Thermodynamic Stability of Oxides under Reducing and Oxidizing Conditions: Ellingham Diagrams, Oxygen Potential, and Stability Limits

A thermodynamic framework for assessing oxide stability as a function of temperature and oxygen partial pressure, including Ellingham diagrams, equilibrium oxygen pressure, predominance diagrams, and practical interpretation.

THERMODYNAMICS OF FUNCTIONAL MATERIALS · OXIDE STABILITY 11

Thermodynamic Stability of Oxides under Reducing and Oxidizing Conditions: Ellingham Diagrams, Oxygen Potential, and Stability Limits

The stability of an oxide is not an intrinsic yes-or-no property independent of its environment. It depends on temperature, oxygen chemical potential, gas composition, and the chemical potentials of the other elements present. An oxide that is stable in air may be reduced in a hydrogen-containing atmosphere, while a reduced oxide may reoxidize when exposed to a sufficiently high oxygen partial pressure.

Thermodynamic analysis provides a framework for predicting these changes. The central quantities are the Gibbs energy of formation, the oxygen chemical potential, and the equilibrium oxygen partial pressure. Ellingham diagrams and predominance diagrams organize these quantities into practical maps of oxidation and reduction conditions.

This article develops these tools with particular attention to functional oxides used in solid-state ionics, thermoelectrics, oxygen electrodes, and hydrogen-energy devices. The emphasis is on interpreting stability limits correctly rather than treating a single diagram as a universal prediction of material behaviour.

1. What does oxide stability mean?

Consider the formation of an oxide from its elements:

[ ]

The standard Gibbs energy of formation is

[ f G^{}(T) = G^_{}-G{}-^{} ]

For a specified temperature and standard states, a negative value indicates that formation of the oxide is thermodynamically favourable relative to the chosen elemental reference states.

However, the actual reaction Gibbs energy depends on the oxygen pressure:

[ _rG = _rG^+RTQ ]

For pure condensed phases with unit activity and an ideal oxygen gas,

[ Q=()^{-1/2} ]

so that

[ ]

The oxide is favoured in the forward direction when (_rG<0), and reduction is favoured when the reverse reaction has a negative Gibbs energy.

Thus, oxide stability must always be stated together with temperature, oxygen potential, and the assumed phases.

2. Oxygen chemical potential as the environmental control variable

For an ideal oxygen gas,

[ {}(T,p{}) = ^_{}(T) + RT() ]

The chemical potential per mole of oxygen atoms is one-half of the molecular oxygen value:

[ ]

Increasing oxygen partial pressure raises oxygen chemical potential and favours more oxidized states. Decreasing oxygen partial pressure lowers oxygen chemical potential and favours oxygen release, reduction, or formation of oxygen vacancies, depending on the material’s defect chemistry.

For non-ideal gases, fugacity replaces partial pressure:

[ {} = ^{} + RT() ]

The logarithm must contain a dimensionless ratio. Using a dimensional pressure directly inside a logarithm is thermodynamically incorrect.

3. Equilibrium oxygen pressure for oxide formation

At the equilibrium boundary between metal and oxide,

[ _rG=0 ]

For the formation reaction in Section 1,

[ 0 = fG^{} -() ]

Rearranging gives

[ ]

or

[ ]

This expression applies to the stated reaction and standard-state convention. If the oxide stoichiometry or reaction coefficients differ, the factor multiplying the Gibbs energy changes accordingly.

At a fixed temperature:

• Above the equilibrium oxygen pressure, the oxide is favoured relative to the metal.
• Below the equilibrium oxygen pressure, the metal is favoured relative to the oxide.
• At the boundary, the two phases can coexist at equilibrium.

A very negative formation Gibbs energy gives a very low equilibrium oxygen pressure, indicating that the oxide remains stable under strongly reducing conditions.

4. Ellingham diagrams: Gibbs energy versus temperature

An Ellingham diagram plots the standard Gibbs energy change of an oxidation reaction against temperature. The vertical coordinate is commonly (G^), while temperature is on the horizontal axis.

For a reaction with approximately constant reaction enthalpy and entropy over a limited temperature range,

[ G^(T) = H^-TS^ ]

The slope is

[ ]

and the intercept in this simplified linear representation is related to (H^).

