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5.03:_Enthalpy
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<div class="box-objectives"> <span id="Learning_Objectives"></span><h5 class="box-legend"><span class="lt-icon-default"><span style="display:none;"> </span>Learning Objectives</span></h5> <ul> <li class="lt-chem-21723" style="background-color: unset;">To know the key features of a state function.</li> <li class="lt-chem-21723" style="background-color: unset;">To use Hess’s law and thermochemical cycles to calculate enthalpy changes of chemical reactions.</li> </ul> </div> <p class="lt-chem-21723" style="background-color: unset;">Both heat and work represent energy transfer mechanisms. As discussed previously, the term "heat" refers to a <em>process </em>in which a body (the contents of a tea kettle, for example) acquires or loses energy as a direct consequence of its having a <em>different temperature </em>than its surroundings. Similarly, work refers to the transfer of energy that does not involve temperature differences. Hence, work, like energy, can take various forms, the most familiar being mechanical and electrical.</p> <ul> <li class="lt-chem-21723" style="background-color: unset;"><em>Mechanical work</em> arises when an object moves a distance <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="100" style="font-size: 85%; position: relative;"><svg width="3.179ex" height="1.645ex" role="img" focusable="false" viewbox="0 -716 1405 727" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-101-NCM-N-394" d="M759 0C776 0 785 4 785 13L445 696C438 709 429 716 416 716C403 716 394 709 387 696L52 25C49 18 47 14 47 12C47 4 56 0 73 0M384 604L644 83L124 83Z"></path><path id="MJX-101-NCM-N-2062" d=""></path><path id="MJX-101-NCM-I-1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 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data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:5" data-semantic-id="28" data-semantic-parent="32" data-semantic-attributes="latex:Δ" data-semantic-level-number="4" data-speech-node="true"><use data-c="394" href="#MJX-103-NCM-N-394"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:5" data-semantic-id="31" data-semantic-parent="32" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="4" data-speech-node="true" transform="translate(833,0)"><use data-c="2062" href="#MJX-103-NCM-N-2062"></use></g><g data-mml-node="mi" data-latex="\nonumber" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:5" data-semantic-id="29" data-semantic-parent="32" data-semantic-attributes="latex:\nonumber" data-semantic-level-number="4" data-speech-node="true" transform="translate(833,0)"><use data-c="1D465" href="#MJX-103-NCM-I-1D465"></use></g></g></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="0 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:" data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="37" data-semantic-level-number="2" data-speech-node="true"></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math data-latex=" w_{mechanical} = f Δx \nonumber " display="block" data-semantic-structure="(38 (37 (35 (30 (24 1 (23 2 20 3 21 4 22 (19 5 18 (17 6 12 7 13 8 14 9 15 10 16 11)))) 25 26) 27 (32 28 31 29)) 0))"><mtable displaystyle="true" data-latex="w_{mechanical} = f Δx \nonumber" data-semantic-type="multiline" data-semantic-role="unknown" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="37" data-semantic-attributes="latex: w_{mechanical} = f Δx \nonumber " data-semantic-owns="37" data-semantic-level-number="0" data-speech-node="true"><mlabeledtr data-semantic-type="line" data-semantic-role="multiline" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="35" data-semantic-content="0" data-semantic-parent="38" data-semantic-owns="35 0" data-semantic-level-number="1" data-speech-node="true"><mtd data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="37" data-semantic-level-number="2" data-speech-node="true"></mtd><mtd><mrow data-semantic-added="true" data-semantic-type="punctuated" data-semantic-role="text" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-children="30,27,32" data-semantic-parent="37" data-semantic-collapsed="(35 (c 33 34) 30 27 32)" data-semantic-owns="30 27 32" data-semantic-level-number="2" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:4" data-semantic-="" data-semantic-children="24,26" data-semantic-content="25" data-semantic-parent="35" data-semantic-owns="24 25 26" data-semantic-level-number="3" data-speech-node="true"><msub data-latex="w_{m e c h a n i c a l}" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-children="1,23" data-semantic-parent="30" data-semantic-attributes="latex:w_{m e c h a n i c a l}" data-semantic-owns="1 23" data-semantic-level-number="4" data-speech-node="true"><mi data-latex="w" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:6" data-semantic-="" data-semantic-parent="24" data-semantic-attributes="latex:w" data-semantic-level-number="5" data-speech-node="true">w</mi><mrow data-mjx-texclass="ORD" data-latex="{m e c h a n i c a l}" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:6" data-semantic-="" data-semantic-children="2,3,4,19" data-semantic-content="20,21,22" data-semantic-parent="24" data-semantic-attributes="latex:{m e c h a n i c a l};texclass:ORD" data-semantic-owns="2 20 3 21 4 22 19" data-semantic-level-number="5" data-speech-node="true"><mi data-latex="m" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="23" data-semantic-attributes="latex:m" data-semantic-level-number="6" data-speech-node="true">m</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="23" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="6" data-speech-node="true">⁢</mo><mi data-latex="e" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="23" data-semantic-attributes="latex:e" data-semantic-level-number="6" data-speech-node="true">e</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="23" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="6" data-speech-node="true">⁢</mo><mi data-latex="c" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="23" data-semantic-attributes="latex:c" data-semantic-level-number="6" data-speech-node="true">c</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="23" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="6" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-children="5,17" data-semantic-content="18,5" data-semantic-parent="23" data-semantic-owns="5 18 17" data-semantic-level-number="6" data-speech-node="true"><mi data-latex="h" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:8" data-semantic-="" data-semantic-parent="19" data-semantic-attributes="latex:h" data-semantic-operator="appl" data-semantic-level-number="7" data-speech-node="true">h</mi><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-annotation="nemeth:number;depth:8" data-semantic-="" data-semantic-parent="19" data-semantic-added="true" data-semantic-operator="appl" data-semantic-level-number="7" data-speech-node="true">⁡</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:8" data-semantic-="" data-semantic-children="6,7,8,9,10,11" data-semantic-content="12,13,14,15,16" data-semantic-parent="19" data-semantic-owns="6 12 7 13 8 14 9 15 10 16 11" data-semantic-level-number="7" data-speech-node="true"><mi data-latex="a" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:a" data-semantic-level-number="8" data-speech-node="true">a</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="8" data-speech-node="true">⁢</mo><mi data-latex="n" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:n" data-semantic-level-number="8" data-speech-node="true">n</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="8" data-speech-node="true">⁢</mo><mi data-latex="i" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:i" data-semantic-level-number="8" data-speech-node="true">i</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="8" data-speech-node="true">⁢</mo><mi data-latex="c" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:c" data-semantic-level-number="8" data-speech-node="true">c</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="8" data-speech-node="true">⁢</mo><mi data-latex="a" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:a" data-semantic-level-number="8" data-speech-node="true">a</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="8" data-speech-node="true">⁢</mo><mi data-latex="l" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:9" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:l" data-semantic-level-number="8" data-speech-node="true">l</mi></mrow></mrow></mrow></msub><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:5" data-semantic-="" data-semantic-parent="30" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="4" data-speech-node="true">=</mo><mi data-latex="f" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:5" data-semantic-="" data-semantic-parent="30" data-semantic-attributes="latex:f" data-semantic-level-number="4" data-speech-node="true">f</mi></mrow><mo data-semantic-type="text" data-semantic-role="space" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:4" data-semantic-="" data-semantic-parent="35" data-semantic-level-number="3" data-speech-node="true"> </mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:4" data-semantic-="" data-semantic-children="28,29" data-semantic-content="31" data-semantic-parent="35" data-semantic-owns="28 31 29" data-semantic-level-number="3" data-speech-node="true"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:5" data-semantic-="" data-semantic-parent="32" data-semantic-attributes="latex:Δ" data-semantic-level-number="4" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="32" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="4" data-speech-node="true">⁢</mo><mi data-latex="\nonumber" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:5" data-semantic-="" data-semantic-parent="32" 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data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="21" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mi data-latex="a" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="21" data-semantic-attributes="latex:a" data-semantic-level-number="5" data-speech-node="true">a</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="21" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mi data-latex="l" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="21" data-semantic-attributes="latex:l" data-semantic-level-number="5" data-speech-node="true">l</mi></mrow></msub><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:4" data-semantic-="" data-semantic-parent="30" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="3" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:4" data-semantic-="" data-semantic-children="24,25,26" data-semantic-content="27,28" data-semantic-parent="30" data-semantic-owns="24 27 25 28 26" data-semantic-level-number="3" data-speech-node="true"><mi data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:5" data-semantic-="" data-semantic-parent="29" data-semantic-attributes="latex:q" data-semantic-level-number="4" data-speech-node="true">q</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="29" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="4" data-speech-node="true">⁢</mo><mi mathvariant="normal" data-latex="\Delta" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:5" data-semantic-="" data-semantic-parent="29" data-semantic-attributes="latex:\Delta" data-semantic-level-number="4" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="29" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="4" data-speech-node="true">⁢</mo><mi data-latex="\nonumber" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:5" data-semantic-="" data-semantic-parent="29" data-semantic-attributes="latex:\nonumber" data-semantic-level-number="4" data-speech-node="true">V</mi></mrow></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></li> </ul> <p class="lt-chem-21723" style="background-color: unset;">The unit of work is Joules. Work, like heat, exists only when energy is being transferred. When two bodies are placed in thermal contact and energy flows from the warmer body to the cooler one, this process is called “heat”. A transfer of energy to or from a system by any means other than heat is called “work”.</p> <span id="Enthalpy_as_a_Composite_Function"></span><h2 style="background-color: unset;" class="lt-chem-21723">Enthalpy as a Composite Function</h2> <p class="lt-chem-21723" style="background-color: unset;">To further understand the relationship between heat flow (<mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="106" style="font-size: 85%; position: relative;"><svg width="1.023ex" height="1.439ex" role="img" focusable="false" viewbox="0 -442 452 636" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-107-NCM-I-1D45E" d="M372 377C352 420 321 442 280 442C215 442 158 409 109 342C63 280 40 216 40 149C40 62 90-13 173-13C209-13 245 4 282 38L241-126C238-141 229-150 215-153C210-154 197-155 174-155C168-156 164-156 162-156C153-157 148-165 148-179L148-183C151-190 157-194 165-194C184-194 244-191 263-191C282-191 345-194 364-194C378-194 385-186 385-170C385-160 375-155 356-155C338-155 313-156 313-144C313-140 314-133 317-123L452 427C452 436 447 441 438 441C420 441 382 391 372 377M340 373C349 352 354 338 354 329C354 328 353 322 351 313L327 216C308 144 299 107 298 105C279 70 224 16 176 16C137 16 117 46 117 105C117 151 148 263 162 300C181 347 225 412 280 412C307 412 327 399 340 373Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="q" data-semantic-structure="0"><g data-mml-node="mi" data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:q" data-semantic-level-number="0" data-speech-node="true"><use data-c="1D45E" href="#MJX-107-NCM-I-1D45E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="q" data-semantic-structure="0"><mi data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:q" data-semantic-level-number="0" data-speech-node="true">q</mi></math></mjx-assistive-mml></mjx-container>) and the resulting change in internal energy (<mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="107" style="font-size: 85%; position: relative;"><svg width="3.604ex" height="1.67ex" role="img" focusable="false" viewbox="0 -716 1593 738" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-108-NCM-N-394" d="M759 0C776 0 785 4 785 13L445 696C438 709 429 716 416 716C403 716 394 709 387 696L52 25C49 18 47 14 47 12C47 4 56 0 73 0M384 604L644 83L124 83Z"></path><path id="MJX-108-NCM-N-2062" d=""></path><path id="MJX-108-NCM-I-1D448" d="M642 680C623 680 559 683 540 683C525 683 518 675 518 659C518 650 525 645 539 644C582 644 603 631 603 605C603 599 602 592 600 585L511 232C496 175 467 126 424 84C377 39 325 17 269 17C195 17 152 67 152 141L153 150C153 169 156 191 163 217L259 602C264 623 273 636 286 641C292 643 309 644 338 644C364 644 375 644 375 668C375 678 369 683 358 683L231 680L104 683C88 683 81 674 81 659C81 652 84 647 89 646C99 645 107 644 112 644C143 643 161 641 166 640C171 639 173 636 173 631C173 629 172 622 169 611L74 230C69 211 67 192 67 172C67 57 150-22 265-22C330-22 390 3 445 54C498 102 532 158 548 223L636 574C649 625 677 643 739 644C753 645 760 653 760 668L760 672C757 679 752 683 743 683C724 683 661 680 642 680Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="ΔU" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:1" data-semantic-id="3" data-semantic-children="0,1" data-semantic-content="2" data-semantic-attributes="latex:ΔU" data-semantic-owns="0 2 1" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(3 0 2 1)"><g data-mml-node="mi" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="3" data-semantic-attributes="latex:Δ" data-semantic-level-number="1" data-speech-node="true"><use data-c="394" href="#MJX-108-NCM-N-394"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:2" data-semantic-id="2" data-semantic-parent="3" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="1" data-speech-node="true" transform="translate(833,0)"><use data-c="2062" href="#MJX-108-NCM-N-2062"></use></g><g data-mml-node="mi" data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-id="1" data-semantic-parent="3" data-semantic-attributes="latex:U" data-semantic-level-number="1" data-speech-node="true" transform="translate(833,0)"><use data-c="1D448" href="#MJX-108-NCM-I-1D448"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="ΔU" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:1" data-semantic-="" data-semantic-children="0,1" data-semantic-content="2" data-semantic-attributes="latex:ΔU" data-semantic-owns="0 2 1" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(3 0 2 1)"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:Δ" data-semantic-level-number="1" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="1" data-speech-node="true">⁢</mo><mi data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:U" data-semantic-level-number="1" data-speech-node="true">U</mi></math></mjx-assistive-mml></mjx-container>), we can look at two sets of limiting conditions: reactions that occur at constant volume and reactions that occur at constant pressure. We will assume that <span class="acronym-ignore">PV</span> work is the only kind of work possible for the system, so we can substitute its definition <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="108" style="font-size: 85%; position: relative;"><svg width="1.62ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 716 1000" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="w=-P\Delta V" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="10" data-semantic-children="0,9" data-semantic-content="1" data-semantic-attributes="latex:w=-P\Delta V" data-semantic-owns="0 1 9" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(10 0 1 (9 2 (8 3 6 4 7 5)))"><g data-mml-node="mi" data-latex="w" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="10" data-semantic-attributes="latex:w" data-semantic-level-number="1" data-speech-node="true"><use data-c="1D464" href="#MJX-109-NCM-I-1D464"></use></g></g></g><defs><path id="MJX-109-NCM-I-1D464" d="M590 391C590 382 595 372 606 362C629 340 641 313 641 281C641 264 635 237 623 198C586 78 539 18 482 18C435 18 412 45 412 100C412 119 416 142 423 171C441 243 460 314 477 387C479 394 480 398 480 401C480 421 469 431 447 431C427 431 414 421 407 401L367 245C352 187 341 167 341 115C341 107 341 101 342 98C319 45 290 18 255 18C205 18 180 47 180 104C180 141 197 204 231 293C242 323 248 344 248 357C248 406 212 442 163 442C118 442 84 417 59 367C39 328 29 301 29 287C29 278 34 273 45 273C58 273 60 279 64 293C88 373 120 413 160 413C174 413 181 403 181 384C181 369 175 347 164 317C127 220 108 154 108 117C108 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</mjx-break><svg width="9.481ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 4190.8 1000" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="w=-P\Delta V" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="10" data-semantic-children="0,9" data-semantic-content="1" data-semantic-attributes="latex:w=-P\Delta V" data-semantic-owns="0 1 9" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(10 0 1 (9 2 (8 3 6 4 7 5)))"><g data-mml-node="mo" data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="1" data-semantic-parent="10" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="1" data-speech-node="true"><use data-c="3D" href="#MJX-109-NCM-N-3D"></use></g><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="prefixop" data-semantic-role="negative" data-semantic-annotation="depth:2" data-semantic-id="9" data-semantic-children="8" data-semantic-content="2" data-semantic-parent="10" data-semantic-owns="2 8" data-semantic-level-number="1" data-speech-node="true" transform="translate(1055.8,0)"><g data-mml-node="mo" data-latex="-" data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-annotation="nemeth:number;depth:3" data-semantic-id="2" data-semantic-parent="9" data-semantic-attributes="latex:-" data-semantic-operator="prefixop,−" data-semantic-level-number="2" data-speech-node="true"><use data-c="2212" href="#MJX-109-NCM-N-2212"></use></g><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:3" data-semantic-id="8" data-semantic-children="3,4,5" data-semantic-content="6,7" data-semantic-parent="9" data-semantic-owns="3 6 4 7 5" data-semantic-level-number="2" data-speech-node="true" transform="translate(778,0)"><g