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<title>Computing_gibbs_Granada</title>
</head>
<body class='typora-export'><div class='typora-export-content'>
<div id='write' class=''><h1 id='the-apparent-gibbs-energy-equation-from-the-elements'><span>The (apparent) Gibbs energy equation from the elements</span></h1><p><span>These notes are compiled mostly following the notes from the James D. Connolly course at the ETH (</span><a href='http://www.perplex.ethz.ch/thermo_course' target='_blank' class='url'>www.perplex.ethz.ch/thermo_course</a><span>), the Perple_X website (</span><a href='https://www.perplex.ethz.ch' target='_blank' class='url'>https://www.perplex.ethz.ch</a><span>) and Holland and Powell (1998, Journal of Metamorphic Geology). Any error or misunderstanding is mine. Rather than a comprehensive derivation of thermodynamic quantities, these notes are intended as a cheat sheet, that could be useful for your project this week. </span></p><p><span>There is only </span><strong><span>one</span></strong><span> number (scalar) that rules all phase diagrams (assuming your variables are strictly pressure and temperature only), and I am still perplexed about the beauty resulting from it (</span><em><span>nothing but a scalar</span></em><span>). Although in detail the computation is rather sophisticated, the only important thing that feeds the algorithm is the Gibbs energy of the potential phases at the specific </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.593ex" height="1.532ex" role="img" focusable="false" viewBox="0 -677 704 677" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-92-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></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="1D447" xlink:href="#MJX-92-TEX-I-1D447"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">T</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.699ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 751 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-93-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></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="1D443" xlink:href="#MJX-93-TEX-I-1D443"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">P</script><span> that can be stable given the compositional space of your system. The trick is then to find those phases (and their proportion) having [1] the lowest Gibbs energy and [2] satisfying a mass balance conservation equation </span><strong><span>constrained</span></strong><span> by the composition of the system. So the bricks of all this construction are the Gibbs energies of the phases that can be computed by integrating the differential equation, </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n4" cid="n4" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="1" data-math-labels="[]"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 28.744ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex; min-width: 28.744ex;"><defs><path id="MJX-1-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 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xlink:href="#MJX-2-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(15871.1,0)"><use data-c="1D451" xlink:href="#MJX-2-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(16391.1,0)"><use data-c="1D447" xlink:href="#MJX-2-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(17317.4,0)"><use data-c="2B" xlink:href="#MJX-2-TEX-N-2B"></use></g><g data-mml-node="msubsup" transform="translate(18317.6,0)"><g data-mml-node="mo" transform="translate(0 1)"><use data-c="222B" xlink:href="#MJX-2-TEX-LO-222B"></use></g><g data-mml-node="mi" transform="translate(1046.4,1088.1) scale(0.707)"><use data-c="1D443" xlink:href="#MJX-2-TEX-I-1D443"></use></g><g data-mml-node="TeXAtom" transform="translate(589,-896.4) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-2-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(675,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-2-TEX-I-1D45F"></use></g></g></g></g><g data-mml-node="mi" transform="translate(20111.7,0)"><use data-c="1D449" xlink:href="#MJX-2-TEX-I-1D449"></use></g><g data-mml-node="mo" transform="translate(20880.7,0)"><use data-c="28" xlink:href="#MJX-2-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(21269.7,0)"><use data-c="1D447" xlink:href="#MJX-2-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(21973.7,0)"><use data-c="2C" xlink:href="#MJX-2-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(22418.4,0)"><use data-c="1D443" xlink:href="#MJX-2-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(23169.4,0)"><use data-c="29" xlink:href="#MJX-2-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(23558.4,0)"><use data-c="1D451" xlink:href="#MJX-2-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(24078.4,0)"><use data-c="1D443" xlink:href="#MJX-2-TEX-I-1D443"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1539.5 1 2578.9"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:Gibbs" transform="translate(0,718.5)"><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-2-TEX-N-28"></use><use data-c="32" xlink:href="#MJX-2-TEX-N-32" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-2-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><mi>G</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>G</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mi>T</mi></msubsup><mi>S</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mi>d</mi><mi>T</mi><mo>+</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>P</mi><mi>r</mi></msub></mrow><mi>P</mi></msubsup><mi>V</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo><mi>d</mi><mi>P</mi></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>where the subscript </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.02ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 451 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-94-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 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="mi"><use data-c="1D45F" xlink:href="#MJX-94-TEX-I-1D45F"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">r</script><span> denotes reference conditions (298.15 and 1 bar). It is implicit that the first integral is done at constant pressure (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.362ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1043.9 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-117-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-117-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 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="1D443" xlink:href="#MJX-117-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(675,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-117-TEX-I-1D45F"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">P_r</script><span>) and the volume integral at a constant temperature different from </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.231ex" height="1.889ex" role="img" focusable="false" viewBox="0 -677 985.9 834.