For many oxidation reactions, oxygen gas is consumed. The decrease in gas entropy can make the reaction entropy negative, giving a positive slope of (G^) versus temperature. This is a common trend, not a universal rule: the actual slope is determined by the reaction entropy and can change when heat capacities or phase transitions become important.

An Ellingham diagram is useful for comparing oxidation reactions at a common temperature. It does not, by itself, describe all possible phases, nonstoichiometry, solid solutions, or kinetic barriers.

5. Interpreting line intersections and reduction feasibility

If two oxidation reactions are written using consistent oxygen normalization, their relative Gibbs energies can be compared directly. A reducing agent can reduce an oxide when oxidation of the reducing agent is thermodynamically more favourable than oxidation of the metal being reduced, after the complete balanced reaction is constructed.

For example, consider the reduction of a generic monoxide by hydrogen:

[ ]

The reaction Gibbs energy is

[ _rG = rG^ + RT( {p{}/p^} ) ]

For unit activities of the pure solids, reduction is favoured when this expression is negative.

The relevant gas variable is therefore not simply the hydrogen concentration. It is the ratio (p_{}/p_{}), or equivalently the oxygen potential established by the H(_2)/H(_2)O buffer at a specified temperature.

Similar reasoning applies to CO/CO(2) mixtures, where the ratio (p{}/p_{}) controls the oxygen potential under equilibrium assumptions.

6. Oxygen potential imposed by gas buffers

For the equilibrium

[ ]

the equilibrium constant is

[ K_{} = {a_{}a_{}^{1/2}} ]

For ideal gases,

[ ]

The equilibrium constant is temperature dependent. Consequently, the same H(_2)/H(_2)O ratio does not impose the same oxygen partial pressure at every temperature.

A gas mixture may be treated as an oxygen buffer only if gas-phase equilibrium is sufficiently established. Catalytic surfaces, leaks, flow rates, and residence time can cause the actual sample atmosphere to deviate from the nominal equilibrium value.

7. Stability of oxides with multiple oxidation states

Many transition-metal oxides have more than one stable oxidation state. A generic sequence can be represented as

[ ]

The actual phase sequence and stoichiometry depend on the metal and temperature; the expression above is only a schematic reminder that multiple oxide phases may compete.

For each possible reaction, calculate the Gibbs energy using a balanced equation and consistent standard states. The stable phase at a given oxygen potential is the one with the lowest appropriate Gibbs energy, subject to mass balance and phase coexistence.

A single metal/oxide equilibrium pressure is insufficient when several oxides or nonstoichiometric phases are possible. In that case, a predominance diagram or a full Gibbs-energy minimization is more appropriate.

8. Predominance diagrams and oxygen-potential windows

A predominance diagram maps the thermodynamically stable phase as a function of variables such as temperature and oxygen partial pressure. At fixed temperature, boundaries between phases can often be expressed as equilibrium oxygen pressures.

For a reaction involving a change in oxygen content, the boundary follows from

[ _rG^(T) + RTQ=0 ]

The resulting boundary separates regions in which different phases have the lowest Gibbs energy.

For a multicomponent oxide, additional chemical potentials must be specified. A diagram plotted against (T) and (p_{}) is therefore conditional on the chemical potentials of cations and other gas species, as well as the set of phases included in the calculation.

A useful engineering concept is the oxygen-potential window: the range of oxygen chemical potential over which the desired phase remains stable without reducing or oxidizing into competing phases.

9. Stability of nonstoichiometric oxides

Many functional oxides do not have a fixed oxygen content. Their composition is better written as

[ ]

where () changes with temperature and oxygen partial pressure.

Oxygen loss can be represented by

[ ]

The equilibrium oxygen deficiency is governed by the defect formation free energy, charge neutrality, and the oxygen chemical potential.

The material may remain in the same average crystal structure while its oxygen content and electronic carrier concentration change. Therefore, the onset of reduction does not always correspond to immediate decomposition into a different crystalline phase.

At sufficiently reducing conditions, however, the nonstoichiometric solid solution may reach a phase boundary, form a lower oxide, exsolve a metal, or undergo structural decomposition.

