Biswajit Mukhopadhyay

Garnet-biotite geothermometry revised: New Margules parameters and a natural specimen data set from Maine

The gamet-biotite geothermometer lias been recalibrated using recently obtained Mar- gules parameters for iron-magnesium-calcium garnet. Mn interactions in garnet, and Al interactions in biotite. as well as the Fe oxidation state of both minerals. Fe-Mg and ΔWAl Margules parameters for biotite have been retrieved by combining experimental results on [6]Al-free and [6]Ahbearing biotite using statistical methods. Margules parameters, per mole of biotite are WBt MgFe= 40 719 - 30T J/mole, ΔWBtAl = WBtFeAl - WBtMgAl = 210190 - 245.40 T J/mole, ΔWBtTi = WBtFeTi - WBtMgTi = 310990 - 370.397 T/mole. Based on this model, the exchange reaction ΔH is 41952 J/mol and ΔS is 10.35 J/(K mol). Estimated uncertainty for this geothermometer is 25 °C. This geothermometer was tested on two data sets. The first consisted of 98 specimens containing garnet and biotite from west-central Maine, which formed under reducing fO₂, with graphite, a limited range of P (∼3 to 4.5 kbar). and a moderate range in T (∼550-650 °C). and which were all analyzed on a single microprobe using the same standards. Results indicate that the Maine staurolite zone averages 574 °C compared with 530 °C previously calculated and that the muscovite-breakdown T is consistent with experimental data. The second set consisted of cordierite-garnet granulites without hyperstliene from Ontario. Results here suggest an average T of 662 °C. compared with significantly lower or higher Ts calculated from other geothermometers. This model reproduces the Perchuk and Lavrent‘eva (1983) experimental Ts with a standard deviation of 12 °C and discriminates the assemblages in the Maine data set better than other models.

A statistical model of thermodynamic mixing properties of Ca-Mg-Fe2+ garnets

Abstract At present the most widely used models of thermodynamic mixing properties of garnet are those that are optimized to fit the experimental phase-equilibrium data along various binary garnet joins using the method of mathematical programming. We conducted a thorough weighted regression analysis of the available volumetric, calorimetric, and phase- equilibrium data. This resulted in a much improved set of a-X relationships with the following Margules parameters (in joules, kelvins, bars). WGFeMg = -24166 + 22.09T - 0.034P, WGMgFe = 22265 - 12.40T + 0.050P, WGFeCa= 17526 - 14.51T + 0.135P, WGCaFe= -18113 + 15.51T + 0.040P, WGMgCa= 14306 - 2.49T + 0.140P, WGCaMg= 65182 - 20.82T+ 0.068P (all pfu containing 12 O atoms). The robust regression analyses allowed us to obtain the uncertainties associated with each of the parameters, quantities not available by mathematical programming analysis of phase-equilibrium data. Uncertainty estimates are essential for rigorous quantification of errors associated with mineralogic thermobarometers. From the probabilistic model developed here, we present detailed formulations for ascertaining errors of activity coefficients of the end-member components in any garnet solid solution. This provides an opportunity to evaluate the uncertainties in mineralogic thermobarometers involving garnets of compositions dissimilar to those used in the experiments on which the garnet mixing models are based. The results of this study elicit several significant points. The analysis strongly suggests that there is an excess entropy of mixing along the Fe-Mg join because Fe-Mg mixing in garnet is substantially nonideal. The excess entropy of mixing along the Ca-Mg and Ca-Fe joins is asymmetric and therefore assumption of a large symmetric entropy parameter or no entropy along these joins is an oversimplification. Large uncertainty in the mixing properties exists for Ca-rich garnets. There should be a Ca-Mg-Fe ternary interaction parameter (7110 J/mol).

Cordierite-garnet-sillimanite-quartz equilibrium: I. New experimental calibration in the system FeO−Al2O3−SiO2−H2O and certain P-T-XH2O relations