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-114-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-114-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 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="1D447" xlink:href="#MJX-114-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(617,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-114-TEX-I-1D45F"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">T_r</script><span>. Note that this expression is written differently in most common thermodynamic databases such as the ones from Holland and Powell but the expression [</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-124-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"><a href="#mjx-eqn%3AGibbs"><g data-mml-node="mrow" class=""><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="500" height="666" y="0"></rect><g data-mml-node="mtext"><use data-c="32" xlink:href="#MJX-124-TEX-N-32"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow href="#mjx-eqn%3AGibbs" class="MathJax_ref"><mtext>2</mtext></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\ref{Gibbs}</script><span>] seems more natural when coming from the definition of the Gibbs function to describe isothermal, isobaric chemically closed systems (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="18.553ex" height="1.781ex" role="img" focusable="false" viewBox="0 -705 8200.4 787" xmlns:xlink="http://www.w3.org/1999/xlink" 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mi>U</mi><mo>−</mo><mi>T</mi><mi>S</mi><mo>+</mo><mi>P</mi><mi>V</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">G = U -TS +PV</script><span>).</span></p><p><span>The computed </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.836ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3463.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-99-TEX-I-1D43A" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></path><path id="MJX-99-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-99-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-99-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-99-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-99-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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="1D43A" xlink:href="#MJX-99-TEX-I-1D43A"></use></g><g data-mml-node="mo" transform="translate(786,0)"><use data-c="28" xlink:href="#MJX-99-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1175,0)"><use data-c="1D447" xlink:href="#MJX-99-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(1879,0)"><use data-c="2C" xlink:href="#MJX-99-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2323.7,0)"><use data-c="1D443" xlink:href="#MJX-99-TEX-I-1D443"></use></g><g data-mml-node="mo" transform="translate(3074.7,0)"><use data-c="29" xlink:href="#MJX-99-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">G(T,P)</script><span> values are said to be </span><strong><span>apparent</span></strong><span> because the entropy, volume and heat capacity of the elements forming the phase are not considered. The reason is that it makes computation hairy and because it is unnecessary for phase equilibria computations that need only relative differences. Remember however that the retrieved values are not absolute and therefore cannot be directly compared to other absolute computations such as those done with first principle calculations.</span></p><h2 id='reference-gibbs-energy'><span>Reference Gibbs energy </span></h2><p><span>One common source of confusion comes from different conventions about the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.125ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4475 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-100-TEX-I-1D43A" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></path><path id="MJX-100-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-100-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-100-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-100-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-100-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-100-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-100-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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="1D43A" xlink:href="#MJX-100-TEX-I-1D43A"></use></g><g data-mml-node="mn" transform="translate(819,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-100-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1222.6,0)"><use data-c="28" xlink:href="#MJX-100-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(1611.6,0)"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-100-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(617,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-100-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(2597.5,0)"><use data-c="2C" xlink:href="#MJX-100-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3042.1,0)"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-100-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(675,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-100-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(4086,0)"><use data-c="29" xlink:href="#MJX-100-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">G_0(T_r,P_r)</script><span> in [</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-124-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"><a href="#mjx-eqn%3AGibbs"><g data-mml-node="mrow" class=""><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="500" height="666" y="0"></rect><g data-mml-node="mtext"><use data-c="32" xlink:href="#MJX-124-TEX-N-32"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow href="#mjx-eqn%3AGibbs" class="MathJax_ref"><mtext>2</mtext></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\ref{Gibbs}</script><span>] (referred to in Perple_X as SUPCRT and HSC convection). The confusion comes when even the same authors used different conventions for different versions of the datasets (such as in the case of hp02ver.dat = SUPCRT and hp11ver.dat = HSC, both by Holland and Powell). For the convection used in other datasets see </span><a href='https://www.perplex.ethz.ch/perplex_thermodynamic_data_file.html' target='_blank' class='url'>https://www.perplex.ethz.ch/perplex_thermodynamic_data_file.html</a></p><h5 id='supcrt-convention'><span>SUPCRT convention</span></h5><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="31.538ex" height="2.364ex" role="img" focusable="false" viewBox="0 -750 13939.8 1045" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.667ex;"><defs><path id="MJX-121-TEX-I-1D43A" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></path><path id="MJX-121-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-121-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 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enthalpy and can be measured experimentally at different temperatures. 