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Ellingham diagram

An Ellingham diagram is a thermodynamic plot that displays the standard Gibbs free energy change (ΔG°) for the formation of metal oxides (and sometimes sulfides or other compounds) as a function of temperature, typically normalized to the reaction of one mole of O₂ at standard pressure (1 atm).^[1] It serves as a key tool in extractive metallurgy to evaluate the relative stability of oxides and predict the feasibility of reduction processes, such as determining which metals or reducing agents can liberate another metal from its oxide ore.^[2] Developed by British chemist Harold J. T. Ellingham in 1944, the diagram originated from his analysis of oxide and sulfide reducibility in metallurgical processes, presented in a seminal paper that plotted free energy data to guide industrial reductions during and after World War II.^[3] Ellingham's work built on earlier thermodynamic principles but provided a unified visual framework, which has since become a standard reference in materials science and engineering.^[1] In construction, the diagram features temperature on the horizontal axis (typically 0–2000°C) and ΔG° on the vertical axis (with more negative values indicating greater stability downward). Each line represents a specific reaction, such as 2M + O₂ → 2MO for a metal M, derived from the equation ΔG° = ΔH° - TΔS°, where the y-intercept reflects the enthalpy change (ΔH°) and the slope reflects the entropy change (ΔS°).^[1] Most oxide formation lines slope upward because the reaction consumes a gas (O₂), decreasing entropy (negative ΔS°), though exceptions like carbon oxidation to CO (2C + O₂ → 2CO) slope downward due to increased moles of gas, enhancing its utility as a reductant at higher temperatures.^[2] Phase transitions, such as melting or boiling of metals or oxides, cause kinks or shifts in the lines, accounting for latent heat effects.^[1] The diagram's interpretive power lies in its ability to compare reactions: a metal can reduce the oxide of another if its formation line lies below the target oxide's line at the operating temperature, as the more stable (lower ΔG°) oxide will form preferentially.^[2] For instance, magnesium (with a low-lying line) can reduce titanium dioxide, while carbon monoxide enables iron oxide reduction above approximately 700°C.^[1] Additional scales on the diagram often include equilibrium oxygen partial pressure (pO₂) and CO/CO₂ ratios, aiding in process optimization for smelting, roasting, and alloy production.^[2] Modern variants extend to other systems, like halides or aqueous environments (e.g., Pourbaix diagrams), but the classic oxide version remains foundational for understanding high-temperature thermodynamics in metallurgy.^[1]

Thermodynamic Foundations

Standard Free Energy and Oxidation Reactions

The standard Gibbs free energy change, denoted as ΔG°, for the formation of metal oxides quantifies the thermodynamic tendency of a metal to oxidize under standard conditions (1 atm pressure and specified temperature). It is defined by the fundamental equation

\Delta G^\circ = \Delta H^\circ - T \Delta S^\circ

where \Delta H^\circ is the standard enthalpy change (typically negative for exothermic oxide formations), T is the absolute temperature in Kelvin, and \Delta S^\circ is the standard entropy change. This expression arises from the definition of Gibbs free energy and allows prediction of reaction behavior as temperature varies.^[1]^[4] In the context of oxidation reactions, \Delta G^\circ determines the spontaneity: a negative value indicates the reaction proceeds forward under standard conditions, while a positive value suggests the reverse (reduction) is favored. At equilibrium, \Delta G^\circ = 0, corresponding to the point where the reaction neither favors oxidation nor reduction. For comparability across different metals, Ellingham diagrams normalize all reactions to involve the consumption of one mole of O₂ gas, such as in the general form \frac{2}{x} \mathrm{M} + \mathrm{O_2} \rightarrow \frac{2}{x} \mathrm{MO}_{x/2}, where M is the metal; this standardization facilitates direct assessment of relative oxide stabilities without scaling differences.^[1]^[4] (Note: Original Ellingham reference for normalization practice.) Specific examples illustrate this for common elements. For carbon oxidation to carbon monoxide, the reaction is $2\mathrm{C} + \mathrm{O_2} \rightarrow 2\mathrm{CO}, with \Delta G^\circ becoming more negative at higher temperatures due to a positive \Delta S^\circ from the increase in gas moles (1 mole O₂ to 2 moles CO). In contrast, for magnesium, $2\mathrm{Mg} + \mathrm{O_2} \rightarrow 2\mathrm{MgO} shows \Delta G^\circ less negative with rising temperature, reflecting a negative \Delta S^\circ as gaseous O₂ is consumed to form solid MgO. These \Delta G^\circ values are derived from tabulated thermodynamic data for standard states.^[1]^[4] The role of entropy changes (\Delta S^\circ) is central to understanding temperature dependence in oxidation reactions. Most metal oxidations involve a net decrease in gas moles (e.g., from O₂ to solid oxide), yielding negative \Delta S^\circ and thus positive slopes in \Delta G^\circ versus T plots, meaning oxides become less stable at higher temperatures. This entropy penalty for gas-consuming processes underscores why many reductions require elevated temperatures to shift equilibria favorably.^[1]^[4]

