Molekulaarinen jakaumafunktio

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Näistä kolme ensiksimainittua energiaa koostuvat erillisistä energiatiloista, joiden välisiin siirtymiin perustuvat mm. <a href="/wiki/Spektri" title="Spektri">spektriviivat</a>. Kaikkia näitä energioita voidaan tarkastella <b>molekulaarisella jakaumafunktiolla</b>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="fi" dir="ltr"><h2 id="mw-toc-heading">Sisällys</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Molekulaarinen_jakaumafunktio"><span class="tocnumber">1</span> <span class="toctext">Molekulaarinen jakaumafunktio</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Translaatiojakaumafunktio"><span class="tocnumber">1.1</span> <span class="toctext">Translaatiojakaumafunktio</span></a></li> <li class="toclevel-2 tocsection-3"><a href="#Rotaatiojakaumafunktio"><span class="tocnumber">1.2</span> <span class="toctext">Rotaatiojakaumafunktio</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Vibraatiojakaumafunktio"><span class="tocnumber">1.3</span> <span class="toctext">Vibraatiojakaumafunktio</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Elektroninen_jakaumafunktio"><span class="tocnumber">1.4</span> <span class="toctext">Elektroninen jakaumafunktio</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Lisätieto"><span class="tocnumber">2</span> <span class="toctext">Lisätieto</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Katso_myös"><span class="tocnumber">3</span> <span class="toctext">Katso myös</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Lähteet"><span class="tocnumber">4</span> <span class="toctext">Lähteet</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Molekulaarinen_jakaumafunktio">Molekulaarinen jakaumafunktio</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&veaction=edit&section=1" title="Muokkaa osiota Molekulaarinen jakaumafunktio" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&action=edit&section=1" title="Muokkaa osion lähdekoodia: Molekulaarinen jakaumafunktio"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Moniatomisen molekyylin energia on jaettavissa neljään vapausasteeseen, joihin molekyylin molekulaarinen kokonaisjakaumafunktio perustuu. Nämä vapausasteet ovat translaatio, rotaatio, vibraatio, ja elektroninen. Jos oletetaan, että nämä vapausasteet eivät ole toisiinsa kytkeytyneet, niin molekyylin energia voidaan jakaa translaatio-, rotaatio-, vibraatio, ja elektronienergiaan: </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E_{t}+E_{r}+E_{v}+E_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{t}+E_{r}+E_{v}+E_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bca5809edde17517200139efdec5a3647616a241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.084ex; height:2.509ex;" alt="{\displaystyle E=E_{t}+E_{r}+E_{v}+E_{e}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p><a href="/wiki/Alkeisreaktio#Palautuvan_reaktion_tasapainovakio" title="Alkeisreaktio">Boltzmannin lain</a> mukaan voidaan kirjoittaa:<a href="#Lisätieto"><sup>B</sup></a> </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\sum _{i}g_{t}g_{r}g_{v}g_{e}\,e^{\frac {-(E_{t}+E_{r}+E_{v}+E_{e})}{k_{B}T}}=Q_{t}Q_{r}Q_{v}Q_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\sum _{i}g_{t}g_{r}g_{v}g_{e}\,e^{\frac {-(E_{t}+E_{r}+E_{v}+E_{e})}{k_{B}T}}=Q_{t}Q_{r}Q_{v}Q_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b3ef37ea8c7130a9efcc6757adf1a6ec62150d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.777ex; height:7.176ex;" alt="{\displaystyle Q=\sum _{i}g_{t}g_{r}g_{v}g_{e}\,e^{\frac {-(E_{t}+E_{r}+E_{v}+E_{e})}{k_{B}T}}=Q_{t}Q_{r}Q_{v}Q_{e}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Translaatiojakaumafunktio">Translaatiojakaumafunktio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&veaction=edit&section=2" title="Muokkaa osiota Translaatiojakaumafunktio" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&action=edit&section=2" title="Muokkaa osion lähdekoodia: Translaatiojakaumafunktio"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Translaatioliikkeessä ei erillisiä energiatasoja tavallisesti oteta tarkasteluun, koska ne ovat niin lähellä toisiaan. <a href="/wiki/Kvanttimekaniikka" title="Kvanttimekaniikka">Kvanttimekaniikan</a> mukaan ne ovat laskettavissa esim. molekyylille, joka liikkuu tunnetussa laskennollisesti yksinkertaisessa tilavuudessa. Kemiallisessa kinetiikassa otetaan lähtökohdaksi klassinen liike ja yhtenäinen energia-alue, jolloin Boltzmannin laissa oleva summaus korvataan integraalilla. Translaatiojakaumafunktioksi yksiulotteisessa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>-pituisessa laatikossa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>-massaiselle molekyylille saadaan (yksi ulottuvuus): </p> <table class="toccolours collapsible" width="100%" style="text-align:left"> <tbody><tr> <th>Translaatiojakaumafunktion johtaminen<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td>Boltzmannin jakaumalaki on <dl><dd><dl><dd>(1)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad {\frac {N_{i}}{N}}\,=\,{\frac {g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}{\sum _{i}e^{-{\frac {E_{i}}{k_{B}T}}}}}\,=\,{\frac {g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}{Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> <mi>Q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad {\frac {N_{i}}{N}}\,=\,{\frac {g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}{\sum _{i}e^{-{\frac {E_{i}}{k_{B}T}}}}}\,=\,{\frac {g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}{Q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8084b455f5e8953d9c9c18fd080189ac74500496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:35.923ex; height:10.676ex;" alt="{\displaystyle \qquad {\frac {N_{i}}{N}}\,=\,{\frac {g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}{\sum _{i}e^{-{\frac {E_{i}}{k_{B}T}}}}}\,=\,{\frac {g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}{Q}}}"></span></dd></dl></dd></dl> <p>Translaatioenergiatilojen tapauksessa peräkkäisten energiatasojen väli on niin pieni, että nämä energiatilat muodostavat näennäisen jatkumon. Tällöin energiatilojen miehittymistä tarkasteltaessa kvanttiteoriassa Planckin vakion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> dimensiot ovat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{(matka) x (momentti)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(matka) x (momentti)</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{(matka) x (momentti)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef49d5788b75d67b10c502fda818b78a71db88d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.216ex; height:2.843ex;" alt="{\displaystyle {\text{(matka) x (momentti)}}}"></span>. Jos piirretään kuvaaja, jossa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> (matka) on y-akseli ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5055ed65713825b48aa6ee05118c072e6f026a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.431ex; height:2.009ex;" alt="{\displaystyle p_{x}}"></span> (momentti) on x-akseli, niin tässä faasitasossa (engl. phase plane) diskreetti tila tai solu miehittää pinta-alan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>. Tilojen lukumäärä, joilla on momentti välillä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x}\to p_{x}+dp_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x}\to p_{x}+dp_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad5dfad083845faae77e6ecd646f3b455959fdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.785ex; height:2.509ex;" alt="{\displaystyle p_{x}\to p_{x}+dp_{x}}"></span> ja etäisyydellä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to x+dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mo>+</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to x+dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/599084bdea8769b28c078cb58d6d4e85ad679cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.659ex; height:2.343ex;" alt="{\displaystyle x\to x+dx}"></span> on </p> <dl><dd><dl><dd>(2)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad g_{i}\,=\,{\frac {dx\,dp_{x}}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad g_{i}\,=\,{\frac {dx\,dp_{x}}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ced8d04b1685c04bec40c30649607e46e43319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.753ex; height:5.509ex;" alt="{\displaystyle \qquad g_{i}\,=\,{\frac {dx\,dp_{x}}{h}}}"></span></dd></dl></dd></dl> <p>Tässä derivaatta liikemäärästä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dp_{x}=m\,dv_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dp_{x}=m\,dv_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7719a02e653c1b637119994784fca82ba024430d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.6ex; height:2.509ex;" alt="{\displaystyle dp_{x}=m\,dv_{x}}"></span>. Translaatiojakaumafunktio, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{t}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{t}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e3ce3af9818f992520843af7a2fe4f6de35fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.803ex; height:2.843ex;" alt="{\displaystyle Q_{t}(x)}"></span>, yhdessä ulottuvuudessa on: </p> <dl><dd><dl><dd>(3)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad Q_{t}(x)\,=\,\sum {\frac {m\,dv_{x}\,dx}{h}}e^{-{\frac {m\,v_{x}^{2}}{2k_{B}T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mi>h</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mspace width="thinmathspace" /> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad Q_{t}(x)\,=\,\sum {\frac {m\,dv_{x}\,dx}{h}}e^{-{\frac {m\,v_{x}^{2}}{2k_{B}T}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c887bab8c05e2bd3b5b211e8c2d7d1d043c8c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.654ex; height:6.509ex;" alt="{\displaystyle \qquad Q_{t}(x)\,=\,\sum {\frac {m\,dv_{x}\,dx}{h}}e^{-{\frac {m\,v_{x}^{2}}{2k_{B}T}}}}"></span></dd></dl></dd></dl> <p>Translaatioenergiatasot muodostavat käytännössä jatkumon (engl. continuum), joten summaus voidaan korvata integroimisella. Jos kaasu on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>-pituisessa laatikossa, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> on saa arvot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826b3d12efe96889f7f22819daf2200aef85f301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle 0\to a}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704b7ad1ece77840fde455daa6d2e51e64282b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.009ex;" alt="{\displaystyle v_{x}}"></span> saa arvot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty \to +\infty \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty \to +\infty \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0de428794f195ed39aee10eb6c56ec54846876d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:12.265ex; height:2.176ex;" alt="{\displaystyle -\infty \to +\infty \,\!