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data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-3-TEX-N-28"></use><use data-c="33" xlink:href="#MJX-3-TEX-N-33" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-3-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>(3)</mtext></mtd><mtd><msub><mi>C</mi><mi>P</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mrow><mi>∂</mi><mi>H</mi></mrow><mrow><mi>∂</mi><mi>T</mi></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mi>P</mi><mi>r</mi></msub></mrow></msub><mo>≡</mo><mfrac><mrow><mi>d</mi><mi>H</mi></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span> We can relate this experimental value to entropy by noting that at constant pressure </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="17.812ex" height="1.781ex" role="img" focusable="false" viewBox="0 -705 7873 787" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-108-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 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VdP</script><span> becomes </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.431ex" height="1.781ex" role="img" focusable="false" viewBox="0 -705 4610.6 787" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-109-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 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data-mml-node="mfrac" transform="translate(2498.6,0)"><g data-mml-node="mrow" transform="translate(220,394) scale(0.707)"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-110-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(520,0)"><use data-c="1D43B" xlink:href="#MJX-110-TEX-I-1D43B"></use></g></g><g data-mml-node="mi" transform="translate(468.9,-345) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-110-TEX-I-1D447"></use></g><rect width="1195.6" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>S</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>H</mi></mrow><mi>T</mi></mfrac></math></mjx-assistive-mml></mjx-container><script type="math/tex">dS = \frac{dH}{T}</script><span>, therefore we can express the heat capacity in terms of entropy variations,</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block 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displaystyle="true"><mlabeledtr><mtd><mtext>(5)</mtext></mtd><mtd><mi>d</mi><mi>S</mi><mo>=</mo><mfrac><msub><mi>C</mi><mi>P</mi></msub><mi>T</mi></mfrac><mi>d</mi><mi>T</mi></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span> if a reference </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.818ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3897.5 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-112-TEX-I-1D446" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 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transform="translate(3508.5,0)"><use data-c="29" xlink:href="#MJX-112-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">S(T_r,P_r)</script><span> is known. Contrary to the references in thermodynamics, the reference for entropy is absolute because at 0 K the entropy of a perfect crystal is zero. This can be used to define the third law of entropy (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.042ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 3996.6 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-113-TEX-I-1D446" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 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data-c="1D436" xlink:href="#MJX-7-TEX-I-1D436"></use></g><g data-mml-node="mi" transform="translate(748,-150) scale(0.707)"><use data-c="1D443" xlink:href="#MJX-7-TEX-I-1D443"></use></g></g><g data-mml-node="mi" transform="translate(532.5,-686)"><use data-c="1D447" xlink:href="#MJX-7-TEX-I-1D447"></use></g><rect width="1529" height="60" x="120" y="220"></rect></g><g data-mml-node="mi" transform="translate(12424.6,0)"><use data-c="1D451" xlink:href="#MJX-7-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(12944.6,0)"><use data-c="1D447" xlink:href="#MJX-7-TEX-I-1D447"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1489.3 1 2478.7"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:7" transform="translate(0,672.6)"><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-7-TEX-N-28"></use><use data-c="37" xlink:href="#MJX-7-TEX-N-37" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-7-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>(7)</mtext></mtd><mtd><mi>S</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mi>S</mi><mrow data-mjx-texclass="ORD"><mi>c</mi><mi>o</mi><mi>n</mi><mi>f</mi></mrow></msup><mo>+</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mn>0</mn><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow></msubsup><mfrac><msub><mi>C</mi><mi>P</mi></msub><mi>T</mi></mfrac><mi>d</mi><mi>T</mi></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>because </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.007ex" height="1.934ex" role="img" focusable="false" viewBox="0 -705 1329 855" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-134-TEX-I-1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path><path id="MJX-134-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></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="1D436" xlink:href="#MJX-134-TEX-I-1D436"></use></g><g data-mml-node="mi" transform="translate(748,-150) scale(0.707)"><use data-c="1D443" xlink:href="#MJX-134-TEX-I-1D443"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>P</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">C_P</script><span> is measured at ambient conditions (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.362ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1043.9 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-117-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-117-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 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="1D443" xlink:href="#MJX-117-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(675,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-117-TEX-I-1D45F"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">P_r</script><span>), it is implicit that the above integration is done over a constant pressure </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.362ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1043.9 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-117-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-117-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 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 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stretchy="false">)</mo><mi>d</mi><mi>T</mi><mo>=</mo><mo>−</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mi>T</mi></msubsup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mi>T</mi></msubsup><mfrac><msub><mi>C</mi><mi>P</mi></msub><mi>T</mi></mfrac><mi>d</mi><mi>T</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow><mi>d</mi><mi>T</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>·</mo><msub><mi>S</mi><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub></mrow></msub><mo>−</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msubsup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msubsup><mfrac><msub><mi>C</mi><mi>p</mi></msub><mi>T</mi></mfrac><mi>d</mi><mi>T</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow><mi>d</mi><mi>T</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> - \int_{T_r}^T