10. Defect stability is not the same as phase stability

Two different questions must be distinguished:

1. Defect equilibrium: What are the equilibrium concentrations of vacancies, electrons, holes, and other point defects within a specified phase?
2. Phase stability: Is that phase itself more stable than all competing phases at the imposed chemical potentials?

A Brouwer diagram can describe defect concentrations within an assumed phase. It does not prove that the phase remains stable across the entire plotted oxygen-pressure range.

For example, a defect model may predict increasing oxygen-vacancy concentration as (p_{}) decreases. Before that concentration becomes arbitrarily large, the material may cross a phase boundary or undergo a change in defect association, ordering, or crystal structure.

The defect model must therefore be constrained by the thermodynamic stability range of the host phase.

11. Oxide stability in electrochemical devices

In solid oxide fuel cells, electrolysis cells, oxygen-permeation membranes, and protonic ceramic devices, different parts of a component can experience different oxygen chemical potentials.

An electrolyte may be exposed to a high oxygen potential on one side and a low oxygen potential on the other. The resulting gradient can drive changes in defect concentration and, if the stability limit is exceeded, chemical reduction or decomposition.

For an oxide electrolyte, the practical stability window must consider:

• Oxidation and reduction limits of the electrolyte.
• Electronic conductivity under both electrode atmospheres.
• Chemical compatibility with electrodes and interconnects.
• Cation interdiffusion and secondary-phase formation.
• Temperature-dependent phase stability.
• Local gas composition and electrochemical overpotential.

A material can be thermodynamically stable as a bulk phase yet unsuitable as an electrolyte if its electronic conductivity becomes too high under one operating condition.

12. Chemical compatibility between oxides

When two oxides are in contact, stability is not determined only by their separate stability in oxygen. They may react to form a new compound even when each parent oxide is individually stable in the same atmosphere.

For a generic solid-state reaction,

[ ]

the reaction Gibbs energy is

[ rG = G{} - G_{} - G_{} ]

For pure solid phases, the activities are often approximated as unity, and the sign of (_rG) indicates the thermodynamic driving force for reaction under the selected conditions.

This type of compatibility analysis is important for electrolyte/electrode interfaces, ceramic composites, protective coatings, and sintering additives. Oxygen partial pressure may alter the oxidation states and therefore change which interfacial reactions are possible.

13. Worked framework: estimating an equilibrium oxygen pressure

Suppose a monoxide formation reaction has a standard Gibbs energy of formation of (-200 ) at a selected temperature of (1000 K):

[ ]

Using

[ () = ]

with (R=8.314 ),

[ () = ]

[ () ]

Therefore,

[ ]

This is an illustrative calculation, not a property value for a named oxide. It assumes pure solid phases, ideal oxygen gas, and a known standard Gibbs energy at the stated temperature. For a real material, the correct (_fG^(T)), phase identity, and standard-state data must be used.

The result indicates that the oxide is stable relative to the metal at oxygen pressures above the calculated boundary, under the stated equilibrium model.

14. Temperature dependence and heat-capacity corrections

The linear Ellingham approximation assumes that reaction enthalpy and entropy do not vary strongly with temperature. More generally,

[ _rG^(T) = _rH^(T)-T_rS(T) ]

The temperature dependence of enthalpy and entropy is obtained from the reaction heat capacity:

[ _rC_P^ = _iiC{P,i}^ ]

If (_rC_P^) is treated as constant over a temperature interval,

[ _rH^(T) = _rH^(T_0) + _rC_P^(T-T_0) ]

and

[ _rS^(T) = _rS^(T_0) + _rC_P^() ]

These corrections can be important over wide temperature ranges or near phase transitions. Reliable stability calculations should use thermodynamic data valid for the phases and temperature interval of interest.

15. Thermodynamic stability versus kinetic persistence

A phase may persist outside its equilibrium stability range because decomposition or reduction requires nucleation, diffusion, or reconstructive structural changes.

This distinction is essential:

• Thermodynamic stability identifies the state with the lowest Gibbs energy.
• Metastability describes a local free-energy minimum that is not the global minimum.
• Kinetic persistence describes how slowly a transformation proceeds because of activation barriers or limited mass transport.

