁺, K⁺, Cl⁻, NO₃⁻, and H₂O. Due to the reciprocal relationships among the salts (where any combination of the two cations and two anions can form) and the electroneutrality constraint in the solution phase, the effective number of independent components reduces to four for the purpose of applying the phase rule. This reduction arises because the compositions of the phases can be expressed using three independent composition variables (e.g., the concentrations of two cations relative to the anions), plus the solvent, ensuring the rule accurately predicts the degrees of freedom.^[15] A key application of the phase rule in such systems occurs at univariant points, where multiple solid phases precipitate simultaneously from the solution. In quaternary reciprocal systems like Na-K-Cl-NO₃-H₂O, a univariant equilibrium can exist with four solid phases (e.g., NaCl, KCl, NaNO₃, and KNO₃) coexisting with the saturated liquid solution, totaling five phases (P=5). With an effective C=4, the degrees of freedom F = C - P + 2 = 1, meaning this equilibrium is univariant, occurring at a specific temperature for a given pressure (or along a curve in temperature-pressure space), with the composition fixed at the equilibrium point independent of overall system composition variations. Such points mark the boundaries where the solution becomes supersaturated with respect to all four solids, leading to simultaneous precipitation without changing the intensive variables along the univariant line. Phase diagrams for these multicomponent electrolyte solutions are often represented using polythermal projections, which plot solubility curves as a function of temperature in a reduced compositional space (e.g., a triangular diagram for the three effective salt components), revealing univariant curves and invariant points under fixed pressure. Compatibility triangles in these diagrams delineate regions of stable solid-liquid equilibria, showing which combinations of solid phases can coexist with the solution without violating the phase rule; for instance, triangles may connect NaCl, KNO₃, and the solution phase. These diagrams extend the principles from simpler binary systems to higher dimensions, allowing prediction of precipitation sequences under varying conditions. The historical study of the Na-K-Cl-NO₃-H₂O system, pioneered by H.W.B. Roozeboom in the late 19th century, exemplifies the phase rule's utility in electrolytes, demonstrating up to five phases at univariant points through experimental solubility determinations and graphical representations. Complications such as common ion effects—where the presence of one ion suppresses the solubility of another sharing the same charge—can influence phase boundaries but do not alter the rule's applicability, provided components are defined in terms of independent ions and constraints like charge balance are properly incorporated.

Constant Pressure Conditions

In systems where pressure is held constant, such as under isobaric conditions, the Gibbs phase rule is modified to account for the fixed pressure, which eliminates it as a degree of freedom. The resulting form is F' = C - P + 1, where F' is the number of remaining degrees of freedom, C is the number of components, and P is the number of phases. This adaptation reduces the variance by one compared to the general rule F = C - P + 2, as pressure no longer varies independently.^[16] For a single-component system (C = 1) at constant pressure, the implications are straightforward. With one phase (P = 1), F' = 1, allowing temperature to be varied independently while maintaining equilibrium, such as in the liquid state of a pure substance. With two phases (P = 2), F' = 0, resulting in an invariant condition where temperature is fixed, exemplified by the melting or boiling point of the substance at that pressure. These fixed points represent transitions where the system cannot tolerate changes in temperature without altering the phase count.^[16] In binary systems (C = 2) at fixed pressure, three-phase equilibria become invariant (F' = 0), occurring at specific temperatures and compositions, such as the eutectic point where solid, liquid, and another solid phase coexist at a fixed eutectic temperature. Here, univariant lines in the full phase diagram project to specific points in the isobaric temperature-composition plane, marking invariant transitions. Two-phase regions, in contrast, are univariant (F' = 1), traced as lines in these diagrams where temperature or composition can vary along the equilibrium boundary. This modified rule finds extensive application in constructing isobaric phase diagrams, which plot temperature against composition at constant pressure, commonly 1 atm in metallurgy to predict alloy behaviors during casting or heat treatment. For instance, in binary metal alloys, these diagrams delineate regions of phase stability and invariant points like eutectics, guiding processes such as solidification where pressure variations are negligible. It is particularly useful in atmospheric pressure experiments, where the full rule's pressure term is irrelevant, simplifying analysis of condensed phases.^[2]

