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annotate texts/XML/diverse/de/Einst_Therm_de_1912.xml @ 27:8dce37600d38
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author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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date | Wed, 30 Jul 2014 15:58:21 +0200 |
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1 <?xml version="1.0" encoding="utf-8" ?> |
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5 <html> |
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7 <head><title></title> |
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8 <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> |
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" /> |
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" /> |
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11 <!-- xhtml,html --> |
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12 <meta name="src" content="Einst_Therm_de_1912.tex" /> |
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13 <meta name="date" content="2005-03-02 20:12:00" /> |
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14 <link rel="stylesheet" type="text/css" href="Einst_Therm_de_1912.css" /> |
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15 </head><body > |
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17 <!--l. 13--><p class="noindent"><pb/></p> |
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18 <div class="center" > |
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19 |
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20 <!--l. 14--><p class="noindent"> |
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21 </p><!--l. 15--><p class="noindent"><span |
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22 class="cmr-12">11. </span><span |
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23 class="cmbxti-10x-x-120">Thermodynamische Begr</span><span |
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24 class="cmbxti-10x-x-120">ündung </span> <br/><span |
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25 class="cmbxti-10x-x-120">des photochemischen</span> |
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26 <span |
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27 class="cmbxti-10x-x-120">Äquivalentgesetzes; </span> <br/><span |
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28 class="cmbxti-10x-x-120">von A. Einstein.</span></p></div> |
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29 <div class="center" > |
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30 |
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31 <!--l. 19--><p class="noindent"> |
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32 </p><!--l. 20--><p class="noindent">---------</p></div> |
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33 <!--l. 23--><p class="indent"> Im folgenden wird auf wesentlich thermodynamischem <br/>Wege gleichzeitig das Wiensche Strahlungsgesetz und das <br/>photochemische Äquivalentgesetz abgeleitet. Unter dem letz-<br/>teren verstehe ich den Satz, daß es zur Zersetzung eines |
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34 <br/>Grammäquivalentes durch einen photochemischen Vorgang der <br/>absorbierten Strahlungsenergie <span |
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35 class="cmmi-10">N h<img |
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36 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /> </span>bedarf, falls man mit <span |
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37 class="cmmi-10">N </span> <br/>die Zahl der Moleküle im Gramm-Mol, mit <span |
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38 class="cmmi-10">h </span>die bekannte <br/>Konstante in Plancks Strahlungsformel, mit <span |
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39 class="cmmi-10"><img |
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40 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /> </span>die Frequenz <br/>der wirksamen Strahlung bezeichnet.<sup ><span |
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41 class="cmr-7">1</span></sup>) Das Gesetz erscheint <br/>im wesentlichen als eine konsequenz der Voraussetzung, daß <br/>die Zahl der pro Zeiteinheit zersetzten Moleküle der Dichte <br/>der wirksamen Strahlung proportional ist; doch ist hervor-<br/>zuheben, daß die thermodynamischen Zusammenhänge und das |
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42 <br/>Strahlungsgesetz es nicht gestatten, diese Annahme durch eine <br/>beliebige andere zu ersetzen, wie am Schlusse der Arbeit kurz <br/>gezeigt wird. |
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43 </p><!--l. 42--><p class="indent"> Aus dem Folgenden geht ferner klar hervor, daß das <br/>Äquivalentgesetz bzw. die zu demselben führenden Annahmen <br/>nur so lange gelten, als die wirksame Strahlung dem Gültig-<br/>keitsbereiche des Wienschen Gesetzes angehört, Für solche <br/>Strahlung aber ist nun an der Gültigkeit des Gesetzes kaum <br/>mehr zu zweifeln. |
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44 </p> |
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45 <div class="center" > |
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46 |
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47 <!--l. 49--><p class="noindent"> |
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48 </p><!--l. 50--><p class="noindent"><span |