The equilibrium in which hydrous Fe-cordierite breaks down to almandine, sillimanite, quartz, and water was previously experimentally determined by Richardson (1968) and Holdaway and Lee (1977) using QMF buffer and by Weisbrod (1973) using QIF buffer. All these studies yielded similar results — a negative dP/dT slope for the equilibrium curve. However, based on theoretical arguments, Martignole and Sisi (1981), and based on Fe-Mg partitioning experiments on coexisting cordierite and garnet in equilibrium with sillimanite and quartz, Aranovich and Podlesskii (1983) suggested that this equilibrium curve has a positive dP/dT slope and its position depends on the water content of the equilibrium cordierite. We have redetermined this equilibrium using a much improved tecnique of detecting reaction direction, and cordierite starting material that contained virtually no hercynite. Hercynite was present as a contaminant in the cordierites of previous experimental studies and possibly reacted with quartz during the experimental runs to expand the apparent stability field of Fe-cordierite. We synthesized Fe-cordierite from reagent grade oxides at 710°C and 2 kbar (using QMF buffer) with two intermediate stages of grinding and mixing. The cordierite has a unit cell volume of 1574.60 Å3 (molar volume=23.706 J/bar) and no Fe3+ as indicated by X-ray diffraction and room temperature Mössbauer studies respectively. Reaction direction was concluded by noting≥20% change of the ratios of intensities of two key X-ray diffraction peaks of cordierite and almandine. Our results show that the four-phase equilibrium curve passes through the points 2.1 kbar, 650°C and 2.5 kbar, 750°C. This disagrees with all previous experimental studies. H2O in the Fe-cordierite, equilibrated at 2.2 kbar and 700°C and determined by H-extraction line in the stable isotope laboratory, is 1.13 wt% (n=0.41 moles). H2O content of pure Mg-cordierite equilibrated under identical conditions and determined by thermogravimentric conditions and determined by thermogravimetric analysis is 1.22 wt% (n=0.40). Similar determinations on Fe-cordierite and Mg-cordierite equilibrated at 2.0 kbar and 650°C show 1.27 wt% (n=0.46) and 1.47 wt% (n=0.48) of H2O respectively. Thus, H2O content appears to be independent of Fe/Mg ratio in cordierite, a conclusion which supports previous experimental determinations. The experimentally determined equilibrium curve represents conditions of PH2O=Ptotal. From this we calculated the anhydrous curve representing equilibrium under conditions of XH2OV=0.0. A family of calculated equilibrium curves of constant nH2OCord cut the experimentally determined curve at a very small angle indicating a slight variation in nH2OCord in cordierite in equilibrium with almandine, sillimanite, and quartz under the conditions of constant XH2OV. Ancther set of calculated equilibrium curves, each representing constant aH2OV demonstrate that the slopes of the curves vary with XH2OV, and are all positive in the full range of 0.0≤XH2OV≤1.0.

A discussion of Margules-type formulations for multicomponent solutions with a generalized approach

Abstract The generalized mathematical method for obtaining Margules-type formulations for the excess free energy of solution, Gxs or any other excess molar thermochemical property, and RT ln (γi), or the corresponding partial molar quantity in multicomponent solutions, is presented in this paper. For the ncomponent system, the Gxs function is approximated by a pth-order Taylor series involving (n − 1) independent compositional variables. The expression for Gxs is differentiated with respect to each compositional variable, and the resultant partial derivatives are evaluated at each of the compositional extremes to obtain the Margules parameters (activity coefficients at infinite dilution) replacing the constants in the Taylor series which have no thermodynamic meaning. When p = 2, the solutions are symmetric (strictly regular) and no ternary and higher order interaction parameters exist in ternary and higher order systems. For p = 3, the solutions are asymmetric (subregular) and no quaternary and higher order parameters exist in quarternary and higher order systems. As a special case of the latter system (p = 3) the constituent binaries can exhibit symmetric behavior, but the expressions for Gxsand RT ln (γi) can contain ternary interaction parameters. Thus, the existence of nonbinary interaction parameters is a result of the complexity of the Taylor series approximation to the Gxs function. In general, ternary interaction parameters cannot be completely defined by the binary interaction parameters. Simple expressions for the excess function and the activity coefficients in symmetric and asymmetric multicomponent solutions have been derived. Previously published binary, ternary, and quarternary symmetric and asymmetric solution models are discussed and compared to the solution models derived herein.