Construction and Interpretation

The Ellingham diagram was developed by Harold J. T. Ellingham in 1944 for the British Non-Ferrous Metals Research Association to analyze the reducibility of oxides and sulfides in metallurgical processes.^[4] This graphical tool compiles thermodynamic data for oxidation reactions, allowing for efficient prediction of reaction feasibility without repeated calculations.^[4] Construction begins with plotting the standard Gibbs free energy change (ΔG°) for oxide formation reactions on the y-axis, in units of kJ/mol, against absolute temperature (T) on the x-axis, typically spanning 0 to 2000°C or 300 to 2200 K. The data are sourced from reliable thermochemical compilations, such as the JANAF Thermochemical Tables, which provide values for ΔG° under standard conditions.^[4] Each line represents the temperature dependence of ΔG° for a specific reaction, often appearing nearly linear because ΔH° and ΔS° vary little with temperature except at phase transitions.^[1]^[4] A fundamental convention in construction is normalizing all reactions to the consumption of one mole of O₂ (e.g., 2M + O₂ → 2MO), which standardizes the scale and enables direct comparison of oxide stabilities regardless of the metal's valence or stoichiometry.^[4]^[1] This per-mole-of-O₂ basis ensures that the y-axis values reflect equivalent oxygen affinities, with the slope of each line influenced by the entropy change (ΔS°), typically positive for reactions producing gases and thus upward-sloping.^[4] Interpretation relies on the relative positions of the lines: those lower on the diagram indicate more stable oxides, as their more negative ΔG° signifies a stronger driving force for formation and greater resistance to reduction.^[4]^[1] Intersection points between lines mark the equilibrium temperatures where the ΔG° values are equal, allowing the metal corresponding to the lower line (at higher temperatures) to serve as a reducing agent for the oxide on the upper line, as the overall reaction would then have a negative ΔG°.^[4]^[1] The diagram operates under assumptions of standard states, including pure solids/liquids for metals and oxides, and a fixed oxygen partial pressure of 1 atm (or 1 bar in modern versions), focusing solely on equilibrium thermodynamics.^[4]^[1] Limitations include neglect of kinetic barriers, which may prevent reactions from reaching equilibrium, and exclusion of non-ideal effects such as solution thermodynamics in alloys or varying activities in impure systems.^[4]