data-mml-node="mi" data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:4" data-semantic-id="3" data-semantic-parent="8" data-semantic-attributes="latex:P" data-semantic-level-number="3" data-speech-node="true"><use data-c="1D443" href="#MJX-109-NCM-I-1D443"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:4" data-semantic-id="6" data-semantic-parent="8" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="3" data-speech-node="true" transform="translate(754,0)"><use data-c="2062" href="#MJX-109-NCM-N-2062"></use></g><g data-mml-node="mi" data-latex="\Delta" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:4" data-semantic-id="4" data-semantic-parent="8" data-semantic-attributes="latex:\Delta" data-semantic-level-number="3" data-speech-node="true" transform="translate(754,0)"><use data-c="394" href="#MJX-109-NCM-N-394"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:4" data-semantic-id="7" data-semantic-parent="8" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="3" data-speech-node="true" transform="translate(1587,0)"><use data-c="2062" href="#MJX-109-NCM-N-2062"></use></g><g data-mml-node="mi" data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:4" data-semantic-id="5" data-semantic-parent="8" data-semantic-attributes="latex:V" data-semantic-level-number="3" data-speech-node="true" transform="translate(1587,0)"><use data-c="1D449" href="#MJX-109-NCM-I-1D449"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="w=-P\Delta V" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,9" data-semantic-content="1" data-semantic-attributes="latex:w=-P\Delta V" data-semantic-owns="0 1 9" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(10 0 1 (9 2 (8 3 6 4 7 5)))"><mi data-latex="w" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="10" data-semantic-attributes="latex:w" data-semantic-level-number="1" data-speech-node="true">w</mi><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="10" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="1" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="prefixop" data-semantic-role="negative" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="8" data-semantic-content="2" data-semantic-parent="10" data-semantic-owns="2 8" data-semantic-level-number="1" data-speech-node="true"><mo data-latex="-" data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-annotation="nemeth:number;depth:3" data-semantic-="" data-semantic-parent="9" data-semantic-attributes="latex:-" data-semantic-operator="prefixop,−" data-semantic-level-number="2" data-speech-node="true">−</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:3" data-semantic-="" data-semantic-children="3,4,5" data-semantic-content="6,7" data-semantic-parent="9" data-semantic-owns="3 6 4 7 5" data-semantic-level-number="2" data-speech-node="true"><mi data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:4" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:P" data-semantic-level-number="3" data-speech-node="true">P</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:4" data-semantic-="" data-semantic-parent="8" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="3" data-speech-node="true">⁢</mo><mi mathvariant="normal" data-latex="\Delta" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:4" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:\Delta" data-semantic-level-number="3" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:4" data-semantic-="" data-semantic-parent="8" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="3" data-speech-node="true">⁢</mo><mi data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:4" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:V" data-semantic-level-number="3" data-speech-node="true">V</mi></mrow></mrow></math></mjx-assistive-mml></mjx-container> into the first Law of Thermodynamics (<mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="109" style="font-size: 85%; position: relative;"><svg width="3.604ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1593 1000" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Delta U = q + w" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="9" data-semantic-children="7,8" data-semantic-content="2" data-semantic-attributes="latex:\Delta U = q + w" data-semantic-owns="7 2 8" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(9 (7 0 6 1) 2 (8 3 4 5))"><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:2" data-semantic-id="7" data-semantic-children="0,1" data-semantic-content="6" data-semantic-parent="9" data-semantic-owns="0 6 1" data-semantic-level-number="1" data-speech-node="true"><g data-mml-node="mi" data-latex="\Delta" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="7" data-semantic-attributes="latex:\Delta" data-semantic-level-number="2" data-speech-node="true"><use data-c="394" href="#MJX-110-NCM-N-394"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:3" data-semantic-id="6" data-semantic-parent="7" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="2" data-speech-node="true" transform="translate(833,0)"><use data-c="2062" href="#MJX-110-NCM-N-2062"></use></g><g data-mml-node="mi" data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" 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640C171 639 173 636 173 631C173 629 172 622 169 611L74 230C69 211 67 192 67 172C67 57 150-22 265-22C330-22 390 3 445 54C498 102 532 158 548 223L636 574C649 625 677 643 739 644C753 645 760 653 760 668L760 672C757 679 752 683 743 683C724 683 661 680 642 680Z"></path><path id="MJX-110-NCM-N-3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"></path><path id="MJX-110-NCM-I-1D45E" d="M372 377C352 420 321 442 280 442C215 442 158 409 109 342C63 280 40 216 40 149C40 62 90-13 173-13C209-13 245 4 282 38L241-126C238-141 229-150 215-153C210-154 197-155 174-155C168-156 164-156 162-156C153-157 148-165 148-179L148-183C151-190 157-194 165-194C184-194 244-191 263-191C282-191 345-194 364-194C378-194 385-186 385-170C385-160 375-155 356-155C338-155 313-156 313-144C313-140 314-133 317-123L452 427C452 436 447 441 438 441C420 441 382 391 372 377M340 373C349 352 354 338 354 329C354 328 353 322 351 313L327 216C308 144 299 107 298 105C279 70 224 16 176 16C137 16 117 46 117 105C117 151 148 263 162 300C181 347 225 412 280 412C307 412 327 399 340 373Z"></path><path id="MJX-110-NCM-N-2B" d="M698 274L413 274L413 559C413 575 405 583 389 583C373 583 365 575 365 559L365 274L80 274C64 274 56 266 56 250C56 234 64 226 80 226L365 226L365-59C365-75 373-83 389-83C405-83 413-75 413-59L413 226L698 226C714 226 722 234 722 250C722 263 711 274 698 274Z"></path><path id="MJX-110-NCM-I-1D464" d="M590 391C590 382 595 372 606 362C629 340 641 313 641 281C641 264 635 237 623 198C586 78 539 18 482 18C435 18 412 45 412 100C412 119 416 142 423 171C441 243 460 314 477 387C479 394 480 398 480 401C480 421 469 431 447 431C427 431 414 421 407 401L367 245C352 187 341 167 341 115C341 107 341 101 342 98C319 45 290 18 255 18C205 18 180 47 180 104C180 141 197 204 231 293C242 323 248 344 248 357C248 406 212 442 163 442C118 442 84 417 59 367C39 328 29 301 29 287C29 278 34 273 45 273C58 273 60 279 64 293C88 373 120 413 160 413C174 413 181 403 181 384C181 369 175 347 164 317C127 220 108 154 108 117C108 32 165-11 252-11C294-11 328 10 355 53C375 10 416-11 479-11C541-11 590 30 625 112C648 164 691 302 691 369C691 384 690 394 689 399C685 418 666 442 644 442C618 442 590 417 590 391Z"></path></defs></svg><mjx-break size="4"> </mjx-break><svg width="3.411ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1507.8 1000" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Delta U = q + w" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="9" data-semantic-children="7,8" data-semantic-content="2" data-semantic-attributes="latex:\Delta U = q + w" data-semantic-owns="7 2 8" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(9 (7 0 6 1) 2 (8 3 4 5))"><g data-mml-node="mo" data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="9" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="1" data-speech-node="true"><use data-c="3D" href="#MJX-110-NCM-N-3D"></use></g><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="addition" data-semantic-annotation="depth:2" data-semantic-id="8" data-semantic-children="3,5" data-semantic-content="4" data-semantic-parent="9" data-semantic-owns="3 4 5" data-semantic-level-number="1" data-speech-node="true" transform="translate(1055.8,0)"><g data-mml-node="mi" data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-id="3" data-semantic-parent="8" data-semantic-attributes="latex:q" data-semantic-level-number="2" data-speech-node="true"><use data-c="1D45E" href="#MJX-110-NCM-I-1D45E"></use></g></g></g></g></svg><mjx-break size="3"> </mjx-break><svg width="3.883ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1716.2 1000" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Delta U = q + w" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="9" data-semantic-children="7,8" data-semantic-content="2" data-semantic-attributes="latex:\Delta U = q + w" data-semantic-owns="7 2 8" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(9 (7 0 6 1) 2 (8 3 4 5))"><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="addition" data-semantic-annotation="depth:2" data-semantic-id="8" data-semantic-children="3,5" data-semantic-content="4" data-semantic-parent="9" data-semantic-owns="3 4 5" data-semantic-level-number="1" data-speech-node="true"><g data-mml-node="mo" data-latex="+" data-semantic-type="operator" data-semantic-role="addition" data-semantic-annotation="depth:3" data-semantic-id="4" data-semantic-parent="8" data-semantic-attributes="latex:+" data-semantic-operator="infixop,+" data-semantic-level-number="2" data-speech-node="true"><use data-c="2B" href="#MJX-110-NCM-N-2B"></use></g><g data-mml-node="mi" data-latex="w" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-id="5" data-semantic-parent="8" data-semantic-attributes="latex:w" data-semantic-level-number="2" data-speech-node="true" transform="translate(1000.2,0)"><use data-c="1D464" href="#MJX-110-NCM-I-1D464"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\Delta U = q + w" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="7,8" data-semantic-content="2" data-semantic-attributes="latex:\Delta U = q + w" data-semantic-owns="7 2 8" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(9 (7 0 6 1) 2 (8 3 4 5))"><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:2" data-semantic-="" data-semantic-children="0,1" data-semantic-content="6" data-semantic-parent="9" data-semantic-owns="0 6 1" data-semantic-level-number="1" data-speech-node="true"><mi mathvariant="normal" data-latex="\Delta" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-attributes="latex:\Delta" data-semantic-level-number="2" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="2" data-speech-node="true">⁢</mo><mi data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-attributes="latex:U" data-semantic-level-number="2" data-speech-node="true">U</mi></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="9" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="1" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="addition" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="3,5" data-semantic-content="4" data-semantic-parent="9" data-semantic-owns="3 4 5" data-semantic-level-number="1" data-speech-node="true"><mi data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:q" data-semantic-level-number="2" data-speech-node="true">q</mi><mo data-latex="+" data-semantic-type="operator" data-semantic-role="addition" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:+" data-semantic-operator="infixop,+" data-semantic-level-number="2" data-speech-node="true">+</mo><mi data-latex="w" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:w" data-semantic-level-number="2" data-speech-node="true">w</mi></mrow></math></mjx-assistive-mml></mjx-container>) to obtain the following:</p> <p class="mt-align-center lt-chem-21723" style="background-color: unset;"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" display="true" width="full" tabindex="0" ctxtmenu_counter="110" style="font-size: 85%; min-width: 32.186ex; position: relative;"><svg width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 14226 1000" aria-hidden="true" style="vertical-align: -0.566ex; min-width: 32.186ex;"><defs><path id="MJX-111-NCM-N-394" d="M759 0C776 0 785 4 785 13L445 696C438 709 429 716 416 716C403 716 394 709 387 696L52 25C49 18 47 14 47 12C47 4 56 0 73 0M384 604L644 83L124 83Z"></path><path id="MJX-111-NCM-N-2062" d=""></path><path id="MJX-111-NCM-I-1D448" d="M642 680C623 680 559 683 540 683C525 683 518 675 518 659C518 650 525 645 539 644C582 644 603 631 603 605C603 599 602 592 600 585L511 232C496 175 467 126 424 84C377 39 325 17 269 17C195 17 152 67 152 141L153 150C153 169 156 191 163 217L259 602C264 623 273 636 286 641C292 643 309 644 338 644C364 644 375 644 375 668C375 678 369 683 358 683L231 680L104 683C88 683 81 674 81 659C81 652 84 647 89 646C99 645 107 644 112 644C143 643 161 641 166 640C171 639 173 636 173 631C173 629 172 622 169 611L74 230C69 211 67 192 67 172C67 57 150-22 265-22C330-22 390 3 445 54C498 102 532 158 548 223L636 574C649 625 677 643 739 644C753 645 760 653 760 668L760 672C757 679 752 683 743 683C724 683 661 680 642 680Z"></path><path id="MJX-111-NCM-N-3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 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293 623 290 613L158 82C152 60 142 47 128 42C121 40 103 39 72 39C50 39 40 37 40 16C40 5 46 0 59 0L184 3L248 2C259 2 299 0 312 0C328 0 336 8 336 24C336 34 325 39 304 39C264 39 244 43 244 52C244 52 245 55 247 68L307 312L472 312C537 312 599 332 657 371C722 414 754 467 754 530C754 629 660 683 555 683M524 644C611 644 654 614 654 554C654 498 626 423 597 397C559 363 509 346 447 346L313 346L379 610C387 644 387 644 429 644Z"></path><path id="MJX-111-NCM-I-1D449" d="M671 680C652 680 592 683 573 683C558 683 550 675 550 660C550 650 557 645 570 644C598 643 612 633 612 616C612 607 607 595 598 580L300 107L234 619C234 636 255 644 298 644C318 644 327 651 327 668C327 678 321 683 309 683C287 683 209 680 187 680C167 680 99 683 79 683C64 683 56 675 56 660C56 649 66 644 85 644C102 644 114 642 122 640C138 635 137 633 140 614L218 4C221-13 229-22 242-22C255-22 265-15 273-2L629 564C652 600 674 623 696 632C711 639 730 643 752 644C763 645 768 652 769 667C770 678 764 683 752 683C737 683 686 680 671 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39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path><path id="MJX-111-NCM-N-29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.021719,-0.021719) translate(0, -750)"><g data-mml-node="math" data-latex="ΔU = q − PΔV \label{5.2.5} " data-semantic-structure="(22 (21 (19 (14 5 13 6) 7 (18 8 9 (17 10 15 11 16 12))) (4 (3 0 1 2))))"><g data-mml-node="mtable" data-latex="ΔU = q − PΔV \label{5.2.5}" data-semantic-type="multiline" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="22" data-semantic-children="21" data-semantic-attributes="latex:ΔU = q − PΔV \label{5.2.5} " data-semantic-owns="21" data-semantic-level-number="0" data-speech-node="true" transform="translate(3634,0) translate(-3634,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(46.04)"><svg data-table="true" preserveaspectratio="xMidYMid" viewbox="3479 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" data-semantic-type="line" data-semantic-role="multiline" data-semantic-annotation="depth:2" data-semantic-id="21" data-semantic-children="19" data-semantic-content="4" data-semantic-parent="22" data-semantic-owns="19 4" data-semantic-level-number="1" data-speech-node="true"><g data-mml-node="mtd"><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:3" data-semantic-id="19" data-semantic-children="14,18" data-semantic-content="7" data-semantic-parent="21" data-semantic-owns="14 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data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="12" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mi data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="12" data-semantic-attributes="latex:U" data-semantic-level-number="5" data-speech-node="true">U</mi></mrow></mrow><mtext data-latex="\textrm{constant volume}" data-semantic-type="text" data-semantic-role="text" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:4" data-semantic-="" data-semantic-parent="15" data-semantic-attributes="latex:\textrm{constant volume}" data-semantic-level-number="3" data-speech-node="true">constant volume</mtext></munder></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></p> <blockquote> <p class="lt-chem-21723" style="background-color: unset;">At constant volume, no <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="112" style="font-size: 85%; position: relative;"><svg width="3.448ex" height="1.595ex" role="img" focusable="false" viewbox="0 -683 1524 705" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-113-NCM-I-1D443" d="M555 683L235 683C212 683 201 682 201 660C201 649 212 644 234 644C254 644 294 647 294 631C294 629 293 623 290 613L158 82C152 60 142 47 128 42C121 40 103 39 72 39C50 39 40 37 40 16C40 5 46 0 59 0L184 3L248 2C259 2 299 0 312 0C328 0 336 8 336 24C336 34 325 39 304 39C264 39 244 43 244 52C244 52 245 55 247 68L307 312L472 312C537 312 599 332 657 371C722 414 754 467 754 530C754 629 660 683 555 683M524 644C611 644 654 614 654 554C654 498 626 423 597 397C559 363 509 346 447 346L313 346L379 610C387 644 387 644 429 644Z"></path><path id="MJX-113-NCM-N-2062" d=""></path><path id="MJX-113-NCM-I-1D449" d="M671 680C652 680 592 683 573 683C558 683 550 675 550 660C550 650 557 645 570 644C598 643 612 633 612 616C612 607 607 595 598 580L300 107L234 619C234 636 255 644 298 644C318 644 327 651 327 668C327 678 321 683 309 683C287 683 209 680 187 680C167 680 99 683 79 683C64 683 56 675 56 660C56 649 66 644 85 644C102 644 114 642 122 640C138 635 137 633 140 614L218 4C221-13 229-22 242-22C255-22 265-15 273-2L629 564C652 600 674 623 696 632C711 639 730 643 752 644C763 645 768 652 769 667C770 678 764 683 752 683C737 683 686 680 671 680Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="PV" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:1" data-semantic-id="3" data-semantic-children="0,1" data-semantic-content="2" data-semantic-attributes="latex:PV" data-semantic-owns="0 2 1" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(3 0 2 1)"><g data-mml-node="mi" data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="3" data-semantic-attributes="latex:P" data-semantic-level-number="1" data-speech-node="true"><use data-c="1D443" href="#MJX-113-NCM-I-1D443"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:2" data-semantic-id="2" data-semantic-parent="3" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="1" data-speech-node="true" transform="translate(754,0)"><use data-c="2062" href="#MJX-113-NCM-N-2062"></use></g><g data-mml-node="mi" data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-id="1" data-semantic-parent="3" data-semantic-attributes="latex:V" data-semantic-level-number="1" data-speech-node="true" transform="translate(754,0)"><use data-c="1D449" href="#MJX-113-NCM-I-1D449"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="PV" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:1" data-semantic-="" data-semantic-children="0,1" data-semantic-content="2" data-semantic-attributes="latex:PV" data-semantic-owns="0 2 1" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(3 0 2 1)"><mi data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:P" data-semantic-level-number="1" data-speech-node="true">P</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="1" data-speech-node="true">⁢</mo><mi data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:V" data-semantic-level-number="1" data-speech-node="true">V</mi></math></mjx-assistive-mml></mjx-container> work can be done, and the change in the internal energy of the system is equal to the amount of heat transferred from the system to the surroundings or vice versa.