Metallurgical Thermodynamics: Solutions, Phase Equilibria, Ellingham Diagrams and Electrochemistry

In a solution the chemical potential of component i is written μ_i = μ_i° + RT ln a_i, where μ_i° is its value in a chosen standard state and a_i is its activity; the activity coefficient γ_i = a_i/x_i measures departure from ideality. Raoult’s law, a_i = x_i, holds for the solvent as x_i → 1; Henry’s law, a_i = γ_i° x_i with a constant γ_i°, holds for a solute as x_i → 0. Molten iron and steel use a third standard state, the 1 wt% (Henrian) standard state, in which the activity of a dilute solute equals its weight percent — so the oxygen, sulphur or carbon activity in steel is written directly as [%O], [%S] or [%C] times an interaction-corrected coefficient. The Gibbs-Duhem equation, Σ x_i d ln a_i = 0, ties the components together: in a binary, if the solute obeys Henry’s law the solvent must obey Raoult’s.

An ideal solution has ΔH_mix = 0 and ΔS_mix = −R Σ x_i ln x_i, so ΔG_mix = RT Σ x_i ln x_i is always negative; for an equimolar binary ΔS_mix = R ln 2 = 5.76 J/(mol K). A regular solution keeps the ideal entropy but has ΔH_mix = Ω x_A x_B, with Ω the interaction parameter: Ω < 0 means unlike neighbours are preferred (ordering, compound formation, negative deviation), Ω > 0 means like neighbours are preferred (clustering, positive deviation). The activity coefficients follow directly: RT ln γ_A = Ω x_B² and RT ln γ_B = Ω x_A². When Ω > 0 the G-x curve develops two minima below the critical temperature T_c = Ω/(2R), and the alloy separates into two solutions of different composition — a miscibility gap.

Diatomic gases dissolve in liquid and solid metals as atoms: ½H₂(g) = [H], ½N₂(g) = [N]. The equilibrium constant is K = [%H]/√p_H₂, so the dissolved content is proportional to the square root of the gas partial pressure — Sieverts’ law, [%H] = K_H √p_H₂. It is the whole basis of vacuum degassing: lowering p_H₂ a hundredfold lowers the equilibrium hydrogen tenfold, not a hundredfold. Because the dissolution of hydrogen in iron is endothermic, K_H rises with temperature, which is why hydrogen picked up in liquid steel is rejected on solidification and cooling and causes porosity and flaking.

The Gibbs phase rule counts the intensive variables that can be changed independently without changing the number of phases: F = C − P + 2, with C components and P phases. Metallurgical phase diagrams are drawn at one atmosphere, and pressure has little effect on condensed phases, so the condensed phase rule F = C − P + 1 is used. In a binary at fixed pressure a single-phase field has F = 2 (temperature and composition both free), a two-phase field has F = 1 (fix the temperature and both phase compositions are fixed), and three phases coexist only at an invariant point, F = 0: the eutectic (L → α + β), the peritectic (L + α → β), the eutectoid (γ → α + β), the peritectoid and the monotectic (L₁ → α + L₂).

Inside a two-phase field a horizontal tie line at the temperature of interest ends on the two phase boundaries, which give the compositions of the two phases; the lever rule gives their amounts. For an alloy of overall composition C₀ between phases α (C_α) and β (C_β), the fraction of α is W_α = (C_β − C₀)/(C_β − C_α) and W_β = (C₀ − C_α)/(C_β − C_α) — each phase takes the arm of the lever on the far side. It is a mass balance and nothing more, so it works in weight fractions when compositions are in wt% and in mole fractions when they are in at%. On the iron-carbon diagram (Fe-Fe₃C) the numbers most used are the eutectoid at 727 °C and about 0.76 wt% C (texts quote 0.76 to 0.80), the maximum solubility of carbon in ferrite of about 0.022 wt% at 727 °C, cementite at 6.70 wt% C, the eutectic at 1147 °C and 4.3 wt% C and the maximum solubility in austenite of about 2.1 wt% C.

Every phase diagram is a projection of free-energy versus composition curves, one curve per phase at each temperature. Where one curve lies lowest across a composition range, that phase alone is stable. Where the lowest total G is obtained by a mixture, the common tangent to two curves touches them at the equilibrium compositions of the two phases — which is exactly the statement that μ_A and μ_B are equal in both phases, since the tangent’s intercepts at x_B = 0 and x_B = 1 are μ_A and μ_B. As temperature falls the liquid curve rises relative to the solid curves, the tangent points sweep out the solidus and liquidus, and a single common tangent touching three curves at once is an invariant reaction. A curve with a central hump (Ω > 0) gives a miscibility gap and, inside its inflection points (d²G/dx² < 0), spinodal decomposition.

For a reaction at constant T and P, ΔG = ΔG° + RT ln Q, where Q is the activity quotient; at equilibrium ΔG = 0 and Q becomes the equilibrium constant, ΔG° = −RT ln K. A pure solid or liquid in its standard state has activity 1, and a gas has activity equal to its partial pressure in atm, so for 2M(s) + O₂(g) = 2MO(s) with pure M and MO, K = 1/p_O₂ and the equilibrium oxygen partial pressure is p_O₂ = exp(ΔG°/RT). If the actual p_O₂ in the furnace atmosphere is above this value the metal oxidises; below it the oxide is reduced. Since ΔG° = ΔH° − TΔS°, and ΔH° and ΔS° change slowly with temperature, ΔG° is very nearly a straight line in T.