Key Features of the Diagram

Line Characteristics and Slopes

In Ellingham diagrams, the slope of each line represents the negative of the standard entropy change (-\Delta S^\circ) for the corresponding oxidation reaction, derived from the temperature dependence of the standard Gibbs free energy change (\Delta G^\circ = \Delta H^\circ - T \Delta S^\circ). For most metal oxidation reactions, such as 2M + O_2 \rightarrow 2MO (where M is a metal), the slope is positive because \Delta S^\circ is negative; this occurs due to a decrease in the number of gas moles, as one mole of O_2 gas is consumed to form a solid oxide, reducing the system's entropy.^[1] In contrast, reactions that increase the number of gas moles exhibit negative slopes; a prominent example is the formation of carbon monoxide, 2C + O_2 \rightarrow 2CO, where the production of two moles of CO gas from one mole of O_2 leads to an entropy increase, resulting in a downward-sloping line.^[4] The y-intercept of each line, evaluated at T = 0 K, corresponds to the standard enthalpy of formation (\Delta H^\circ) of the oxide, providing insight into the reaction's exothermicity. Since most metal oxide formations are highly exothermic (\Delta H^\circ < 0), the intercepts lie well below the \Delta G^\circ = 0 axis, indicating the thermodynamic stability of these compounds even at absolute zero. This intercept value underscores the energetic favorability of oxide formation, with more negative \Delta H^\circ values placing lines lower on the diagram and signifying greater oxide stability.^[1] Many lines for metal oxidations appear nearly parallel because their \Delta S^\circ values are similar, typically around -120 J/mol·K, owing to the common involvement of consuming one mole of O_2 gas in the reaction stoichiometry. This uniformity in entropy change arises from the comparable phase transitions (gas to solid) across different metals, minimizing variations in slope unless interrupted by phase changes like melting or boiling of the metal or oxide. Such parallelism simplifies comparisons of oxide stabilities, as differences in line positions primarily reflect variations in \Delta H^\circ rather than entropy effects.^[4] As temperature increases, the lines' slopes dictate shifts in reaction feasibility: positive slopes (negative \Delta S^\circ) cause \Delta G^\circ to become less negative (or more positive), potentially destabilizing oxides at high temperatures, while negative slopes (positive \Delta S^\circ), as in CO formation, make \Delta G^\circ more negative, enhancing the reaction's driving force. This temperature dependence favors reduction processes at elevated temperatures when the reducing agent's line has a steeper negative slope than the oxide's line, such as in carbon-based reductions where CO production becomes increasingly viable. Phase transitions can abruptly alter slopes, introducing kinks that reflect entropy changes from melting or vaporization.^[1]

Identification of Reducing Agents

Ellingham diagrams facilitate the identification of suitable reducing agents by comparing the standard free energy changes (ΔG°) for the formation of metal oxides, allowing metallurgists to predict which substances can effectively reduce a target oxide to its elemental form. The core principle is that a reducing agent, represented by a line lower on the diagram (indicating a more negative ΔG° for its oxidation reaction), can reduce the oxide corresponding to a line above it, provided the temperature is sufficient for their intersection or crossing point. This intersection temperature marks the point where the reduction becomes thermodynamically feasible, as the overall ΔG° for the displacement reaction approaches zero or becomes negative.^[1]^[5] Common reducing agents include carbon (often via the formation of CO), hydrogen, and active metals such as aluminum or magnesium. For instance, the line for the reaction 2C + O₂ → 2CO exhibits a downward slope due to the increase in gaseous moles, enhancing its reducing power at higher temperatures and enabling it to reduce many metal oxides like iron oxide above approximately 1020 K. Hydrogen, with its line for 2H₂ + O₂ → 2H₂O positioned relatively low, serves as an effective agent for oxides above it, while aluminum and magnesium, whose formation lines lie even lower, are used for highly stable oxides in processes like the thermite reaction. These agents are selected based on their positions relative to the target oxide's line, ensuring the reduction reaction's spontaneity without requiring independent thermodynamic calculations.^[1]^[5]^[6] The feasibility of reduction is determined by the criterion that ΔG° for the overall reaction must be negative, which is directly inferred from the relative positions and slopes of the lines on the diagram at a given temperature. If the reducing agent's line is below and does not intersect the oxide's line within practical temperature ranges, the reduction is viable across the diagram's temperature scale. However, practical selection also considers factors such as cost, availability, and environmental impact; for example, carbon remains prevalent in blast furnaces due to its low cost and abundant supply from coal, despite generating significant CO₂ emissions (approximately 1.8–2.0 tons per ton of steel produced), whereas hydrogen is increasingly favored for its cleaner profile, producing water as a byproduct and potentially reducing global CO₂ emissions by 2.3 Gt annually if scaled, though its production requires green energy sources that may limit availability.^[1]^[5]^[6]