</p> </blockquote> <p class="lt-chem-21723" style="background-color: unset;">Many chemical reactions are not, however, carried out in sealed containers at constant volume but in open containers at a more or less constant pressure of about 1 atm. The heat flow under these conditions is given the symbol <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="113" style="font-size: 85%; position: relative;"><svg width="2.002ex" height="1.65ex" role="img" focusable="false" viewbox="0 -442 884.7 729.2" aria-hidden="true" style="vertical-align: -0.65ex;"><defs><path id="MJX-114-NCM-I-1D45E" d="M372 377C352 420 321 442 280 442C215 442 158 409 109 342C63 280 40 216 40 149C40 62 90-13 173-13C209-13 245 4 282 38L241-126C238-141 229-150 215-153C210-154 197-155 174-155C168-156 164-156 162-156C153-157 148-165 148-179L148-183C151-190 157-194 165-194C184-194 244-191 263-191C282-191 345-194 364-194C378-194 385-186 385-170C385-160 375-155 356-155C338-155 313-156 313-144C313-140 314-133 317-123L452 427C452 436 447 441 438 441C420 441 382 391 372 377M340 373C349 352 354 338 354 329C354 328 353 322 351 313L327 216C308 144 299 107 298 105C279 70 224 16 176 16C137 16 117 46 117 105C117 151 148 263 162 300C181 347 225 412 280 412C307 412 327 399 340 373Z"></path><path id="MJX-114-NCM-I-1D45D" d="M355 442C311 442 269 419 228 373C217 419 186 442 137 442C96 442 66 409 45 344C35 311 30 292 30 286C30 277 35 272 46 272C51 272 54 273 57 275C62 284 65 291 66 298C84 374 107 412 134 412C152 412 161 398 161 371C161 356 159 339 154 321L44-118C35-151 34-155-5-155C-23-155-32-163-32-178C-32-189-26-194-15-194C0-194 52-191 67-191C86-191 146-194 165-194C180-194 187-186 187-170C187-160 178-155 159-155C141-155 114-156 114-144C114-131 155 26 159 43C179 6 209-13 249-13C314-13 371 20 421 87C467 148 490 213 490 280C490 367 438 442 355 442M352 412C391 412 411 382 411 323C411 296 405 260 394 215C371 128 342 71 307 42C286 25 267 16 248 16C219 16 199 29 188 56C179 77 174 92 174 100L225 309C230 331 248 354 276 377C304 400 329 412 352 412Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="q_p" data-semantic-structure="(2 0 1)"><g data-mml-node="msub" data-latex="q_p" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:1" data-semantic-id="2" data-semantic-children="0,1" data-semantic-attributes="latex:q_p" data-semantic-owns="0 1" data-semantic-level-number="0" data-speech-node="true"><g data-mml-node="mi" data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="2" data-semantic-attributes="latex:q" data-semantic-level-number="1" data-speech-node="true"><use data-c="1D45E" href="#MJX-114-NCM-I-1D45E"></use></g><g data-mml-node="mi" transform="translate(479,-150) scale(0.707)" data-latex="p" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-id="1" data-semantic-parent="2" data-semantic-attributes="latex:p" data-semantic-level-number="1" data-speech-node="true"><use data-c="1D45D" href="#MJX-114-NCM-I-1D45D"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="q_p" data-semantic-structure="(2 0 1)"><msub data-latex="q_p" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,1" data-semantic-attributes="latex:q_p" data-semantic-owns="0 1" data-semantic-level-number="0" data-speech-node="true"><mi data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="2" data-semantic-attributes="latex:q" data-semantic-level-number="1" data-speech-node="true">q</mi><mi data-latex="p" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-="" data-semantic-parent="2" data-semantic-attributes="latex:p" data-semantic-level-number="1" data-speech-node="true">p</mi></msub></math></mjx-assistive-mml></mjx-container> to indicate constant pressure. Replacing q in Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="114" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-115-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 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556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{5.3.3}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#mjx-eqn:5.3.3"><g data-mml-node="mrow" data-latex="\ref{5.3.3}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="2056" height="688" x="0" y="-22"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{5.3.3};href:#mjx-eqn%3A5.3.3" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="35" href="#MJX-115-NCM-N-35"></use><use data-c="2E" href="#MJX-115-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-115-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-115-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="32" href="#MJX-115-NCM-N-32" transform="translate(1556,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{5.3.3}" data-semantic-structure="0"><mrow href="#mjx-eqn%3A5.3.3" data-latex="\ref{5.3.3}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{5.3.3};href:#mjx-eqn%3A5.3.3" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">5.3.2</mtext></mrow></math></mjx-assistive-mml></mjx-container> by <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="115" style="font-size: 85%; position: relative;"><svg width="2.002ex" height="1.65ex" role="img" focusable="false" viewbox="0 -442 884.7 729.2" aria-hidden="true" style="vertical-align: -0.65ex;"><defs><path id="MJX-116-NCM-I-1D45E" d="M372 377C352 420 321 442 280 442C215 442 158 409 109 342C63 280 40 216 40 149C40 62 90-13 173-13C209-13 245 4 282 38L241-126C238-141 229-150 215-153C210-154 197-155 174-155C168-156 164-156 162-156C153-157 148-165 148-179L148-183C151-190 157-194 165-194C184-194 244-191 263-191C282-191 345-194 364-194C378-194 385-186 385-170C385-160 375-155 356-155C338-155 313-156 313-144C313-140 314-133 317-123L452 427C452 436 447 441 438 441C420 441 382 391 372 377M340 373C349 352 354 338 354 329C354 328 353 322 351 313L327 216C308 144 299 107 298 105C279 70 224 16 176 16C137 16 117 46 117 105C117 151 148 263 162 300C181 347 225 412 280 412C307 412 327 399 340 373Z"></path><path id="MJX-116-NCM-I-1D45D" d="M355 442C311 442 269 419 228 373C217 419 186 442 137 442C96 442 66 409 45 344C35 311 30 292 30 286C30 277 35 272 46 272C51 272 54 273 57 275C62 284 65 291 66 298C84 374 107 412 134 412C152 412 161 398 161 371C161 356 159 339 154 321L44-118C35-151 34-155-5-155C-23-155-32-163-32-178C-32-189-26-194-15-194C0-194 52-191 67-191C86-191 146-194 165-194C180-194 187-186 187-170C187-160 178-155 159-155C141-155 114-156 114-144C114-131 155 26 159 43C179 6 209-13 249-13C314-13 371 20 421 87C467 148 490 213 490 280C490 367 438 442 355 442M352 412C391 412 411 382 411 323C411 296 405 260 394 215C371 128 342 71 307 42C286 25 267 16 248 16C219 16 199 29 188 56C179 77 174 92 174 100L225 309C230 331 248 354 276 377C304 400 329 412 352 412Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="q_p" data-semantic-structure="(2 0 1)"><g data-mml-node="msub" data-latex="q_p" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:1" data-semantic-id="2" data-semantic-children="0,1" data-semantic-attributes="latex:q_p" data-semantic-owns="0 1" data-semantic-level-number="0" data-speech-node="true"><g data-mml-node="mi" data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="2" data-semantic-attributes="latex:q" data-semantic-level-number="1" data-speech-node="true"><use data-c="1D45E" href="#MJX-116-NCM-I-1D45E"></use></g><g data-mml-node="mi" transform="translate(479,-150) scale(0.707)" data-latex="p" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-id="1" data-semantic-parent="2" data-semantic-attributes="latex:p" data-semantic-level-number="1" data-speech-node="true"><use data-c="1D45D" href="#MJX-116-NCM-I-1D45D"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="q_p" data-semantic-structure="(2 0 1)"><msub data-latex="q_p" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,1" data-semantic-attributes="latex:q_p" data-semantic-owns="0 1" data-semantic-level-number="0" data-speech-node="true"><mi data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="2" data-semantic-attributes="latex:q" data-semantic-level-number="1" data-speech-node="true">q</mi><mi data-latex="p" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-="" data-semantic-parent="2" data-semantic-attributes="latex:p" data-semantic-level-number="1" data-speech-node="true">p</mi></msub></math></mjx-assistive-mml></mjx-container> and rearranging to solve for q<sub>p</sub>,</p> <p class="mt-align-center lt-chem-21723" style="background-color: unset;"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" display="true" width="full" tabindex="0" ctxtmenu_counter="116" style="font-size: 85%; min-width: 33.249ex; position: relative;"><svg width="100%" height="4.245ex" role="img" focusable="false" data-mjx-viewbox="0 -1188.1 14696.2 1876.2" aria-hidden="true" style="vertical-align: -1.557ex; min-width: 33.249ex;"><defs><path id="MJX-117-NCM-I-1D45E" d="M372 377C352 420 321 442 280 442C215 442 158 409 109 342C63 280 40 216 40 149C40 62 90-13 173-13C209-13 245 4 282 38L241-126C238-141 229-150 215-153C210-154 197-155 174-155C168-156 164-156 162-156C153-157 148-165 148-179L148-183C151-190 157-194 165-194C184-194 244-191 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href="#MJX-117-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-117-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-117-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="33" href="#MJX-117-NCM-N-33" transform="translate(1556,0)"></use></g><g data-mml-node="mtext" data-latex="\text{)}" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="general:text;depth:5" data-semantic-id="2" data-semantic-parent="3" data-semantic-attributes="latex:\text{)}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true" transform="translate(2445,0)"><use data-c="29" href="#MJX-117-NCM-N-29"></use></g></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math data-latex="\underset{\textrm{constant pressure}}{q_{\textrm p}=\Delta U+P\Delta V} \label{5.3.4} " display="block" data-semantic-structure="(26 (25 (23 (21 (7 5 6) 8 (20 (16 9 15 10) 11 (19 12 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data-semantic-attributes="latex:U" data-semantic-level-number="6" data-speech-node="true">U</mi></mrow><mo data-latex="+" data-semantic-type="operator" data-semantic-role="addition" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="20" data-semantic-attributes="latex:+" data-semantic-operator="infixop,+" data-semantic-level-number="5" data-speech-node="true">+</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:6" data-semantic-="" data-semantic-children="12,13,14" data-semantic-content="17,18" data-semantic-parent="20" data-semantic-owns="12 17 13 18 14" data-semantic-level-number="5" data-speech-node="true"><mi data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="19" data-semantic-attributes="latex:P" data-semantic-level-number="6" data-speech-node="true">P</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="19" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="6" data-speech-node="true">⁢</mo><mi mathvariant="normal" data-latex="\Delta" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="19" data-semantic-attributes="latex:\Delta" data-semantic-level-number="6" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="19" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="6" data-speech-node="true">⁢</mo><mi data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="19" data-semantic-attributes="latex:V" data-semantic-level-number="6" data-speech-node="true">V</mi></mrow></mrow></mrow><mtext data-latex="\textrm{constant pressure}" data-semantic-type="text" data-semantic-role="text" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:4" data-semantic-="" data-semantic-parent="23" data-semantic-attributes="latex:\textrm{constant pressure}" data-semantic-level-number="3" data-speech-node="true">constant pressure</mtext></munder></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></p> <p class="lt-chem-21723" style="background-color: unset;">Thus, at constant pressure, the heat flow for any process is equal to the change in the internal energy of the system plus the PV work done.</p> <p class="lt-chem-21723" style="background-color: unset;">Because 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data-speech-node="true" transform="translate(754,0)"><use data-c="1D449" href="#MJX-118-NCM-I-1D449"></use></g></g></g></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="0 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:" data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="12" data-semantic-level-number="2" data-speech-node="true"></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math data-latex="H =U + PV \nonumber " display="block" data-semantic-structure="(13 (12 (10 1 2 (9 3 4 (8 5 7 6))) 0))"><mtable displaystyle="true" data-latex="H =U + PV \nonumber" data-semantic-type="multiline" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="12" data-semantic-attributes="latex:H =U + PV \nonumber " data-semantic-owns="12" data-semantic-level-number="0" data-speech-node="true"><mlabeledtr data-semantic-type="line" data-semantic-role="multiline" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="10" data-semantic-content="0" data-semantic-parent="13" data-semantic-owns="10 0" data-semantic-level-number="1" data-speech-node="true"><mtd data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="12" data-semantic-level-number="2" data-speech-node="true"></mtd><mtd><mrow data-semantic-added="true" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-children="1,9" data-semantic-content="2" data-semantic-parent="12" data-semantic-owns="1 2 9" data-semantic-level-number="2" data-speech-node="true"><mi data-latex="H" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:4" data-semantic-="" data-semantic-parent="10" data-semantic-attributes="latex:H" data-semantic-level-number="3" data-speech-node="true">H</mi><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:4" data-semantic-="" data-semantic-parent="10" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="3" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="addition" data-semantic-annotation="depth:4" data-semantic-="" data-semantic-children="3,8" data-semantic-content="4" data-semantic-parent="10" data-semantic-owns="3 4 8" data-semantic-level-number="3" data-speech-node="true"><mi data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:5" data-semantic-="" data-semantic-parent="9" data-semantic-attributes="latex:U" data-semantic-level-number="4" data-speech-node="true">U</mi><mo data-latex="+" data-semantic-type="operator" data-semantic-role="addition" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="9" data-semantic-attributes="latex:+" data-semantic-operator="infixop,+" data-semantic-level-number="4" data-speech-node="true">+</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:5" data-semantic-="" data-semantic-children="5,6" data-semantic-content="7" data-semantic-parent="9" data-semantic-owns="5 7 6" data-semantic-level-number="4" data-speech-node="true"><mi data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:P" data-semantic-level-number="5" data-speech-node="true">P</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="8" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mi data-latex="\nonumber" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:\nonumber" data-semantic-level-number="5" data-speech-node="true">V</mi></mrow></mrow></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></p> <p class="lt-chem-21723" style="background-color: unset;">At constant pressure, the change in the enthalpy of a system is as follows:</p> <p class="mt-align-center 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data-speech-node="true">V</mi></mrow><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="25" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">)</mo></mrow></mrow></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:4" data-semantic-="" data-semantic-parent="40" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="3" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="addition" data-semantic-annotation="depth:4" data-semantic-="" data-semantic-children="29,38" data-semantic-content="19" data-semantic-parent="40" data-semantic-owns="29 19 38" data-semantic-level-number="3" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple;depth:5" data-semantic-="" data-semantic-children="17,18" data-semantic-content="28,17" data-semantic-parent="39" data-semantic-owns="17 28 18" data-semantic-level-number="4" data-speech-node="true"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:6" data-semantic-="" data-semantic-parent="29" data-semantic-attributes="latex:Δ" data-semantic-operator="appl" data-semantic-level-number="5" data-speech-node="true">Δ</mi><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-annotation="nemeth:number;depth:6" data-semantic-="" data-semantic-parent="29" data-semantic-added="true" data-semantic-operator="appl" data-semantic-level-number="5" data-speech-node="true">⁡</mo><mi data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:6" data-semantic-="" data-semantic-parent="29" data-semantic-attributes="latex:U" data-semantic-level-number="5" data-speech-node="true">U</mi></mrow><mo data-latex="+" data-semantic-type="operator" data-semantic-role="addition" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="39" data-semantic-attributes="latex:+" data-semantic-operator="infixop,+" data-semantic-level-number="4" data-speech-node="true">+</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-children="20,27" data-semantic-content="37" data-semantic-parent="39" data-semantic-owns="20 37 27" data-semantic-level-number="4" data-speech-node="true"><mi data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="38" data-semantic-attributes="latex:P" data-semantic-level-number="5" data-speech-node="true">P</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="38" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-children="21,22" data-semantic-content="26,21" data-semantic-parent="38" data-semantic-owns="21 26 22" data-semantic-level-number="5" data-speech-node="true"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:7" data-semantic-="" data-semantic-parent="27" data-semantic-attributes="latex:Δ" data-semantic-operator="appl" data-semantic-level-number="6" data-speech-node="true">Δ</mi><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-annotation="nemeth:number;depth:7" data-semantic-="" data-semantic-parent="27" data-semantic-added="true" data-semantic-operator="appl" data-semantic-level-number="6" data-speech-node="true">⁡</mo><mi data-latex="\label{5.3.5}" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:7" data-semantic-="" data-semantic-parent="27" data-semantic-attributes="latex:\label{5.3.5}" data-semantic-level-number="6" data-speech-node="true">V</mi></mrow></mrow></mrow></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></p> <p class="lt-chem-21723" style="background-color: unset;">Comparing the previous two equations shows that at constant pressure, the change in the enthalpy of a system is equal to the heat flow: <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="119" style="font-size: 85%; position: relative;"><svg width="3.878ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1714 1000" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="ΔH = q_p" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="8" data-semantic-children="7,5" data-semantic-content="2" data-semantic-attributes="latex:ΔH = q_p" data-semantic-owns="7 2 5" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(8 (7 0 6 1) 2 (5 3 4))"><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:2" data-semantic-id="7" data-semantic-children="0,1" data-semantic-content="6" data-semantic-parent="8" data-semantic-owns="0 6 1" data-semantic-level-number="1" data-speech-node="true"><g data-mml-node="mi" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="7" data-semantic-attributes="latex:Δ" data-semantic-level-number="2" data-speech-node="true"><use data-c="394" href="#MJX-120-NCM-N-394"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:3" data-semantic-id="6" data-semantic-parent="7" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="2" 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154 321L44-118C35-151 34-155-5-155C-23-155-32-163-32-178C-32-189-26-194-15-194C0-194 52-191 67-191C86-191 146-194 165-194C180-194 187-186 187-170C187-160 178-155 159-155C141-155 114-156 114-144C114-131 155 26 159 43C179 6 209-13 249-13C314-13 371 20 421 87C467 148 490 213 490 280C490 367 438 442 355 442M352 412C391 412 411 382 411 323C411 296 405 260 394 215C371 128 342 71 307 42C286 25 267 16 248 16C219 16 199 29 188 56C179 77 174 92 174 100L225 309C230 331 248 354 276 377C304 400 329 412 352 412Z"></path></defs></svg><mjx-break size="4"> </mjx-break><svg width="4.39ex" height="2.347ex" role="img" focusable="false" viewbox="0 -750 1940.5 1037.2" style="vertical-align: -0.65ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="ΔH = q_p" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="8" data-semantic-children="7,5" data-semantic-content="2" data-semantic-attributes="latex:ΔH = q_p" data-semantic-owns="7 2 5" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(8 (7 0 6 1) 2 (5 3 4))"><g data-mml-node="mo" data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="8" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="1" data-speech-node="true"><use data-c="3D" href="#MJX-120-NCM-N-3D"></use></g><g data-mml-node="msub" data-latex="q_p" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:2" data-semantic-id="5" data-semantic-children="3,4" data-semantic-parent="8" data-semantic-attributes="latex:q_p" data-semantic-owns="3 4" data-semantic-level-number="1" data-speech-node="true" transform="translate(1055.8,0)"><g data-mml-node="mi" data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-id="3" data-semantic-parent="5" data-semantic-attributes="latex:q" data-semantic-level-number="2" data-speech-node="true"><use data-c="1D45E" href="#MJX-120-NCM-I-1D45E"></use></g><g data-mml-node="mi" transform="translate(479,-150) scale(0.707)" data-latex="p" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-id="4" data-semantic-parent="5" data-semantic-attributes="latex:p" data-semantic-level-number="2" data-speech-node="true"><use data-c="1D45D" href="#MJX-120-NCM-I-1D45D"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="ΔH = q_p" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="7,5" data-semantic-content="2" data-semantic-attributes="latex:ΔH = q_p" data-semantic-owns="7 2 5" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(8 (7 0 6 1) 2 (5 3 4))"><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:2" data-semantic-="" data-semantic-children="0,1" data-semantic-content="6" data-semantic-parent="8" data-semantic-owns="0 6 1" data-semantic-level-number="1" data-speech-node="true"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-attributes="latex:Δ" data-semantic-level-number="2" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="2" data-speech-node="true">⁢</mo><mi data-latex="H" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-attributes="latex:H" data-semantic-level-number="2" data-speech-node="true">H</mi></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="1" data-speech-node="true">=</mo><msub data-latex="q_p" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="3,4" data-semantic-parent="8" data-semantic-attributes="latex:q_p" data-semantic-owns="3 4" data-semantic-level-number="1" data-speech-node="true"><mi data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:q" data-semantic-level-number="2" data-speech-node="true">q</mi><mi data-latex="p" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:p" data-semantic-level-number="2" data-speech-node="true">p</mi></msub></math></mjx-assistive-mml></mjx-container>. This expression is consistent with our definition of enthalpy, where we stated that enthalpy is the heat absorbed or produced during any process that occurs at constant pressure.