Solid solution — aqueous solution equilibria as a function of pH

The equations relating element distribution and pH are derived for systems containing an ideal solid solution in equilibrium with an ideal aqueous solution, assuming no polymeric complexes form in the aqueous solution and the solid solution does not contain molecular units with multiple atoms of the substituting elements. These expressions demonstrate that the ratio of the partition coefficients describing element distribution for a system containing a multi-component solid solution is inversely proportional to the solubilities of the end member components at any given pH raised to the power equal to the ratio of the sum of the stoichiometric coefficients of the end-member salt to the stoichiometric coefficient of the substituting radical. The coefficient describing distribution between the aqueous phase and a two-component solid solution is equal to the inverse of the ratio of the end member solubilities raised to the above power. Element distribution between the two phases will be homogenous at any pH resulting in identical solubilities for the two end-member components, and a reversal in relative solubilities will result in a corresponding reversal in the element preferentially incorporated into the solid solution. Because of the dependence of element distribution on pH, a crystal could develop both zoning and reverse zoning as a result of changes in pH. The distribution coefficient could provide information regarding the pH of the aqueous solution at the time of mineral formation if independent evidence establishes the ratio of end-member components in the aqueous phase. The equations describing element distribution may be expressed in terms of the solubility products of the end-member components and the ionization constants of the substituting radicals. Based on the relative values of the ionization constants, pH intervals can be established in which only the concentration of a single complex for each substituting radical need be considered. Within such an interval, the curve of the log of the distribution coefficient vs. pH is linear with a slope equal to the difference in the charges of the two complexes. This approach to the examination of element distribution is developed in some detail for the geologically important case of a two component solid solution having composition (A²⁺, B²⁺) X^2?.     

Thermochemistry of the high structural state plagioclases

The enthalpies of solution of a suite of 19 high-structural state synthetic plagioclases were measured in a Pb₂B₂O₅ melt at 970 K. The samples were crystallized from analyzed glasses at 1200°C and 20 kbar pressure in a piston-cylinder apparatus. A number of runs were also made on Amelia albite and Amelia albite synthetically disordered at 1050–1080°C and one bar for one month and at 1200°C and 20 kbar for 10 hr. The component oxides of anorthite, CaO, Al₂O₃ and SiO₂, were remeasured.The ΔH of disorder of albite inferred in the present study from albite crystallized from glass is 3.23 kcal, which agrees with the 3.4 found by Holm and Kleppa (1968). It is not certain whether this value includes the ΔH of a reversible displacive transition to monoclinic symmetry, as suggested by Helgesonet al. (1978) for the Holm-Kleppa results. The enthalpy of solution value for albite accepted for the solid solution series is based on the heat-treated Amelia albite and is 2.86 kcal less than for untreated Amelia albite.The enthalpy of formation from the oxides at 970 K of synthetic anorthite is ?24.06 ± 0.31 kcal, significantly higher than the ?23.16 kcal found by Charluet al. (1978), and in good agreement with the value of ?23.89 ± 0.82 given by Robieet al. (1979), based on acid calorimetry.The excess enthalpy of mixing in high plagioclase can be represented by the expression, valid at 970 K: ΔH^ex(±0.16 kcal) = 6.7461 X_abX²_An + 2.0247 X_AnX²_Ab where X_Ab and X_An are, respectively, the mole fractions of NaAlSi₃O₈ and CaAl₂Si₂O₈. This ΔH^ex, together with the mixing entropy of Kerrick and Darken's (1975) Al-avoidance model, reproduces almost perfectly the free energy of mixing found by Orville (1972) in aqueous cation-exchange experiments at 700°C. It is likely that Al-avoidance is the significant stabilizing factor in the high plagioclase series, at least for X_An≥ 0.3. At high temperatures the plagioclases have nearly the free energies of ideal one-site solid solutions. The Al-avoidance model leads to the following Gibbs energy of mixing for the high plagioclase series: $\text{ΔG}{}^{\mathrm{mix}}\text{= ΔH}{}^{\mathrm{ex}}\text{+ RT X}{}_{\mathrm{Ab}}\text{ln[X}{}^2{}_{\mathrm{Ab}}\text{(2 ? X}{}_{\mathrm{Ab}}\text{)]+ X}{}_{\mathrm{An}}\text{ln}\text{[X}{}_{\mathrm{An}}\text{(1+X}{}_{\mathrm{An}}\text{)}{}^2\text{]}\text{4}$. The entropy and enthalpy of mixing should be very nearly independent of temperature because of the unlikelihood of excess heat capacity in the albite-anorthite join.     