}"></span>. Nämä muodostavat integrointirajat: </p> <dl><dd><dl><dd>(4)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad Q_{t}(x)\,=\,{\frac {m}{h}}\int _{0}^{a}dx\,\int _{-\infty }^{+\infty }e^{-{\frac {m\,v_{x}^{2}}{2k_{B}T}}}dv_{x}\,=\,{\frac {a}{h}}{\sqrt {2\pi \,m\,k_{B}\,T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>h</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mspace width="thinmathspace" /> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mi>d</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>h</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mi>m</mi> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>T</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad Q_{t}(x)\,=\,{\frac {m}{h}}\int _{0}^{a}dx\,\int _{-\infty }^{+\infty }e^{-{\frac {m\,v_{x}^{2}}{2k_{B}T}}}dv_{x}\,=\,{\frac {a}{h}}{\sqrt {2\pi \,m\,k_{B}\,T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25bea70185435bd655d66012aa7df183cc5fa9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.652ex; height:7.009ex;" alt="{\displaystyle \qquad Q_{t}(x)\,=\,{\frac {m}{h}}\int _{0}^{a}dx\,\int _{-\infty }^{+\infty }e^{-{\frac {m\,v_{x}^{2}}{2k_{B}T}}}dv_{x}\,=\,{\frac {a}{h}}{\sqrt {2\pi \,m\,k_{B}\,T}}}"></span></dd></dl></dd></dl> <p>Huomaa: standardi-integraali <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{+\infty }e^{-\beta x^{2}}dx={\sqrt {\frac {\pi }{\beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>π</mi> <mi>β</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{+\infty }e^{-\beta x^{2}}dx={\sqrt {\frac {\pi }{\beta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a41c29fd453a061a7d2a20e054294244ae9e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.449ex; height:6.509ex;" alt="{\displaystyle \int _{-\infty }^{+\infty }e^{-\beta x^{2}}dx={\sqrt {\frac {\pi }{\beta }}}}"></span> </p> </td></tr></tbody></table> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{t}\,=\,{\frac {a}{h}}{\sqrt {2\pi \,m\,k_{B}\,T}}\,=\,{\frac {a}{\Lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>h</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mi>m</mi> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>T</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi mathvariant="normal">Λ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{t}\,=\,{\frac {a}{h}}{\sqrt {2\pi \,m\,k_{B}\,T}}\,=\,{\frac {a}{\Lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb10a1e6db4ee309b7ad1c23576a53bbeec3702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.381ex; height:4.843ex;" alt="{\displaystyle Q_{t}\,=\,{\frac {a}{h}}{\sqrt {2\pi \,m\,k_{B}\,T}}\,=\,{\frac {a}{\Lambda }}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> on <a href="/wiki/De_Broglien_aallonpituus" title="De Broglien aallonpituus">terminen de Broglie-aallonpituus</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> on <a href="/wiki/Planckin_vakio" title="Planckin vakio">Planckin vakio</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f38f7b73e53fd7b5d9ca64bec3a1438cc0eade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.691ex; height:2.509ex;" alt="{\displaystyle k_{B}}"></span> on <a href="/wiki/Boltzmannin_vakio" title="Boltzmannin vakio">Boltzmannin vakio</a> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> on lämpötila Kelvin-asteissa. Suorakulmaisessa <a href="/wiki/Suuntaiss%C3%A4rmi%C3%B6" title="Suuntaissärmiö">suuntaissärmiössä</a>, jonka tilavuus on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, liikkuvalle samalle molekyylille saadaan (kolme ulottuvuutta, nämä translaatiovapausasteet otetaan toisistaan erillisinä): </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{t}\,=\,{\frac {V}{h^{3}}}{\Big (}{2\pi m\,k_{B}\,T}{\Big )}^{\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>m</mi> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>T</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{t}\,=\,{\frac {V}{h^{3}}}{\Big (}{2\pi m\,k_{B}\,T}{\Big )}^{\frac {3}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc26508b7d3a812a6af1cd429e80e555c8b6cb2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.914ex; height:6.176ex;" alt="{\displaystyle Q_{t}\,=\,{\frac {V}{h^{3}}}{\Big (}{2\pi m\,k_{B}\,T}{\Big )}^{\frac {3}{2}}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Rotaatiojakaumafunktio">Rotaatiojakaumafunktio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&veaction=edit&section=3" title="Muokkaa osiota Rotaatiojakaumafunktio" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&action=edit&section=3" title="Muokkaa osion lähdekoodia: Rotaatiojakaumafunktio"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rotaatioliikkeessä energiatasot ovat lähellä toisiaan, mutta erotettavissa toisistaan. Huoneen lämpötilassa, n. 298 K, useat rotaatiotilat ovat miehittyneet, koska rotaatioenergialtaan molekyyli on täysin virittynyt. Rotaatiojakaumafunktio lineaariselle molekyylille (esim. vety, H-H), joka oletetaan jäykäksi pyörijäksi ja jolla on kaksi vapausastetta on: </p> <table class="toccolours collapsible" width="100%" style="text-align:left"> <tbody><tr> <th>Rotaatiojakaumafunktion johtaminen<sup id="cite_ref-Steinfeld1998_4-0" class="reference"><a href="#cite_note-Steinfeld1998-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td>Lineaarisen jäykän pyörijän rotaatioenergiat ovat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}=i(i+1)\mathrm {P} ={\frac {i(i+1)\hbar ^{2}}{2I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>I</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}=i(i+1)\mathrm {P} ={\frac {i(i+1)\hbar ^{2}}{2I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5576df08fc43ceafa2719e25002aeb7f621b254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.341ex; height:5.843ex;" alt="{\displaystyle E_{i}=i(i+1)\mathrm {P} ={\frac {i(i+1)\hbar ^{2}}{2I}}}"></span>. Rotaatiojakaumafunktion johtamisessa käytetään Boltzmannin jakaumalakia, johon sijoitetaan rotaatioenergiat ja energiatasojen multiplisiteetti: <dl><dd><dl><dd>(1)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad Q_{r}\,=\,\sum _{i=0}^{\infty }i(i+1)e^{-{\frac {i(i+1)\hbar ^{2}}{2I\,k_{B}T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>I</mi> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad Q_{r}\,=\,\sum _{i=0}^{\infty }i(i+1)e^{-{\frac {i(i+1)\hbar ^{2}}{2I\,k_{B}T}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b989238189dc4e050560359e0f132cb7a057bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.144ex; height:7.509ex;" alt="{\displaystyle \qquad Q_{r}\,=\,\sum _{i=0}^{\infty }i(i+1)e^{-{\frac {i(i+1)\hbar ^{2}}{2I\,k_{B}T}}}}"></span></dd></dl></dd></dl> <p>Jos rotaatioenergiatasojen väli on pieni verrattuna <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{B}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{B}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b4baed050c25a0d97fc74e8da32d37dcecaf50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.327ex; height:2.509ex;" alt="{\displaystyle k_{B}T}"></span>:een, niin summa voidaan korvata integraalilla. </p> <dl><dd><dl><dd>(2)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad Q_{r}\,=\,\int _{i=0}^{\infty }i(i+1)e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}di}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mi>d</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad Q_{r}\,=\,\int _{i=0}^{\infty }i(i+1)e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}di}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b866fc046fdb98358bf036b19b4ada3300aeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.496ex; height:6.509ex;" alt="{\displaystyle \qquad Q_{r}\,=\,\int _{i=0}^{\infty }i(i+1)e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}di}"></span></dd></dl></dd></dl> <p>Integroimisessa on huomioitavissa, että <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d{\Big (}e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}{\Big )}}{di}}=-{\frac {\mathrm {P} (2i\,+\,1)}{k_{B}T}}e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mrow> <mrow> <mi>d</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d{\Big (}e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}{\Big )}}{di}}=-{\frac {\mathrm {P} (2i\,+\,1)}{k_{B}T}}e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4aef1e4b22c6360d84dbf43658e3cceef6089d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.713ex; height:9.176ex;" alt="{\displaystyle {\frac {d{\Big (}e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}{\Big )}}{di}}=-{\frac {\mathrm {P} (2i\,+\,1)}{k_{B}T}}e^{-{\frac {i(i+1)\mathrm {P} }{k_{B}T}}}}"></span>, joten yhtälöstä (2) saadaan </p> <dl><dd><dl><dd>(3)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad Q_{r}\,=\,\int _{i=0}^{\infty }{\frac {-k_{B}T}{\mathrm {P} }}{\frac {d}{di}}{\Big (}e^{-{\frac {i(i\,+\,1)\mathrm {P} }{k_{B}T}}}{\Big )}di\,=\,{\frac {-k_{B}T}{\mathrm {P} }}{\Bigg |}_{i=0}^{\infty }e^{-{\frac {i(i\,+\,1)\mathrm {P} }{k_{B}T}}}\,=\,{\frac {k_{B}T}{\mathrm {P} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mi>d</mi> <mi>i</mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad Q_{r}\,=\,\int _{i=0}^{\infty }{\frac {-k_{B}T}{\mathrm {P} }}{\frac {d}{di}}{\Big (}e^{-{\frac {i(i\,+\,1)\mathrm {P} }{k_{B}T}}}{\Big )}di\,=\,{\frac {-k_{B}T}{\mathrm {P} }}{\Bigg |}_{i=0}^{\infty }e^{-{\frac {i(i\,+\,1)\mathrm {P} }{k_{B}T}}}\,=\,{\frac {k_{B}T}{\mathrm {P} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/364712401c8aa41eb103c96966340e8e6be0ff57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:71.66ex; height:6.843ex;" alt="{\displaystyle \qquad Q_{r}\,=\,\int _{i=0}^{\infty }{\frac {-k_{B}T}{\mathrm {P} }}{\frac {d}{di}}{\Big (}e^{-{\frac {i(i\,+\,1)\mathrm {P} }{k_{B}T}}}{\Big )}di\,=\,{\frac {-k_{B}T}{\mathrm {P} }}{\Bigg |}_{i=0}^{\infty }e^{-{\frac {i(i\,+\,1)\mathrm {P} }{k_{B}T}}}\,=\,{\frac {k_{B}T}{\mathrm {P} }}}"></span></dd></dl></dd></dl> <p>Yhtälöön (3) lisätään nimittäjään symmetrialuku molekyylin symmetrian takia. </p> </td></tr></tbody></table> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{r}={\frac {8\pi ^{2}Ik_{B}T}{h^{2}\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>I</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> <mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>σ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{r}={\frac {8\pi ^{2}Ik_{B}T}{h^{2}\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffede94389a1efbeeb798dbc37e752154dc6a387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.796ex; height:6.009ex;" alt="{\displaystyle Q_{r}={\frac {8\pi ^{2}Ik_{B}T}{h^{2}\sigma }}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> molekyylin <a href="/wiki/Hitausmomentti" title="Hitausmomentti">hitausmomentti</a> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> on symmetrialuku (so. samanlaisten suuntautumisien lukumäärä)<a href="#Lisätieto"><sup>C</sup></a>. Epälineaariselle molekyylille (esim. metaani) on huomioitava kolme vapausastetta ja täten kolme eri hitausmomenttia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}"></span>. Tällöin rotaatiojakaumafunktio on: </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{r}=8\pi ^{2}{\frac {(8\pi ^{3}ABC)^{\frac {1}{2}}(k_{B}T)^{\frac {3}{2}}}{h^{3}\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>8</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>8</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>A</mi> <mi>B</mi> <mi>C</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>σ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{r}=8\pi ^{2}{\frac {(8\pi ^{3}ABC)^{\frac {1}{2}}(k_{B}T)^{\frac {3}{2}}}{h^{3}\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b95ef0ffb418a62f9aa0b0ac9854cd46512de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.539ex; height:7.176ex;" alt="{\displaystyle Q_{r}=8\pi ^{2}{\frac {(8\pi ^{3}ABC)^{\frac {1}{2}}(k_{B}T)^{\frac {3}{2}}}{h^{3}\sigma }}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Jäykän pyörijän energiatasot ovat </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}=i(i+1)\mathrm {P} ={\frac {i(i+1)h^{2}}{2I(2\pi )^{2}}}={\frac {i(i+1)\hbar ^{2}}{2I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>I</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>I</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}=i(i+1)\mathrm {P} ={\frac {i(i+1)h^{2}}{2I(2\pi )^{2}}}={\frac {i(i+1)\hbar ^{2}}{2I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ebb0ec0127cc807c77747f7385ced59cfc7be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.086ex; height:6.676ex;" alt="{\displaystyle E_{i}=i(i+1)\mathrm {P} ={\frac {i(i+1)h^{2}}{2I(2\pi )^{2}}}={\frac {i(i+1)\hbar ^{2}}{2I}}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72172888980d0d3565baec875a4c3e8eed50ed26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle \mathrm {P} }"></span> on rotaatiovakio (pyörimisvakio) ja rotaatiokvanttiluku <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=0,1,2,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=0,1,2,...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c9f5498bc4d20b8b01b0656c3d5ad97872b4da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.205ex; height:2.509ex;" alt="{\displaystyle i=0,1,2,...}"></span>. Jokaisen rotaatioenergiatason monikerrainnaisuus (multiplisiteetti) on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2i+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2i+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f077f73c2ecdf3c6e29a120f948a7255c0a65da1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.968ex; height:2.343ex;" alt="{\displaystyle 2i+1}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Vibraatiojakaumafunktio">Vibraatiojakaumafunktio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&veaction=edit&section=4" title="Muokkaa osiota Vibraatiojakaumafunktio" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&action=edit&section=4" title="Muokkaa osion lähdekoodia: Vibraatiojakaumafunktio"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vibraatioliikkeessä energiatasot ovat selvästi erillään toisistaan, joten vain muutama alin tila on miehittynyt. Tämän vuoksi jakaumafunktio on laskettava summaamalla. <a href="/wiki/Harmoninen_v%C3%A4r%C3%A4htelij%C3%A4" title="Harmoninen värähtelijä">Harmoniselle värähtelijälle</a><a href="#Lisätieto"><sup>D</sup></a>, jolla on vain yksi vapausaste (esim. kaksiatominen molekyyli) vibraatiojakaumafunktio on: </p> <table class="toccolours collapsible collapsed" width="100%" style="text-align:left"> <tbody><tr> <th>Vibraatiojakaumafunktion johtaminen<sup id="cite_ref-Steinfeld1998_4-1" class="reference"><a href="#cite_note-Steinfeld1998-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td>Vibraatiojakaumafunktion laskemisessa käytetään Boltzmannin jakaumalakia <dl><dd><dl><dd>(1)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad Q_{v}\,=\,\sum _{i}g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad Q_{v}\,=\,\sum _{i}g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d082c06f6d74e07f2158d92880e7779dd5df19e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.635ex; height:7.176ex;" alt="{\displaystyle \qquad Q_{v}\,=\,\sum _{i}g_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}"></span></dd></dl></dd></dl> <p>Harmonisen värähtelijän vibraatioenergiatasoilla ei ole degeneraatiota, joten vibraatioenergia on </p> <dl><dd><dl><dd>(2)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad E_{v}\,=\,{\Big (}v+{\frac {1}{2}}{\Big )}h\,c\,{\tilde {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>v</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mi>h</mi> <mspace width="thinmathspace" /> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad E_{v}\,=\,{\Big (}v+{\frac {1}{2}}{\Big )}h\,c\,{\tilde {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69abdfe17060d91ecf66f6f7e64c0b7db936793c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.352ex; height:5.176ex;" alt="{\displaystyle \qquad E_{v}\,=\,{\Big (}v+{\frac {1}{2}}{\Big )}h\,c\,{\tilde {v}}}"></span></dd></dl></dd></dl> <p>Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> on valon nopeus tyhjiössä ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d80868ab74f365f110a675c9cc440ce6b3b14d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.227ex; height:2.176ex;" alt="{\displaystyle {\tilde {v}}}"></span> on yhden värähdystason harmoninen taajuus (yksikössä cm<sup>-1</sup> ). Jokaisen energiatason multiplisiteetti on 1. Vibraatiojakaumafunktion laskemiseksi kaikki energiatilat pitää summata. Yhtälöstä (1) saadaan sijoittamalla siihen yhtälö (2) </p> <dl><dd><dl><dd>(3)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad Q_{v}\,=\,\sum _{v=0}^{\infty }exp{\Bigg [}{-{\frac {{\Big (}v+{\frac {1}{2}}{\Big )}h\,c\,{\tilde {v}}}{k_{B}T}}}{\Bigg ]}\,=\,e^{-{\frac {h\,c\,{\tilde {v}}}{2k_{B}T}}}\sum _{v=0}^{\infty }e^{-{\frac {h\,c\,{\tilde {v}}\,v}{k_{B}T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>e</mi> <mi>x</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>v</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mi>h</mi> <mspace width="thinmathspace" /> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mspace width="thinmathspace" /> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mspace width="thinmathspace" /> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>v</mi> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad Q_{v}\,=\,\sum _{v=0}^{\infty }exp{\Bigg [}{-{\frac {{\Big (}v+{\frac {1}{2}}{\Big )}h\,c\,{\tilde {v}}}{k_{B}T}}}{\Bigg ]}\,=\,e^{-{\frac {h\,c\,{\tilde {v}}}{2k_{B}T}}}\sum _{v=0}^{\infty }e^{-{\frac {h\,c\,{\tilde {v}}\,v}{k_{B}T}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc66865e4d9fba0731e9161addd9cbd407cef10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.012ex; height:8.843ex;" alt="{\displaystyle \qquad Q_{v}\,=\,\sum _{v=0}^{\infty }exp{\Bigg [}{-{\frac {{\Big (}v+{\frac {1}{2}}{\Big )}h\,c\,{\tilde {v}}}{k_{B}T}}}{\Bigg ]}\,=\,e^{-{\frac {h\,c\,{\tilde {v}}}{2k_{B}T}}}\sum _{v=0}^{\infty }e^{-{\frac {h\,c\,{\tilde {v}}\,v}{k_{B}T}}}}"></span></dd></dl></dd></dl> <p>Summalausekkeen sarjakehitelmä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{v=0}^{\infty }e^{-vax}\simeq {\frac {1}{1-e^{-ax}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>v</mi> <mi>a</mi> <mi>x</mi> </mrow> </msup> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{v=0}^{\infty }e^{-vax}\simeq {\frac {1}{1-e^{-ax}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e6b154ab87d56cb08687b7bbba3fd646848a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.285ex; height:6.843ex;" alt="{\displaystyle \sum _{v=0}^{\infty }e^{-vax}\simeq {\frac {1}{1-e^{-ax}}}}"></span>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> joten vibraatiojakaumafunktioksi (NPE on mukana) saadaan: </p> <dl><dd><dl><dd>(4)<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad Q_{v}\,=\,{\frac {e^{-{\frac {h\,c\,{\tilde {v}}}{2k_{B}T}}}}{1\,-\,e^{-{\frac {h\,c\,{\tilde {v}}}{k_{B}T}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mspace width="thinmathspace" /> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mrow> <mn>1</mn> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mspace width="thinmathspace" /> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad Q_{v}\,=\,{\frac {e^{-{\frac {h\,c\,{\tilde {v}}}{2k_{B}T}}}}{1\,-\,e^{-{\frac {h\,c\,{\tilde {v}}}{k_{B}T}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfec9b08545ac53f5740a5cf506a3118dee490d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:23.282ex; height:10.343ex;" alt="{\displaystyle \qquad Q_{v}\,=\,{\frac {e^{-{\frac {h\,c\,{\tilde {v}}}{2k_{B}T}}}}{1\,-\,e^{-{\frac {h\,c\,{\tilde {v}}}{k_{B}T}}}}}}"></span></dd></dl></dd></dl> <p>Kun käytännön syistä alin energiataso merkitään <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1c21cab5a83c62e7611d29260196ead8771d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.03ex; height:2.509ex;" alt="{\displaystyle E_{0}=0}"></span>, niin yhtälön (4) osoittajaksi tulee 1. Täten muiden vibraatioenergiatasoja energioita verrataan tähän energiatasoon. </p> </td></tr></tbody></table> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{v}={\Big (}1-e^{-{\frac {h\nu }{k_{B}T}}}{\Big )}^{-1}=Q_{v,0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>ν</mi> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{v}={\Big (}1-e^{-{\frac {h\nu }{k_{B}T}}}{\Big )}^{-1}=Q_{v,0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb555101a46f34fbc387916915e32398562fa54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.534ex; height:5.509ex;" alt="{\displaystyle Q_{v}={\Big (}1-e^{-{\frac {h\nu }{k_{B}T}}}{\Big )}^{-1}=Q_{v,0}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> on harmonisen värähtelijän värähdystaajuus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{v,0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{v,0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7a4075c790db5ac30f6bf650b34c0f421f9b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.147ex; height:2.843ex;" alt="{\displaystyle Q_{v,0}}"></span> on tarkka merkintä siitä, että kyseessä on vibraatiojakaumafunktio suhteessa molekyylin alimpaan vibratioenergiatasoon (se merkitään nollaksi).<a href="#Lisätieto"><sup>E</sup></a> Kun molekyylillä on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>-kappaletta vibraatiovapausastetta, vibraatiojakaumafunktioksi saadaan: </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{v,0}=\prod _{i=1}^{s}{\Big (}1-e^{-{\frac {h\nu _{i}}{k_{B}T}}}{\Big )}^{-1}=Q_{v,0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{v,0}=\prod _{i=1}^{s}{\Big (}1-e^{-{\frac {h\nu _{i}}{k_{B}T}}}{\Big )}^{-1}=Q_{v,0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e455e8ff75eb390d90edf0d904c10a327fe83ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.17ex; height:7.176ex;" alt="{\displaystyle Q_{v,0}=\prod _{i=1}^{s}{\Big (}1-e^{-{\frac {h\nu _{i}}{k_{B}T}}}{\Big )}^{-1}=Q_{v,0}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb0577728049599bceabd5ed148f426e9d44308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.948ex; height:2.009ex;" alt="{\displaystyle \nu _{i}}"></span> ovat eri värähdystaajuudet. Harmonisen värähtelijän energiatasot ovat </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{v}={\Big (}v+{\frac {1}{2}}{\Big )}h\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>v</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mi>h</mi> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{v}={\Big (}v+{\frac {1}{2}}{\Big )}h\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8a8ab7d864247988ab0c08cd800fa2389d2f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.157ex; height:5.176ex;" alt="{\displaystyle E_{v}={\Big (}v+{\frac {1}{2}}{\Big )}h\nu }"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> on värähtelytaajuus (yksikössä s<sup>-1</sup>) ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=0,1,2,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=0,1,2,...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d96a12264711669479db7726e47a6b11f3e1a85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.53ex; height:2.509ex;" alt="{\displaystyle v=0,1,2,...