Practical Applications in Reduction Processes

Reduction of Hematite with Carbon

The reduction of hematite (Fe₂O₃), the primary iron ore, to metallic iron using carbon is a cornerstone of pyrometallurgical processes, as analyzed through the Ellingham diagram. This diagram reveals the thermodynamic feasibility by plotting standard free energy changes (ΔG°) for oxidation reactions against temperature, allowing prediction of conditions under which carbon can serve as a reducing agent for iron oxides. In the blast furnace, hematite undergoes stepwise reduction: first to magnetite (Fe₃O₄), then to wüstite (FeO), and finally to iron (Fe). The overall process relies on both indirect reduction by carbon monoxide (CO) in the upper zones and direct reduction by carbon in the lower, hotter regions.^[2] A critical step in the direct reduction is the reaction:

$2\mathrm{FeO} + \mathrm{C} \rightarrow 2\mathrm{Fe} + \mathrm{CO}

The Ellingham diagram shows that ΔG° for this reaction becomes negative above approximately 700°C, marking the temperature threshold where the process is spontaneous. This is determined by the intersection of the FeO/Fe oxidation line (2Fe + O₂ → 2FeO) with the C/CO line (2C + O₂ → 2CO), as the downward slope of the latter—due to the positive entropy change from gas production—allows carbon to reduce iron oxide at elevated temperatures. Below this intersection, the reaction is not favorable, emphasizing the need for sufficient heat in the furnace to drive reduction.^[7]^[1] In industrial steelmaking, coke—derived from coal—acts as both the carbon source for reduction and a fuel to maintain high temperatures (up to 2000°C at the tuyeres). Limestone (CaCO₃) is added as a flux, decomposing to CaO, which reacts with silica (SiO₂) and other impurities in the ore to form a molten slag (primarily CaSiO₃) that separates from the molten iron, enabling purification. This setup produces pig iron containing 4–4.5% carbon, which is further refined into steel. The Ellingham diagram aids in selecting optimal conditions, such as CO/CO₂ ratios, to maximize efficiency and minimize energy use.^[2] The carbon-based blast furnace process has formed the basis of large-scale iron production since the 18th century, following Abraham Darby's innovation of using coke in 1709, which overcame charcoal shortages and spurred industrial growth. The development of the Ellingham diagram in 1944 provided deeper thermodynamic insights, enabling post-war optimizations in furnace design, reducing agent efficiency, and process control to enhance yield and sustainability in steelmaking.^[8]^[4]

Limitations for Chromium Oxide Reduction

The Ellingham diagram reveals that the line for the formation of Cr₂O₃ from chromium and oxygen lies above the line for 2C + O₂ → 2CO at temperatures exceeding approximately 1225°C, indicating that the standard Gibbs free energy change (ΔG°) for the carbothermic reduction reaction Cr₂O₃ + 3C → 2Cr + 3CO becomes negative above this threshold, rendering the process thermodynamically spontaneous at sufficiently high temperatures.^[1] This crossing point arises because the Cr₂O₃ line has a positive slope due to the entropy decrease in oxide formation, while the C/CO line slopes downward owing to the increased entropy from gas production.^[9] However, the stability of Cr₂O₃ stems from the strong Cr-O bonds, with ΔG° values for its formation remaining more negative than those for less stable oxides like FeO at lower temperatures, limiting feasibility below the intersection.^[7] Despite thermodynamic viability at elevated temperatures, practical limitations hinder direct carbon reduction of Cr₂O₃, particularly for producing high-purity chromium. The requirement for temperatures around 1500–1600°C in electric arc furnaces results in substantial energy consumption, often exceeding 3000–4500 kWh per ton of alloy, contributing to high operational costs and environmental impacts from CO emissions.^[10] Moreover, carbon incorporation during reduction introduces impurities, forming chromium carbides that complicate purification and degrade material properties for applications requiring low carbon content.^[11] In ferrochrome production, carbothermic reduction remains the dominant method, but it is adapted with ferrosilicon addition at high temperatures to facilitate partial decarburization and achieve low-carbon variants, though this increases complexity and costs.^[11] For pure chromium, aluminothermic reduction—where aluminum displaces chromium via the exothermic reaction Cr₂O₃ + 2Al → Al₂O₃ + 2Cr—is preferred, as the Ellingham diagram shows the Al₂O₃ line below Cr₂O₃, ensuring ΔG° < 0 without carbon contamination.^[12] Modern refining employs vacuum distillation or plasma-arc processes to further mitigate impurities and energy demands, enabling higher purity levels up to 99.99%.^[13]