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49 class="cmsy-10">§ </span>1. Über das thermodynamische Gleichgewicht <br/>zwischen Strahlung und einem teilweise dissoziierten Gase vom <br/>Standpunkt des Massenwirkungsgesetzes.</p></div> |
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50 <!--l. 54--><p class="indent"> Es sei in einem Volumen <span |
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51 class="cmmi-10">V </span>eine Mischung dreier chemisch <br/>verschiedener Gase mit den Molekulargewichten <span |
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52 class="cmmi-10">m</span><sub ><span |
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53 class="cmr-7">1</span></sub>, <span |
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54 class="cmmi-10">m</span><sub ><span |
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55 class="cmr-7">2</span></sub>, <span |
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56 class="cmmi-10">m</span><sub ><span |
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57 class="cmr-7">3</span> <br/></sub> <br/>----------- |
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58 </p><!--l. 59--><p class="indent"> 1) Vgl. A. Einstein, Ann. d. Phys. <span |
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59 class="cmbx-10">4. </span>(17). p. 132. <pb/> |
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60 </p><!--l. 65--><p class="indent"> |
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61 |
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62 </p><!--l. 66--><p class="noindent">vorhanden. <span |
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63 class="cmmi-10">n</span><sub ><span |
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64 class="cmr-7">1</span></sub> sei die Anzahl g-Mole des ersten, <span |
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65 class="cmmi-10">n</span><sub ><span |
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66 class="cmr-7">2</span></sub> die des <br/>zweiten, <span |
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67 class="cmmi-10">n</span><sub ><span |
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68 class="cmr-7">3</span></sub> die des dritten Gases.<sup ><span |
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69 class="cmr-7">1</span></sup>) Zwischen diesen drei <br/>Molekülarten sei eine Reaktion möglich, darin bestehend, daß <br/>ein Molekül erster Art zerfällt in ein Molekül zweiter und ein <br/>Molekül dritter Art. Bei thermodynamischem Gleichgewichte <br/>besteht gleiche Häufigkeit der Reaktionen |
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70 </p> |
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71 <center class="par-math-display" > |
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72 <img |
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73 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19120x.png" alt="m1 --> m2 + m3 " class="par-math-display" /></center> |
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74 <!--l. 78--><p class="nopar"> |
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75 </p><!--l. 82--><p class="noindent">und |
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76 </p> |
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77 <center class="par-math-display" > |
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78 <img |
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79 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19121x.png" alt="m2 + m3 --> m1. " class="par-math-display" /></center> |
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80 <!--l. 88--><p class="nopar"> |
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81 </p><!--l. 92--><p class="indent"> Wir wollen den Fall ins Auge fassen, daß der Zerfall von <br/>Molekülen <span |
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82 class="cmmi-10">m</span><sub ><span |
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83 class="cmr-7">1</span></sub> |
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84 ausschließlich durch die Wirkung der Wärme-<br/>strahlung erfolge, und zwar unter der Wirkung eines Teiles <br/>der Wärmestrahlung, dessen Frequenz sich wenig von einer |
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85 <br/>gewissen Frequenz <span |
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86 class="cmmi-10"><img |
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87 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span><sub ><span |
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88 class="cmr-7">0</span></sub> unterscheidet. Die bei einem derartigen <br/>Zerfall im Mittel absorbierte Strahlungsenergie sei <span |
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89 class="cmmi-10"><img |
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90 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /></span>. In diesem <br/>Falle muß umgekehrt bei dem Prozeß der Vereinigung von <span |
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91 class="cmmi-10">m</span><sub ><span |
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92 class="cmr-7">2</span></sub> <br/>und <span |
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93 class="cmmi-10">m</span><sub ><span |
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94 class="cmr-7">3</span></sub> zu <span |
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95 class="cmmi-10">m</span><sub ><span |
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96 class="cmr-7">1</span></sub> Strahlung von der Frequenzgegend <span |
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97 class="cmmi-10"><img |