Thermochemistry of forsterite-fayalite olivine solutions

The enthalpies of solution of synthetic Mg₂SiO₄-Fe₂SiO₄ olivine solid solutions have been measured in Pb₂B₂O₅ melt at 970 K. The heat of solution of forsterite was found to be 15.62 ± 0.3 kcal mol^?1 and that of fayalite 9.39 ± 0.14 kcal mol^?1. Solid solutions between these end-members exhibit small positive deviations from mixing ideality, asymmetric towards the Fe end-member. In terms of the sub-regular solution model, excess enthalpies of intermediate olivine are adequately represented by the equation H^xs = 2(1000 + 1000X_Fe) X_FeX_MgThe enthalpies of solution at 970 K are consistent with high temperature phase equilibrium measurements of activity-composition relationships in the olivine series. Excess entropy terms are not needed to relate the phase equilibrium data to the calorimetric data presented here.The enthalpy of solution of FeSiO₃ ferrosilite at 970 K was found to be 4.36 ± 0.10 kcal mol^?1. This value, when taken together with calorimetric measurements on fayalite and quartz, is consistent with phase equilibrium investigations of the reaction: 2FeSiO₃ = Fe₂SiO₄ + SiO₂ Ferrosilite Fayalite QuartzThese provide a check on the internal consistency of the calorimetric data presented here.     

Miscibility in the CaSO4·2H2O–CaSeO4·2H2O system: Implications for the crystallisation and dehydration behaviour

SeO₄²⁻ ions can substitute for sulphate in the gypsum structure. In this work crystals of different Ca(SO₄,SeO₄)·2H₂O solid solutions were precipitated by mixing a CaCl₂ solution with solutions containing different ratios of Na₂SO₄ and Na₂SeO₄. The compositions of the precipitates were analysed by EDS and the cell parameters were determined by X-ray powder diffraction. Moreover, a comparative study on dehydration behaviour of selenate rich and sulfate rich Ca(SO₄,SeO₄)·2H₂O solid solutions was carried out by thermogravimetry.The experimental results show that the Ca(SO₄,SeO₄)·2H₂O solid solution presents a symmetric miscibility gap for compositions ranging from X_CaSO4·2H2O = 0.23 to X_CaSO4·2H2O = 0.77. By considering a regular solution model a Guggenheim parameter a₀ = 2.238 was calculated. The solid phase activity coefficients obtained with this parameter were used to calculate a Lippmann diagram for the system Ca(SO₄,SeO₄)·2H₂O–H₂O.     

Experimentally determined compositions of diopside-jadeite pyroxene in equilibrium with albite and quartz at 1200–1350°C and 15–34 kbar

Equilibrium compositions of diopside-jadeite pyroxene coexisting with albite and quartz were experimentally determined at 25 different P-T conditions, using an electron microprobe for analysis. The new data and the 600°C data of Holland (1983) provided the following mixing properties of the diopside (Di)-jadeite (Jd) solid solution (J, K): G^xs = X_JdX_Di[12600 ? 9.45T + (12600 ? 7.6T)(X_Jd ? X_Di) ? (21400 ? 16.2T)(X_Jd ? X_Di)²]. The Di-Jd solution is close to ideal above 1000°C but immiscible below 565°C. The Di-Jd solvus is slightly asymmetric with the crest at composition Di_42.4Jd_57.6. Excess enthalpy is positive but smaller than indicated by the enthalpy of solution measurements of Woodet al. (1980). Disorder in the Di-Jd solution is significantly smaller than complete disorder implied by the ionic two-site model.     

Compositions and phase relations of calcic amphiboles in epidote- and clinopyroxene-bearing rocks of the amphibolite and lower granulite facies,central Massachusetts,USA

Calcic amphiboles coexisting with epidotegroup minerals (zoisite, clinozoisite, epidote) and/or clinopyroxene±plagioclase±quartz±garnet occur in amphibolites and calc-silicate rocks that underwent amphibolite to lower granulite-facies metamorphism in the Acadian metamorphic high of central Massachusetts, USA. Across the region, peak metamorphic conditions range from about 580° C and 6.2 kbar to 730° C and 6.3 kbar. The coexistence of most Ca-amphiboles with Fe³⁺-rich epidote-group minerals suggests the presence of Fe³⁺ in most of these amphiboles. An empirical Fe³⁺ estimation for the microprobe analyses is based on two constraints: the Na?Ca content of the M4 sites of Ca-saturated, gravimetrically analyzed hornblendes gives the relation: Ca(M4) ^c =-1.479 Na(M4) ^c +2 (c=corrected). The second constraint is the stoichiometric equation Ca(M4)+Na(M4)+FM=15, where FM is the sum of all cations exclusive of Ca, Na, and K. Solving the two equations simultaneously gives: 20.185=0.479 Ca(M4)+1.479 ΣFM. Starting with the uncorrected values of Ca(M4) ^u and ΣFM(M4) ^u (u = uncorrected) of the all ferrous formula, the normalization factor NF for calculating the corrected cations of the ferric formulas is: 20.185/(0.478 Ca(M4) ^u +1.479 ΣFM ^u ). From the deficient oxygen the Fe³⁺ content which is equal to 2(23-ΣOX) can be calculated. Determinations of Fe³⁺ contents of four hornblende separates by Mössbauer spectroscopy are in agreement with the calculated values. The Ca-amphiboles show systematic changes in composition with increasing grade of metamorphism within the amphibolite and lower granulite-facies zones: increasing edenite and tschermakite substitution, increasing Ti content, and increasing Fe²⁺/(Fe²⁺+Mg) ratio. In addition, the coexisting clinopyroxenes are also characterized by an increase in Fe²⁺/(Fe²⁺+Mg) ratio. In quartz-free rocks with coexisting Ca-amphibole and plagioclase there is an increase in the ratio X _Ab/X _Ed, where X _Ab=Na/(Na+Ca) in plagioclase and X _Ed=Na in the amphibole A-site. These chemical changes in mineral composition together with the disappearance of epidote at the transition to granulite-facies metamorphic conditions are attributed to the continuous reaction: albite+epidote+Fe-Mg hornblende→Fe?Mg clinopyroxene+anorthite+(NaAlSi_-1)^Hbl+H₂O.     