Aluminothermic Reduction of Metals

Aluminothermic reduction involves the exothermic reaction of aluminum with metal oxides to produce the corresponding metals or alloys, as exemplified by the reduction of chromium(III) oxide:

\mathrm{Cr_2O_3 + 2Al \rightarrow Al_2O_3 + 2Cr}

This process generates intense heat, reaching temperatures of 1800–2500°C, which sustains the reaction without external heating once initiated by ignition, and enables applications such as thermite welding for joining rails and structural components.^[14]^[1] Ellingham diagrams predict the viability of this reduction by showing the Al/Al₂O₃ line positioned well below those of many refractory metal oxides, such as Cr₂O₃, due to the high stability of Al₂O₃ and its steep negative slope from entropy effects, resulting in a favorable negative ΔG° for reductions even at ambient temperatures.^[1] This positioning confirms aluminum's ability to displace metals like chromium, manganese, and vanadium from their oxides spontaneously under standard conditions.^[14] Developed as the Goldschmidt process in 1895 by German chemist Hans Goldschmidt, this method predates the Ellingham diagram but aligns with its thermodynamic principles, providing a retrospective validation for selecting aluminum as a reductant.^[15] It is widely applied in producing pure metals such as chromium (>98% Cr) and low-carbon ferroalloys such as ferrochromium (65–75% Cr), ferromanganese (80–85% Mn), and ferrovanadium (≥50% V) with low carbon content (<0.1%), essential for specialty steels and ferroalloys.^[14] The process yields products of exceptional purity by avoiding carbon contamination, and its self-sustaining exothermicity makes it efficient for small-scale or on-site operations like welding.^[14] However, it is energy-intensive overall due to the high cost and electrolytic production of aluminum, and the rapid heat release poses safety risks, including burns and explosion hazards from molten metal and slag.^[14]

Extensions and Advanced Uses

Inclusion of Gas-Phase Equilibria

Ellingham diagrams are extended to incorporate gas-phase equilibria by plotting the standard Gibbs free energy changes (ΔG°) for reactions involving gaseous reactants and products, beyond the standard metal-oxygen oxidations. A key example is the reaction for water formation, given by

$2\mathrm{H_2} + \mathrm{O_2} \rightarrow 2\mathrm{H_2O},

which appears as a line with a positive slope on the diagram due to the decrease in the number of gas moles, reflecting an unfavorable entropy change. This positioning demonstrates that hydrogen exhibits increasing reducing potential with rising temperature; for instance, it can thermodynamically reduce oxides of metals with less stable oxides, such as copper or nickel, at temperatures above their line intersections. For iron oxides, reduction requires non-standard conditions with high H₂/H₂O ratios, typically above 800–1000°C in industrial direct reduction processes.^[16] Similarly, the oxidation of sulfur,