}"></span> on vibraatiokvanttiluku. Jokainen vibraatiotila voi olla vain kertaalleen miehittynyt (ei degeneraatiota). </p><p>Kaksiatomisella molekyylillä on vibraatioenergiaan suhteutettuja vapausasteita <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3N-5=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>N</mi> <mo>−</mo> <mn>5</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3N-5=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc1725db43addcf0d5de09c5eae0c15e6d4050c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.49ex; height:2.343ex;" alt="{\displaystyle 3N-5=1}"></span> kappaletta. Epälineaariselle molekyylillä niitä on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3N-6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>N</mi> <mo>−</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3N-6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75ea2e0a32f1b8db6dd8f42cc30bf84b694e77d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.229ex; height:2.343ex;" alt="{\displaystyle 3N-6}"></span> kappaletta. </p> <div class="mw-heading mw-heading3"><h3 id="Elektroninen_jakaumafunktio">Elektroninen jakaumafunktio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&veaction=edit&section=5" title="Muokkaa osiota Elektroninen jakaumafunktio" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&action=edit&section=5" title="Muokkaa osion lähdekoodia: Elektroninen jakaumafunktio"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elektroninen jakaumafunktio on Boltzmannin lain mukainen: </p> <dl><dd><dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{e,0}=\sum _{i}g_{i}e^{\frac {-E_{i}}{k_{B}T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{e,0}=\sum _{i}g_{i}e^{\frac {-E_{i}}{k_{B}T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f830c7bed1523725deb2b830361d1da8c461b605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.798ex; height:7.176ex;" alt="{\displaystyle Q_{e,0}=\sum _{i}g_{i}e^{\frac {-E_{i}}{k_{B}T}}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce36142a0a1c6660e82bdf3ef3f1551317efe0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle g_{i}}"></span> on degeneraatio ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span> on molekyylin alimpaan elektroniseen tilaan suhteutettu tilan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> energia. Elektroniset energiatasot ovat usein selvästi erillään toisistaan, joten elektronisen perustilan lisäksi vain jotkut viritystilat pitää ottaa huomioon molekyylin ominaisuuksista. Usein kun peräkkäisten elektronienergiatasojen väli on hyvin suuri, siis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{e,0}\gg k_{B}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>≫</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{e,0}\gg k_{B}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2beabfdeebb947b679caa9999dcff34dd7a524c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.934ex; height:2.843ex;" alt="{\displaystyle E_{e,0}\gg k_{B}T}"></span>, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{e,0}\simeq g_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>≃</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{e,0}\simeq g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189e394d365b0edb84611d4c68056f1a6cebb2ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.378ex; height:2.843ex;" alt="{\displaystyle Q_{e,0}\simeq g_{0}}"></span> tai peruselektronitilan degeneraatio. Tässä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d13273b9af4564fa2c421c96d039c414db8628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{0}}"></span> on elektroniselta energialtaan alimman elektronitilan miehittyminen. Kemiallisessa reaktiossa, jossa mukana on duplettitila (esim. radikaalilla) tai triplettitila (esim. aromaattiset molekyylit), spinnin degeneroituminen otetaan huomioon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce36142a0a1c6660e82bdf3ef3f1551317efe0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.909ex; height:2.009ex;" alt="{\displaystyle g_{i}}"></span>:lla. </p> <div class="mw-heading mw-heading2"><h2 id="Lisätieto"><span id="Lis.C3.A4tieto"></span>Lisätieto</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&veaction=edit&section=6" title="Muokkaa osiota Lisätieto" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&action=edit&section=6" title="Muokkaa osion lähdekoodia: Lisätieto"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><sup>A</sup> Suomalaiset vastineet ovat: pyörimisenergia ja värähtelyenergia. Translaatioenergia on liike-energiaa. </p><p><sup>B</sup> Termisessä tasapainotilassa elektronisen jakaumafunktion arvo on tyypillisesti 1, koska usein selvästi erillään olevista elektronisista energiatasoista vain alin taso on käytössä (so. miehittynyt) ja se on singlettitila. Reaktioissa, joissa on mukana dupletti- tai triplettitilat, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aefe939a50242cd2141bb6937a11a543fc4645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.107ex; height:2.009ex;" alt="{\displaystyle g_{e}}"></span> määräytyy spinnikerrannaisuuden mukaan. - Molekulaarisessa kokonaisjakaumafunktiossa on mukana myös ydin<a href="/wiki/Spin" title="Spin">spin</a>. Ydinspinnin osuus kemiallisessa reaktiossa on hyvin pieni. </p><p><sup>C</sup> Esimerkiksi ammoniakilla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> on 3 ja metaanilla 12. </p><p><sup>D</sup> Käytännössä kaksiatominen (so. diatominen) molekyyli on epäharmoninen värähtelijä ja sen <a href="/wiki/Potentiaalienergia" title="Potentiaalienergia">potentiaalienergiaa</a> pitää kuvata esim. <a href="/wiki/Morse-potentiaali" title="Morse-potentiaali">morse-potentiaalilla</a>. </p><p><sup>E</sup> Alimman vibraatioenergiatason arvo on nolla huolimatta siitä, että se on puoli värähtelykvanttia harmonisen