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Margules.html
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Margules.html
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<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: "Lucida Console",Consolas,"Courier",monospace; --title-bar-height: 20px; }
.mac-os-11 { --title-bar-height: 28px; }
html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }
body { margin: 0px; padding: 0px; height: auto; bottom: 0px; top: 0px; left: 0px; right: 0px; font-size: 1rem; line-height: 1.42857143; overflow-x: hidden; background-image: inherit; background-size: inherit; background-attachment: inherit; background-origin: inherit; background-clip: inherit; background-color: inherit; tab-size: 4; background-position: inherit inherit; background-repeat: inherit inherit; }
iframe { margin: auto; }
a.url { word-break: break-all; }
a:active, a:hover { outline: 0px; }
.in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); }
#write { margin: 0px auto; height: auto; width: inherit; word-break: normal; word-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; }
#write.first-line-indent p { text-indent: 2em; }
#write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; }
#write.first-line-indent li { margin-left: 2em; }
.for-image #write { padding-left: 8px; padding-right: 8px; }
body.typora-export { padding-left: 30px; padding-right: 30px; }
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#write { padding-left: 20px; padding-right: 20px; }
}
#write li > figure:last-child { margin-bottom: 0.5rem; }
#write ol, #write ul { position: relative; }
img { max-width: 100%; vertical-align: middle; }
button, input, select, textarea { color: inherit; font-family: inherit; font-size: inherit; font-style: inherit; font-variant-caps: inherit; font-weight: inherit; font-stretch: inherit; line-height: inherit; }
input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; }
*, ::after, ::before { box-sizing: border-box; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; }
p { line-height: inherit; }
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h2 { font-size: 1.8rem; }
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sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.701961); color: rgb(85, 85, 85); border-top-left-radius: 4px; border-top-right-radius: 4px; border-bottom-right-radius: 4px; border-bottom-left-radius: 4px; cursor: pointer; }
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#write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; }
figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; }
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thead, tr { break-inside: avoid; break-after: auto; }
thead { display: table-header-group; }
table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; }
table.md-table td { min-width: 32px; }
.CodeMirror-gutters { border-right-width: 0px; background-color: inherit; }
.CodeMirror-linenumber { }
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.CodeMirror pre { padding: 0px 4px; }
.CodeMirror-lines { padding: 0px; }
div.hr:focus { cursor: none; }
#write pre { white-space: pre-wrap; }
#write.fences-no-line-wrapping pre { white-space: pre; }
#write pre.ty-contain-cm { white-space: normal; }
.CodeMirror-gutters { margin-right: 4px; }
.md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; background-image: inherit; background-size: inherit; background-attachment: inherit; background-origin: inherit; background-clip: inherit; background-color: inherit; position: relative !important; background-position: inherit inherit; background-repeat: inherit inherit; }
.md-fences-adv-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; }
#write .md-fences.mock-cm { white-space: pre-wrap; }
.md-fences.md-fences-with-lineno { padding-left: 0px; }
#write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; }
.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; }
.CodeMirror-line, twitterwidget { break-inside: avoid; }
svg { break-inside: avoid; }
.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; }
.footnotes + .footnotes { margin-top: 0px; }
.md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; background-position: 0px 0px; background-repeat: initial initial; }
li div { padding-top: 0px; }
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li .mathjax-block, li p { margin: 0.5rem 0px; }
li blockquote { margin: 1rem 0px; }
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.footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; }
#write .footnote-line { white-space: pre-wrap; }
@media print {
body, html { border: 1px solid transparent; height: 99%; break-after: avoid-page; break-before: avoid-page; font-variant-ligatures: no-common-ligatures; }
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pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background-color: rgb(204, 204, 204); display: block; overflow-x: hidden; background-position: initial initial; background-repeat: initial initial; }
p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; }
#write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; }
p > .md-image:only-child { display: inline-block; width: 100%; }
#write .MathJax_Display { margin: 0.8em 0px 0px; }
.md-math-block { width: 100%; }
.md-math-block:not(:empty)::after { display: none; }
.MathJax_ref { fill: currentcolor; }
[contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; }
.md-task-list-item { position: relative; list-style-type: none; }
.task-list-item.md-task-list-item { padding-left: 0px; }
.md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; }
.math { font-size: 1rem; }