S(T,P_r)dT = -\int_{T_r}^T \left [S(T_r,P_r) + \int_{T_r}^T \frac{C_P}{T}dT \right]dT = - (T-T_r)·S_{T_r,P_r} - \int_{T_r}^{T}\left[\int_{T_r}^{T} \frac{C_p}{T}dT \right]dT</script></p><p><span>integrating by parts the double integral results in</span></p><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="73.478ex" 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data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-9-TEX-N-28"></use><use data-c="39" xlink:href="#MJX-9-TEX-N-39" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-9-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>(9)</mtext></mtd><mtd><mi>G</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>H</mi><mi>f</mi></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo>·</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msubsup><mfrac><msub><mi>C</mi><mi>p</mi></msub><mi>T</mi></mfrac><mi>d</mi><mi>T</mi><mo>+</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msubsup><msub><mi>C</mi><mi>p</mi></msub><mi>d</mi><mi>T</mi><mo>+</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>P</mi><mi>r</mi></msub></mrow><mi>P</mi></msubsup><mi>V</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo><mi>d</mi><mi>P</mi></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>I think the perversion comes from the fact that it might be not so obvious how to reach [</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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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"><a href="#mjx-eqn%3AGibbs"><g data-mml-node="mrow" class=""><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="500" height="666" y="0"></rect><g data-mml-node="mtext"><use data-c="32" xlink:href="#MJX-124-TEX-N-32"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow href="#mjx-eqn%3AGibbs" class="MathJax_ref"><mtext>2</mtext></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\ref{Gibbs}</script><span>], specially to people like me that forgot how to make integration by parts to solve for the double entropy integral. 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data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msubsup><mfrac><msub><mi>C</mi><mi>p</mi></msub><mi>T</mi></mfrac><mi>d</mi><mi>T</mi><mo>−</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msubsup><msub><mi>C</mi><mi>p</mi></msub><mi>d</mi><mi>T</mi></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h2 id='the-volume-term'><span>The volume term</span><span> </span></h2><p><span>We need still to bring the volume term to pressures higher than 1 bar. This is accomplished by using different Equations of States (EoS). Here I derive the most simple one but in practice, we will use another popular one in geosciences (Murnaghan EoS).</span></p><h4 id='isobaric-expansion'><span>Isobaric expansion</span></h4><p><span>It is defined as (in units of 1/K) </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n48" cid="n48" mdtype="math_block" data-math-tag-before="11" data-math-tag-after="12" data-math-labels="[]"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 27.314ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="5.76ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.314ex; min-width: 27.314ex;"><defs><path id="MJX-12-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 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xlink:href="#MJX-12-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D463" xlink:href="#MJX-12-TEX-I-1D463"></use></g></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-12-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D447" xlink:href="#MJX-12-TEX-I-1D447"></use></g></g><rect width="1470" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2446,0) translate(0 -0.5)"><use data-c="29" xlink:href="#MJX-12-TEX-S3-29"></use></g></g><g data-mml-node="TeXAtom" transform="translate(3215,-984.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-12-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(675,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-12-TEX-I-1D45F"></use></g></g></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1778 -1523 1 2545.9"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:12" transform="translate(0,823.5)"><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-12-TEX-N-28"></use><use data-c="31" xlink:href="#MJX-12-TEX-N-31" transform="translate(389,0)"></use><use data-c="32" xlink:href="#MJX-12-TEX-N-32" transform="translate(889,0)"></use><use data-c="29" xlink:href="#MJX-12-TEX-N-29" transform="translate(1389,0)"></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>(12)</mtext></mtd><mtd><mi>α</mi><mo>=</mo><mfrac><mn>1</mn><mi>v</mi></mfrac><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mrow><mi>∂</mi><mi>v</mi></mrow><mrow><mi>∂</mi><mi>T</mi></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mi>P</mi><mi>r</mi></msub></mrow></msub></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>The simplest approach is to assume that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.448ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 640 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-133-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></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="1D6FC" xlink:href="#MJX-133-TEX-I-1D6FC"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\alpha</script><span> and </span><strong><span>1 bar</span></strong><span> is constant and independent of temperature. If so, we can solve for </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.097ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 485 454" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-128-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></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="1D463" xlink:href="#MJX-128-TEX-I-1D463"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">v</script><span> (here the small letters denote specific properties, molar volume in this case). Remember also that volume needs to be expressed in energy units (J/bar/mol) so when multiplied by pressure gives Joules.