</p> <blockquote> <p class="lt-chem-21723" style="background-color: unset;">At constant pressure, the change in the enthalpy of a system is equal to the heat flow: <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="120" style="font-size: 85%; position: relative;"><svg width="3.878ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1714 1000" aria-hidden="true" style="vertical-align: -0.566ex;"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="ΔH = q_p" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="8" data-semantic-children="7,5" data-semantic-content="2" data-semantic-attributes="latex:ΔH = q_p" data-semantic-owns="7 2 5" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(8 (7 0 6 1) 2 (5 3 4))"><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:2" data-semantic-id="7" data-semantic-children="0,1" data-semantic-content="6" data-semantic-parent="8" data-semantic-owns="0 6 1" data-semantic-level-number="1" data-speech-node="true"><g data-mml-node="mi" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="7" data-semantic-attributes="latex:Δ" data-semantic-level-number="2" data-speech-node="true"><use data-c="394" href="#MJX-121-NCM-N-394"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" 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data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-id="8" data-semantic-children="7,5" data-semantic-content="2" data-semantic-attributes="latex:ΔH = q_p" data-semantic-owns="7 2 5" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(8 (7 0 6 1) 2 (5 3 4))"><g data-mml-node="mo" data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="8" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="1" data-speech-node="true"><use data-c="3D" href="#MJX-121-NCM-N-3D"></use></g><g data-mml-node="msub" data-latex="q_p" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:2" data-semantic-id="5" data-semantic-children="3,4" data-semantic-parent="8" data-semantic-attributes="latex:q_p" data-semantic-owns="3 4" data-semantic-level-number="1" data-speech-node="true" transform="translate(1055.8,0)"><g data-mml-node="mi" data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-id="3" data-semantic-parent="5" data-semantic-attributes="latex:q" data-semantic-level-number="2" data-speech-node="true"><use data-c="1D45E" href="#MJX-121-NCM-I-1D45E"></use></g><g data-mml-node="mi" transform="translate(479,-150) scale(0.707)" data-latex="p" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-id="4" data-semantic-parent="5" data-semantic-attributes="latex:p" data-semantic-level-number="2" data-speech-node="true"><use data-c="1D45D" href="#MJX-121-NCM-I-1D45D"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="ΔH = q_p" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="7,5" data-semantic-content="2" data-semantic-attributes="latex:ΔH = q_p" data-semantic-owns="7 2 5" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(8 (7 0 6 1) 2 (5 3 4))"><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:2" data-semantic-="" data-semantic-children="0,1" data-semantic-content="6" data-semantic-parent="8" data-semantic-owns="0 6 1" data-semantic-level-number="1" data-speech-node="true"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-attributes="latex:Δ" data-semantic-level-number="2" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="2" data-speech-node="true">⁢</mo><mi data-latex="H" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-="" data-semantic-parent="7" data-semantic-attributes="latex:H" data-semantic-level-number="2" data-speech-node="true">H</mi></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="8" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="1" data-speech-node="true">=</mo><msub data-latex="q_p" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="3,4" data-semantic-parent="8" data-semantic-attributes="latex:q_p" data-semantic-owns="3 4" data-semantic-level-number="1" data-speech-node="true"><mi data-latex="q" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:3" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:q" data-semantic-level-number="2" data-speech-node="true">q</mi><mi data-latex="p" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:3" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:p" data-semantic-level-number="2" data-speech-node="true">p</mi></msub></math></mjx-assistive-mml></mjx-container>.</p> </blockquote> <div class="box-example"> <span id="Example_.5C(.5CPageIndex.7B1.7D.5C)"></span><h5 class="box-legend"><span class="lt-icon-default">Example <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="121" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-122-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-122-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-122-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-122-NCM-N-31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\PageIndex{1}" data-semantic-structure="(5 0 1 2 3 4)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="1}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-id="5" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{1};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><g data-mml-node="mn" data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true"><use data-c="35" href="#MJX-122-NCM-N-35"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(500,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="1" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-122-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true" transform="translate(778,0)"><use data-c="33" href="#MJX-122-NCM-N-33"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(1278,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="3" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-122-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="1" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="4" data-semantic-parent="5" data-semantic-attributes="latex:1" data-semantic-level-number="1" data-speech-node="true" transform="translate(1556,0)"><use data-c="31" href="#MJX-122-NCM-N-31"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\PageIndex{1}" data-semantic-structure="(5 0 1 2 3 4)"><mrow data-mjx-texclass="ORD" data-latex="1}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{1};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><mn data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true">5</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true">3</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="1" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:1" data-semantic-level-number="1" data-speech-node="true">1</mn></mrow></math></mjx-assistive-mml></mjx-container></span></h5> <p class="lt-chem-21723" style="background-color: unset;">The molar enthalpy of fusion for ice at 0.0°C and a pressure of 1.00 atm is 6.01 kJ, and the molar volumes of ice and water at 0°C are 0.0197 L and 0.0180 L, respectively. Calculate ΔH and ΔU for the melting of ice at 0.0°C.</p> <p class="lt-chem-21723" style="background-color: unset;"><strong>Given</strong>: enthalpy of fusion for ice, pressure, and molar volumes of ice and water </p> <p class="lt-chem-21723" style="background-color: unset;"><strong>Asked for</strong>: ΔH and ΔU for ice melting at 0.0°C</p> <p class="lt-chem-21723" style="background-color: unset;"><strong>Strategy</strong>:</p> <ol start="1" style="list-style-type: upper-alpha;"> <li class="lt-chem-21723" style="background-color: unset;">Determine the sign of q and set this value equal to ΔH.</li> <li class="lt-chem-21723" style="background-color: unset;">Calculate Δ(PV) from the information given.</li> <li class="lt-chem-21723" style="background-color: unset;">Determine ΔU by substituting the calculated values into Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="122" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.581ex" role="img" focusable="false" viewbox="0 -677 2056 699" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-123-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-123-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-123-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-123-NCM-N-34" d="M353 677C344 677 336 672 330 663L28 199L28 163L289 163L289 81C289 63 285 51 278 46C271 41 252 39 219 39L194 39L194 0C223 2 269 3 331 3C393 3 439 2 468 0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{5.3.5}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#mjx-eqn:5.3.5"><g data-mml-node="mrow" data-latex="\ref{5.3.5}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="2056" height="699" x="0" y="-22"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="35" href="#MJX-123-NCM-N-35"></use><use data-c="2E" href="#MJX-123-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-123-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-123-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="34" href="#MJX-123-NCM-N-34" transform="translate(1556,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{5.3.5}" data-semantic-structure="0"><mrow href="#mjx-eqn%3A5.3.5" data-latex="\ref{5.3.5}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">5.3.4</mtext></mrow></math></mjx-assistive-mml></mjx-container>.</li> </ol> <span id="Solution"></span><h6>Solution</h6> <p class="lt-chem-21723" style="background-color: unset;">A Because 6.01 kJ of heat is absorbed from the surroundings when 1 mol of ice melts, q = +6.01 kJ. When the process is carried out at constant pressure, q = q<sub>p</sub> = ΔH = 6.01 kJ.</p> <p class="lt-chem-21723" style="background-color: unset;">B To find ΔU using Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="123" style="font-size: 85%; position: relative;"><svg width="3.204ex" height="1.595ex" role="img" focusable="false" viewbox="0 -705 1416 705" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-124-NCM-N-3F" d="M226 705C140 705 56 652 56 570C56 535 72 518 105 518C138 518 154 535 154 568C154 598 137 614 102 616C127 655 168 675 223 675C297 675 326 646 326 573C326 555 324 540 321 528C313 501 312 502 294 482C236 419 207 344 207 257L207 213C207 194 212 185 222 185C233 185 239 195 239 216L239 250C239 331 281 404 366 467C399 492 415 525 415 568C415 662 328 705 226 705M278 56C278 87 253 113 222 113C191 113 167 87 167 56C167 26 192 0 222 0C253 0 278 25 278 56Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{18.11}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#"><g data-mml-node="mrow" data-latex="\ref{18.11}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="1416" height="705" x="0" y="0"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{18.11};href:#" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="3F" href="#MJX-124-NCM-N-3F"></use><use data-c="3F" href="#MJX-124-NCM-N-3F" transform="translate(472,0)"></use><use data-c="3F" href="#MJX-124-NCM-N-3F" transform="translate(944,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{18.11}" data-semantic-structure="0"><mrow href="#" data-latex="\ref{18.11}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{18.11};href:#" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">???</mtext></mrow></math></mjx-assistive-mml></mjx-container>, we need to calculate Δ(PV). 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data-latex="\begin{align}\Delta(PV)&=P\Delta V=P(V_{\textrm f}-V)=(1.00\textrm{ atm})(\textrm{0.0180 L}-\textrm{0.0197 L}) \\ &=(-1.7\times10^{-3}\;\mathrm{L\cdot atm})(101.3\;\mathrm{J/L\cdot atm})=-0.0017\textrm{ J}\end{align} \nonumber " data-semantic-structure="(114 (58 (15 (14 5 13 (12 6 (11 7 10 8) 9))) (57 (56 49 17 (51 18 50 (48 19 47 20)) 21 (53 22 52 (41 23 (40 (26 24 25) 27 28) 29)) 30 (55 (44 31 (43 32 33) 34) 54 (46 35 (45 36 37 38) 39)))) (4 (3 0 1 2))) (113 64 (112 (111 (109 105 66 (107 (101 67 (100 (97 68 69) 70 (99 (75 71 (74 72 73)) 98 (80 77 78 79))) 81) 106 (104 82 (103 83 102 (91 (90 85 86 87) 88 89)) 92)) 93 (108 94 95)) 96)) (63 (62 59 60 61))))"><g data-mml-node="mtable" data-latex="\begin{align}\Delta(PV)&=P\Delta V=P(V_{\textrm f}-V)=(1.00\textrm{ atm})(\textrm{0.0180 L}-\textrm{0.0197 L}) \\ &=(-1.7\times10^{-3}\;\mathrm{L\cdot atm})(101.3\;\mathrm{J/L\cdot atm})=-0.0017\textrm{ J}\end{align} \nonumber" data-semantic-type="table" data-semantic-role="unknown" 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data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true" transform="translate(4221,0)"><use data-c="29" href="#MJX-125-NCM-N-29"></use></g></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-id="54" data-semantic-parent="55" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true" transform="translate(4610,0)"><use data-c="2062" href="#MJX-125-NCM-N-2062"></use></g><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:6" data-semantic-id="46" data-semantic-children="45" data-semantic-content="35,39" data-semantic-parent="55" data-semantic-owns="35 45 39" data-semantic-level-number="5" data-speech-node="true" transform="translate(4610,0)"><g data-mml-node="mo" data-latex="(" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:7" data-semantic-id="35" data-semantic-parent="46" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true"><use data-c="28" href="#MJX-125-NCM-N-28"></use></g><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-annotation="depth:7" data-semantic-id="45" data-semantic-children="36,38" data-semantic-content="37" data-semantic-parent="46" data-semantic-owns="36 37 38" data-semantic-level-number="6" data-speech-node="true" transform="translate(389,0)"><g data-mml-node="mtext" data-latex="\textrm{0.0180 L}" data-semantic-type="text" data-semantic-role="text" data-semantic-font="normal" data-semantic-annotation="factor:operator;clearspeak:unit;depth:8" data-semantic-id="36" data-semantic-parent="45" data-semantic-attributes="latex:\textrm{0.0180 L}" data-semantic-level-number="7" 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data-c="28" href="#MJX-125-NCM-N-28"></use></g><g data-mml-node="mtext" data-latex="\text{5.3.5}" data-semantic-type="text" data-semantic-role="annotation" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-id="1" data-semantic-parent="3" data-semantic-attributes="latex:\text{5.3.5}" data-semantic-level-number="4" data-speech-node="true" transform="translate(389,0)"><use data-c="35" href="#MJX-125-NCM-N-35"></use><use data-c="2E" href="#MJX-125-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-125-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-125-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="35" href="#MJX-125-NCM-N-35" transform="translate(1556,0)"></use></g><g data-mml-node="mtext" data-latex="\text{)}" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="general:text;depth:5" data-semantic-id="2" data-semantic-parent="3" data-semantic-attributes="latex:\text{)}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true" transform="translate(2445,0)"><use data-c="29" href="#MJX-125-NCM-N-29"></use></g></g></g></g><g data-mml-node="mtd" id="mjx-eqn:5.3.6" data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="depth:3" data-semantic-id="63" data-semantic-children="62" data-semantic-parent="113" data-semantic-operator="row" data-semantic-owns="62" data-semantic-level-number="2" data-speech-node="true" transform="translate(0,31)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mrow" data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:4" data-semantic-id="62" data-semantic-children="60" data-semantic-content="59,61" data-semantic-parent="63" data-semantic-owns="59 60 61" data-semantic-level-number="3" data-speech-node="true"><g data-mml-node="mtext" data-latex="\text{(}" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="general:text;depth:5" data-semantic-id="59" data-semantic-parent="62" data-semantic-attributes="latex:\text{(}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true"><use data-c="28" href="#MJX-125-NCM-N-28"></use></g><g data-mml-node="mtext" data-latex="\text{5.3.6}" data-semantic-type="text" data-semantic-role="annotation" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-id="60" data-semantic-parent="62" data-semantic-attributes="latex:\text{5.3.6}" data-semantic-level-number="4" data-speech-node="true" transform="translate(389,0)"><use data-c="35" href="#MJX-125-NCM-N-35"></use><use data-c="2E" href="#MJX-125-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-125-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-125-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="36" href="#MJX-125-NCM-N-36" transform="translate(1556,0)"></use></g><g data-mml-node="mtext" data-latex="\text{)}" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="general:text;depth:5" data-semantic-id="61" data-semantic-parent="62" data-semantic-attributes="latex:\text{)}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true" transform="translate(2445,0)"><use data-c="29" href="#MJX-125-NCM-N-29"></use></g></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math data-latex="\begin{align}\Delta(PV)&=P\Delta V=P(V_{\textrm f}-V)=(1.00\textrm{ atm})(\textrm{0.0180 L}-\textrm{0.0197 L}) \\ &=(-1.7\times10^{-3}\;\mathrm{L\cdot atm})(101.3\;\mathrm{J/L\cdot atm})=-0.0017\textrm{ J}\end{align} \nonumber " display="block" data-semantic-structure="(114 (58 (15 (14 5 13 (12 6 (11 7 10 8) 9))) (57 (56 49 17 (51 18 50 (48 19 47 20)) 21 (53 22 52 (41 23 (40 (26 24 25) 27 28) 29)) 30 (55 (44 31 (43 32 33) 34) 54 (46 35 (45 36 37 38) 39)))) (4 (3 0 1 2))) (113 64 (112 (111 (109 105 66 (107 (101 67 (100 (97 68 69) 70 (99 (75 71 (74 72 73)) 98 (80 77 78 