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98 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span><sub ><span |
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99 class="cmr-7">0</span></sub> |
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100 emittiert <br/>werden, und zwar ausschließlich Strahlung von der Frequenz-<br/>gegend <span |
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101 class="cmmi-10"><img |
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102 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span><sub ><span |
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103 class="cmr-7">0</span></sub>, und es muß die bei einem Wiedervereinigungsprozeß <br/>emittierte Strahlungsenergie im Mittel ebenfalls gleich <span |
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104 class="cmmi-10"><img |
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105 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /> </span>sein, <br/>da sonst das Strahlungsgleichgewicht durch die Existenz des <br/>Gases gestört würde; denn die Zahl der Zerfallprozesse ist <br/>gleich der Zahl der Vereinigungsprozesse. |
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106 </p><!--l. 104--><p class="indent"> Besitzt das Gasgemisch die Temperatur <span |
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107 class="cmmi-10">T</span>, so wird thermo-<br/>dynamisches Gleichgewicht des Systems jedenfalls bestehen <br/>können, wenn die im Raum befindliche Strahlung in der Um-<br/>gebung der Frequenz <span |
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108 class="cmmi-10"><img |
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109 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span><sub ><span |
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110 class="cmr-7">0</span></sub> diejenige (monochromatische) Dichte <span |
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111 class="cmmi-10"><img |
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112 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span> <br/>besitzt, welche zur Wärmestrahlung der Temperatur <span |
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113 class="cmmi-10">T </span>gehört. |
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114 <br/>Wir analysieren nun die beiden einander gerade aufhebenden <br/>Reaktionen genauer, indem wir über den Mechanismus der-<br/>selben gewisse Annahmen machen. |
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115 </p><!--l. 114--><p class="indent"> Der Zerfall eines Moleküls erster Art geschehe so, wie <br/>wenn die übrigen Moleküle nicht da wären (Annahme I). Daraus <br/>folgt, daß wir die Zahl der pro Zeiteinheit zerfallenden Mole-<br/>küle erster Art deren Anzahl (<span |
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116 class="cmmi-10">n</span><sub ><span |
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117 class="cmr-7">1</span></sub>) unter sonst gleichen Um-<br/>ständen proportional, und daß wir die Zahl der pro Zeiteinheit |
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118 <br/>----------- |
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119 </p><!--l. 123--><p class="indent"> 1) Natürlich kann eines der Gase mit den Indizes 2 und 3 aus <br/>Elektronen bestehen. <pb/> |
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120 </p><!--l. 129--><p class="indent"> |
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121 |
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122 </p><!--l. 130--><p class="noindent">zerfallenden Moleküle als von den drei Gasdichten unabhängig <br/>anzusetzen haben. Außerdem nehmen wir an, daß die Wahr-<br/>scheinlichkeit dafür, daß ein Molekül erster Art in einem Zeit-<br/>teilchen zerfalle, der monochromatischen Strahlungsdichte <span |
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123 class="cmmi-10"><img |
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124 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /></span> |
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125 <br/>proportional sei (Annahme II). |
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126 </p><!--l. 137--><p class="indent"> Hauptsächlich von der zweiten dieser Annahmen muß <br/>hervorgehoben werden, daß ihre Richtigkeit durchaus nicht <br/>selbstverständlich ist. Sie enthält die Aussage, daß die <br/>chemische Wirkung einer auf einen Körper fallenden Strahlung <br/>nur von der Gesamtmenge der wirkenden Strahlung abhänge, <br/>aber nicht von der Bestrahlungsintensität; die Existenz einer <br/>unteren Wirksamkeitsschwelle der Strahlung wird durch diese <br/>Annahme vollkommen ausgeschlossen. Wir setzen uns durch <br/>letztere in Widerspruch mit den Ergebnissen zweier Arbeiten |
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127 <br/>von E. Warburg<sup ><span |
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128 class="cmr-7">1</span></sup>), durch die ich die Anregung für die vor-<br/>liegende Arbeit empfing. |
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129 </p><!--l. 150--><p class="indent"> Aus den beiden Annahmen folgt, daß die Zahl <span |
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130 class="cmmi-10">Z </span>der pro <br/>Zeiteinheit zerfallenden Moleküle erster Art gegeben ist durch <br/>den Ausdruck</p> |
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131 <table width="100%" |
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132 class="equation"><tr><td><a |
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133 id="x1-2r1"></a> |