A sample that appears unchanged after a short anneal is not necessarily thermodynamically stable. Conversely, a thermodynamically favourable transformation may be too slow to occur during the experiment or device lifetime.

Experimental stability claims should therefore state annealing time, temperature, gas composition, heating/cooling history, and the sensitivity of the characterization methods.

16. Practical workflow for evaluating an oxide’s stability

A defensible assessment can follow this sequence:

1. Define the composition and candidate phases. Include plausible lower oxides, competing compounds, and decomposition products.
2. Balance all reactions. Check elemental and charge balance before evaluating Gibbs energies.
3. Specify temperature and gas atmosphere. Use measured or controlled gas ratios, not only nominal labels such as “reducing”.
4. Obtain thermodynamic data. Use consistent standard states and temperature-dependent data for all phases.
5. Calculate reaction Gibbs energies. Evaluate the relevant reaction quotients for the actual gas composition.
6. Determine equilibrium boundaries. Calculate oxygen pressures or chemical-potential limits for competing phases.
7. Check point-defect equilibria inside each phase. Apply charge neutrality and mass action only within the phase’s valid stability domain.
8. Assess kinetics and compatibility. Consider diffusion, surface exchange, interfacial reactions, and possible metastability.
9. Validate experimentally. Use XRD, thermogravimetry, conductivity, microscopy, and controlled-atmosphere annealing as appropriate.

For complex multicomponent systems, Gibbs-energy minimization using a validated thermodynamic database is generally more reliable than comparing a small number of isolated reactions.

17. Common errors when using Ellingham diagrams

• Comparing reactions with different oxygen normalization: reaction coefficients must be made consistent before comparing Gibbs-energy values.
• Ignoring gas composition: a nominally reducing gas does not specify oxygen potential without temperature and gas ratios.
• Using room-temperature data at high temperature: heat capacities and phase transitions can change the Gibbs energy substantially.
• Assuming unit activity for every solid solution: pure condensed phases may have activity near unity, but components in solid solutions generally do not.
• Treating a line intersection as a complete phase diagram: it compares selected reactions, not every possible competing phase.
• Ignoring nonstoichiometry: many oxides accommodate oxygen vacancies and electronic defects over finite composition ranges.
• Equating thermodynamic stability with practical durability: kinetics and microstructure determine how quickly degradation occurs.

18. Key equations

For oxide formation,

[ ]

the reaction Gibbs energy is

[ ]

The equilibrium oxygen pressure is

[ ]

The oxygen chemical potential per mole of oxygen atoms is

[ ]

The Ellingham slope is

[ ]

These expressions form the basis for assessing oxidation and reduction boundaries. Their use requires balanced reactions, consistent standard states, and the correct thermodynamic data for each phase.

19. Further reading

The following articles develop related concepts used in oxide-stability analysis:

• Chemical Potential and Activity in Solid Materials
• Gibbs Free Energy and Phase Stability in Functional Materials
• Thermodynamic Equilibrium and the Mass Action Law in Solids
• Temperature Dependence of Equilibrium Constants: The van ’t Hoff Equation
• Defect Formation Energy and Equilibrium Concentrations in Oxides
• Oxygen Chemical Potential and pO2-Dependent Defect Chemistry
• Oxygen Nonstoichiometry Measurement
• Chemical Expansion and Thermochemical Strain

20. Final perspective

Oxide stability is governed by the competition between Gibbs energies at a specified temperature and set of chemical potentials. Oxygen partial pressure is a powerful control variable because it directly sets oxygen chemical potential, but it is only one part of a multicomponent thermodynamic description.

Ellingham diagrams provide a compact comparison of oxidation reactions, while equilibrium oxygen pressures and predominance diagrams identify the conditions under which phases can coexist or transform. For functional ceramics, these tools must be combined with defect chemistry, nonstoichiometry, phase compatibility, and kinetic analysis.

The most reliable conclusion is not simply that an oxide is “stable” or “unstable”, but that a specified phase is thermodynamically favoured within a clearly defined temperature and oxygen-potential window.