79))) 81) 106 (104 82 (103 83 102 (91 (90 85 86 87) 88 89)) 92)) 93 (108 94 95)) 96)) (63 (62 59 60 61))))"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt" data-break-align="bottom top" data-latex="\begin{align}\Delta(PV)&=P\Delta V=P(V_{\textrm f}-V)=(1.00\textrm{ atm})(\textrm{0.0180 L}-\textrm{0.0197 L}) \\ &=(-1.7\times10^{-3}\;\mathrm{L\cdot atm})(101.3\;\mathrm{J/L\cdot atm})=-0.0017\textrm{ J}\end{align} \nonumber" data-semantic-type="table" data-semantic-role="unknown" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="58,113" data-semantic-attributes="latex:\begin{align}\Delta(PV)&=P\Delta V=P(V_{\textrm f}-V)=(1.00\textrm{ atm})(\textrm{0.0180 L}-\textrm{0.0197 L}) \\ &=(-1.7\times10^{-3}\\0003B\mathrm{L\cdot atm})(101.3\\0003B\mathrm{J/L\cdot atm})=-0.0017\textrm{ J}\end{align} \nonumber " data-semantic-owns="58 113" data-semantic-level-number="0" data-speech-node="true"><mlabeledtr data-semantic-type="row" data-semantic-role="table" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="15,57" data-semantic-content="4" data-semantic-parent="114" data-semantic-owns="15 57 4" data-semantic-level-number="1" data-speech-node="true"><mtd data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="depth:3" data-semantic-id="4" data-semantic-children="3" data-semantic-parent="58" data-semantic-operator="row" data-semantic-owns="3" data-semantic-level-number="2" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:4" data-semantic-="" data-semantic-children="1" data-semantic-content="0,2" data-semantic-parent="4" data-semantic-owns="0 1 2" data-semantic-level-number="3" data-speech-node="true"><mtext data-latex="\text{(}" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="general:text;depth:5" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:\text{(}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true">(</mtext><mtext data-latex="\text{5.3.5}" data-semantic-type="text" data-semantic-role="annotation" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:\text{5.3.5}" data-semantic-level-number="4" data-speech-node="true">5.3.5</mtext><mtext data-latex="\text{)}" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="general:text;depth:5" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:\text{)}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true">)</mtext></mrow></mtd><mtd data-semantic-type="cell" data-semantic-role="table" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-children="14" data-semantic-parent="58" data-semantic-owns="14" data-semantic-level-number="2" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple;depth:4" data-semantic-="" data-semantic-children="5,12" data-semantic-content="13,5" data-semantic-parent="15" data-semantic-owns="5 13 12" data-semantic-level-number="3" data-speech-node="true"><mi mathvariant="normal" data-latex="\Delta" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:5" data-semantic-="" data-semantic-parent="14" data-semantic-attributes="latex:\Delta" data-semantic-operator="appl" data-semantic-level-number="4" data-speech-node="true">Δ</mi><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-annotation="nemeth:number;depth:5" data-semantic-="" data-semantic-parent="14" data-semantic-added="true" data-semantic-operator="appl" data-semantic-level-number="4" data-speech-node="true">⁡</mo><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:5" data-semantic-="" data-semantic-children="11" data-semantic-content="6,9" data-semantic-parent="14" data-semantic-owns="6 11 9" data-semantic-level-number="4" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="12" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="5" data-speech-node="true">(</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:6" data-semantic-="" data-semantic-children="7,8" data-semantic-content="10" data-semantic-parent="12" data-semantic-owns="7 10 8" data-semantic-level-number="5" data-speech-node="true"><mi data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="11" data-semantic-attributes="latex:P" data-semantic-level-number="6" data-speech-node="true">P</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="11" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="6" data-speech-node="true">⁢</mo><mi data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="11" data-semantic-attributes="latex:V" data-semantic-level-number="6" data-speech-node="true">V</mi></mrow><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="12" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="5" data-speech-node="true">)</mo></mrow></mrow></mtd><mtd data-semantic-type="cell" data-semantic-role="table" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-children="56" data-semantic-parent="58" data-semantic-owns="56" data-semantic-level-number="2" data-speech-node="true"><mstyle indentshift="2em"><mi></mi><mrow data-semantic-added="true" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:4" data-semantic-="" data-semantic-children="49,51,53,55" data-semantic-content="17,21,30" data-semantic-parent="57" data-semantic-owns="49 17 51 21 53 30 55" data-semantic-level-number="3" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="empty" data-semantic-role="unknown" data-semantic-annotation="nemeth:number;depth:5" data-semantic-="" data-semantic-parent="56" data-semantic-level-number="4" data-speech-node="true"></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:5" data-semantic-="" data-semantic-parent="56" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="4" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-children="18,48" data-semantic-content="50" data-semantic-parent="56" data-semantic-owns="18 50 48" data-semantic-level-number="4" data-speech-node="true"><mi data-latex="P" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:6" data-semantic-="" data-semantic-parent="51" data-semantic-attributes="latex:P" data-semantic-level-number="5" data-speech-node="true">P</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="51" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-children="19,20" data-semantic-content="47,19" data-semantic-parent="51" data-semantic-owns="19 47 20" data-semantic-level-number="5" data-speech-node="true"><mi mathvariant="normal" data-latex="\Delta" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:7" data-semantic-="" data-semantic-parent="48" data-semantic-attributes="latex:\Delta" data-semantic-operator="appl" data-semantic-level-number="6" data-speech-node="true">Δ</mi><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-annotation="nemeth:number;depth:7" data-semantic-="" data-semantic-parent="48" data-semantic-added="true" data-semantic-operator="appl" data-semantic-level-number="6" data-speech-node="true">⁡</mo><mi data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:7" data-semantic-="" data-semantic-parent="48" data-semantic-attributes="latex:V" data-semantic-level-number="6" data-speech-node="true">V</mi></mrow></mrow><mo 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data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="53" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:6" data-semantic-="" data-semantic-children="40" data-semantic-content="23,29" data-semantic-parent="53" data-semantic-owns="23 40 29" data-semantic-level-number="5" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="41" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">(</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-children="26,28" data-semantic-content="27" data-semantic-parent="41" data-semantic-owns="26 27 28" data-semantic-level-number="6" data-speech-node="true"><msub data-latex="V_{\textrm f}" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-children="24,25" data-semantic-parent="40" data-semantic-attributes="latex:V_{\textrm f}" data-semantic-owns="24 25" data-semantic-level-number="7" data-speech-node="true"><mi data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:9" data-semantic-="" data-semantic-parent="26" data-semantic-attributes="latex:V" data-semantic-level-number="8" data-speech-node="true">V</mi><mrow data-mjx-texclass="ORD" data-latex="{\textrm f}"><mtext data-latex="\textrm f" data-semantic-type="text" data-semantic-role="latinletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:9" data-semantic-="" data-semantic-parent="26" data-semantic-attributes="latex:{\textrm f};texclass:ORD" data-semantic-level-number="8" data-speech-node="true">f</mtext></mrow></msub><mo data-latex="-" data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="40" data-semantic-attributes="latex:-" data-semantic-operator="infixop,−" data-semantic-level-number="7" data-speech-node="true">−</mo><mi data-latex="V" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="40" data-semantic-attributes="latex:V" data-semantic-level-number="7" data-speech-node="true">V</mi></mrow><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="41" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">)</mo></mrow></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:5" data-semantic-="" data-semantic-parent="56" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="4" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-children="44,46" data-semantic-content="54" data-semantic-parent="56" data-semantic-owns="44 54 46" data-semantic-level-number="4" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:6" data-semantic-="" data-semantic-children="43" data-semantic-content="31,34" data-semantic-parent="55" data-semantic-owns="31 43 34" data-semantic-level-number="5" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="44" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">(</mo><mrow data-semantic-added="true" data-semantic-type="punctuated" data-semantic-role="text" data-semantic-annotation="clearspeak:unit;depth:7" data-semantic-="" data-semantic-children="32,33" data-semantic-parent="44" data-semantic-collapsed="(43 (c 42) 32 33)" data-semantic-owns="32 33" data-semantic-level-number="6" data-speech-node="true"><mn data-latex="1.00" data-semantic-type="number" data-semantic-role="float" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:8" data-semantic-="" data-semantic-parent="43" data-semantic-attributes="latex:1.00" data-semantic-level-number="7" data-speech-node="true">1.00</mn><mtext data-latex="\textrm{ atm}" data-semantic-type="text" data-semantic-role="text" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:8" data-semantic-="" data-semantic-parent="43" data-semantic-attributes="latex:\textrm{ atm}" data-semantic-level-number="7" data-speech-node="true"> atm</mtext></mrow><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="44" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">)</mo></mrow><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="55" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:6" data-semantic-="" data-semantic-children="45" data-semantic-content="35,39" data-semantic-parent="55" data-semantic-owns="35 45 39" data-semantic-level-number="5" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="46" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">(</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-children="36,38" data-semantic-content="37" data-semantic-parent="46" data-semantic-owns="36 37 38" data-semantic-level-number="6" data-speech-node="true"><mtext data-latex="\textrm{0.0180 L}" data-semantic-type="text" data-semantic-role="text" data-semantic-font="normal" data-semantic-annotation="factor:operator;clearspeak:unit;depth:8" data-semantic-="" data-semantic-parent="45" data-semantic-attributes="latex:\textrm{0.0180 L}" data-semantic-level-number="7" data-speech-node="true">0.0180 L</mtext><mo data-latex="-" data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="45" data-semantic-attributes="latex:-" data-semantic-operator="infixop,−" data-semantic-level-number="7" data-speech-node="true">−</mo><mtext data-latex="\textrm{0.0197 L}" data-semantic-type="text" data-semantic-role="text" data-semantic-font="normal" data-semantic-annotation="factor:operator;clearspeak:unit;depth:8" data-semantic-="" data-semantic-parent="45" data-semantic-attributes="latex:\textrm{0.0197 L}" data-semantic-level-number="7" data-speech-node="true">0.0197 L</mtext></mrow><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="46" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">)</mo></mrow></mrow></mrow></mstyle></mtd></mlabeledtr><mlabeledtr data-semantic-type="row" data-semantic-role="table" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="64,112" data-semantic-content="63" data-semantic-parent="114" data-semantic-owns="64 112 63" data-semantic-level-number="1" data-speech-node="true"><mtd data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="depth:3" data-semantic-id="63" data-semantic-children="62" data-semantic-parent="113" data-semantic-operator="row" data-semantic-owns="62" data-semantic-level-number="2" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:4" data-semantic-="" data-semantic-children="60" data-semantic-content="59,61" data-semantic-parent="63" data-semantic-owns="59 60 61" data-semantic-level-number="3" data-speech-node="true"><mtext data-latex="\text{(}" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="general:text;depth:5" data-semantic-="" data-semantic-parent="62" data-semantic-attributes="latex:\text{(}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true">(</mtext><mtext data-latex="\text{5.3.6}" data-semantic-type="text" data-semantic-role="annotation" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-parent="62" data-semantic-attributes="latex:\text{5.3.6}" data-semantic-level-number="4" data-speech-node="true">5.3.6</mtext><mtext data-latex="\text{)}" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="general:text;depth:5" data-semantic-="" data-semantic-parent="62" data-semantic-attributes="latex:\text{)}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true">)</mtext></mrow></mtd><mtd data-semantic-type="cell" data-semantic-role="table" data-semantic-annotation="nemeth:number;depth:3" data-semantic-="" data-semantic-parent="113" data-semantic-level-number="2" data-speech-node="true"></mtd><mtd data-semantic-type="cell" data-semantic-role="table" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-children="111" data-semantic-parent="113" data-semantic-owns="111" data-semantic-level-number="2" data-speech-node="true"><mstyle indentshift="2em"><mi></mi><mrow data-semantic-added="true" data-semantic-type="punctuated" data-semantic-role="text" data-semantic-annotation="depth:4" data-semantic-="" data-semantic-children="109,96" data-semantic-parent="112" data-semantic-collapsed="(111 (c 110) 109 96)" data-semantic-owns="109 96" data-semantic-level-number="3" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-children="105,107,108" data-semantic-content="66,93" data-semantic-parent="111" data-semantic-owns="105 66 107 93 108" data-semantic-level-number="4" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="empty" data-semantic-role="unknown" data-semantic-annotation="nemeth:number;depth:6" data-semantic-="" data-semantic-parent="109" data-semantic-level-number="5" data-speech-node="true"></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:6" data-semantic-="" data-semantic-parent="109" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="5" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:6" data-semantic-="" data-semantic-children="101,104" data-semantic-content="106" data-semantic-parent="109" data-semantic-owns="101 106 104" data-semantic-level-number="5" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:7" data-semantic-="" data-semantic-children="100" data-semantic-content="67,81" data-semantic-parent="107" data-semantic-owns="67 100 81" data-semantic-level-number="6" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="101" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">(</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="multiplication" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-children="97,99" data-semantic-content="70" data-semantic-parent="101" data-semantic-owns="97 70 99" data-semantic-level-number="7" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="prefixop" data-semantic-role="negative" data-semantic-annotation="clearspeak:simple;depth:9" data-semantic-="" data-semantic-children="69" data-semantic-content="68" data-semantic-parent="100" data-semantic-owns="68 69" data-semantic-level-number="8" data-speech-node="true"><mo data-latex="-" data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-annotation="depth:10" data-semantic-="" data-semantic-parent="97" data-semantic-attributes="latex:-" data-semantic-operator="prefixop,−" data-semantic-level-number="9" data-speech-node="true">−</mo><mn data-latex="1.7" data-semantic-type="number" data-semantic-role="float" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:10" data-semantic-="" data-semantic-parent="97" data-semantic-attributes="latex:1.7" data-semantic-level-number="9" data-speech-node="true">1.7</mn></mrow><mo data-latex="\times" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:9" data-semantic-="" data-semantic-parent="100" data-semantic-attributes="latex:\times" data-semantic-operator="infixop,×" data-semantic-level-number="8" data-speech-node="true">×</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:9" data-semantic-="" data-semantic-children="75,80" data-semantic-content="98" data-semantic-parent="100" data-semantic-owns="75 98 80" data-semantic-level-number="8" data-speech-node="true"><msup data-latex="10^{-3}" data-semantic-type="superscript" data-semantic-role="integer" data-semantic-annotation="depth:10" data-semantic-="" data-semantic-children="71,74" data-semantic-parent="99" data-semantic-attributes="latex:10^{-3}" data-semantic-owns="71 74" data-semantic-level-number="9" data-speech-node="true"><mn data-latex="10" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:11" data-semantic-="" data-semantic-parent="75" data-semantic-attributes="latex:10" data-semantic-level-number="10" data-speech-node="true">10</mn><mrow data-mjx-texclass="ORD" data-latex="{-3}" data-semantic-type="prefixop" data-semantic-role="negative" data-semantic-annotation="clearspeak:simple;depth:11" data-semantic-="" data-semantic-children="73" data-semantic-content="72" data-semantic-parent="75" data-semantic-attributes="latex:{-3};texclass:ORD" data-semantic-owns="72 73" data-semantic-level-number="10" data-speech-node="true"><mo data-latex="-" data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-annotation="depth:12" data-semantic-="" data-semantic-parent="74" data-semantic-attributes="latex:-" data-semantic-operator="prefixop,−" data-semantic-level-number="11" data-speech-node="true">−</mo><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:12" data-semantic-="" data-semantic-parent="74" data-semantic-attributes="latex:3" data-semantic-level-number="11" data-speech-node="true">3</mn></mrow></msup><mspace width="0.278em" data-latex="\;"></mspace><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:10" data-semantic-="" data-semantic-parent="99" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="9" data-speech-node="true">⁢</mo><mrow data-mjx-texclass="ORD" data-latex="\mathrm{L\cdot atm}" data-semantic-type="infixop" data-semantic-role="multiplication" data-semantic-annotation="depth:10" data-semantic-="" data-semantic-children="77,79" data-semantic-content="78" data-semantic-parent="99" data-semantic-attributes="latex:\mathrm{L\cdot atm};texclass:ORD" data-semantic-owns="77 78 79" data-semantic-level-number="9" data-speech-node="true"><mi mathvariant="normal" data-latex="L" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:11" data-semantic-="" data-semantic-parent="80" data-semantic-attributes="latex:L" data-semantic-level-number="10" data-speech-node="true">L</mi><mo data-latex="\cdot" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:11" data-semantic-="" data-semantic-parent="80" data-semantic-attributes="latex:\cdot" data-semantic-operator="infixop,⋅" data-semantic-level-number="10" data-speech-node="true">⋅</mo><mi data-mjx-auto-op="false" data-latex="atm" data-semantic-type="identifier" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="depth:11" data-semantic-="" data-semantic-parent="80" data-semantic-attributes="latex:atm" data-semantic-level-number="10" data-speech-node="true">atm</mi></mrow></mrow></mrow><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="101" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">)</mo></mrow><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="107" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="6" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:7" data-semantic-="" data-semantic-children="103" data-semantic-content="82,92" data-semantic-parent="107" data-semantic-owns="82 