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E. R. Vance coauthored research on the crystal chemistry and phase control of Synroc. The scientific problem is to distribute waste elements among stable ceramic phases with suitable crystal sites.

Changing the overall composition can change which phases form and which ions each phase incorporates. This links phase equilibrium and crystal structure to the chemical durability of a waste form.^[ref]

Liquid air is a cryogenic liquid mixture produced from atmospheric gases, chiefly nitrogen and oxygen. It does not behave as a pure substance with a single composition-independent boiling temperature.

Because its components have different volatilities, evaporation changes the mixture's composition. Fractional distillation uses that difference to separate major atmospheric gases, linking phase equilibrium with industrial gas production.

Liquid immiscibility is the failure of a mixture to form one homogeneous liquid at the specified composition and conditions. In a two-liquid phase equilibrium, each thermodynamic phase can contain some of both components, but their equilibrium compositions differ. “Immiscible” therefore need not mean that no molecules of either component dissolve in the other.

Hexane and water provide a concrete example of limited mutual solubility. A NIST-compiled experimental record describes overnight shaking, microsyringe sampling of an aqueous solution, and gas chromatography with flame-ionization detection to measure the dissolved hydrocarbon. Detecting a small dissolved amount does not imply that every bulk mixing ratio gives one liquid phase.^[ref]

An original lever rule example shows how phase amounts differ from phase compositions. Suppose a binary system at fixed temperature and pressure has coexisting liquid compositions $x_{\alpha }=0.1$ and $x_{\beta }=0.8$, where $x$ is the mole fraction of componentB. These stipulated values are not measurements of hexane/water. If the overall mole fraction is $z=0.4$, material balance givesHere $\lambda$ is the fraction of the total amount of substance in phaseβ. A100-mole sample therefore contains $300/7\simeq 42.86$ moles in phaseβ and $400/7\simeq 57.14$ moles in phaseα. The phase compositions stay0.8 and0.1; neither is the overall composition0.4.

Shaking the sample can create dispersed droplets without making a single molecularly homogeneous phase. Equilibration, temperature, pressure and composition must be stated when testing liquid-liquid equilibrium. The lever-rule fraction is a mole-amount fraction, not necessarily the fraction of vessel height or volume occupied by one layer.

The molar Gibbs free energy is the Gibbs free energy per amount of substance, $G_m=G/n$, with units $\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}$. For a homogeneous pure substance, extensivity gives $G=n\mu$, so $G_m$ equals its chemical potential $\mu$.

For a mixture, $G/n$ is an average over the components; the separate chemical potentials are the partial derivatives ${\mu }_i=(\partial G/\partial n_i{)}_{T,p,n_{j\ne i}}$. Equality of a component's chemical potential in two phases is a condition for phase equilibrium. This explains why comparing molar free energies predicts which phase of a pure substance is stable at specified temperature and pressure.

Phase equilibrium minimizes Gibbs free energy at specified temperature, pressure and composition. Coexisting phases have equal temperature, pressure and chemical potential. For a pure substance, a first-order transition has continuous equilibrium Gibbs free energy but a jump in its first derivatives, such as entropy or volume; the latent heat is $T\Delta S$. Continuous transitions lack that latent heat and can instead have singular or discontinuous higher responses.

The van der Waals molar equationcombines an excluded-volume correction with attractive interactions. Its inflection conditions giveBelow $T_c$, the oscillatory isotherm contains a mechanically unstable segment with $(\partial p/\partial v{)}_T>0$. Equilibrium instead replaces the loop by a constant-pressure coexistence line. Equality of chemical potentials gives the Maxwell construction,On isothermal compression the gas reaches saturation, then converts progressively into liquid at fixed pressure. The mean molar volume obeys $v=(1-x)v_{\ell }+x{v}_g$, where $x$ is the gas fraction. Metastable superheated or supercooled states can persist when nucleation is delayed; they are not the global equilibrium selected by this construction.

A pressure–temperature phase diagram contains coexistence curves separating solid, liquid and gas. Three phases meet at a triple point; the liquid–gas coexistence curve ends at a thermodynamic critical point, above which liquid and gas can be connected without crossing a first-order phase boundary. A generic solid–liquid line continues beyond the liquid–gas critical temperature. For one component the Gibbs phase rule is $F=3-P$, so a three-phase equilibrium fixes both temperature and pressure.

Along a two-phase coexistence curve, $d\mu =-sdT+vdp$ for each phase givesThis is the Clapeyron equation, valid with consistent molar or specific quantities. For vaporization, neglecting the liquid volume and treating the vapour as ideal gives $\Delta v\simeq RT/p$, hence $d\ln p/dT=L/(R{T}^2)$. With approximately constant molar latent heat,This Clausius–Clapeyron approximation explains the rapid rise of vapour pressure with temperature. It fails close to the thermodynamic critical point, where phase volumes converge, and is not a universal expression for melting. The sign of a melting-line slope depends on its volume change; water's denser liquid gives a familiar negative slope.

Notation. The source uses $p_s$ for both the hypothetical dry surface pressure and the actual surface pressure after evaporation. Distinguish them as $P_d$ and $P$, respectively, and interpret the specified pressure thickness $\Delta p$ as fixed. This conserves the original hydrogen inventory and permits $P$ to vary. Reading both occurrences as one fixed number would be inconsistent with adding water vapor to a closed atmospheric column mass budget.

Let $E(T)$ denote the saturation pressure of water, let $x$ be the mole fraction of water vapor in the mixed layer, and let $M_d,M_v$ be the molar masses of hydrogen and water. Surface phase equilibrium and the ideal gas law giveBecause the layer has constant temperature, its relative humidity isThus every level strictly above the surface is subsaturated: hydrostatic balance makes $p<P$, whereas the saturation pressure stays constant. The result does not require a dilute water vapor approximation.