.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-top-left-radius: 10px; border-top-right-radius: 10px; border-bottom-right-radius: 10px; border-bottom-left-radius: 10px; }
.md-toc-content { position: relative; margin-left: 0px; }
.md-toc-content::after, .md-toc::after { display: none; }
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.md-toc-inner:hover { text-decoration: underline; }
.md-toc-inner { display: inline-block; cursor: pointer; }
.md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; }
.md-toc-h2 .md-toc-inner { margin-left: 2em; }
.md-toc-h3 .md-toc-inner { margin-left: 4em; }
.md-toc-h4 .md-toc-inner { margin-left: 6em; }
.md-toc-h5 .md-toc-inner { margin-left: 8em; }
.md-toc-h6 .md-toc-inner { margin-left: 10em; }
@media screen and (max-width: 48em) {
.md-toc-h3 .md-toc-inner { margin-left: 3.5em; }
.md-toc-h4 .md-toc-inner { margin-left: 5em; }
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.md-toc-h6 .md-toc-inner { margin-left: 8em; }
}
a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; }
.footnote-line a:not(.reversefootnote) { color: inherit; }
.md-attr { display: none; }
.md-fn-count::after { content: "."; }
code, pre, samp, tt { font-family: var(--monospace); }
kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background-color: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; background-position: initial initial; background-repeat: initial initial; }
.md-comment { color: rgb(162, 127, 3); opacity: 0.6; font-family: var(--monospace); }
code { text-align: left; }
a.md-print-anchor { white-space: pre !important; border: none !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; text-shadow: initial !important; background-position: 0px 0px !important; background-repeat: initial initial !important; }
.os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; }
.md-diagram-panel > svg { max-width: 100%; }
[lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; }
[lang="mermaid"] .node text { font-size: 1rem; }
table tr th { border-bottom-width: 0px; }
video { max-width: 100%; display: block; margin: 0px auto; }
iframe { max-width: 100%; width: 100%; border: none; }
.highlight td, .highlight tr { border: 0px; }
mark { background-color: rgb(255, 255, 0); color: rgb(0, 0, 0); background-position: initial initial; background-repeat: initial initial; }
.md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; }
.md-expand mark .md-meta { opacity: 0.3 !important; }
mark .md-meta { color: rgb(0, 0, 0); }
@media print {
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}
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.md-fences.md-fences-math { font-size: 1em; }
.md-fences-advanced:not(.md-focus) { padding: 0px; white-space: nowrap; border: 0px; }
.md-fences-advanced:not(.md-focus) { background-image: inherit; background-size: inherit; background-attachment: inherit; background-origin: inherit; background-clip: inherit; background-color: inherit; background-position: inherit inherit; background-repeat: inherit inherit; }
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:root {
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--control-text-color: #777;
}
@include-when-export url(https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext);
/* open-sans-regular - latin-ext_latin */
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</style><title>Margules</title>
</head>
<body class='typora-export'><div class='typora-export-content'>
<div id='write' class=''><h3 id='a-brief-introduction-to-ideal-and-regular-solutions'><span>A brief introduction to ideal and regular solutions</span></h3><p><span>If one wants not to get lost with solid solutions is better to start from definitions (I am copying here the notes from the course of James Connolly). The most fundamental equation is the gibbs energy of a (non-ideal) mixture (in units of J/mol, here is mol of the solid solution as given in the arbitrarily defined structural formulae,)</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n24" cid="n24" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="1"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 46.863ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.628ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.748ex; min-width: 46.863ex;"><defs><path id="MJX-48-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 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transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1)</mtext></mtd><mtd><mi>g</mi><mo>=</mo><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>m</mi><mi>e</mi><mi>c</mi><mi>h</mi><mi>a</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi></mrow></msup><mo>+</mo><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>c</mi><mi>o</mi><mi>n</mi><mi>f</mi><mi>i</mi><mi>g</mi><mi>u</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>a</mi><mi>l</mi></mrow></msup><mo>+</mo><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>e</mi><mi>x</mi><mi>c</mi><mi>e</mi><mi>s</mi><mi>s</mi></mrow></msup></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>The real complication here (and source of confusion) is the configuration energy. It is confusing because </span><em><span>ideal</span></em><span> mixing is far from being obvious as it is not known </span><em><span>a priori</span></em><span> if you are mixing in crystallographic sites or you are mixing molecules (where the original definition of activity and classic thermodynamics is coming). 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transform="translate(4985.5,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-58-TEX-LO-2211"></use></g><g data-mml-node="TeXAtom" transform="translate(148.2,-1087.