The thermodynamic properties of reciprocal solid solutions

A multisite solid solution of the type (A, B) (X, Y) has the four possible components AX, AY, BX, BY. Taking the standard state to be the pure phase at the pressure and temperature of interest, the mixing of these components is shown not to be ideal unless the condition: $$\Delta G^0 = (\mu _{AX}^0 + \mu _{BY}^0 - \mu _{AY}^0 - \mu _{BX}^0 = 0$$ applies. Even for the case in which mixing on each of the individual sublattices is ideal, ΔG ⁰ contributes terms of the following form to the activity coefficients of the constituent components: $$RT\ln \gamma _{AX} = - X_{B_1 } X_{Y_2 } \Delta G^0$$ (X _Ji refers to the atomic fraction of J on sublattice i). The above equation, which assumes complete disorder on (A, B) sites and on (X, Y) sites is extended to the general n-component case. Methods of combining the “cross-site” or reciprocal terms with non-ideal terms for each of the individual sites are also described. The reciprocal terms appear to be significant in many geologically important solid solutions, and clinopyroxene, garnet and spinel solid solutions are all used as examples. Finally, it is shown that the assumption of complete disorder only applies under the condition: $$\Delta G^0 \ll zn_1 RT$$ where z is the number of nearest-neighbour (X, Y) sites around A and n ₁ is the number of (A, B) sites in the formula unit. If ΔG ⁰ is relatively large, then substantial short range oder must occur and the activity coefficient is given by (ignoring individual site terms): $$\gamma _{AX} = \left( {\frac{{1 - X'_{Y_2 } }}{{1 - X_{Y_2 } }}} \right)^{zn_1 }$$ where X′_Y2 is the equilibrium atomic fraction of Y atoms surrounding A atoms in the structure. The ordered model may be developed for multicomponent solutions and individual site interactions added, but numerical methods are needed to solve the simultaneous equations involved.     

Refinements in a site-mixing model for illites: local electrostatic balance and the quasi-chemical approximation

A quasi-chemical model for illites has been derived, and local electrostatic balance has been added to a random regular solution site-mixing model for illites (Stoessell, 1979). Each model assumes similar order-disorder conditions for both the end-members micas and the solid solution. Thermodynamic properties of illites predicted by the random, electrostatic, and quasi-chemical models are compared as a function of composition. For natural illite compositions, molar entropies of mixing in the electrostatic model are about 1 entropy unit less than those in the random model. Intermediate values are given by the quasi-chemical model. Each model predicts an increased entropy of mixing in dominantly trioctahedral illites as compared to dioctahedral illites. Each model also predicts destabilization of trioctahedral illites using absolute molar exchange energies greater than 2 RT/Z_x, where Z_x is the number of adjacent cation interactions per site in the Xth site class. The most negative free energies of mixing are predicted by the quasi-chemical model. Intermediate values predicted by the random model are apparently the result of error cancellation due to overestimation of both the entropy and enthalpy of mixing.     

Empirical garnet-muscovite-plagioclase-quartz geobarometry in medium- to high-grade metapelites