103 92" data-semantic-level-number="6" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="104" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">(</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:8" data-semantic-="" data-semantic-children="83,91" data-semantic-content="102" data-semantic-parent="104" data-semantic-owns="83 102 91" data-semantic-level-number="7" data-speech-node="true"><mn data-latex="101.3" data-semantic-type="number" data-semantic-role="float" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:9" data-semantic-="" data-semantic-parent="103" data-semantic-attributes="latex:101.3" data-semantic-level-number="8" data-speech-node="true">101.3</mn><mspace width="0.278em" data-latex="\;"></mspace><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:9" data-semantic-="" data-semantic-parent="103" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="8" data-speech-node="true">⁢</mo><mrow data-mjx-texclass="ORD" data-latex="\mathrm{J/L\cdot atm}" data-semantic-type="infixop" data-semantic-role="multiplication" data-semantic-annotation="depth:9" data-semantic-="" data-semantic-children="90,89" data-semantic-content="88" data-semantic-parent="103" data-semantic-attributes="latex:\mathrm{J/L\cdot atm};texclass:ORD" data-semantic-owns="90 88 89" data-semantic-level-number="8" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="division" data-semantic-annotation="depth:10" data-semantic-="" data-semantic-children="85,87" data-semantic-content="86" data-semantic-parent="91" data-semantic-owns="85 86 87" data-semantic-level-number="9" data-speech-node="true"><mi mathvariant="normal" data-latex="J" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:11" data-semantic-="" data-semantic-parent="90" data-semantic-attributes="latex:J" data-semantic-level-number="10" data-speech-node="true">J</mi><mo data-latex="/" data-semantic-type="operator" data-semantic-role="division" data-semantic-annotation="depth:11" data-semantic-="" data-semantic-parent="90" data-semantic-attributes="latex:/" data-semantic-operator="infixop,/" data-semantic-level-number="10" data-speech-node="true">/</mo><mi mathvariant="normal" data-latex="L" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:11" data-semantic-="" data-semantic-parent="90" data-semantic-attributes="latex:L" data-semantic-level-number="10" data-speech-node="true">L</mi></mrow><mo data-latex="\cdot" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:10" data-semantic-="" data-semantic-parent="91" data-semantic-attributes="latex:\cdot" data-semantic-operator="infixop,⋅" data-semantic-level-number="9" data-speech-node="true">⋅</mo><mi data-mjx-auto-op="false" data-latex="atm" data-semantic-type="identifier" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="depth:10" data-semantic-="" data-semantic-parent="91" data-semantic-attributes="latex:atm" data-semantic-level-number="9" data-speech-node="true">atm</mi></mrow></mrow><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="104" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">)</mo></mrow></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:6" data-semantic-="" data-semantic-parent="109" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="5" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="prefixop" data-semantic-role="negative" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-children="95" data-semantic-content="94" data-semantic-parent="109" data-semantic-owns="94 95" data-semantic-level-number="5" data-speech-node="true"><mo data-latex="-" data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="108" data-semantic-attributes="latex:-" data-semantic-operator="prefixop,−" data-semantic-level-number="6" data-speech-node="true">−</mo><mn data-latex="0.0017" data-semantic-type="number" data-semantic-role="float" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:7" data-semantic-="" data-semantic-parent="108" data-semantic-attributes="latex:0.0017" data-semantic-level-number="6" data-speech-node="true">0.0017</mn></mrow></mrow><mtext data-latex="\textrm{ J}" data-semantic-type="text" data-semantic-role="text" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:5" data-semantic-="" data-semantic-parent="111" data-semantic-attributes="latex:\textrm{ J}" data-semantic-level-number="4" data-speech-node="true"> J</mtext></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div> <p class="lt-chem-21723" style="background-color: unset;">C Substituting the calculated values of ΔH and PΔV into Equation 18.11,</p> <p class="mt-align-center lt-chem-21723" style="background-color: unset;">ΔU = ΔH − PΔV = 6010 J − (−0.0017 J) = 6010 J = 6.01 kJ</p> </div> <div class="box-exercise"> <span id="Exercise_.5C(.5CPageIndex.7B1.7D.5C)"></span><h5 class="box-legend"><span class="lt-icon-default">Exercise <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="125" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-126-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-126-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-126-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-126-NCM-N-31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\PageIndex{1}" data-semantic-structure="(5 0 1 2 3 4)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="1}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-id="5" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{1};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><g data-mml-node="mn" data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true"><use data-c="35" href="#MJX-126-NCM-N-35"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(500,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="1" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-126-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true" transform="translate(778,0)"><use data-c="33" href="#MJX-126-NCM-N-33"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(1278,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="3" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-126-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="1" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="4" data-semantic-parent="5" data-semantic-attributes="latex:1" data-semantic-level-number="1" data-speech-node="true" transform="translate(1556,0)"><use data-c="31" href="#MJX-126-NCM-N-31"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\PageIndex{1}" data-semantic-structure="(5 0 1 2 3 4)"><mrow data-mjx-texclass="ORD" data-latex="1}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{1};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><mn data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true">5</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true">3</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="1" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:1" data-semantic-level-number="1" data-speech-node="true">1</mn></mrow></math></mjx-assistive-mml></mjx-container></span></h5> <p class="lt-chem-21723" style="background-color: unset;">At 298 K and 1 atm, the conversion of graphite to diamond requires the input of 1.850 kJ of heat per mole of carbon. The molar volumes of graphite and diamond are 0.00534 L and 0.00342 L, respectively. Calculate ΔH and ΔU for the conversion of C (graphite) to C (diamond) under these conditions.</p> <dl> <dt><strong class="emphasis bold">Answer a</strong></dt> <dd> <p class="lt-chem-21723" style="background-color: unset;">ΔH = 1.85 kJ/mol</p> </dd> <dt><strong class="emphasis bold">Answer b</strong></dt> <dd> <p class="lt-chem-21723" style="background-color: unset;">ΔU = 1.85 kJ/mol</p> </dd> </dl> </div> <span id="The_Relationship_between_.CE.94H_and_.CE.94U"></span><h2 style="background-color: unset;" class="lt-chem-21723">The Relationship between ΔH and ΔU</h2> <p class="lt-chem-21723" style="background-color: unset;">If ΔH for a reaction is known, we can use the change in the enthalpy of the system (Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="126" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.581ex" role="img" focusable="false" viewbox="0 -677 2056 699" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-127-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-127-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-127-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-127-NCM-N-34" d="M353 677C344 677 336 672 330 663L28 199L28 163L289 163L289 81C289 63 285 51 278 46C271 41 252 39 219 39L194 39L194 0C223 2 269 3 331 3C393 3 439 2 468 0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{5.3.5}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#mjx-eqn:5.3.5"><g data-mml-node="mrow" data-latex="\ref{5.3.5}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="2056" height="699" x="0" y="-22"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="35" href="#MJX-127-NCM-N-35"></use><use data-c="2E" href="#MJX-127-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-127-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-127-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="34" href="#MJX-127-NCM-N-34" transform="translate(1556,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{5.3.5}" data-semantic-structure="0"><mrow href="#mjx-eqn%3A5.3.5" data-latex="\ref{5.3.5}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">5.3.4</mtext></mrow></math></mjx-assistive-mml></mjx-container>) to calculate its change in internal energy. When a reaction involves only solids, liquids, liquid solutions, or any combination of these, the volume does not change appreciably (ΔV = 0). Under these conditions, we can simplify Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="127" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.581ex" role="img" focusable="false" viewbox="0 -677 2056 699" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-128-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-128-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-128-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-128-NCM-N-34" d="M353 677C344 677 336 672 330 663L28 199L28 163L289 163L289 81C289 63 285 51 278 46C271 41 252 39 219 39L194 39L194 0C223 2 269 3 331 3C393 3 439 2 468 0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{5.3.5}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#mjx-eqn:5.3.5"><g data-mml-node="mrow" data-latex="\ref{5.3.5}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="2056" height="699" x="0" y="-22"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="35" href="#MJX-128-NCM-N-35"></use><use data-c="2E" href="#MJX-128-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-128-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-128-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="34" href="#MJX-128-NCM-N-34" transform="translate(1556,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{5.3.5}" data-semantic-structure="0"><mrow href="#mjx-eqn%3A5.3.5" data-latex="\ref{5.3.5}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">5.3.4</mtext></mrow></math></mjx-assistive-mml></mjx-container> to ΔH = ΔU. If gases are involved, however, ΔH and ΔU can differ significantly. We can calculate ΔU from the measured value of ΔH by using the right side of Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="128" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.581ex" role="img" focusable="false" viewbox="0 -677 2056 699" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-129-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-129-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-129-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-129-NCM-N-34" d="M353 677C344 677 336 672 330 663L28 199L28 163L289 163L289 81C289 63 285 51 278 46C271 41 252 39 219 39L194 39L194 0C223 2 269 3 331 3C393 3 439 2 468 0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{5.3.5}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#mjx-eqn:5.3.5"><g data-mml-node="mrow" data-latex="\ref{5.3.5}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="2056" height="699" x="0" y="-22"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="35" href="#MJX-129-NCM-N-35"></use><use data-c="2E" href="#MJX-129-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-129-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-129-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="34" href="#MJX-129-NCM-N-34" transform="translate(1556,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{5.3.5}" data-semantic-structure="0"><mrow href="#mjx-eqn%3A5.3.5" data-latex="\ref{5.3.5}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">5.3.4</mtext></mrow></math></mjx-assistive-mml></mjx-container> together with the ideal gas law, PV = nRT. Recognizing that Δ(PV) = Δ(nRT), we can rewrite Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="129" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.581ex" role="img" focusable="false" viewbox="0 -677 2056 699" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-130-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 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0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{5.3.5}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#mjx-eqn:5.3.5"><g data-mml-node="mrow" data-latex="\ref{5.3.5}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="2056" height="699" x="0" y="-22"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="35" href="#MJX-130-NCM-N-35"></use><use data-c="2E" href="#MJX-130-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-130-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-130-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="34" href="#MJX-130-NCM-N-34" transform="translate(1556,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{5.3.5}" data-semantic-structure="0"><mrow href="#mjx-eqn%3A5.3.5" data-latex="\ref{5.3.5}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{5.3.5};href:#mjx-eqn%3A5.3.5" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">5.3.4</mtext></mrow></math></mjx-assistive-mml></mjx-container> as follows:</p> <p class="mt-align-center lt-chem-21723" style="background-color: unset;"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" display="true" width="full" tabindex="0" ctxtmenu_counter="130" style="font-size: 85%; min-width: 54.5ex; position: relative;"><svg width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 24089 1000" aria-hidden="true" style="vertical-align: -0.566ex; min-width: 54.5ex;"><defs><path id="MJX-131-NCM-N-394" d="M759 0C776 0 785 4 785 13L445 696C438 709 429 716 416 716C403 716 394 709 387 696L52 25C49 18 47 14 47 12C47 4 56 0 73 0M384 604L644 83L124 83Z"></path><path id="MJX-131-NCM-N-2061" d=""></path><path id="MJX-131-NCM-I-1D43B" d="M881 668C881 678 875 683 863 683L736 680L609 683C593 683 586 674 586 659C586 652 589 647 594 646C604 645 612 644 617 644C648 643 665 641 670 640C675 639 678 636 678 631C677 628 676 622 674 613L615 375L321 375L379 602C384 623 393 636 406 641C413 643 430 644 458 644C485 644 496 644 496 668C496 678 490 683 478 683L351 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d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-131-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-131-NCM-N-37" d="M475 604C482 613 485 626 485 644L243 644C174 644 135 648 128 657C125 660 122 667 120 676L89 676L55 464L88 464C98 520 106 550 112 555C115 558 146 560 205 560L401 560L295 410C214 295 174 171 174 36C174-3 190-22 223-22C256-22 272-3 272 36L272 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data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:6" data-semantic-="" data-semantic-parent="40" data-semantic-attributes="latex:U" data-semantic-level-number="5" data-speech-node="true">U</mi></mrow><mo data-latex="+" data-semantic-type="operator" data-semantic-role="addition" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="43" data-semantic-attributes="latex:+" data-semantic-operator="infixop,+" data-semantic-level-number="4" data-speech-node="true">+</mo><mrow data-semantic-added="true" data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple;depth:5" data-semantic-="" data-semantic-children="11,28" data-semantic-content="37,11" data-semantic-parent="43" data-semantic-owns="11 37 28" data-semantic-level-number="4" data-speech-node="true"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="simple function" 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data-semantic-level-number="5" data-speech-node="true">Δ</mi><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-annotation="nemeth:number;depth:6" data-semantic-="" data-semantic-parent="36" data-semantic-added="true" data-semantic-operator="appl" data-semantic-level-number="5" data-speech-node="true">⁡</mo><mi data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:6" data-semantic-="" data-semantic-parent="36" data-semantic-attributes="latex:U" data-semantic-level-number="5" data-speech-node="true">U</mi></mrow><mo data-latex="+" data-semantic-type="operator" data-semantic-role="addition" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="44" data-semantic-attributes="latex:+" data-semantic-operator="infixop,+" data-semantic-level-number="4" data-speech-node="true">+</mo><mrow data-semantic-added="true" 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data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:6" data-semantic-="" data-semantic-children="31" data-semantic-content="21,25" data-semantic-parent="34" data-semantic-owns="21 31 25" data-semantic-level-number="5" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="32" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">(</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:7" data-semantic-="" data-semantic-children="22,23,24" data-semantic-content="29,30" data-semantic-parent="32" data-semantic-owns="22 29 23 30 24" data-semantic-level-number="6" data-speech-node="true"><mi data-latex="n" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="31" data-semantic-attributes="latex:n" data-semantic-level-number="7" data-speech-node="true">n</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="31" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="7" data-speech-node="true">⁢</mo><mi data-latex="R" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="31" data-semantic-attributes="latex:R" data-semantic-level-number="7" data-speech-node="true">R</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="31" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="7" data-speech-node="true">⁢</mo><mi data-latex="T" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="31" data-semantic-attributes="latex:T" data-semantic-level-number="7" data-speech-node="true">T</mi></mrow><mo data-latex="\label{5.3.6}" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="32" data-semantic-attributes="latex:\label{5.3.6}" data-semantic-operator="fenced" data-semantic-level-number="6" data-speech-node="true">)</mo></mrow></mrow></mrow></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></p> <p class="lt-chem-21723" style="background-color: unset;">At constant temperature, Δ(nRT) = RTΔn, where Δn is the difference between the final and initial numbers of moles of gas. 