\mathrm{S} + \mathrm{O_2} \rightarrow \mathrm{SO_2},

is included to compare the relative stabilities of sulfides and oxides, allowing assessment of reactions where sulfur gases participate in equilibrium with condensed phases.^[17] These gas-phase inclusions enable practical predictions in metallurgical processes involving volatile species. In the roasting of sulfide ores, such as those of copper or zinc, the diagram helps evaluate the oxidation to metal oxides and SO₂ release by comparing the ΔG° lines for sulfide formation (e.g., 2M + S₂ → 2MS) against oxygen-consuming reactions, confirming feasibility at typical roasting temperatures around 500–900°C where oxide lines lie below those for sulfides.^[18] For hydrogen-based reductions, the diagram illustrates equilibria in the water-gas shift reaction (CO + H₂O ⇌ CO₂ + H₂), where the relative positions of the H₂/H₂O and CO/CO₂ lines indicate the temperature-dependent favorability of hydrogen production or consumption, aiding optimization in direct reduction of ores with syngas mixtures.^[2] Modifications to the standard diagram account for gas-phase partial pressures through the relation

\Delta G = \Delta G^\circ + RT \ln Q,

where Q incorporates the partial pressures of gaseous species (e.g., p_{O_2} or p_{SO_2}), shifting the effective line position to reflect non-unit activities. Despite this, conventional Ellingham diagrams retain ΔG° plots assuming standard states (unit fugacity for gases) for straightforward comparisons, with partial pressure adjustments applied separately for specific process conditions. The inclusion of gas-phase equilibria evolved from Harold Ellingham's original 1944 oxide-focused diagrams, with significant adaptations in the 1950s by researchers like John Chipman to address non-ferrous metallurgy, incorporating sulfur and hydrogen systems for processes like roasting and gas reductions in copper and zinc extraction.^[1]

Variations for Non-Oxide Systems

Ellingham diagrams have been adapted for sulfide systems to assess the stability of metal sulfides and the feasibility of processes like roasting, where sulfides are oxidized to oxides and sulfur dioxide. These diagrams typically plot the standard Gibbs free energy change (ΔG°) as a function of temperature for reactions such as MS + 3/2 O₂ → MO + SO₂, normalized per mole of oxygen, allowing comparison of sulfide stabilities similar to oxide lines.^[19] The lines in sulfide Ellingham diagrams often show that common heavy metal sulfides (e.g., FeS, Cu₂S) have relatively close positions, indicating similar thermodynamic stabilities, which complicates selective oxidation during roasting.^[19] Sulfur's interference is evident as high partial pressures of S₂ stabilize sulfides, shifting phase boundaries and hindering complete reduction to metals unless oxygen activity is sufficiently high.^[1] For chloride systems, Ellingham diagrams are constructed by plotting ΔG° versus temperature for formation or decomposition reactions, such as 2MCl → 2M + Cl₂, normalized to one mole of Cl₂ gas.^[20] These variants are particularly useful in predicting the viability of molten salt electrolysis for metal production, where the diagram's lines reveal the temperature at which chloride decomposition becomes thermodynamically favorable based on the slope and position relative to other halides.^[20] For instance, in chloride salt reactors, such diagrams, generated from databases like SGTE, help rank metal-chlorine affinities and identify potential precipitates or corrosion risks for alloys like those containing Cr, Fe, and Ni.^[21] Since the 2000s, computational extensions have enhanced non-oxide Ellingham diagrams by integrating density functional theory (DFT) calculations with thermodynamic databases to incorporate alloy effects and high-pressure conditions. Tools like FactSage and VASP enable the generation of custom diagrams for complex systems, such as Pt-based alloys under oxidation, by computing virtual oxygen chemical potentials that account for multi-component interactions without relying solely on experimental gas-phase data.^[22] These software-driven diagrams, drawing from CALPHAD methods and databases like MSTDB-TC, allow for predictions of phase stability in non-standard states, including pressure-dependent shifts in line positions for sulfide or chloride equilibria. As of 2024, machine learning enhancements to CALPHAD methods have enabled more precise predictions of phase stabilities in multicomponent non-oxide systems.^[23]^[21] Despite these advances, non-oxide Ellingham diagrams are less standardized than their oxide counterparts, often requiring bespoke entropy and enthalpy data due to limited experimental thermodynamic compilations for sulfides and chlorides.^[4] This variability arises from challenges in measuring reliable free energy values across wide temperature ranges, leading to potential inaccuracies in lines for less-studied compounds.^[1]

References

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