potentiaalikäyrän minimin yläpuolella. Täten molekyylin nollapiste-energia (NPE) on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}h\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>h</mi> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}h\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c5e994ae615b2d21be2c67e6adb7d65c66027f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.57ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}h\nu }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Katso_myös"><span id="Katso_my.C3.B6s"></span>Katso myös</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&veaction=edit&section=7" title="Muokkaa osiota Katso myös" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&action=edit&section=7" title="Muokkaa osion lähdekoodia: Katso myös"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="div-col columns column-count column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2;"> <ul><li><a href="/wiki/Reaktiokinetiikka" title="Reaktiokinetiikka">Reaktiokinetiikka</a></li> <li><a href="/wiki/Arrheniuksen_yht%C3%A4l%C3%B6" title="Arrheniuksen yhtälö">Arrheniuksen yhtälö</a></li> <li><a href="/wiki/Ketjureaktio_(kinetiikka)" title="Ketjureaktio (kinetiikka)">Ketjureaktio</a></li> <li><a href="/wiki/T%C3%B6rm%C3%A4ysteoria" title="Törmäysteoria">Törmäysteoria</a></li> <li><a href="/wiki/Unimolekulaarinen_reaktio" title="Unimolekulaarinen reaktio">Unimolekulaarinen reaktio</a></li> <li><a href="/wiki/Siirtym%C3%A4tilateoria" title="Siirtymätilateoria">Siirtymätilateoria</a></li> <li><a href="/wiki/RRKM" class="mw-redirect" title="RRKM">RRKM</a></li> <li><a href="/wiki/Morse-potentiaali" title="Morse-potentiaali">Morse-potentiaali</a></li> <li><a href="/wiki/Lennard-Jonesin_potentiaali" title="Lennard-Jonesin potentiaali">Lennard-Jones (6,12)-potentiaali</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Lähteet"><span id="L.C3.A4hteet"></span>Lähteet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&veaction=edit&section=8" title="Muokkaa osiota Lähteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Molekulaarinen_jakaumafunktio&action=edit&section=8" title="Muokkaa osion lähdekoodia: Lähteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="viitteet-malline" class="viitteet-malline" style="list-style-type:decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">John W. Moore ja Ralph G. Pearson, Kinetics and Mechanism, 3. painos, (1981), John Wiley & Sons, <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0471035580" class="internal mw-magiclink-isbn">ISBN 0-471-03558-0</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Jeffrey I. Steinfeld, Joseph S. Francisco, ja William L. Hase, Chemical Kinetics and Dynamics, 2. painos, (1998), Prentice Hall, <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0137371232" class="internal mw-magiclink-isbn">ISBN 0-13-737123-2</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Frank Wilkinson, Chemical Kinetics and Reaction Mechanisms, sivu 98, (1980), van Nostrand Reinhold Company, <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0442302487" class="internal mw-magiclink-isbn">ISBN 0-442-30248-7</a></span> </li> <li id="cite_note-Steinfeld1998-4"><span class="mw-cite-backlink">↑ <a href="#cite_ref-Steinfeld1998_4-0"><sup><i>a</i></sup></a> <a href="#cite_ref-Steinfeld1998_4-1"><sup><i>b</i></sup></a></span> <span class="reference-text">Jeffrey I. Steinfeld, Joseph S. Francisco, ja William L. Hase, Chemical Kinetics and Dynamics, 2. painos, sivu 325, (1998), Prentice Hall, <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0137371232" class="internal mw-magiclink-isbn">ISBN 0-13-737123-2</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Thomas Engel ja Philip Reid, Thermodynamics, Statistical Thermodynamics and Kinetics, (2006), s. 338, Pearson, <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0805338446" class="internal mw-magiclink-isbn">ISBN 0-8053-3844-6</a></span> </li> </ol> </div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Noudettu kohteesta ”<a dir="ltr" href="https://fi.wikipedia.org/w/index.php?title=Molekulaarinen_jakaumafunktio&oldid=22468977">https://fi.wikipedia.org/w/index.php?title=Molekulaarinen_jakaumafunktio&oldid=22468977</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Toiminnot:Luokat" title="Toiminnot:Luokat">Luokat</a>: <ul><li><a href="/wiki/Luokka:Funktiot" title="Luokka:Funktiot">Funktiot</a></li><li><a href="/wiki/Luokka:Molekyylimallinnus" title="Luokka:Molekyylimallinnus">Molekyylimallinnus</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Piilotettu luokka: <ul><li><a href="/wiki/Luokka:Sivut,_jotka_k%C3%A4ytt%C3%A4v%C3%A4t_ISBN-taikalinkkej%C3%A4" title="Luokka:Sivut, jotka käyttävät ISBN-taikalinkkejä">Sivut, jotka käyttävät ISBN-taikalinkkejä</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigointivalikko</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Henkilökohtaiset työkalut</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="IP-osoitteesi käyttäjäsivu">Et ole kirjautunut</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Toiminnot:Oma_keskustelu" title="Keskustelu tämän IP-osoitteen muokkauksista [n]" accesskey="n"><span>Keskustelu</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Toiminnot:Omat_muokkaukset" title="Luettelo tästä IP-osoitteesta tehdyistä muokkauksista [y]" accesskey="y"><span>Muokkaukset</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:Luo_tunnus&returnto=Molekulaarinen+jakaumafunktio&returntoquery=section%3D7%26veaction%3Dedit" title="On suositeltavaa luoda käyttäjätunnus ja kirjautua sisään. 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Molekulaarinen jakaumafunktio – Wikipedia