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n50" cid="n50" mdtype="math_block" data-math-tag-before="12" data-math-tag-after="13" data-math-labels="[]"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 22.169ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="4.676ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.773ex; min-width: 22.169ex;"><defs><path id="MJX-13-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 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55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-129-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-129-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-129-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-129-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-129-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-129-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></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="1D463" xlink:href="#MJX-129-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(485,0)"><use data-c="28" xlink:href="#MJX-129-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(874,0)"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-129-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(617,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-129-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(1859.9,0)"><use data-c="2C" xlink:href="#MJX-129-TEX-N-2C"></use></g><g data-mml-node="mn" transform="translate(2304.6,0)"><use data-c="31" xlink:href="#MJX-129-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(2804.6,0)"><use data-c="29" xlink:href="#MJX-129-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3471.3,0)"><use data-c="3D" xlink:href="#MJX-129-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(4527.1,0)"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-129-TEX-I-1D463"></use></g><g data-mml-node="mn" transform="translate(518,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-129-TEX-N-30"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">v(T_r,1) = v_0</script><span>, we have,</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n53" cid="n53" mdtype="math_block" data-math-tag-before="14" data-math-tag-after="15" data-math-labels="[]"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 40.445ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex; min-width: 40.445ex;"><defs><path id="MJX-15-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path><path id="MJX-15-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-15-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-15-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path 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d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z"></path><path id="MJX-15-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-15-TEX-N-2061" d=""></path><path id="MJX-15-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-15-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-15-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 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192T84 210T107 214Q132 214 148 197T164 157Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.019382,-0.019382) translate(0, -750)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2578,0) translate(-2578,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(51.6)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="6360.5 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mi"><use data-c="1D6FC" xlink:href="#MJX-15-TEX-I-1D6FC"></use></g><g data-mml-node="mo" transform="translate(640,0)"><use data-c="28" xlink:href="#MJX-15-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1029,0)"><use data-c="1D447" xlink:href="#MJX-15-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(1955.2,0)"><use data-c="2212" xlink:href="#MJX-15-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(2955.4,0)"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-15-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(617,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-15-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(3941.3,0)"><use data-c="29" xlink:href="#MJX-15-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4608.1,0)"><use data-c="3D" xlink:href="#MJX-15-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(5663.9,0)"><use data-c="6C" xlink:href="#MJX-15-TEX-N-6C"></use><use data-c="6E" xlink:href="#MJX-15-TEX-N-6E" transform="translate(278,0)"></use></g><g data-mml-node="mo" transform="translate(6497.9,0)"><use data-c="2061" xlink:href="#MJX-15-TEX-N-2061"></use></g><g data-mml-node="mi" transform="translate(6664.6,0)"><use data-c="1D463" xlink:href="#MJX-15-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(7149.6,0)"><use data-c="28" xlink:href="#MJX-15-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(7538.6,0)"><use data-c="31" xlink:href="#MJX-15-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(8038.6,0)"><use data-c="2C" xlink:href="#MJX-15-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(8483.2,0)"><use data-c="1D447" xlink:href="#MJX-15-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(9187.2,0)"><use data-c="29" xlink:href="#MJX-15-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(9798.5,0)"><use data-c="2212" xlink:href="#MJX-15-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(10798.7,0)"><use data-c="6C" xlink:href="#MJX-15-TEX-N-6C"></use><use data-c="6E" xlink:href="#MJX-15-TEX-N-6E" transform="translate(278,0)"></use></g><g data-mml-node="mo" transform="translate(11632.7,0)"><use data-c="2061" xlink:href="#MJX-15-TEX-N-2061"></use></g><g data-mml-node="msub" transform="translate(11799.3,0)"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-15-TEX-I-1D463"></use></g><g data-mml-node="mn" transform="translate(518,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-15-TEX-N-30"></use></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1778 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:15"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-15-TEX-N-28"></use><use data-c="31" xlink:href="#MJX-15-TEX-N-31" transform="translate(389,0)"></use><use data-c="35" xlink:href="#MJX-15-TEX-N-35" transform="translate(889,0)"></use><use data-c="29" xlink:href="#MJX-15-TEX-N-29" transform="translate(1389,0)"></use></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>(15)</mtext></mtd><mtd><mi>α</mi><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo data-mjx-texclass="NONE"></mo><mi>v</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>−</mo><mi>ln</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>v</mi><mn>0</mn></msub></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>Then the volume at </span><strong><span>1 bar</span></strong><span> and T is</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n55" cid="n55" mdtype="math_block" data-math-tag-before="15" data-math-tag-after="16" data-math-labels="[]"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 38.746ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 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display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(16)</mtext></mtd><mtd><mi>v</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub><mi>exp</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h4 id='isothermal-compressibility'><span>Isothermal compressibility </span></h4><p><span>We can proceed similarly for the isothermal compressibility (i.e. assuming </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.034ex" role="img" focusable="false" viewBox="0 -705 1146.8 899" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 