Based on the net transfer reactions among garnet, muscovite, plagioclase, and quartz (for both Mg and Fe end-member models), the garnet-muscovite-plagioclase-quartz (GMPQ) geobarometry was empirically calibrated under the physical conditions of P=1.0-11.4 kbar and T=505-745 °C for 128 natural metapelitic rock samples collected from the literature. The input temperatures and pressures were simultaneously determined by the garnet-biotite thermometer and the garnet-aluminosilicate-plagioclase-quartz (GASP) barometer. The GMPQ calibrations adopted the same asymmetric quaternary solid solution model of garnet and the same Al-avoidance asymmetric ternary model of plagioclase as the calibrations of the garnet-biotite geothermometer and the GASP geobarometer. A symmetric Fe-Mg-Al^VI ternary solid solution model of muscovite was adopted, and the Margules parameters of muscovite were obtained through regression. The Mg and Fe model reactions, along with the assumption of whether the ferric iron content in muscovite is 0% or 50%, resulted in four GMPQ barometry formulae. The GMPQ barometry formulae reproduce the input GASP pressures well within ±1.0 kbar (mostly within ±0.5 kbar). For both aluminosilicate-bearing and aluminosilicate-absent samples, the GMPQ barometry formulae yield identical pressures for every sample, whether the sample was included or not in calibrating the barometers. For each of the Mg or Fe model reaction, the formulae gave identical pressures within ±40 bars. The random error of the GMPQ barometry may be expected as ±1.4 kbar. The dP/dT slopes of these GMPQ formulae are close to that of the GASP barometer in the P-T space. Applications of the GMPQ barometry to aluminosilicate-absent metapelites within a limited geographic area without postmetamorphic structural discontinuity generally show no pressure difference. It may be concluded that the GMPQ barometry formulae derived in this work may be used as practical tools for metamorphic pelites under the conditions of 505-745 °C and 1-11.4 kbar, in the composition range of X_gros>3% in garnet and X_An>17% in plagioclase.     

Constraints on the thermodynamic mixing properties of plagioclase feldspars

Taking account of the Cˉ1/Iˉ1 (Al/Si order/disorder) transformation at high temperatures in the albite-anorthite solid solution leads to a simple model for the mixing properties of the high structural state plagioclase feldspars. The disordered (Cˉ1) solid solution can be treated as ideal (constant activity coefficient) and, for anorthite-rich compositions, deviations from ideality can be ascribed to cation ordering. Values of the activity coefficient for anorthite in the Cˉ1 solid solution (γ _An ^Cˉ1 ) are then controlled by the free energy difference between Cˉ1 and Iˉ1 anorthite at the temperature (T) of interest according to the relation: ΔˉG _ord ^Iˉ1 ⇌Cˉ1 =RT ln γ _An ^Cˉ1 . If the Iˉ1⇌Cˉ1 transformation in pure anorthite is treated, to a first approximation, as first order and the enthalpy and entropy of ordering are taken as 3.7±0.6 kcal/mole (extrapolated from calorimetric data) and 1.4–2.2 cal/mole (using an equilibrium order/disorder temperature for An₁₀₀ of 2,000–2,250 K), a crude estimate of γ _An ^Cˉ1 for all temperatures can be made. The activity coefficient of albite in the Cˉ1 solid solution (γ _Ab ^Cˉ1 ) can be taken as 1.0. The possible importance of this model lies in its identification of the principal constraints on the mixing properties rather than in the actual values of γ _An ^Cˉ1 and γ _Ab ^Cˉ1 obtained. In particular it is recognised that γ _An ^Cˉ1 depends critically on ordering in anorthite as well as, at lower temperatures, any ordering in the Cˉ1 solid solution. A brief review of activity-composition data, from published experiments involving ranges of plagioclase compositions and from the combined heats of mixing plus Al-avoidance entropy model (Newton et al. 1980), reveals some inconsistencies. The values of γ _An ^Cˉ1 calculated using the approach of Newton et al. (1980), although consistent with Orville's (1972) ion exchange data, are slightly lower than values derived from experiments by Windom and Boettcher (1976) and Goldsmith (1982) or from ion-exchange experiments of Kotel'nikov et al. (1981). Based on the Cˉ1/Iˉ1 transformation model, values of γ _An ^Cˉ1 <1.0 are unlikely. Discrepancies between the experimental data sets are attributed to incomplete (non-equilibrium) Al/Si order attained during the experiments. It is suggested that the choice of activity coefficients remains somewhat subjective. The development of accurate mixing models would be greatly assisted by better thermodynamic data for ordering in pure anorthite and by more thorough characterisation of the state of order in plagioclase crystals used for phase equilibrium experiments.     

On calculating activity coefficients and other excess functions from the intracrystalline exchange properties of a double-site phase