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-58-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="3D" xlink:href="#MJX-58-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-58-TEX-N-31"></use></g></g><g data-mml-node="mi" transform="translate(509.9,1150) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-58-TEX-I-1D45B"></use></g></g><g data-mml-node="munderover" transform="translate(6596.2,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-58-TEX-LO-2211"></use></g><g data-mml-node="TeXAtom" transform="translate(124.5,-1087.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-58-TEX-I-1D457"></use></g><g 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data-mml-node="TeXAtom" transform="translate(498,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-58-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D457" xlink:href="#MJX-58-TEX-I-1D457"></use></g></g></g><g data-mml-node="mi" transform="translate(10229.8,0)"><use data-c="6C" xlink:href="#MJX-58-TEX-N-6C"></use><use data-c="6E" xlink:href="#MJX-58-TEX-N-6E" transform="translate(278,0)"></use></g><g data-mml-node="mo" transform="translate(11063.8,0)"><use data-c="2061" xlink:href="#MJX-58-TEX-N-2061"></use></g><g data-mml-node="msub" transform="translate(11230.4,0)"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-58-TEX-I-1D467"></use></g><g data-mml-node="TeXAtom" transform="translate(498,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-58-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D457" xlink:href="#MJX-58-TEX-I-1D457"></use></g></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1708.6 1 2917.3"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:2" transform="translate(0,873.5)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-58-TEX-N-28"></use><use data-c="32" xlink:href="#MJX-58-TEX-N-32" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-58-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(2)</mtext></mtd><mtd><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>c</mi><mi>o</mi><mi>n</mi><mi>f</mi></mrow></msup><mo>=</mo><mi>R</mi><mi>T</mi><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>m</mi><mi>i</mi></msub></mrow></munderover><msub><mi>q</mi><mi>i</mi></msub><msub><mi>z</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mi>ln</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>z</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>where </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.749ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 773 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-25-TEX-I-1D45E" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-25-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45E" xlink:href="#MJX-25-TEX-I-1D45E"></use></g><g data-mml-node="mi" transform="translate(479,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-25-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">q_i</script><span> is the number of sites of the i-th type (also called mutiplicity), </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.726ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 1205 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-26-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-26-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-26-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(911,-150) scale(0.707)"><use 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596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-29-TEX-I-1D467"></use></g><g data-mml-node="TeXAtom" transform="translate(498,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-29-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D457" xlink:href="#MJX-29-TEX-I-1D457"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">z_{ij}</script><span> is the atomic fraction of the j-th type of atom mixing on the i-th identisite. </span><strong><span>The problem here is that the definition is not given in terms of mole fraction of the endmember </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.848ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 817 647" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-28-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-28-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-28-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-28-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">y_i</script><span> but atomic site fractions </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.451ex" height="1.666ex" role="img" focusable="false" viewBox="0 -442 1083.3 736.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.666ex;"><defs><path id="MJX-29-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path id="MJX-29-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-29-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-29-TEX-I-1D467"></use></g><g data-mml-node="TeXAtom" transform="translate(498,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-29-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D457" xlink:href="#MJX-29-TEX-I-1D457"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">z_{ij}</script><span>. </span></strong></p><p><span>For olivine, if you assumed that there is only one type of octahedral site (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.324ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 4563.1 765" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-30-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-30-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-30-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-30-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-30-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(622.8,0)"><use data-c="3D" xlink:href="#MJX-30-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1678.6,0)"><use