Extensions and Limitations

Colloidal and Non-Ideal Mixtures

In colloidal systems, the Gibbs phase rule is applied by recognizing the dispersed phase—typically consisting of particles, droplets, or bubbles—and the continuous dispersion medium as distinct phases, which effectively increases the number of phases (P) compared to homogeneous systems. This distinction is crucial for systems like sols (where solid or liquid particles are dispersed in a liquid medium) and gels (where a solid network traps a liquid), allowing the rule to predict the degrees of freedom (F = C - P + 2) once these phases are properly defined. For instance, a simple lyophilic sol may be treated as involving two phases, elevating the effective component count or phase tally and thus permitting more variance in intensive variables like temperature and composition before equilibrium is fixed.^[17]^[18] Non-ideal behaviors in colloidal and other mixtures arise from interactions such as electrostatic repulsion or van der Waals forces, which are quantified using activity coefficients to correct chemical potentials in equilibrium conditions; however, the phase rule itself remains applicable as it relies solely on the invariance of intensive variables at equilibrium, independent of ideality assumptions. Limitations emerge when interfacial effects dominate, as in highly dispersed colloids, where surface tension introduces an additional constraint akin to an extra variable, potentially reducing the predicted F if not accounted for in phase identification. In such cases, the rule's predictions may deviate unless surface contributions are incorporated into the thermodynamic analysis.^[19]^[20] A representative example is lyophobic colloids, such as gold sols in water, where adsorption of stabilizing ions onto particle surfaces prevents coagulation; here, the degrees of freedom may require adjustment to reflect surface-specific equilibria, with the Gibbs adsorption isotherm providing essential context by relating changes in surface tension (dγ) to the surface excess concentration (Γ_i) of adsorbed species via dγ = -∑ Γ_i dμ_i, highlighting how interfacial adsorption influences overall system stability without altering the core phase rule structure. In some gel systems, the dispersed solid network and continuous liquid medium are explicitly counted as two phases, yielding a higher P value and correspondingly lower F, which aligns with the constrained equilibrium observed in gelation under fixed external conditions. Wolfgang Ostwald's foundational work in colloid chemistry extended phase rule applications by emphasizing colloids as quasi-independent phases, facilitating more accurate counting in heterogeneous dispersions like gels.^[20]^[21] The phase rule presupposes bulk phases where surface-to-volume ratios are negligible, but in colloidal systems with nanoscale particles, this assumption breaks down, leading to failures when dominant interfacial energies, gravitational sedimentation, or diffusive fluxes are overlooked, as these introduce unaccounted variances that prevent true equilibrium. Consequently, for very fine dispersions, empirical modifications or complementary models, such as those incorporating capillary effects, are often necessary to extend the rule's utility.^[22]

Historical Development and Scope

The phase rule was first formulated by American physicist and mathematician Josiah Willard Gibbs in his seminal two-part memoir "On the Equilibrium of Heterogeneous Substances," published in 1876 and 1878, respectively.^[23] This work established the thermodynamic foundations for analyzing equilibrium in multiphase systems, deriving the relation between the number of components, phases, and degrees of freedom, and was independently explored by Pierre Duhem in the 1880s.^[24] Gibbs' formulation formalized the conditions for phase stability, providing a rigorous framework that integrated the first and second laws of thermodynamics with the concept of chemical potentials across phases.^[23] Despite its groundbreaking nature, Gibbs' contributions, including the phase rule, were initially overlooked in Europe and the broader scientific community, remaining largely unrecognized for over two decades due to limited circulation and the memoir's mathematical complexity.^[25] The rule gained prominence in the 1890s through the efforts of Dutch physical chemist Hendrik Willem Bakhuis Roozeboom, who applied it experimentally to construct detailed phase diagrams for binary and ternary systems, as detailed in his multi-volume treatise Die heterogenen Gleichgewichte vom Standpunkte der Phasenlehre (starting 1899).^[26] Roozeboom's visualizations and empirical validations popularized the rule among chemists and engineers, demonstrating its utility in predicting phase behaviors under varying conditions. Concurrently, Norwegian geologist and petrologist Johan Herman Lie Vogt extended its applications to igneous rocks and mineral assemblages in the late 1890s and early 1900s, using it to model crystallization processes in silicate systems and slags, thereby bridging thermodynamics with petrological interpretations.^[27] The phase rule's scope encompasses thermodynamic equilibrium in diverse fields, including classical chemistry for vapor-liquid equilibria, materials science for alloy design, and geology for analyzing mineral parageneses in rocks where the number of coexisting phases is constrained by system components.^[28] In its generalized form, it accounts for additional intensive variables such as electric or magnetic fields, incrementing the degrees of freedom by one per external field to describe systems like electrochemical cells.^[28] Modern extensions include computational approaches like the CALPHAD (CALculation of PHAse Diagrams) method, developed in the 1970s by Larry Kaufman and others, which extrapolates experimental data to predict multicomponent phase diagrams for advanced materials.^[29] However, the rule applies strictly to stable equilibrium states and does not address metastable configurations or kinetic barriers that hinder phase transitions in real-world processes.^[29]