For a phase at equilibrium in which two cation species are partitioned ideally between two sub-lattice sites, the excess functions of mixing (free energy, enthalpy and entropy) are directly related to the bulk composition of the phase and ΔG_E°(T, P), the standard-state intra- crystalline exchange free energy. If the phase is not at equilibrium internally, an additional ordering parameter is necessary to fix the excess free energy of mixing, G_mix^EX, unambiguously. Conversely, for any fixed G_mix^EX there exists an infinity of possible intracrystalline cation dis- tributions, only one of which is the equilibrium distribution for the specified temperature and pressure. As ideal intraphase cation ordering becomes more pronounced, G_mix^EX decreases. In response, the total free energy of mixing for the phase decreases progressively for non-end member compositions, approaching, at the limits of ordering, values appropriate for stabilizing compounds of intermediate composition.The model-dependent activity coefficient for component A in the phase, γ_A^T, can be calculated for any bulk composition, X_A^T, either from G_mix^EX directly or from more basic equations involving the interrelation of chemical potentials at equilibrium. A general form for γ_A^T is ln $\text{γ}{}_{\mathrm{A}}{}^{\mathrm{T}}\text{= 1n[}\text{2(X}{}_{\mathrm{A}}{}^{\mathrm{\alpha }}\text{X}{}_{\mathrm{A}}{}^{\mathrm{\beta }}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/}\text{(X}{}_{\mathrm{A}}{}^{\mathrm{\alpha }}\text{+X}{}_{\mathrm{A}}{}^{\mathrm{\beta }}\text{)}\text{]+Y}$, where Xj^κ denotes the mole fraction of species j in site κ. The first term on the right-hand side of this equation is the contribution to γ_A^T from ideal intracrystalline partitioning, and is common to the several theories lately presented to model intraphase cation partitioning. It can be shown rigorously that this term contributes to a negative deviation from ideality for the bulk phase. The second term is the contribution to the macroscopic activity coefficient from non-ideal intraphase partitioning, and is related to an enthalpy of mixing, H_mix^N in excess of that resulting from ideal inter-site cation ordering. While the expression represented by Y can take several functional forms, the additional enthalpy can be evaluated explicitly for specific non-ideal partitioning models from the relation $\text{H}{}_{\mathrm{mix}}{}^{\mathrm{N}}\text{= 2RT(1? X}{}_{\mathrm{A}}{}^{\mathrm{T}}\text{) ∝}\text{Y}\text{(1 ? X}{}_{\mathrm{A}}{}^{\mathrm{T}}\text{)}{}^2\text{dX}{}_{\mathrm{A}}{}^{\mathrm{T}}$.In those cases, G_mix^EX can also be determined exactly.     

Solid solutions of trace Eu(III) in calcite: Thermodynamic evaluation of experimental data over a wide range of pH and pCO2

The thermodynamics of dilute Eu-calcite solid solutions formed under widely different pH-pCO₂ conditions at T = 25°C and p = 1 bar were investigated using three sets of Eu(III) uptake experiments, two of which were taken from the literature: (a) recrystallization in synthetic cement pore water at pH ∼ 13 and pCO₂ ∼ 10⁻¹³ bar (this work); (b) coprecipitation in 0.1 M NaClO₄ at pH ∼ 6 and pCO₂ ∼ 1 bar; (c) coprecipitation in synthetic seawater at pH ∼ 8 and pCO₂ ranging from 3 × 10⁻⁴ to 0.3 bar.Solid solution formation was modeled using the Gibbs energy minimization (GEM) method. In a first step (“forward” modeling), we tested ideal binary solid solution models between calcite and the Eu end-members Eu₂(CO₃)_3, EuNa(CO₃)₂, Eu(OH)CO₃ or Eu(OH)₃, for which solids with independently measured solubility products exist. None of these four binary solid solutions was capable of reproducing all three experimental datasets simultaneously. In a second step (“inverse” modeling), ideal binary solid solutions were constructed between calcite and the candidate Eu end-members EuO(OH), EuH(CO₃)₂ and EuO(CO₃)_0.5, for which no independent solubility products are available. For each single data point and each of these end-members, a free energy of formation with inherent activity coefficient term ( = G_α^o + RT lnγ_α) was estimated from “dual thermodynamic” GEM calculations. The statistical mean of was then calculated for each of the three datasets. A specific end-member was considered to be acceptable if a standard deviation of ± 2 kJ mol⁻¹ or less resulted for each single dataset, and if the mean -values calculated for the three datasets coincided. No binary solid solution with any of the seven above mentioned end-members proved to satisfy these criteria.The third step in our analysis involved consideration of ternary solid solutions with CaCO₃ as the major end-member and any two of the seven considered Eu trace end-members. It was found that the three datasets can only be reproduced simultaneously with the ternary ideal solid solution EuH(CO₃)₂ - EuO(OH) - CaCO₃, setting = −1773 kJ mol⁻¹ and = −955 kJ mol⁻¹, whereas all other end-member combinations failed. Our results are consistent with time-resolved laser fluorescence data for Cm(III) and Eu(III) indicating that two distinct species are incorporated in calcite: one partially hydrated, the other completely dehydrated. In conclusion, our study shows that substitution of trivalent for divalent cations in carbonate crystal structures is a more complex process than the classical isomorphic divalent-divalent substitution and may need consideration of multicomponent solid solution models.     