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data-semantic-annotation="general:text;depth:5" data-semantic-id="0" data-semantic-parent="3" data-semantic-attributes="latex:\text{(}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true"><use data-c="28" href="#MJX-132-NCM-N-28"></use></g><g data-mml-node="mtext" data-latex="\text{5.3.8}" data-semantic-type="text" data-semantic-role="annotation" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-id="1" data-semantic-parent="3" data-semantic-attributes="latex:\text{5.3.8}" data-semantic-level-number="4" data-speech-node="true" transform="translate(389,0)"><use data-c="35" href="#MJX-132-NCM-N-35"></use><use data-c="2E" href="#MJX-132-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-132-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-132-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="38" href="#MJX-132-NCM-N-38" transform="translate(1556,0)"></use></g><g data-mml-node="mtext" data-latex="\text{)}" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="general:text;depth:5" data-semantic-id="2" data-semantic-parent="3" data-semantic-attributes="latex:\text{)}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true" transform="translate(2445,0)"><use data-c="29" href="#MJX-132-NCM-N-29"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math data-latex="ΔU = ΔH − RTΔn \label{5.3.7} " display="block" data-semantic-structure="(27 (26 (24 (16 5 15 6) 7 (23 (18 8 17 9) 10 (22 11 19 12 20 13 21 14))) (4 (3 0 1 2))))"><mtable displaystyle="true" data-latex="ΔU = ΔH − RTΔn \label{5.3.7}" data-semantic-type="multiline" data-semantic-role="equality" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="26" data-semantic-attributes="latex:ΔU = ΔH − RTΔn \label{5.3.7} " data-semantic-owns="26" data-semantic-level-number="0" data-speech-node="true"><mlabeledtr data-semantic-type="line" data-semantic-role="multiline" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="24" data-semantic-content="4" data-semantic-parent="27" data-semantic-owns="24 4" data-semantic-level-number="1" data-speech-node="true"><mtd data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="depth:3" data-semantic-id="4" data-semantic-children="3" data-semantic-parent="26" data-semantic-owns="3" data-semantic-level-number="2" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:4" data-semantic-="" data-semantic-children="1" data-semantic-content="0,2" data-semantic-parent="4" data-semantic-owns="0 1 2" data-semantic-level-number="3" data-speech-node="true"><mtext data-latex="\text{(}" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="general:text;depth:5" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:\text{(}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true">(</mtext><mtext data-latex="\text{5.3.8}" data-semantic-type="text" data-semantic-role="annotation" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:\text{5.3.8}" data-semantic-level-number="4" data-speech-node="true">5.3.8</mtext><mtext data-latex="\text{)}" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="general:text;depth:5" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:\text{)}" data-semantic-operator="fenced" data-semantic-level-number="4" data-speech-node="true">)</mtext></mrow></mtd><mtd><mrow data-semantic-added="true" data-semantic-type="relseq" data-semantic-role="equality" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-children="16,23" data-semantic-content="7" data-semantic-parent="26" data-semantic-owns="16 7 23" data-semantic-level-number="2" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:4" data-semantic-="" data-semantic-children="5,6" data-semantic-content="15" data-semantic-parent="24" data-semantic-owns="5 15 6" data-semantic-level-number="3" data-speech-node="true"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:5" data-semantic-="" data-semantic-parent="16" data-semantic-attributes="latex:Δ" data-semantic-level-number="4" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="16" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="4" data-speech-node="true">⁢</mo><mi data-latex="U" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:5" data-semantic-="" data-semantic-parent="16" data-semantic-attributes="latex:U" data-semantic-level-number="4" data-speech-node="true">U</mi></mrow><mo data-latex="=" data-semantic-type="relation" data-semantic-role="equality" data-semantic-annotation="nemeth:number;depth:4" data-semantic-="" data-semantic-parent="24" data-semantic-attributes="latex:=" data-semantic-operator="relseq,=" data-semantic-level-number="3" data-speech-node="true">=</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="subtraction" data-semantic-annotation="depth:4" data-semantic-="" 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data-semantic-parent="18" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mi data-latex="H" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="18" data-semantic-attributes="latex:H" data-semantic-level-number="5" data-speech-node="true">H</mi></mrow><mo data-latex="−" data-semantic-type="operator" data-semantic-role="subtraction" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="23" data-semantic-attributes="latex:−" data-semantic-operator="infixop,−" data-semantic-level-number="4" data-speech-node="true">−</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-children="11,12,13,14" data-semantic-content="19,20,21" data-semantic-parent="23" data-semantic-owns="11 19 12 20 13 21 14" data-semantic-level-number="4" data-speech-node="true"><mi data-latex="R" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="22" data-semantic-attributes="latex:R" data-semantic-level-number="5" data-speech-node="true">R</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="22" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mi data-latex="T" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="22" data-semantic-attributes="latex:T" data-semantic-level-number="5" data-speech-node="true">T</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="22" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="22" data-semantic-attributes="latex:Δ" data-semantic-level-number="5" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="22" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mi data-latex="\label{5.3.7}" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="22" data-semantic-attributes="latex:\label{5.3.7}" data-semantic-level-number="5" data-speech-node="true">n</mi></mrow></mrow></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></p> <p class="lt-chem-21723" style="background-color: unset;">For reactions that result in a net production of gas, Δn > 0, so ΔU < ΔH. Conversely, endothermic reactions (ΔH > 0) that result in a net consumption of gas have Δn < 0 and ΔU > ΔH. The relationship between ΔH and ΔU for systems involving gases is illustrated in Example <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="132" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-133-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-133-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-133-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-133-NCM-N-32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\PageIndex{2}" data-semantic-structure="(5 0 1 2 3 4)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="2}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-id="5" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{2};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><g data-mml-node="mn" data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true"><use data-c="35" href="#MJX-133-NCM-N-35"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(500,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="1" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-133-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true" transform="translate(778,0)"><use data-c="33" href="#MJX-133-NCM-N-33"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(1278,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="3" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-133-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="4" data-semantic-parent="5" data-semantic-attributes="latex:2" data-semantic-level-number="1" data-speech-node="true" transform="translate(1556,0)"><use data-c="32" href="#MJX-133-NCM-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\PageIndex{2}" data-semantic-structure="(5 0 1 2 3 4)"><mrow data-mjx-texclass="ORD" data-latex="2}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{2};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><mn data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true">5</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true">3</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:2" data-semantic-level-number="1" data-speech-node="true">2</mn></mrow></math></mjx-assistive-mml></mjx-container>.</p> <div class="box-note"> <p class="lt-chem-21723" style="background-color: unset;">For reactions that result in a net production of gas, ΔU < ΔH. For endothermic reactions that result in a net consumption of gas, ΔU > ΔH.</p> </div> <div class="box-example"> <span id="Example_.5C(.5CPageIndex.7B2.7D.5C):_The_Relationship_Between_.CE.94H_and_.CE.94U_in_Gas_Evolving_Reactions"></span><h5 class="box-legend"><span class="lt-icon-default">Example <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="133" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-134-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-134-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-134-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-134-NCM-N-32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\PageIndex{2}" data-semantic-structure="(5 0 1 2 3 4)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="2}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-id="5" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{2};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><g data-mml-node="mn" data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true"><use data-c="35" href="#MJX-134-NCM-N-35"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(500,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="1" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-134-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true" transform="translate(778,0)"><use data-c="33" href="#MJX-134-NCM-N-33"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(1278,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="3" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-134-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="4" data-semantic-parent="5" data-semantic-attributes="latex:2" data-semantic-level-number="1" data-speech-node="true" transform="translate(1556,0)"><use data-c="32" href="#MJX-134-NCM-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\PageIndex{2}" data-semantic-structure="(5 0 1 2 3 4)"><mrow data-mjx-texclass="ORD" data-latex="2}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{2};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><mn data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true">5</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true">3</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:2" data-semantic-level-number="1" data-speech-node="true">2</mn></mrow></math></mjx-assistive-mml></mjx-container>: The Relationship Between ΔH and ΔU in Gas Evolving Reactions</span></h5> <p class="lt-chem-21723" style="background-color: unset;">The combustion of graphite to produce carbon dioxide is described by the equation </p> <p class="mt-align-center lt-chem-21723" style="background-color: unset;"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" display="true" width="full" tabindex="0" ctxtmenu_counter="134" style="font-size: 85%; min-width: 31.245ex; position: relative;"><svg width="100%" height="2.49ex" role="img" focusable="false" data-mjx-viewbox="0 -800.3 13810.2 1100.7" aria-hidden="true" style="vertical-align: -0.68ex; min-width: 31.245ex;"><defs><path id="MJX-135-NCM-I-1D436" d="M148 218C148 346 204 476 267 548C318 607 406 666 505 666C609 666 660 587 660 479C660 470 657 438 657 429C657 420 663 415 676 415C681 415 685 416 688 417C693 424 696 431 698 438L760 691C760 700 755 705 745 705C741 705 735 701 727 692L662 619C623 676 568 705 497 705C442 705 388 692 333 667C222 615 141 533 89 421C63 366 50 310 50 253C50 173 75 108 126 56C177 4 242-22 322-22C401-22 473 7 538 64C565 87 588 115 607 146C634 191 648 222 648 241C648 250 643 255 632 255C623 255 618 251 616 242C595 177 562 125 517 88C460 41 400 17 338 17C218 17 148 98 148 218Z"></path><path id="MJX-135-NCM-N-28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 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data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true" transform="translate(866,0)"><use data-c="29" href="#MJX-135-NCM-N-29"></use></g></g></g></g></g></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="0 -800.3 1 1100.7"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:" data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="55" data-semantic-level-number="2" data-speech-node="true" transform="translate(0,50.3)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,0)"></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math data-latex="C_{(graphite, s)} + O_{2(g)} → CO_{2(g)} \nonumber " display="block" data-semantic-structure="(56 (55 (53 (50 (28 1 (27 2 (26 (25 (23 3 22 (21 4 19 5 20 6)) 24 (18 7 17 (16 8 14 9 15 10))) 11 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data-semantic-parent="35" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">(</mo><mi data-latex="g" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="35" data-semantic-attributes="latex:g" data-semantic-level-number="7" data-speech-node="true">g</mi><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="35" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">)</mo></mrow></mrow></msub></mrow><mo data-latex="→" stretchy="false" data-semantic-type="relation" data-semantic-role="arrow" data-semantic-annotation="nemeth:number;depth:4" data-semantic-="" data-semantic-parent="53" data-semantic-attributes="latex:→" data-semantic-operator="relseq,→" data-semantic-level-number="3" data-speech-node="true">→</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:4" data-semantic-="" data-semantic-children="40,49" data-semantic-content="51" data-semantic-parent="53" data-semantic-owns="40 51 49" data-semantic-level-number="3" data-speech-node="true"><mi data-latex="C" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:5" data-semantic-="" data-semantic-parent="52" data-semantic-attributes="latex:C" data-semantic-level-number="4" data-speech-node="true">C</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-parent="52" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="4" data-speech-node="true">⁢</mo><msub data-latex="O_{2(g)}" data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-annotation="depth:5" data-semantic-="" data-semantic-children="41,48" data-semantic-parent="52" data-semantic-attributes="latex:O_{2(g)}" data-semantic-owns="41 48" data-semantic-level-number="4" data-speech-node="true"><mi data-latex="O" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-parent="49" data-semantic-attributes="latex:O" data-semantic-level-number="5" data-speech-node="true">O</mi><mrow data-mjx-texclass="ORD" data-latex="{2(g)}" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:6" data-semantic-="" data-semantic-children="42,46" data-semantic-content="47" data-semantic-parent="49" data-semantic-attributes="latex:{2(g)};texclass:ORD" data-semantic-owns="42 47 46" data-semantic-level-number="5" data-speech-node="true"><mn data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:7" data-semantic-="" data-semantic-parent="48" data-semantic-attributes="latex:2" data-semantic-level-number="6" data-speech-node="true">2</mn><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-parent="48" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="6" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:7" data-semantic-="" data-semantic-children="44" data-semantic-content="43,45" data-semantic-parent="48" data-semantic-owns="43 44 45" data-semantic-level-number="6" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="46" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">(</mo><mi data-latex="g" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="46" data-semantic-attributes="latex:g" data-semantic-level-number="7" data-speech-node="true">g</mi><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="46" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">)</mo></mrow></mrow></msub></mrow></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></p> <p class="lt-chem-21723" style="background-color: unset;">At 298 K and 1.0 atm, ΔH = −393.5 kJ/mol of graphite for this reaction, and the molar volume of graphite is 0.0053 L. What is ΔU for the reaction?</p> <p class="lt-chem-21723" style="background-color: unset;"><strong>Given</strong>: balanced chemical equation, temperature, pressure, ΔH, and molar volume of reactant</p> <p class="lt-chem-21723" style="background-color: unset;"><strong>Asked for</strong>: ΔU</p> <p class="lt-chem-21723" style="background-color: unset;"><strong>Strategy</strong>:</p> <ol start="1" style="list-style-type: upper-alpha;"> <li class="lt-chem-21723" style="background-color: unset;">Use the balanced chemical equation to calculate the change in the number of moles of gas during the reaction.</li> <li class="lt-chem-21723" style="background-color: unset;">Substitute this value and the data given into Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="135" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-136-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-136-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-136-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-136-NCM-N-38" d="M250 666C201 666 158 650 122 618C86 586 68 546 68 497C68 456 83 419 113 386C120 378 142 361 179 335C88 288 42 227 42 153C42 100 64 57 107 24C147-7 194-22 249-22C305-22 354-4 395 32C436 68 457 115 457 170C457 216 441 257 408 294C396 307 365 330 316 361C393 402 431 454 431 515C431 606 344 666 250 666M379 515C379 463 348 418 286 381L167 459C136 479 120 505 120 536C120 596 185 633 249 633C318 633 379 584 379 515M250 14C168 14 99 73 99 153C99 220 136 274 210 315L328 240C376 209 400 174 400 134C400 62 325 14 250 14Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{5.3.7}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#mjx-eqn:5.3.7"><g data-mml-node="mrow" data-latex="\ref{5.3.7}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="2056" height="688" x="0" y="-22"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{5.3.7};href:#mjx-eqn%3A5.3.7" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="35" href="#MJX-136-NCM-N-35"></use><use data-c="2E" href="#MJX-136-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-136-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-136-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="38" href="#MJX-136-NCM-N-38" transform="translate(1556,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{5.3.7}" data-semantic-structure="0"><mrow href="#mjx-eqn%3A5.3.7" data-latex="\ref{5.3.7}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{5.3.7};href:#mjx-eqn%3A5.3.7" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">5.3.8</mtext></mrow></math></mjx-assistive-mml></mjx-container> to obtain ΔU.</li> </ol> <span id="Solution_2"></span><h6>Solution</h6> <p class="lt-chem-21723" style="background-color: unset;">A In this reaction, 1 mol of gas (CO<sub>2</sub>) is produced, and 1 mol of gas (O<sub>2</sub>) is consumed. Thus Δn = 1 − 1 = 0.</p> <p class="lt-chem-21723" style="background-color: unset;">B Substituting this calculated value and the given values into Equation <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="136" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-137-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-137-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-137-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-137-NCM-N-38" d="M250 666C201 666 158 650 122 618C86 586 68 546 68 497C68 456 83 419 113 386C120 378 142 361 179 335C88 288 42 227 42 153C42 100 64 57 107 24C147-7 194-22 249-22C305-22 354-4 395 32C436 68 457 115 457 170C457 216 441 257 408 294C396 307 365 330 316 361C393 402 431 454 431 515C431 606 344 666 250 666M379 515C379 463 348 418 286 381L167 459C136 479 120 505 120 536C120 596 185 633 249 633C318 633 379 584 379 515M250 14C168 14 99 73 99 153C99 220 136 274 210 315L328 240C376 209 400 174 400 134C400 62 325 14 250 14Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\ref{5.3.7}" data-semantic-structure="0"><a href="/Sandboxes/johnnyphung/chemistry_central/05:_Thermochemistry/5.03:_Enthalpy#mjx-eqn:5.3.7"><g data-mml-node="mrow" data-latex="\ref{5.3.7}" class="MathJax_ref"><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="2056" height="688" x="0" y="-22"></rect><g data-mml-node="mtext" data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-id="0" data-semantic-attributes="latex:\ref{5.3.7};href:#mjx-eqn%3A5.3.7" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true"><use data-c="35" href="#MJX-137-NCM-N-35"></use><use data-c="2E" href="#MJX-137-NCM-N-2E" transform="translate(500,0)"></use><use data-c="33" href="#MJX-137-NCM-N-33" transform="translate(778,0)"></use><use data-c="2E" href="#MJX-137-NCM-N-2E" transform="translate(1278,0)"></use><use data-c="38" href="#MJX-137-NCM-N-38" transform="translate(1556,0)"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\ref{5.3.7}" data-semantic-structure="0"><mrow href="#mjx-eqn%3A5.3.7" data-latex="\ref{5.3.7}" class="MathJax_ref"><mtext data-semantic-type="text" data-semantic-role="unknown" data-semantic-font="normal" data-semantic-annotation="clearspeak:unit;nemeth:number;depth:1" data-semantic-="" data-semantic-attributes="latex:\ref{5.3.7};href:#mjx-eqn%3A5.3.7" data-semantic-postfix="link" data-semantic-level-number="0" data-speech-node="true">5.3.8</mtext></mrow></math></mjx-assistive-mml></mjx-container>,</p> <div class="mt-align-center">ΔU=ΔH−RTΔn=(−393.5 kJ/mol)−[8.314 J/(mol⋅K)](298 K)(0)</div> <div class="mt-indent-1" style="text-align:center;">=(−393.5 kJ/mol)−(0 J/mol)=−393.5 kJ/mol</div> <p class="lt-chem-21723" style="background-color: unset;">To understand why only the change in the volume of the gases needs to be considered, notice that the molar volume of graphite is only 0.0053 L. A change in the number of moles of gas corresponds to a volume change of 22.4 L/mol of gas at standard temperature and pressure (STP), so the volume of gas consumed or produced in this case is (1)(22.4 L) = 22.4 L, which is much, much greater than the volume of 1 mol of a solid such as graphite.