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134 <center class="math-display" > |
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135 <img |
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136 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19122x.png" alt="Z = A rn1. " class="math-display" /></center></td><td width="5%">(1)</td></tr></table> |
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137 <!--l. 156--><p class="nopar"> |
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138 </p><!--l. 159--><p class="noindent">Der Proportionalitätsfaktor <span |
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139 class="cmmi-10">A </span>kann nach dem Gesagten nur <br/>von der Gastemperatur |
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140 <span |
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141 class="cmmi-10">T </span>abhängen. Nach dem Vorangehenden <br/>gilt die Gleichung auch in dem Falle, daß die Strahlungs-<br/>dichte <span |
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142 class="cmmi-10"><img |
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143 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>(bei der Frequenz <span |
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144 class="cmmi-10"><img |
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145 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span><sub ><span |
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146 class="cmr-7">0</span></sub>) eine andere ist, als zur Tem-<br/>peratur <span |
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147 class="cmmi-10">T </span>des Gases gehört. |
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148 </p><!--l. 166--><p class="indent"> Von dem Wiedervereinigungsprozeß nehmen wir an, daß <br/>es ein gewöhnlicher Vorgang zweiter Ordnung im Sinne des <br/>Massenwirkungsgesetzes sei, daß also die Zahl der pro Volum-<br/>einheit und Zeiteinheit sich bildenden Moleküle erster Art dem <br/>Produkt der Konzentrationen <span |
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149 class="cmmi-10">n</span><sub ><span |
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150 class="cmr-7">2</span></sub><span |
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151 class="cmmi-10">/V</span> und <span |
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152 class="cmmi-10">n</span><sub ><span |
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153 class="cmr-7">3</span></sub><span |
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154 class="cmmi-10">/V </span>proportional sei, |
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155 <br/>wobei der Proportionalitätskoeffizient nur von der Gastemperatur, <br/>aber nicht von der Dichte der vorhandenen Strahlung abhänge <br/>(Annahme III). Die Zahl <span |
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156 class="cmmi-10">Z</span><span |
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157 class="cmsy-10">' </span>der sich in der Zeiteinheit bildenden <br/>Moleküle erster Art ist also</p> |
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158 <table width="100%" |
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159 class="equation"><tr><td><a |
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160 id="x1-3r2"></a> |
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161 <center class="math-display" > |
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162 |
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163 <img |
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164 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19123x.png" alt="Z'= A'.V .n2.n3-. V V " class="math-display" /></center></td><td width="5%">(2)</td></tr></table> |
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165 <!--l. 182--><p class="nopar"> |
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166 </p><!--l. 185--><p class="noindent">----------- |
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167 </p><!--l. 188--><p class="indent"> 1) E. Warburg, Verh. d. Deutsch. Physik. Ges. <span |
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168 class="cmbx-10">9. </span>p. 24. 1908 und <br/>9. p. 21. 1909. |
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169 <pb/> |
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170 </p><!--l. 194--><p class="indent"> |
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171 |
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172 </p><!--l. 195--><p class="indent"> Das von uns betrachtete, aus Strahlung und Gasgemisch <br/>bestehende System befindet sich stets im thermodynamischen <br/>Gleichgewicht, wenn die Zahl <span |
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173 class="cmmi-10">Z</span> |
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174 der Zerfallsprozesse gleich <br/>ist der Zahl <span |
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175 class="cmmi-10">Z</span><span |
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176 class="cmsy-10">' </span>der Vereinigungsprozesse; denn es bleibt in <br/>diesem Falle nicht nur die Menge einer jeden Gasart, sondern |
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177 <br/>auch die Menge der vorhandenen Strahlung ungeändert.<sup ><span |
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178 class="cmr-7">1</span></sup>) Diese <br/>Bedingung lautet</p> |
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179 <table width="100%" |
