A new activity model for Mg–Al biotites was formulated through an integrated approach combining various experimental results (calorimetry, line-broadening in infrared (IR) spectra, analysis of existing phase-equilibrium data) with density functional theory (DFT) calculations. The resulting model has a sound physical-experimental basis. It considers the three end-members phlogopite (Phl, KMg3[(OH)2AlSi3O10]), ordered eastonite (Eas, KMg2Al[(OH)2Al2Si2O10]), and disordered eastonite (dEas) and, thus, includes Mg–Al order–disorder. The DFT-derived disordering enthalpy, ΔHdis, associated with the disordering of Mg and Al on the M sites of Eas amounts to 34.5 ± 3 kJ/mol. Various biotite compositions along the Phl–Eas join were synthesised hydrothermally at 700 °C and 4 kbar. The most Al-rich biotite synthesized had the composition XEas = 0.77. The samples were characterised by X-ray diffraction (XRD), microprobe analysis and IR spectroscopy. The samples were studied further using relaxation calorimetry to measure their heat capacities (Cp) at temperatures from 2 to 300 K and by differential scanning calorimetry between 282 and 760 K. The calorimetric (vibrational) entropy of Phl at 298.15 K, determined from the low-T Cp measurements on a pure synthetic sample, is Scal = 319.4 ± 2.2 J/(mol K). The standard entropy, So, for Phl is 330.9 ± 2.2 J/(mol K), which is obtained by adding a configurational entropy term, Scfg, of 11.53 J/(mol K) due to tetrahedral Al-Si disorder. This value is ~1% larger than those in different data bases, which rely on older calorimetrical data measured on a natural near-Phl mica. Re-analysing phase-equilibrium data on Phl + quartz (Qz) stability with this new So, gives a standard enthalpy of formation of Phl, ΔHfo,Phl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta H^{\text{o}}_{\text{f}} ,_{\text{Phl}}$$\end{document} = − 6209.83 ± 1.10 kJ/mol, which is 7–8 kJ/mol less negative than published values. The superambient Cp of Phl is given by the polynomial [J/(mol K)] as follows: Cp=667.37±7-3914.50±258·T-0.5-1.52396±0.15×107·T-2+2.17269±0.25×109·T-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\text{p}} = 667.37\left( { \pm 7} \right) - 3914.50\left( { \pm 258} \right) \cdot T^{ - 0.5} - 1.52396\left( { \pm 0.15} \right) \times 10^{7} \cdot T^{ - 2} + \, 2.17269\left( { \pm 0.25} \right) \times 10^{9} \cdot T^{ - 3}$$\end{document}. Calorimetric entropies at 298.15 K vary linearly with composition along the Phl–Eas join, indicating ideal vibrational entropies of mixing in this binary. The linear extrapolation of these results to Eas composition gives So = 294.5 ± 3.0 J/(mol K) for this end-member. This value is in excellent agreement with its DFT-derived So, but ~ 8% smaller than values as appearing in thermodynamic data bases. The DFT-computed superambient Cp of Eas is given by the polynomial [in J/(mol K)] as follows: Cp=656.91±14-3622.01±503·T-0.5-1.70983±0.33×107·T-2+2.31802±0.59×109·T-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\text{p}} = 656.91\left( { \pm 14} \right) - 3622.01\left( { \pm 503} \right) \cdot T^{ - 0.5} - 1.70983\left( { \pm 0.33} \right) \times 10^{7} \cdot T^{ - 2} + \, 2.31802\left( { \pm 0.59} \right) \times 10^{9} \cdot T^{ - 3}$$\end{document}. A maximum excess enthalpy of mixing, ΔHex, of ~6 kJ/mol was derived for the Phl–Eas binary using line-broadening from IR spectra (wavenumber region 400–600 cm−1), which is in accordance with ΔHex determined from published solution-calorimetry data. The mixing behaviour can be described by a symmetric interaction parameter WPhl,EasH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{\text{H}}_{{{\text{Phl}},{\text{Eas}}}}$$\end{document} = 25.4 kJ/mol. Applying this value to published phase-equilibrium data that were undertaken to experimentally determine the Al-saturation level of biotite in the assemblage (Mg–Al)-biotite-sillimanite-sanidine-Qz, gives a ΔHf,Easo\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta H^{\text{o}}_{{{\text{f}},{\text{Eas}}}}$$\end{document} = − 6358.5 ± 1.4 kJ/mol in good agreement with the independently DFT-derived value of ΔHf,EasoDFT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta H^{\rm o}_{\rm f,Eas}}^{\rm DFT}$$\end{document} = − 6360.5 kJ/mol. Application examples demonstrate the effect of the new activity model and thermodynamic standard state data, among others, on the stability of Mg–Al biotite + Qz.