COMPARISON OF STATISTICAL METHODS FOR ESTIMATION OF NUTRIENT LOAD TO SURFACE RESERVOIRS FOR SPARSE DATA SET: APPLICATION WITH A MODIFIED MODEL FOR PHOSPHORUS AVAILABILITY

Abstract Nutrient budget models for lakes and reservoirs critically respond to the input pollutant loading, yet little consensus exists on how to estimate the load, particularly for the common but challenging case of sparse nutrient concentration measurements and abundant input flow data. A statistical load calculation using cluster (in this case, annual) mean concentration and stratified (monthly) flow was compared to estimates by sample mean and ratio estimator methods for phosphorus loading to Whitney Reservoir in North Central Texas. The results varied considerably for the various estimator methods during the six-year study period with the cluster and stratified mean approach estimating extreme high loading periods not captured by the other methods. The variable loading patterns were then tested in phosphorus budget model simulations for Whitney Reservoir that considered vertical stratification of the water column, water–sediment phosphorus interaction, and seasonal variations in water quality. For independently determined settling, interlayer dispersion, recycling rates, and sediment burial rates estimated for the respective loading calculation, the cluster and stratified mean loading pattern provided a better statistical fit of phosphorus concentration measurements in the epilimnion than when ratio estimator load calculations were used. The two loading functions described hypolimnion concentration data equally well. The lesson of this exercise is that various methods of load estimation should be examined in order to develop as reliable a management model as possible when only a sparse data set is available for calibration.

Acid–base chemistry of albite surfaces in aqueous solutions at standard temperature and pressure

Abstract Surface acidity of albite has been determined by potentiometric titration of water-washed and unwashed powders at 23.5°C and pH range of 2–9.5. Using NH 4 Cl as the background electrolyte the surface was titrated with HCl and NH 4 OH in both forward and backward directions. These titrants have the advantage of possessing the same acid- and base-radicals as those of the background electrolyte salt. Acidimetric forward titration of unwashed albite, which is also independent of dissociation of NH 4 + , shows that titration curves of various ionic strengths intersect at a pH near 3.93 (point of zero salt effect or pH PZSE ). The back titrations did not yield a unique pH PZSE . The pH values of point of zero charge (pH ZPC ) calculated from alkalimetric back titration experiments range from 6.75 to 8.14 depending on the prewashed or unwashed nature of the albite and the ionic strength of the solution. During acidimetric back titration, the pH ZPC values vary from 4.15 to 7.14 depending upon the same factors. Published reports on feldspar surface chemistry have relied upon back titration experiments, but these experiments yielded doubtful and discrepant values of pH where the surface charge solely due to protonation is zero (point of zero net proton charge or pH PZNPC ). These resulted due to a combination of factors that included differential dissolution of the mineral, precipitation of Al(OH) 3 , adsorption of ions from the electrolyte salt, and the nature of prewashing. The two contrasting pretreatments produce a Si-rich feldspar surface at acidic pH and an Al-rich feldspar surface at alkaline pH that partly control the difference in results obtained from the back titrations that proceed from the two opposite ends of the pH scale. Therefore, only acidimetric forward titrations with unwashed albite were used in this study, since these yield the most meaningful information on the acid–base chemistry of feldspar surface. The surface charge ( σ S ) arising simply due to protonation is given by molar concentrations of surface-adsorbed hydrogen ions, [H Ad + ] since σ S = F ×[H Ad + ]. However, σ S obtained from calculated values of surface-reacted total proton charge ( σ SR ) is significantly affected by charged species of Al, Si, Ca, and CO 2 that are present in the solutions and Na + ⇔H + exchange that occurs on feldspar surface in solution. For all compositions, in the pH range studied, the net charge of the aqueous species that must be algebraically added to σ SR to obtain σ S , is negative. Thus, the real values of [H Ad + ] must be lower than the apparent values of measured concentrations of surface-reacted H + or [H S + ]. The magnitude of this correction is dependent upon the pH at which the solution compositions are determined. When the net charges of the dissolved species are estimated in the forward acid titration with unwashed albite, the corrected surface charge versus pH curves of various ionic strengths show pH PZNPC =4.38–4.72. The correction procedures assume bulk electrical neutrality of the solution and quasi-equilibrium conditions within the time frame of the experiments and hence disregard the kinetics of dissolution. Since the estimates of the charge in this case are based on solutions collected at alkaline pH, the corrections for charged aqueous species give slight overestimation of surface charge. Hence, for the unwashed albite, pH PZNPC =pH PZSE =3.93±0.05. Below this pH, the amount of surface adsorbed [H + ] is independent of pH. Consistent with this observation is a theoretically calculated adsorption isotherm with an assumed value of pH PZNPC =3.93. The theoretically constructed curve further supports that the amount of [H Ad + ] is exceedingly small below pH PZNPC .