The crystal chemistry of synthetic potassium-bearing neighborite, (Na1− x K x )MgF3

A series of fluoride perovskites related to neighborite was investigated using X-ray and neutron diffraction techniques, and Rietveld profile refinement of powder diffraction data. The series (Na_{1? x} K _x )MgF₃ comprises orthorhombic (Pbnm, a?≈? , b?≈? , c?≈?2a _p , Z=4) perovskites in the compositional range 0?≤?x?≤?0.30, tetragonal perovkites (P4/mbm, a?≈? , c?≈?a _p , Z=2) in the range 0.40?≤?x?≤?0.46, and cubic phases (Pm3¯m, Z=1) for x?>?0.50. The structure of the orthorhombic neighborite is derived from the perovskite aristotype by rotation of MgF₆ octahedra about the [110] and [001] axes of the cubic subcell. The degree of rotation, measured as a composite tilt Φ about the triad axis, varies from 18.2° at x=0 to 11.2° at x=0.30 (as determined from the fractional atomic coordinates). Orthorhombic neighborite also shows a significant displacement of Na and K from the “ideal” position (≤0.25?Å). The tetragonal members of the neighborite series exhibit only in-phase tilting about the [001] axis of the cubic subcell (φ) ranging from 4.5° to 4.8° (determined from the atomic coordinates). The solid solution (Na_{1? x} K _x )MgF₃, shows a regular variation of unit-cell dimensions with composition from 3.8347?Å for the end-member NaMgF₃ (reduced to pseudocubic subcell, a _p ) to 3.9897?Å for KMgF₃. This variation is accompanied by increasing volumes of the A-site polyhedra, whereas the volume of MgF₆ octahedra initially decreases (up to x=0.40), and then increases concomitantly with K content. The polyhedral volume ratio, V _A /V _B , gradually increases towards the tetragonal structural range, in agreement with diminishing octahedral rotation in the structure. The P4/mbm-type neighborite has an “anomalous” polyhedral volume ratio (ca. 5.04) owing to the critical compression of MgF₆ polyhedra.     

High-resolution vertical profiles of groundwater electrical conductivity (EC) and chloride from direct-push EC logs

Elevated groundwater salinity associated with produced water, leaching from landfills or secondary salinity can degrade arable soils and potable water resources. Direct-push electrical conductivity (EC) profiling enables rapid, relatively inexpensive, high-resolution in-situ measurements of subsurface salinity, without requiring core collection or installation of groundwater wells. However, because the direct-push tool measures the bulk EC of both solid and liquid phases (EC_a), incorporation of EC_a data into regional or historical groundwater data sets requires the prediction of pore water EC (EC_w) or chloride (Cl^?) concentrations from measured EC_a. Statistical linear regression and physically based models for predicting EC_w and Cl^? from EC_a profiles were tested on a brine plume in central Saskatchewan, Canada. A linear relationship between EC_a/EC_w and porosity was more accurate for predicting EC_w and Cl^? concentrations than a power-law relationship (Archie’s Law). Despite clay contents of up to 96%, the addition of terms to account for electrical conductance in the solid phase did not improve model predictions. In the absence of porosity data, statistical linear regression models adequately predicted EC_w and Cl^? concentrations from direct-push EC_a profiles (EC_w = 5.48 EC_a + 0.78, R ² = 0.87; Cl^? = 1,978 EC_a – 1,398, R ² = 0.73). These statistical models can be used to predict EC_w in the absence of lithologic data and will be particularly useful for initial site assessments. The more accurate linear physically based model can be used to predict EC_w and Cl^? as porosity data become available and the site-specific EC_w–Cl^? relationship is determined.     