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180 class="equation"><tr><td><a |
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181 id="x1-4r3"></a> |
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182 <center class="math-display" > |
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183 <img |
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184 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19124x.png" alt=" n2n3 V-V- j2j3 A --n1- = -j-- = A'r, V 1 " class="math-display" /></center></td><td width="5%">(3)</td></tr></table> |
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185 <!--l. 209--><p class="nopar"> |
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186 </p><!--l. 213--><p class="noindent">wobei <span |
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187 class="cmmi-10">A </span>und <span |
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188 class="cmmi-10">A</span><span |
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189 class="cmsy-10">' </span>nur von der Temperatur der Gasmischung <br/>abhängen. Eine eigentümliche Konsequenz dieser Betrachtung <br/>ist die, daß bei gegebener Gastemperatur und beliebig ge-<br/>gebener Strahlungsdichte (d. h. auch Strahlungstemperatur) ein <br/>thermodynamisches Gleichgewicht möglich sein soll. Es liegt <br/>aber hierin kein Verstoß gegen den zweiten Hauptsatz, was <br/>damit zusammenhängt, daß mit einem Wärmeübergang von <br/>der Strahlung zum Gase ein bestimmter chemischer Prozeß <br/>zwangläufig verbunden ist; man kann mit Hilfe des von uns <br/>betrachteten Systems kein Perpetuum mobile zweiter Art kon-<br/>struieren. |
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190 </p> |
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191 <div class="center" > |
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192 |
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193 <!--l. 227--><p class="noindent"> |
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194 </p><!--l. 228--><p class="noindent"><span |
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195 class="cmsy-10">§ </span>2. Thermodynamische Gleichgewichtsbedingung für das im <span |
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196 class="cmsy-10">§ </span>1 <br/>betrachtete System.</p></div> |
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197 <!--l. 231--><p class="indent"> Ist <span |
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198 class="cmmi-10">S</span><sub ><span |
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199 class="cmmi-7">s</span></sub> die Entropie der im Volumen <span |
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200 class="cmmi-10">V </span>enthaltenen Strahlung, <br/><span |
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201 class="cmmi-10">S</span><sub ><span |
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202 class="cmmi-7">g</span></sub> diejenige des Gasgemisches, so muß für jeden der im vorigen <br/>Paragraph gefundenen Gleichgewichtszustände die Bedingung <br/>bestehen, daß für jede unendlich kleine virtuelle Änderung der <br/>Zustände von Strahlung und Gas die Änderung der Gesamt-<br/>entropie verschwindet. Die zu betrachtende virtuelle Änderung |
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203 <br/>besteht darin, daß die Energiemenge <span |
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204 class="cmmi-10">N <img |
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205 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /> </span>(aus der Umgebung <br/>von <span |
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206 class="cmmi-10"><img |
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207 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span><sub ><span |
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208 class="cmr-7">0</span></sub>) der Strahlung in Energie des Gasgemisches übergeht <br/>unter gleichzeitigem Zerfall eines Gasmoleküls (g-Mol) erster <br/>Art. Bei einer solchen virtuellen Änderung würde sich die <br/>Temperatur des Gemisches um einen nicht zu vernachlässigenden |
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209 <br/>----------- |
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210 </p><!--l. 242--><p class="indent"> 1) Beim Lesen der Korrektur bemerke ich, daß dieser für das <br/>Folgende wesentliche Schluß nur unter der Voraussetzung gilt, daß bei <br/>gegebener Gastemperatur <span |
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211 class="cmmi-10"><img |
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212 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /> </span>von <span |
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213 class="cmmi-10"><img |
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214 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /></span> |
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215 unabhängig ist. <pb/> |
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216 </p><!--l. 250--><p class="indent"> |
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217 |
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218 </p><!--l. 251--><p class="noindent">Betrag verändern. Um dies zu vermeiden, denken wir uns in <br/>bekannter Weise das Gasgemisch mit einem unendlich großen <br/>Wärmereservoir von derselben Temperatur |
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219 <span |
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220 class="cmmi-10">T </span>in dauernder <br/>wärmeleitender Verbindung. Bei der virtuellen Änderung ändert <br/>sich dann die Temperatur des Gasgemisches nicht; dagegen <br/>ist zu berücksichtigen, daß das Wärmereservoir die Energie <br/><span |