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xlink:href="#MJX-21-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(17607.4,0)"><use data-c="29" xlink:href="#MJX-21-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(17996.4,0)"><use data-c="29" xlink:href="#MJX-21-TEX-N-29"></use></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1778 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:21"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-21-TEX-N-28"></use><use data-c="32" xlink:href="#MJX-21-TEX-N-32" transform="translate(389,0)"></use><use data-c="31" xlink:href="#MJX-21-TEX-N-31" transform="translate(889,0)"></use><use data-c="29" xlink:href="#MJX-21-TEX-N-29" transform="translate(1389,0)"></use></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>(21)</mtext></mtd><mtd><mi>v</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub><mi>exp</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>β</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mo>−</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h2 id='implementation-in-perplex'><span>Implementation in Perple_X</span></h2><p><span>In Perple_X the molar Gibbs energy of a phase (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.079ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 477 647" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-132-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></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="1D454" xlink:href="#MJX-132-TEX-I-1D454"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">g</script><span>) is expressed using several equations of states. In this course, we will use one of the simplest (EoS = 2) with normal polynomials for </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.448ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 640 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-133-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" 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682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path id="MJX-150-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 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displaystyle="true"><mlabeledtr><mtd><mtext>(22)</mtext></mtd><mtd><msub><mi>C</mi><mi>P</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><mo>·</mo><mi>T</mi><mo>+</mo><msub><mi>c</mi><mn>3</mn></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msup><mi>T</mi><mn>2</mn></msup><mo>+</mo><msub><mi>c</mi><mn>5</mn></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msqrt><mi>T</mi></msqrt></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h3 id='murnaghan-equation-for-volume'><span>Murnaghan equation for volume</span></h3><h4 id='isobaric-expansivity'><span>Isobaric expansivity</span></h4><p><span>From an </span><strong><span>experimental</span></strong><span> point of view, the isobaric expansivity is found by fitting volumetric data at 1 bar over a temperature range,</span></p><div 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data-c="35" xlink:href="#MJX-25-TEX-N-35"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(7897.7,0)"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-25-TEX-N-2F"></use></g></g><g data-mml-node="msqrt" transform="translate(8397.7,0)"><g transform="translate(853,0)"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-25-TEX-I-1D447"></use></g></g><g data-mml-node="mo" transform="translate(0,153.8)"><use data-c="221A" xlink:href="#MJX-25-TEX-N-221A"></use></g><rect width="704" height="60" x="853" y="893.8"></rect></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1778 -881.9 1 1263.8"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:25" transform="translate(0,618.1)"><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-25-TEX-N-28"></use><use data-c="32" xlink:href="#MJX-25-TEX-N-32" transform="translate(389,0)"></use><use data-c="35" xlink:href="#MJX-25-TEX-N-35" transform="translate(889,0)"></use><use data-c="29" xlink:href="#MJX-25-TEX-N-29" transform="translate(1389,0)"></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>(25)</mtext></mtd><mtd><mi>α</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo><msub><mi>b</mi><mn>5</mn></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msqrt><mi>T</mi></msqrt></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>Note that in PerpleX </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.958ex" height="1.945ex" role="img" focusable="false" viewBox="0 -694 865.6 859.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.375ex;"><defs><path id="MJX-140-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path id="MJX-140-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 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role="img" focusable="false" viewBox="0 -694 4842.7 859.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.375ex;"><defs><path id="MJX-141-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path id="MJX-141-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 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165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></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="1D44F" xlink:href="#MJX-141-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,-150) scale(0.707)"><use data-c="35" xlink:href="#MJX-141-TEX-N-35"></use></g></g><g data-mml-node="mo" transform="translate(1143.3,0)"><use data-c="3D" xlink:href="#MJX-141-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2199.1,0)"><use data-c="2212" xlink:href="#MJX-141-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2977.1,0)"><use data-c="31" xlink:href="#MJX-141-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-141-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="msub" transform="translate(3977.1,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-141-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-141-TEX-N-31"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mn>5</mn></msub><mo>=</mo><mo>−</mo><mn>10</mn><msub><mi>b</mi><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">b_5 = -10b_1</script><span>, </span></p><p><span>then the volume at any temperature and 1 bar is computed as (note that we use the Taylor expansion approximation for small numbers where </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="14.719ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6506 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" 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unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(26)</mtext></mtd><mtd><mi>V</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mn>0</mn></msub><mo>·</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mn>1</mn><mo>+</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>T</mi><mi>r</mi></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msubsup><mi>α</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mi>d</mi><mi>T</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h4 id='isothermal-bulk-modulus'><span>Isothermal bulk