The Clausius-Clapeyron relation, neglecting the specific volume of liquid water and treating the water vapor as an ideal gas, givesHere $T_t,E_t$ are the triple point values, $L_v$ is the specific latent heat, and $R$ is the molar gas constant. All temperatures in these expressions are absolute. As throughout the question, this is a constant-latent heat, ideal gas model, not an extrapolation valid through the real critical point of water.

For a well-mixed ideal gas mixture, its mean molar mass and water vapor mass fraction areBy hydrostatic balance, the mixed layer has total mass per unit area $\Delta p/g$, of which $w_v\Delta p/g$ is water vapor. The remaining atmosphere contains no water vapor. Consequently conservation of hydrogen mass givesEquivalently the requested quadratic equation isFor each $T_s$, calculate $E$, take the positive root of this quadratic equation, and retain it only if $0\le x<1$ and $P>\Delta p$. The last condition leaves a nonempty dry layer. In particular the dry constituent's surface partial pressure is not generally $P_d$: $P_d$ measures its column weight, and the hydrogen and water vapor have different molar masses.

A convenient parameter is the water vapor mole fraction itself. The mass conservation calculation yieldsBoth functions increase for $0\le x\le 1$. At the endpoint,Therefore the physically admissible hydrogen–water vapor mixture requiresBeyond this bound, the formal positive quadratic equation root requires $x>1$, equivalently a negative hydrogen partial pressure. It is the physical constraints, rather than loss of real roots, that end the solution branch. For a concrete example, put $P_d=\Delta p=1$ in consistent pressure units, $ϵ=9$, and $E=3$. The positive root of $P^2+23P-51=0$ is about $2.037$, giving $x\simeq 1.473$, which is impossible.

The limiting temperature is obtained from $E(T_{\ast })=P_d+\Delta p$:This finite limit exists above $T_t$ if $E_t<P_d+\Delta p<E_t\exp [L_v/(R_v{T}_t)]$. The upper qualification matters mathematically: the stipulated constant-latent heat formula has a finite limiting saturation pressure as $T\to \infty$, so for an arbitrarily large unspecified $P_d+\Delta p$ the printed assertion of a finite breakdown temperature is not unconditional. For a concrete formal counterexample, choose $P_d+\Delta p=10^{15}\mathrm{Pa}$: with the usual triple-point constants the model has $E(\infty )\simeq 2.5\times 10^{11}\mathrm{Pa}$, so it never reaches the proposed inventory limit. This is a counterexample inside the stipulated equations, not a physically credible high-pressure ideal gas ocean. Actual liquid water also ceases to exist above its critical point, before such an extrapolation would make sense.

Physically, the imposed finite-pressure-thickness layer can hold no more than $\Delta p/g$ of water vapor per unit area. As $x\to 1$, it expels all its hydrogen into the dry upper layer. A still larger surface saturation pressure cannot coexist with this geometry. The layer must deepen, the temperature profile must change, or the assumed phase equilibrium must fail. If $P_d<\Delta p$, the additional restriction $P>\Delta p$ can also exclude the low-temperature end of the branch.

The geological record is a collection of thermometers with different sampling biases, not a direct time series of the whole mantle.

Primary melt compositions constrain melting temperatures. Reconstruct primary magma compositions from dated, suitably unaltered basalts by correcting for fractional crystallization and checking equilibrium with olivine. Experimental phase equilibrium constraints and mineral–melt thermometers then constrain the pressure and temperature of melt generation. In laboratory calibrations, a candidate melt and mineral assemblage are equilibrated at controlled pressure and temperature, quenched, and their crystals and glass analyzed. Repetition across adjacent conditions brackets the relevant phase equilibrium boundaries. Correcting for decompression and latent heat then allows comparison as mantle potential temperature. Age-controlled, non-arc basalt suites are especially valuable for separating changes in ambient mantle from exceptional mantle plumes. The 2010 reconstruction by Claude Herzberg and colleagues combines this approach with the temporal distribution of high-Mg parental melts.^[ref]

Komatiites provide evidence for unusually hot melting, with qualifications. Many ancient high-Mg lavas, their olivine compositions and spinifex textures indicate large degrees of melting and high parental temperatures. Their abundance in Archean terrains and rarity in the recent record are consistent with a hotter past. However, a hot mantle plume, dissolved water, unusual source composition or crystal accumulation can change the interpretation. A komatiite temperature is not automatically the ambient mantle temperature, and spinifex texture alone is not a calibrated thermometer.

Melting residues and crustal products provide independent checks. Depleted peridotite mantle xenoliths, preserved mineral chemistry and compatible-element depletion constrain former melt fractions. More extensive melting can produce thicker mafic crust and more refractory residual mantle; these are predictions to compare with preserved rock packages and seismic or petrological constraints on ancient lithosphere. Metamorphic facies and pressure–temperature paths provide contextual evidence about geothermal regimes, but tectonic burial, decompression and preservation can mimic temporal changes. They do not directly measure $T_p$.

The most persuasive inference is therefore agreement between dated melt thermometry, melting residues and their predicted crustal consequences, while separating source fertility, volatile content, mantle plume anomalies, alteration and selective preservation. Those controls explain why hotter ancient melting is better established than one exact global cooling curve.

Petrology studies how rocks form and change, using their composition, texture and mineral assemblages. Crystals and surrounding material record processes such as partial melting, fractional crystallization, deformation and fluid reaction. These observations can be compared with laboratory phase equilibrium measurements and thermal models.

An assemblage constrains a range of conditions rather than a complete history by itself. Minerals can retain older cores, grow new rims or fail to equilibrate fully during a short event. Linking textures with mineral chemistry and dates helps distinguish the original formation of a rock from later metamorphism.