中文翻译：

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通过综合方法确定的 Mg-Al 黑云母的新活性模型

通过将各种实验结果（量热法、红外（IR）光谱的谱线展宽、现有相平衡数据分析）与密度泛函理论（DFT）计算相结合的综合方法，制定了一种新的 Mg-Al 黑云母活性模型。由此产生的模型具有良好的物理实验基础。它考虑了三个端元金云母 (Phl, KMg3[(OH)2AlSi3O10])、有序东长石 (Eas, KMg2Al[(OH)2Al2Si2O10]) 和无序东长石 (dEas)，因此，包括 Mg-Al 有序 -紊乱。DFT 衍生的无序焓 ΔHdis 与 Eas 的 M 位点上的 Mg 和 Al 无序相关，达到 34.5 ± 3 kJ/mol。在 700 °C 和 4 kbar 下，沿 Phl-Eas 连接处的各种黑云母组合物是通过水热合成的。合成的最富含铝的黑云母的成分 XEas = 0.77。通过X射线衍射（XRD）、微探针分析和红外光谱对样品进行表征。使用弛豫量热法进一步研究样品，以测量其在 2 至 300 K 的温度下的热容量 (Cp)，并通过差示扫描量热法在 282 至 760 K 之间进行研究。 Phl 在 298.15 K 的量热（振动）熵，由低-T Cp 对纯合成样品的测量值为 Scal = 319.4 ± 2.2 J/(mol K)。Phl 的标准熵 So 为 330.9 ± 2.2 J/(mol K)，这是通过添加由于四面体 Al-Si 无序导致的 11.53 J/(mol K) 的构型熵项 Scfg 获得的。该值比不同数据库中的值大约 1%，这些数据库依赖于在天然近 Phl 云母上测量的较旧的量热数据。用这个新的 So 重新分析 Phl + 石英 (Qz) 稳定性的相平衡数据，给出了 Phl 的标准形成焓，ΔHfo,Phl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta H^{\text{ o}}_{\text{f}} ,_{\text{Phl}}$$\end{document} = − 6209.83 ± 1.10 kJ/mol，比公布的值低 7–8 kJ/mol。Phl的超环境Cp由多项式[J/(mol K)]给出如下：Cp=667.37±7-3914.50±258·T-0.5-1.52396±0.15×107·T-2+2.17269±0。25×109·T-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\text{p}} = 667.37\left( { \pm 7} \right) - 3914.50\left( { \pm 258} \right ) \cdot T^{ - 0.5} - 1.52396\left( { \pm 0.15} \right) \times 10^{7} \cdot T^{ - 2} + \, 2.17269\left( { \pm 0.25} \右) \times 10^{9} \cdot T^{ - 3}$$\end{document}。298.15 K 处的量热熵随 Phl-Eas 连接处的成分线性变化，表明该二元混合的理想振动熵。这些结果对 Eas 组成的线性外推给出了该端元的 So = 294.5 ± 3.0 J/(mol K)。该值与其 DFT 导出的 So 非常一致，但比热力学数据库中出现的值小约 8%。Eas 的 DFT 计算超环境 Cp 由多项式 [in J/(mol K)] 给出如下： Cp=656.91±14-3622.01±503·T-0.5-1.70983±0.33×107·T-2+2.31802 ±0.59×109·T-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek } \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\text{p}} = 656.91\left( { \pm 14} \right) - 3622.01\left( { \pm 503} \右) \cdot T^{ - 0.5} - 1.70983\left( { \pm 0.33} \right) \times 10^{7} \cdot T^{ - 2} + \, 2.31802\left( { \pm 0.59} \right) \times 10^{9} \cdot T^{ - 3}$$\end{document}。最大过量混合焓，ΔHex，使用 IR 光谱（波数区域 400-600 cm-1）的谱线展宽，Phl-Eas 二元的约 6 kJ/mol 得到了约 6 kJ/mol，这与根据已发表的溶液量热法数据确定的 ΔHex 一致。混合行为可以通过对称交互参数 WPhl,EasH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{ 来描述mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{\text{H}}_{{{\text{Phl}},{\text{Eas} }}}$$\end{document} = 25.4 kJ/mol。将此值应用于已发表的相平衡数据，这些数据是为了通过实验确定 (Mg-Al)-黑云母-硅线石-sanidine-Qz 组合中黑云母的铝饱和度水平，得出 ΔHf，Easo\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin} {-69pt} \begin{document}$$\Delta H^{\text{o}}_{{{\text{f}},{\text{Eas}}}}$$\end{document} = − 6358.5 ± 1.4 kJ/mol 与独立的 DFT 衍生值 ΔHf,EasoDFT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \ usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta H^{\rm o}_{\rm f,Eas}} ^{\rm DFT}$$\end{document} = − 6360.5 kJ/mol。应用示例展示了新活动模型和热力学标准状态数据等的效果，