References

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The Gibbs phase rule links the number of phases, number of components and number of degrees of freedom in a system.
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Jul 7, 2024 · The phase rule applies to equilibrium systems in which any component can move freely between any two phases in which that component is present.
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[9]
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A good example of a system with just one chemical component: H2O. For a one-component system, temperature and pressure alone determine what phase is present at ...
[10]
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[12]
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Application of the Phase Rule to Binary PD. What is the number of DOF at the eutectic point? P=3 (liquid, solid A, B). F = C-P+2 = 2-3+2. F =1. As pressure is ...
[13]
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An example is the Ni-Cu system. Compositions of phases is determined by the tie line. The relative fractions of the phases are determined by the lever rule.
[14]
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### Summary of Gibbs Phase Rule at Constant Pressure (Isobaric Conditions)
[15]
Colloids - Chemistry LibreTexts
Jan 29, 2023 · In colloids, one substance is evenly dispersed in another. The substance being dispersed is referred to as being in the dispersed phase, while ...
[16]
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Sols, Gels, and Aerogels are Colloids. A colloid is a mixture in which at least two different phases are intimately mixed at the nanolevel. The term “phase” ...
[17]
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Gibbs Phase Rule for non-reactive system: F = 2 + C – P where F is the number of degrees of freedom, C is the number of components, and P is the number of ...
[18]
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If phases are in a thermodynamic equilibrium, there is a balanced passing of molecules from both phases into the surface layer and back. Calculations show that ...
[19]
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Lyophobic colloids: Colloidal solution in which the particles of the dispersed phase have no affinity for the dispersion medium are lyophobic colloids.
[20]
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[21]
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HETEROGENEOUS SUBSTANCES. First Part. By J. WILLARD GIBBS,. PROFESSOR OF MATHEMATICAL PHYSICS IN YALE COLLEGE, NEW ...
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Jul 13, 2007 · Among these were the Duhem–Margules and Gibbs–Duhem equations, which deal with reversible processes in thermodynamics as quasi-static limiting ...
[23]
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This article is the first in a three-part series tracing the evolution of CALPHAD from the Gibbs Phase Rule to the first lattice stabilities.
[24]
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Bakhuis Roozeboom, Hendrik Willem, 1854-1907. Die heterogenen gleichgewichte Standpunkte der Phasenlehre / von Dr. H. W. Bakhuis Rooseboom v.3 pt.2 ...
[25]
[PDF] AM32_481.pdf
The study of slags by J. H. L. Vogt was of the utmost importance in orienting the pro- gram of experimental petrology. ... phase rule studies; the ...
[26]
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This brief note calls attention to the literature discussing four pedagogical points relating to the application and teaching of the well-known Gibbs phase ...<|control11|><|separator|>
[27]
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In the earliest days of computer growth, a pre-CALPHAD group of scientists was busy performing what are now considered to be “simple” phase diagram calculations ...