The Ellingham diagram plots ΔG° against T for the formation of each oxide, written per mole of O₂ so that every line can be compared directly. The intercept at 0 K is ΔH° and the slope is −ΔS°. Because a mole of gas is consumed and only condensed phases are formed, ΔS° is about −200 J/(mol K) for most metal oxides and the lines slope upward, nearly parallel. The slope increases at the melting point and much more at the boiling point of the metal, where the entropy lost on forming the oxide grows; it decreases if the oxide melts. Three carbon lines are the reason the diagram matters: C + O₂ = CO₂ is almost horizontal (ΔS° ≈ 0, one mole of gas for one), 2CO + O₂ = 2CO₂ slopes up, and 2C + O₂ = 2CO slopes DOWN (two moles of gas from one, ΔS° > 0). So carbon becomes a stronger reductant the hotter it is, and at a high enough temperature the CO line crosses below any oxide line — which is why iron, manganese and even chromium oxides can be smelted with carbon, while aluminium, magnesium and titanium oxides, whose lines lie far lower, cannot be reduced by carbon at practical temperatures.

• Lower on the diagram = more stable oxide. A metal reduces the oxide of any metal whose line lies ABOVE its own at that temperature (the thermite reaction: aluminium reduces iron oxide).
• The ΔG° of the reduction MO + M′ = M + M′O, per mole O₂, is the vertical gap between the two lines — the larger the gap, the more complete the reduction.
• Nomographic scales around the edge read the equilibrium p_O₂, and the H₂/H₂O and CO/CO₂ ratios, by drawing a line from the O, H or C point on the left through the oxide line at the chosen temperature.
• The same construction exists for sulphides and chlorides; the sulphide diagram explains why roasting precedes reduction for zinc and why copper matte smelting works.

A phase-stability, or predominance-area, diagram asks a different question: at a fixed temperature, which condensed phase is stable as two gas potentials vary together? For the M-S-O system the axes are log p_O₂ and log p_S₂ (or log p_SO₂), and the fields are M, MO, MS and MSO₄. Each boundary is one reaction at equilibrium — for example MS + ½O₂ = MO + ½S₂ gives a straight line whose slope comes from the powers of the pressures in K — so the diagram is built from a handful of ΔG° values and a ruler. It is the tool for designing a roast: an oxidising roast must sit in the MO field, a sulphating roast in the MSO₄ field. The analogous Kellogg-type diagrams for M-C-O and M-Cl-O systems do the same for reduction and chlorination.

A single electrode potential cannot be measured alone; it is quoted against the standard hydrogen electrode (SHE, H⁺ at unit activity, H₂ at 1 atm, platinum), which is assigned 0 V. Standard reduction potentials at 25 °C place Cu²⁺/Cu at +0.34 V, Fe²⁺/Fe at −0.44 V and Zn²⁺/Zn at −0.76 V: the more negative the potential, the more readily the metal gives up electrons. In a galvanic cell the electrode with the lower potential is the anode (oxidation) and the one with the higher potential is the cathode (reduction); the emf is E = E_cathode − E_anode, and the free-energy change is ΔG = −nFE with F = 96 485 C/mol. The Daniell cell, Zn | Zn²⁺ ‖ Cu²⁺ | Cu, has E° = 0.34 − (−0.76) = 1.10 V. An electrolytic cell is the same arrangement driven backwards by an external supply, which is how metals are electrowon and electrorefined.

Away from unit activities the potential follows from ΔG = ΔG° + RT ln Q divided by −nF: the Nernst equation, E = E° − (RT/nF) ln Q. At 298 K, RT/F × ln 10 = 0.0592 V, so E = E° − (0.0592/n) log₁₀ Q. For a single electrode Mⁿ⁺ + ne⁻ = M, E = E° + (0.0592/n) log a_Mⁿ⁺: a tenfold dilution of the ions lowers a divalent electrode’s potential by 29.6 mV. A concentration cell — two identical electrodes in solutions of different activity — has E° = 0 and runs entirely on this term, which is also how a differential-aeration or concentration corrosion cell is set up in a crevice.

A potential-pH (Pourbaix) diagram maps, for a metal in water at 25 °C, which species is thermodynamically stable at each combination of electrode potential and pH, assuming a chosen ion activity (typically 10⁻⁶). Three kinds of boundary appear: horizontal lines for pure electron transfers (Fe²⁺ + 2e⁻ = Fe, independent of pH), vertical lines for pure acid-base reactions (Fe³⁺ + 3H₂O = Fe(OH)₃ + 3H⁺, independent of potential), and sloping lines for reactions involving both. Two dashed lines bound the stability of water itself: below E = 0 − 0.0592 pH hydrogen is evolved, and above E = 1.229 − 0.0592 pH oxygen is evolved. The fields are read as immunity (the metal is stable), corrosion (a soluble ion is stable) and passivation (a solid oxide or hydroxide is stable and may protect). The diagram says what can happen, never how fast; kinetics is the business of polarization, in the next chapter.