modulus</span></h4><p><span>The isothermal bulk modulus at the temperature T and 1bar is,</span></p><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="28.415ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 12559.5 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-143-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path id="MJX-143-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-143-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-143-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-143-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path id="MJX-143-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-143-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 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414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path><path id="MJX-143-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-143-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path id="MJX-143-TEX-N-B7" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-143-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></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="1D43E" xlink:href="#MJX-143-TEX-I-1D43E"></use></g><g data-mml-node="mo" transform="translate(889,0)"><use data-c="28" xlink:href="#MJX-143-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1278,0)"><use data-c="1D447" xlink:href="#MJX-143-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(1982,0)"><use data-c="2C" xlink:href="#MJX-143-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(2426.7,0)"><g data-mml-node="mi"><use data-c="1D443" xlink:href="#MJX-143-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(675,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-143-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(3470.6,0)"><use data-c="29" xlink:href="#MJX-143-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4137.3,0)"><use data-c="3D" xlink:href="#MJX-143-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(5193.1,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-143-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,-150) scale(0.707)"><use data-c="36" xlink:href="#MJX-143-TEX-N-36"></use></g></g><g data-mml-node="mo" transform="translate(6280.9,0)"><use data-c="2B" xlink:href="#MJX-143-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(7281.1,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-143-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,-150) scale(0.707)"><use data-c="37" xlink:href="#MJX-143-TEX-N-37"></use></g></g><g data-mml-node="mo" transform="translate(8368.9,0)"><use data-c="B7" xlink:href="#MJX-143-TEX-N-B7"></use></g><g data-mml-node="mo" transform="translate(8869.1,0)"><use data-c="28" xlink:href="#MJX-143-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(9258.1,0)"><use data-c="1D447" xlink:href="#MJX-143-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(10184.3,0)"><use data-c="2212" xlink:href="#MJX-143-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(11184.6,0)"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-143-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(617,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-143-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(12170.5,0)"><use data-c="29" xlink:href="#MJX-143-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>b</mi><mn>6</mn></msub><mo>+</mo><msub><mi>b</mi><mn>7</mn></msub><mo>·</mo><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mi>r</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">K(T,P_r) = b_6 + b_7·(T-T_r) </script></p><p><span>equivalent to the HP98 expression,</span></p><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="40.778ex" height="2.538ex" role="img" focusable="false" viewBox="0 -871.8 18024 1121.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-144-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 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-141 137 -14T94 250Z"></path><path id="MJX-144-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-144-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-144-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 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data-mml-node="mo" transform="translate(7457,0)"><use data-c="29" xlink:href="#MJX-144-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(7846,0)"><use data-c="28" xlink:href="#MJX-144-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(8235,0)"><use data-c="31" xlink:href="#MJX-144-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(8957.3,0)"><use data-c="2212" xlink:href="#MJX-144-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(9957.5,0)"><use data-c="31" xlink:href="#MJX-144-TEX-N-31"></use><use data-c="2E" xlink:href="#MJX-144-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-144-TEX-N-35" transform="translate(778,0)"></use></g><g data-mml-node="mo" transform="translate(11457.7,0)"><use data-c="B7" xlink:href="#MJX-144-TEX-N-B7"></use></g><g data-mml-node="msup" transform="translate(11957.9,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-144-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-144-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="TeXAtom" transform="translate(1033,393.1) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-144-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="34" xlink:href="#MJX-144-TEX-N-34"></use></g></g></g><g data-mml-node="mo" transform="translate(13944.6,0)"><use data-c="28" xlink:href="#MJX-144-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(14333.6,0)"><use data-c="1D447" xlink:href="#MJX-144-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(15259.8,0)"><use data-c="2212" xlink:href="#MJX-144-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(16260,0)"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-144-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(617,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-144-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(17246,0)"><use data-c="29" xlink:href="#MJX-144-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(17635,0)"><use data-c="29" xlink:href="#MJX-144-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>1.5</mn><mo>·</mo><msup><mn>10</mn><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>4</mn></mrow></msup><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">K(T,P_r) = K(T_r)(1-1.5·10^{-4}(T-T_r)) </script></p><p><span>Note that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.958ex" height="1.945ex" role="img" focusable="false" viewBox="0 -694 865.6 859.