Water and magmas; a mixing model

A model for the mixing of H₂O and silicate melts has been derived from the experimentally determined effects of H₂O on the viscosity (fluidity), volumes, electrical conductivities, and especially the thermodynamic properties of hydrous aluminosilicate melts. It involves primarily the reaction of H₂O with those O^?2 ions of the melt that are shared (bridging) between adjacent (Al, Si)O₄ tetrahedra to produce OH^? ions. However, in those melts that contain trivalent ions in tetrahedral coordination, such as the Al³⁺ ion in feldspathic melts, the model further involves exchange of a proton from H₂O with a non-tetrahedrally coordinated cation that must be present to balance the net charge on the AlO₄ group. This cation exchange reaction, which goes essentially to completion, results in dissociation of the H₂O and is limited only by the availability of H₂O and the number of exchangeable cations per mole of aluminosilicate.In the system NaAlSi₃O₈-H₂O, upon which this thermodynamic model is based, there is 1 mole of exchangeable cations (Na⁺) per mole (GFW) of NaAlSi₃O₈, consequently ion exchange occurs for H₂O contents up to a 1:1 mole ratio (X^m_w = mole fraction H₂O = 0.5). For mole fractions of H₂O greater than 0.5, no further exchange can occur and the reaction with additional bridging oxygens of the melt produces 2 moles of associated OH^? ions per mole of H₂O dissolved. These reactions lead to a linear dependence of the thermodynamic activity of H₂O (a^m_w) on the square of its mole fraction (X^m_w) for values of X^m_w, up to 0.5 and an exponential dependence on X^m_w at higher H₂O contents. Thus, for values of $\text{X}{}^{\mathrm{m}}{}_{\mathrm{w}}\text{? 0.5, a}{}^{\mathrm{m}}{}_{\mathrm{w}}\text{= k(X}{}^{\mathrm{m}}{}_{\mathrm{w}}\text{)}{}^2$, where k is a Henry's law constant for the dissociated solute.Extension of the thermodynamic model for NaAlSi₃O₈-H₂O to predict H₂O solubilities and other behavior of compositionally more complex aluminosilicate melts (magmas) requires placing these melts on an equimolal basis with NaAlSi₃O₈. This is readily accomplished using chemical analyses of quenched glasses by normalizing to the stoichiometric requirements of NaAlSi₃O₈, first in terms of equal numbers of exchangeable cations for mole fractions of H₂O up to 0.5 and secondly in terms of 8 moles of oxygen for higher H₂O contents. Chemical analyses of three igneous-rock glasses, ranging in composition from tholeiitic basalt to lithium-rich pegmatite, were thus recast and the experimental H₂O solubilities were computed on this equimolal basis. The resulting equimolal solubilities are all the same, within experimental error, as the solubility of H₂O in NaAlSi₃O₈ melt calculated from the thermodynamic relations.The equivalence of equimolal solubilities implies that the Henry's law constant (k), which is a function of temperature and pressure, is independent of aluminosilicate composition over a wide range. Moreover, as a consequence of the Gibbs-Duhem relation and the properties of exact differentials, it is clear that the silicate components of the melt, properly defined, mix ideally. Thus, a relatively simple mixing model for H₂O in silicate melts has led to a quantitative thermodynamic model for magmas that has far-reaching consequences in igneous petrogenesis.     

Solution of H2O in diopside melts: a thermodynamic model

Structural similarities between dry diopside melt and superhydrous albite melt (X _w >0.5) — both lack three-dimensional silicate units — suggest that thermodynamic relations may be similar. A model based on that assumption successfully predicts diopside melting relations and H₂O solubilities. For the model, the three partial differential equations describing solution of H₂O in albite melt for X _w >0.5 have been integrated for diopside melt from X _w =0 to X _w at least as large as 0.76, with two exceptions: an alternative partial differential equation for Henrian solution of H₂O in dilute melts was applied for X _w <0.20, and an alternative differential equation for the pressure dependence of a _w at pressures below 2 kbar was developed. The latter alternative equation yields relatively small ¯V_w's at low pressures rather than the large ¯V_w's calculated from the equation from the albite system. Available experimental solubility data are not precise enough to offer a choice between the small-¯V_w and large-¯V_w equations. Integration of all the partial differential equations was constrained solely by the P and T of a single experimentally-determined point on the H₂O-saturated solidus.Solubilities calculated by a Henrian-analogue solution model (a _di=X _di ² ) from the experimental H₂O saturated solidus lie outside experimental solubility constraints for dilute melts. On the other hand, a Henrian model (a _di=X_di) successfully predicts solubilities in dilute melts. The formulation of the Henrian model and magnitudes of model molar entropies of solution are consistent with the hypothesis that H₂O dissolves in diopside melt as an essentially undissociated species with little ordering on melt structural sites. That species could in turn be consistently, if not uniquely, interpreted to be molecular H₂O or a hydroxylation (OH^–) complex formed from nonbridging oxygens.     

Calculations of activity-composition relations in multi-site solid solutions: the example of AB2O4 spinels

A model to calculate activities in multisite solutions like spinels, from a general expression of the Gibbs free energy is developped. The free energy is written as that of a solution with ideal mixing of cations on each sublattice corrected by any suitable higher order terms. It is shown that activities of i^th end-member can be simply written: $${\text{act (}}i{\text{) = (}}\gamma _i {\text{/}}\gamma _i^{\text{0}} {\text{)}}\mathop \prod \limits_j (N_j /N_j^0 )^{P(j,{\text{ }}i)} .$$ N _j are site occupancy fractions; the γ _i are equal to one for the ideal multisite model and depend only on the higher order corrections to this model; ⁰ indicate values for the i ^th end member. The exponents in the matrix P are integers and constants. The activities cannot be expressed explicitly as function of the macroscopic composition. The site occupancy fractions which minimize the Gibbs free energy must be calculated first solving a set of non linear equations which define the internal equilibrium conditions. The (Fe²⁺, Mg) (Al, Cr, Fe³⁺) spinel are used to illustrate these calculations. For multicomponent AB₂O₄ spinels activity expressions derived for the reference ideal multisite mixing model are: $${\text{act (AB}}_{\text{2}} {\text{O}}_{\text{4}} {\text{) = }}\frac{{({\text{A}})[{\text{B}}]^2 }}{{({\text{A}})_0 [{\text{B}}]_0^2 }}$$ (A): fraction of tetrahedral sites occupied by A²⁺; [B]: fraction of octahedral sites occupied by B³⁺. Because the site occupancy fractions at equilibrium are not independent (but related by the internal equilibrium relations) many equivalent expressions of the activities can be obtained. Finally approximations proposed in the literature to obtain simple explicit activity-concentration relationships are discussed.     