data-c="31" xlink:href="#MJX-30-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(2456.3,0)"><use data-c="3D" xlink:href="#MJX-30-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(3512.1,0)"><use data-c="1D440" xlink:href="#MJX-30-TEX-I-1D440"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1</mn><mo>=</mo><mi>M</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">i=1=M</script><span>) (crystallographers will argue that this can not be the case...), you have only one site with a multiplicity of 2 (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.027ex" height="1.946ex" role="img" focusable="false" viewBox="0 -666 3105.7 860" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-31-TEX-I-1D45E" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-31-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path id="MJX-31-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-31-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45E" xlink:href="#MJX-31-TEX-I-1D45E"></use></g><g data-mml-node="mi" transform="translate(479,-150) scale(0.707)"><use data-c="1D440" xlink:href="#MJX-31-TEX-I-1D440"></use></g></g><g data-mml-node="mo" transform="translate(1549.9,0)"><use data-c="3D" xlink:href="#MJX-31-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2605.7,0)"><use data-c="32" xlink:href="#MJX-31-TEX-N-32"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>M</mi></msub><mo>=</mo><mn>2</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">q_M =2</script><span>), </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.004ex" height="1.846ex" role="img" focusable="false" viewBox="0 -666 3537.7 816" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-32-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-32-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path id="MJX-32-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-32-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-32-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(911,-150) scale(0.707)"><use data-c="1D440" xlink:href="#MJX-32-TEX-I-1D440"></use></g></g><g data-mml-node="mo" transform="translate(1981.9,0)"><use data-c="3D" xlink:href="#MJX-32-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3037.7,0)"><use data-c="32" xlink:href="#MJX-32-TEX-N-32"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mi>M</mi></msub><mo>=</mo><mn>2</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">m_M =2</script><span> (two kind of atoms mixing in </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.378ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1051 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-33-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D440" xlink:href="#MJX-33-TEX-I-1D440"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">M</script><span>) and it is straightforward to relate mole fraction of the endmemeber and atomic site fractions. So </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="11.349ex" height="1.986ex" role="img" focusable="false" viewBox="0 -583 5016.4 878" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.667ex;"><defs><path id="MJX-34-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path id="MJX-34-TEX-I-1D440" d="M289 629Q289 635 232 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transform="translate(2846,0)"><use data-c="3D" xlink:href="#MJX-35-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(3901.8,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-35-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-35-TEX-I-1D453"></use></g><g data-mml-node="mi" transform="translate(550,0)"><use data-c="1D45C" xlink:href="#MJX-35-TEX-I-1D45C"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mrow data-mjx-texclass="ORD"><mi>M</mi><mo>,</mo><mi>M</mi><mi>g</mi></mrow></msub><mo>=</mo><msub><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>f</mi><mi>o</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">z_{M,Mg} = y_{fo}</script><span>, that leaves the traditional expression.</span></p><div 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transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(8)</mtext></mtd><mtd><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>f</mi><mi>o</mi></mrow></msub><mo>=</mo><msubsup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>f</mi><mi>o</mi></mrow><mn>2</mn></msubsup><mi>e</mi><mi>x</mi><mi>p</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mrow><mi>W</mi><msubsup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>f</mi><mi>o</mi></mrow><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><mi>W</mi><msub><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>f</mi><mi>o</mi></mrow></msub><mo>+</mo><mi>W</mi></mrow><mrow><mi>R</mi><mi>T</mi></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>That can be 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displaystyle="true"><mlabeledtr><mtd><mtext>(9)</mtext></mtd><mtd><mi>R</mi><mi>T</mi><mi>ln</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>f</mi><mi>o</mi></mrow></msub><mo>=</mo><mi>𝑅</mi><mi>𝑇</mi><mi>ln</mi><mo data-mjx-texclass="NONE"></mo><msubsup><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>f</mi><mi>o</mi></mrow><mn>2</mn></msubsup><mo>+</mo><mi>W</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>y</mi><mrow data-mjx-texclass="ORD"><mi>f</mi><mi>o</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>Which is the same as in the classical papers (please note that sometimes people use W for the exchange per atom, but in the above definition it is taken per mole of atoms exchanged in the structural formula, therefore 2 in olivine)</span></p><p> </p></div></div>
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