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.375ex;"><defs><path id="MJX-145-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path id="MJX-145-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></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="1D44F" xlink:href="#MJX-145-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,-150) scale(0.707)"><use data-c="37" xlink:href="#MJX-145-TEX-N-37"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mn>7</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">b_7</script><span> is also redundant and equal to </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="19.349ex" height="2.347ex" role="img" focusable="false" viewBox="0 -871.8 8552.2 1037.3" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.375ex;"><defs><path id="MJX-146-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path id="MJX-146-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 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178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path id="MJX-146-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-146-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></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="1D44F" xlink:href="#MJX-146-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,-150) 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transform="translate(6065.3,0)"><use data-c="B7" xlink:href="#MJX-146-TEX-N-B7"></use></g><g data-mml-node="msup" transform="translate(6565.6,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-146-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-146-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="TeXAtom" transform="translate(1033,393.1) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-146-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="34" xlink:href="#MJX-146-TEX-N-34"></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>b</mi><mn>7</mn></msub><mo>=</mo><mo>−</mo><msub><mi>b</mi><mn>6</mn></msub><mo>·</mo><mn>1.5</mn><mo>·</mo><msup><mn>10</mn><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>4</mn></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">b_7 = -b_6·1.5 ·10^{-4}</script><span>.</span></p><h3 id='volume-at-pressure-and-temperature'><span>Volume at pressure and temperature</span></h3><p><span>If the pressure derivative of the bulk modulus (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.757ex" height="1.717ex" role="img" focusable="false" viewBox="0 -759 1218.5 759" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-148-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path id="MJX-148-TEX-V-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D43E" xlink:href="#MJX-148-TEX-I-1D43E"></use></g><g data-mml-node="mo" transform="translate(974,363) scale(0.707)"><use data-c="2032" xlink:href="#MJX-148-TEX-V-2032"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>K</mi><mo data-mjx-alternate="1">′</mo></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">K'</script><span>) is positive (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.757ex" height="1.717ex" role="img" focusable="false" viewBox="0 -759 1218.5 759" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-148-TEX-I-1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path><path id="MJX-148-TEX-V-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D43E" xlink:href="#MJX-148-TEX-I-1D43E"></use></g><g data-mml-node="mo" transform="translate(974,363) scale(0.707)"><use data-c="2032" xlink:href="#MJX-148-TEX-V-2032"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>K</mi><mo data-mjx-alternate="1">′</mo></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">K'</script><span> = 4) we retrieve the Murnaghan equation,</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n91" cid="n91" mdtype="math_block" data-math-tag-before="26" data-math-tag-after="27" data-math-labels="[]"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 57.713ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="6.034ex" 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xlink:href="#MJX-27-TEX-N-31"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(500,0)"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-27-TEX-N-2F"></use></g></g><g data-mml-node="mn" transform="translate(1000,0)"><use data-c="34" xlink:href="#MJX-27-TEX-N-34"></use></g></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1778 -1583.4 1 2666.9"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:27" transform="translate(0,626.6)"><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-27-TEX-N-28"></use><use data-c="32" xlink:href="#MJX-27-TEX-N-32" transform="translate(389,0)"></use><use data-c="37" xlink:href="#MJX-27-TEX-N-37" transform="translate(889,0)"></use><use data-c="29" xlink:href="#MJX-27-TEX-N-29" transform="translate(1389,0)"></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>(27)</mtext></mtd><mtd><mi>V</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>·</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>4</mn><mi>P</mi></mrow><mrow><mn>4</mn><mi>P</mi><mo>+</mo><mi>K</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><msub><mi>P</mi><mi>r</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn><mrow 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data-mjx-texclass="OPEN">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>4</mn><mi>P</mi></mrow><mrow><mn>4</mn><mi>P</mi><mo>+</mo><msub><mi>K</mi><mi>T</mi></msub></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>4</mn></mrow></msup></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>Now, in order to compute the energy contribution due to volume at T,P we need to integrate the above expression from 1 bar to P (see Holland and Powell 1998, page 312),</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n95" cid="n95" mdtype="math_block" data-math-tag-before="28" data-math-tag-after="29" data-math-labels="[]"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width: 61.455ex; position: relative;"><svg 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data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>4</mn></mrow></msup><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>Now you have all the necessary knowledge to compute the apparent Gibbs energy of a </span><strong><span>pure</span></strong><span> mineral phase at any desired temperature and pressure conditions (EoS is expected to work at </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.966ex" height="2.045ex" role="img" focusable="false" viewBox="0 -864 3521.1 904" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.09ex;"><defs><path id="MJX-152-TEX-I-1D443" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 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125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="3C" xlink:href="#MJX-153-TEX-N-3C"></use></g><g data-mml-node="mn" transform="translate(1055.8,0)"><use data-c="31" xlink:href="#MJX-153-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-153-TEX-N-30" transform="translate(500,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo><</mo><mn>10</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex"><10</script><span> GPa), for higher pressure other more sophisticated EoS with thermal pressure correction effects, is needed (such as the one implemented by Holland and Powell, 2011, called Modified Tait equation of state). We will not dive into this business. </span></p></div></div>
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