The Variance-Based Cross-Variogram: You Can Add Apples and Oranges

The variance-based cross-variogram between two spatial processes, Z₁ (·) and Z₂ (·), is var (Z₁ ( u ) – Z₂ ( v )), expressed generally as a bivariate function of spatial locations uandv. It characterizes the cross-spatial dependence between Z₁ (·) and Z₂ (·) and can be used to obtain optimal multivariable predictors (cokriging). It has also been called the pseudo cross-variogram; here we compare its properties to that of the traditional (covariance-based) cross-variogram, cov (Z₁ ( u ) – Z₁ ( v ), Z₂ ( u ) – Z₂ ( v )). One concern with the variance-based cross-variogram has been that Z₁ (·) and Z₂ (·) might be measured in different units (apples and oranges). In this note, we show that the cokriging predictor based on variance-based cross-variograms can handle any units used for Z₁ (·) and Z₂ (·); recommendations are given for an appropriate choice of units. We review the differences between the variance-based cross-variogram and the covariance-based cross-variogram and conclude that the former is more appropriate for cokriging. In practice, one often assumes that variograms and cross-variograms are functions of uandv only through the difference u – v. This restricts the types of models that might be fitted to measures of cross-spatial dependence.     

Activity-composition relations in feldspars

Seck's (1971a) compositional data on coexisting feldspars in the Or-Ab-An ternary at 650° C and 1 kb were used to calculate the activity-composition relations in binary alkali feldspar and binary plagioclase. The energy constants in Guggenheim's expression for excess free energy of mixing are A ₀=3920 and A ₁=657 cal/mole for alkali feldspar, in excellent agreement with values obtained by Thompson and Waldbaum (1969), and 1320 and 373 cal/ mole for plagioclase. Using Orville's (1972) data from ion-exchange experiments between plagioclase and Na—Ca chloride solutions at 700° C and 2 kb, we obtained 967 cal/mole for A ₀ and 715 cal/mole for A ₁ in the plagioclase crystalline solution.Activity-composition relations for plagioclase are interpreted in terms of a continuous, random substitution of CaAl for NaSi across the high structural state plagioclase series. This interpretation is consistent with that obtained from a consideration of lattice parameters.     

The distribution of divalent trace elements between sulfides,oxides, silicates and hydrothermal solutions: I. Thermodynamic basis

Experimental data for the standard Gibbs free energies of formation from the elements of a wide variety of metal sulfides and oxides, spinels, olivines and pyroxenes at 25°C and 1 bar define linear correlations, within about ±900 cal·mole^?1, with the corresponding conventional standard partial molal Gibbs free energies of formation of the aqueous M²⁺ cations of the form where a_MaZ and b_MaZ are empirically determined constants characteristic of the structure M_nZ. The only exceptions to correlations of this type are compounds of the heavy alkaline earths Ca, Sr and Ba, which appear to follow correlations with cation radius instead. The linear free energy correlations enable prediction of standard Gibbs free energies of formation of compositional end-members of a particular structure M_nZ provided that a_MaZ and b_MaZ are known accurately. When only the free energy of the Mg end-member is known, the standard Gibbs free energy of formation at 25°C and 1 bar of the Fe endmember, and hence a_MaZ and b_MaZ Can be predicted from the temperature independence of and estimated entropies and heat capacities for the Fe end-member. Using this approach, the free energies of ferrosilite, hedenbergite and annite at 25°C and 1 bar were predicted to within ±1000 cal·mole^?1 of the helgesonet al. (1978) values. Free energies of formation of talc (M₃Si₄O₁₀(OH)₂), clinchlore (M₅Al₂Si₃O₁₀(OH)₈), and tremolite (Ca₂M₅(Si₄O₁₁)₂(OH)₂)-type compounds where M is Mg, Mn, Zn, Fe, Co, or Ni were then predicted at 25°C and 1 bar.Calculation of the equilibrium distribution of Mg, Zn and Sr between galena and hydrothermal solution, and Zn, Mg, Fe and Mn between chlorite and hydrothermal solution demonstrates: (1) that the Sr contents of low temperature galenas (e.g. Mississippi Valley-type) should be negligible (reported analyses of Sr content and Sr isotopic composition of such galenas are probably attributable to fluid inclusions or carbonate inclusions); and (2), that the Zn contents of hydrothermal chlorites in a model of the midoceanic ridge hydrothermal systems are sensitive to temperature, to complexing in the aqueous phase, and to the overall Fe/Mg ratio of the chlorite.