</p> </div> <div class="box-exercise"> <span id="Exercise_.5C(.5CPageIndex.7B2.7D.5C)"></span><h5 class="box-legend"><span class="lt-icon-default">Exercise <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="137" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-138-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-138-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-138-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-138-NCM-N-32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\PageIndex{2}" data-semantic-structure="(5 0 1 2 3 4)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="2}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-id="5" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{2};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><g data-mml-node="mn" data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true"><use data-c="35" href="#MJX-138-NCM-N-35"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(500,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="1" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-138-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true" transform="translate(778,0)"><use data-c="33" href="#MJX-138-NCM-N-33"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(1278,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="3" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-138-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="4" data-semantic-parent="5" data-semantic-attributes="latex:2" data-semantic-level-number="1" data-speech-node="true" transform="translate(1556,0)"><use data-c="32" href="#MJX-138-NCM-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="\PageIndex{2}" data-semantic-structure="(5 0 1 2 3 4)"><mrow data-mjx-texclass="ORD" data-latex="2}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{2};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><mn data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true">5</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true">3</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:2" data-semantic-level-number="1" data-speech-node="true">2</mn></mrow></math></mjx-assistive-mml></mjx-container></span></h5> <p class="lt-chem-21723" style="background-color: unset;">Calculate <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="138" style="font-size: 85%; position: relative;"><svg width="3.604ex" 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href="#MJX-140-NCM-N-29"></use></g></g></g></g></g><g data-mml-node="mo" data-latex="\nonumber" data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:4" data-semantic-id="16" data-semantic-parent="28" data-semantic-attributes="latex:\nonumber" data-semantic-operator="punctuated" data-semantic-level-number="3" data-speech-node="true" transform="translate(7464.7,0)"><use data-c="2E" href="#MJX-140-NCM-N-2E"></use></g></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="0 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:" data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="30" data-semantic-level-number="2" data-speech-node="true"></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math data-latex="3O_2(g) \rightarrow 2O_3(g). \nonumber " display="block" data-semantic-structure="(31 (30 (28 (27 (24 1 23 (22 (4 2 3) 21 (17 5 6 7))) 8 (26 9 25 (20 (12 10 11) 19 (18 13 14 15)))) 16) 0))"><mtable displaystyle="true" data-latex="3O_2 (g) \rightarrow 2O_3 (g). \nonumber" data-semantic-type="multiline" data-semantic-role="unknown" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="30" data-semantic-attributes="latex:3O_2(g) \rightarrow 2O_3(g). \nonumber " data-semantic-owns="30" data-semantic-level-number="0" data-speech-node="true"><mlabeledtr data-semantic-type="line" data-semantic-role="multiline" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-children="28" data-semantic-content="0" data-semantic-parent="31" data-semantic-owns="28 0" data-semantic-level-number="1" data-speech-node="true"><mtd data-semantic-type="cell" data-semantic-role="label" data-semantic-annotation="nemeth:number;depth:3" data-semantic-id="0" data-semantic-parent="30" data-semantic-level-number="2" data-speech-node="true"></mtd><mtd><mrow data-semantic-added="true" data-semantic-type="punctuated" data-semantic-role="endpunct" data-semantic-annotation="depth:3" data-semantic-="" data-semantic-children="27,16" data-semantic-content="16" data-semantic-parent="30" data-semantic-owns="27 16" data-semantic-level-number="2" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="relseq" data-semantic-role="arrow" data-semantic-annotation="depth:4" data-semantic-="" data-semantic-children="24,26" data-semantic-content="8" data-semantic-parent="28" data-semantic-owns="24 8 26" data-semantic-level-number="3" data-speech-node="true"><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-children="1,22" data-semantic-content="23" data-semantic-parent="27" data-semantic-owns="1 23 22" data-semantic-level-number="4" data-speech-node="true"><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:6" data-semantic-="" data-semantic-parent="24" data-semantic-attributes="latex:3" data-semantic-level-number="5" data-speech-node="true">3</mn><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="24" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-children="4,17" data-semantic-content="21,2" data-semantic-parent="24" data-semantic-owns="4 21 17" data-semantic-level-number="5" data-speech-node="true"><msub data-latex="O_2" data-semantic-type="subscript" data-semantic-role="simple function" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-children="2,3" data-semantic-parent="22" data-semantic-attributes="latex:O_2" data-semantic-owns="2 3" data-semantic-level-number="6" data-speech-node="true"><mi data-latex="O" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:8" data-semantic-="" data-semantic-parent="4" data-semantic-attributes="latex:O" data-semantic-operator="appl" data-semantic-level-number="7" data-speech-node="true">O</mi><mn data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="4" data-semantic-attributes="latex:2" data-semantic-level-number="7" data-speech-node="true">2</mn></msub><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-annotation="nemeth:number;depth:7" data-semantic-="" data-semantic-parent="22" data-semantic-added="true" data-semantic-operator="appl" data-semantic-level-number="6" data-speech-node="true">⁡</mo><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:7" data-semantic-="" data-semantic-children="6" data-semantic-content="5,7" data-semantic-parent="22" data-semantic-owns="5 6 7" data-semantic-level-number="6" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">(</mo><mi data-latex="g" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:g" data-semantic-level-number="7" data-speech-node="true">g</mi><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="17" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">)</mo></mrow></mrow></mrow><mo stretchy="false" data-latex="\rightarrow" data-semantic-type="relation" data-semantic-role="arrow" data-semantic-annotation="nemeth:number;depth:5" data-semantic-="" data-semantic-parent="27" data-semantic-attributes="latex:\rightarrow" data-semantic-operator="relseq,→" data-semantic-level-number="4" data-speech-node="true">→</mo><mrow data-semantic-added="true" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:unit;depth:5" data-semantic-="" data-semantic-children="9,20" data-semantic-content="25" data-semantic-parent="27" data-semantic-owns="9 25 20" data-semantic-level-number="4" data-speech-node="true"><mn data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:6" data-semantic-="" data-semantic-parent="26" data-semantic-attributes="latex:2" data-semantic-level-number="5" data-speech-node="true">2</mn><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:6" data-semantic-="" data-semantic-parent="26" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="5" data-speech-node="true">⁢</mo><mrow data-semantic-added="true" data-semantic-type="appl" data-semantic-role="simple function" data-semantic-annotation="clearspeak:simple;depth:6" data-semantic-="" data-semantic-children="12,18" data-semantic-content="19,10" data-semantic-parent="26" data-semantic-owns="12 19 18" data-semantic-level-number="5" data-speech-node="true"><msub data-latex="O_3" data-semantic-type="subscript" data-semantic-role="simple function" data-semantic-annotation="depth:7" data-semantic-="" data-semantic-children="10,11" data-semantic-parent="20" data-semantic-attributes="latex:O_3" data-semantic-owns="10 11" data-semantic-level-number="6" data-speech-node="true"><mi data-latex="O" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:8" data-semantic-="" data-semantic-parent="12" data-semantic-attributes="latex:O" data-semantic-operator="appl" data-semantic-level-number="7" data-speech-node="true">O</mi><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="12" data-semantic-attributes="latex:3" data-semantic-level-number="7" data-speech-node="true">3</mn></msub><mo data-semantic-type="punctuation" data-semantic-role="application" data-semantic-annotation="nemeth:number;depth:7" data-semantic-="" data-semantic-parent="20" data-semantic-added="true" data-semantic-operator="appl" data-semantic-level-number="6" data-speech-node="true">⁡</mo><mrow data-semantic-added="true" data-semantic-type="fenced" data-semantic-role="leftright" data-semantic-annotation="fences:paren;depth:7" data-semantic-="" data-semantic-children="14" data-semantic-content="13,15" data-semantic-parent="20" data-semantic-owns="13 14 15" data-semantic-level-number="6" data-speech-node="true"><mo data-latex="(" stretchy="false" data-semantic-type="fence" data-semantic-role="open" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="18" data-semantic-attributes="latex:(" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">(</mo><mi data-latex="g" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:8" data-semantic-="" data-semantic-parent="18" data-semantic-attributes="latex:g" data-semantic-level-number="7" data-speech-node="true">g</mi><mo data-latex=")" stretchy="false" data-semantic-type="fence" data-semantic-role="close" data-semantic-annotation="depth:8" data-semantic-="" data-semantic-parent="18" data-semantic-attributes="latex:)" data-semantic-operator="fenced" data-semantic-level-number="7" data-speech-node="true">)</mo></mrow></mrow></mrow></mrow><mo data-latex="\nonumber" data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:4" data-semantic-="" data-semantic-parent="28" data-semantic-attributes="latex:\nonumber" data-semantic-operator="punctuated" data-semantic-level-number="3" data-speech-node="true">.</mo></mrow></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></p> <p class="lt-chem-21723" style="background-color: unset;">The value of <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="140" style="font-size: 85%; position: relative;"><svg width="3.878ex" height="1.62ex" role="img" focusable="false" viewbox="0 -716 1714 716" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-141-NCM-N-394" d="M759 0C776 0 785 4 785 13L445 696C438 709 429 716 416 716C403 716 394 709 387 696L52 25C49 18 47 14 47 12C47 4 56 0 73 0M384 604L644 83L124 83Z"></path><path id="MJX-141-NCM-N-2062" d=""></path><path id="MJX-141-NCM-I-1D43B" d="M881 668C881 678 875 683 863 683L736 680L609 683C593 683 586 674 586 659C586 652 589 647 594 646C604 645 612 644 617 644C648 643 665 641 670 640C675 639 678 636 678 631C677 628 676 622 674 613L615 375L321 375L379 602C384 623 393 636 406 641C413 643 430 644 458 644C485 644 496 644 496 668C496 678 490 683 478 683L351 680L223 683C207 683 200 674 200 659C200 652 203 647 209 646C219 645 227 644 232 644C263 643 281 641 286 640C291 639 293 636 293 631C293 629 292 623 289 613L156 82C151 60 141 47 126 42C119 40 101 39 70 39C48 39 39 37 39 15C39 5 45 0 57 0L183 3L246 2C257 2 299 0 310 0C326 0 334 8 334 24C334 34 323 39 302 39C262 39 242 43 242 52C242 52 243 55 245 68L312 336L605 336L538 68C535 53 523 43 503 40C498 39 481 39 452 39C433 39 424 31 424 15C424 5 430 0 442 0L568 3L631 2C642 2 683 0 696 0C712 0 720 8 720 24C720 34 709 39 688 39C647 39 627 43 627 52C627 52 628 55 630 68L764 602C769 623 778 636 791 641C797 643 814 644 843 644C869 644 881 644 881 668Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="ΔH" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:1" data-semantic-id="3" data-semantic-children="0,1" data-semantic-content="2" data-semantic-attributes="latex:ΔH" data-semantic-owns="0 2 1" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(3 0 2 1)"><g data-mml-node="mi" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="3" data-semantic-attributes="latex:Δ" data-semantic-level-number="1" data-speech-node="true"><use data-c="394" href="#MJX-141-NCM-N-394"></use></g><g data-mml-node="mo" data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:2" data-semantic-id="2" data-semantic-parent="3" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="1" data-speech-node="true" transform="translate(833,0)"><use data-c="2062" href="#MJX-141-NCM-N-2062"></use></g><g data-mml-node="mi" data-latex="H" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-id="1" data-semantic-parent="3" data-semantic-attributes="latex:H" data-semantic-level-number="1" data-speech-node="true" transform="translate(833,0)"><use data-c="1D43B" href="#MJX-141-NCM-I-1D43B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math data-latex="ΔH" data-semantic-type="infixop" data-semantic-role="implicit" data-semantic-annotation="clearspeak:simple;clearspeak:unit;depth:1" data-semantic-="" data-semantic-children="0,1" data-semantic-content="2" data-semantic-attributes="latex:ΔH" data-semantic-owns="0 2 1" data-semantic-level-number="0" data-speech-node="true" data-semantic-structure="(3 0 2 1)"><mi mathvariant="normal" data-latex="Δ" data-semantic-type="identifier" data-semantic-role="greekletter" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:Δ" data-semantic-level-number="1" data-speech-node="true">Δ</mi><mo data-semantic-type="operator" data-semantic-role="multiplication" data-semantic-annotation="depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-added="true" data-semantic-operator="infixop,⁢" data-semantic-level-number="1" data-speech-node="true">⁢</mo><mi data-latex="H" data-semantic-type="identifier" data-semantic-role="simple function" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple;depth:2" data-semantic-="" data-semantic-parent="3" data-semantic-attributes="latex:H" data-semantic-level-number="1" data-speech-node="true">H</mi></math></mjx-assistive-mml></mjx-container> for the reaction is 285.4 kJ.</p> <dl> <dt><strong class="emphasis bold">Answer</strong></dt> <dd> <p class="lt-chem-21723" style="background-color: unset;">288 kJ</p> </dd> </dl> </div> <p class="lt-chem-21723" style="background-color: unset;">As Example <mjx-container class="MathJax CtxtMenu_Attached_0" jax="SVG" overflow="linebreak" tabindex="0" ctxtmenu_counter="141" style="font-size: 85%; position: relative;"><svg width="4.652ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 2056 688" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-142-NCM-N-35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"></path><path id="MJX-142-NCM-N-2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"></path><path id="MJX-142-NCM-N-33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"></path><path id="MJX-142-NCM-N-32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\PageIndex{2}" data-semantic-structure="(5 0 1 2 3 4)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="2}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-id="5" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{2};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><g data-mml-node="mn" data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="0" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true"><use data-c="35" href="#MJX-142-NCM-N-35"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" data-latex="{.}" transform="translate(500,0)"><g data-mml-node="mo" data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-id="1" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true"><use data-c="2E" href="#MJX-142-NCM-N-2E"></use></g></g><g data-mml-node="mn" data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-id="2" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true" transform="translate(778,0)"><use data-c="33" 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display="inline"><math data-latex="\PageIndex{2}" data-semantic-structure="(5 0 1 2 3 4)"><mrow data-mjx-texclass="ORD" data-latex="2}" data-semantic-type="punctuated" data-semantic-role="sequence" data-semantic-annotation="depth:1" data-semantic-="" data-semantic-children="0,1,2,3,4" data-semantic-content="1,3" data-semantic-attributes="latex:\PageIndex{2};texclass:ORD" data-semantic-owns="0 1 2 3 4" data-semantic-level-number="0" data-speech-node="true"><mn data-latex="5" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:5" data-semantic-level-number="1" data-speech-node="true">5</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="3" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:3" data-semantic-level-number="1" data-speech-node="true">3</mn><mrow data-mjx-texclass="ORD" data-latex="{.}"><mo data-latex="." data-semantic-type="punctuation" data-semantic-role="fullstop" data-semantic-annotation="nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:{.};texclass:ORD" data-semantic-operator="punctuated" data-semantic-level-number="1" data-speech-node="true">.</mo></mrow><mn data-latex="2" data-semantic-type="number" data-semantic-role="integer" data-semantic-font="normal" data-semantic-annotation="clearspeak:simple;nemeth:number;depth:2" data-semantic-="" data-semantic-parent="5" data-semantic-attributes="latex:2" data-semantic-level-number="1" data-speech-node="true">2</mn></mrow></math></mjx-assistive-mml></mjx-container> illustrates, the magnitudes of ΔH and ΔU for reactions that involve gases are generally rather similar, even when there is a net production or consumption of gases. </p> <span id="Summary"></span><h2 style="background-color: unset;" class="lt-chem-21723">Summary</h2> <p class="lt-chem-21723" style="background-color: unset;">Enthalpy is a state function, and the change in enthalpy of a system is equal to the sum of the change in the internal energy of the system and the PV work done. Enthalpy is a state function whose change indicates the amount of heat transferred from a system to its surroundings or vice versa, at constant pressure. The change in the internal energy of a system is the sum of the heat transferred and the work done. At constant pressure, heat flow (q) and internal energy (U) are related to the system’s enthalpy (H). The heat flow is equal to the change in the internal energy of the system plus the PV work done. 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