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221 class="cmsy-10">-</span><img |
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222 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19125x.png" alt="(dEs +d Eg)" class="left" align="middle" /> bei der virtuellen Änderung in Form von Wärme <br/>aufnimmt, falls man mit <span |
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223 class="cmmi-10">E</span><sub ><span |
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224 class="cmmi-7">s</span></sub> die Energie der Strahlung, mit |
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225 <br/><span |
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226 class="cmmi-10">E</span><sub ><span |
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227 class="cmmi-7">g</span></sub> diejenige des Gases bezeichnet. Die Gleichgewichtsbedin-<br/>gung lautet deshalb</p> |
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228 <table width="100%" |
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229 class="equation"><tr><td><a |
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230 id="x1-5r4"></a> |
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231 <center class="math-display" > |
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232 <img |
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233 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19126x.png" alt="dSs + d Sg- dEs-+-dEg-= 0 . T " class="math-display" /></center></td><td width="5%">(4)</td></tr></table> |
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234 <!--l. 269--><p class="nopar"> |
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235 </p><!--l. 272--><p class="noindent">Wir haben nun die einzelnen Glieder dieser Gleichung zu <br/>berechnen. Es ist zunächst für die von uns betrachtete virtuelle <br/>Anderung |
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236 </p> |
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237 <center class="par-math-display" > |
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238 <img |
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239 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19127x.png" alt="dEs = -N e, dS = -N-e, s Ts " class="par-math-display" /></center> |
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240 <!--l. 283--><p class="nopar"> |
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241 </p><!--l. 287--><p class="noindent">wenn man mit <span |
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242 class="cmmi-10">T</span><sub ><span |
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243 class="cmmi-7">s</span></sub> die zur Strahlungsdichte <span |
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244 class="cmmi-10"><img |
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245 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>gehörige Tem-<br/>peratur bezeichnet. Die auf das Gas bezüglichen Variationen <br/>berechnen wir nach in der Thermodynamik geläufigen Methoden, <br/>wobei wir -- was für das Folgende nicht wesentlich ist --<br/>die spezifischen Wärmen als von der Temperatur unabhängig <br/>behandeln. Man erhält zunächst |
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246 </p> |
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247 <center class="par-math-display" > |
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248 <img |
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249 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19128x.png" alt=" sum |
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250 E0 = sum n1{{cn1T + b1}, n1} S0 = n1 cn1 lgT + c1 - R lg V . " class="par-math-display" /></center> |
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251 <!--l. 303--><p class="nopar"> |
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252 |
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253 </p><!--l. 307--><p class="indent"> Dabei bedeutet |
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254 </p><!--l. 309--><p class="indent"> <span |
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255 class="cmmi-10">c</span><sub ><span |
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256 class="cmmi-7"><img |
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257 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span><sub ><span |
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258 class="cmr-5">1</span></sub></sub> die Wärmekapazität pro g-Mol bei konstantem Volum, |
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259 </p><!--l. 311--><p class="indent"> <span |
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260 class="cmmi-10">b</span><sub ><span |
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261 class="cmr-7">1</span></sub> die Energie pro g-Mol der 1. Gasart bei <span |
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262 class="cmmi-10">T </span>= 0<span |
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263 class="cmmi-10">,</span> |
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264 </p><!--l. 313--><p class="indent"> <span |
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265 class="cmmi-10">c</span><sub ><span |
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266 class="cmr-7">1</span></sub> eine Integrationskonstante der Entropie der ersten Gasart. |
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267 </p><!--l. 315--><p class="indent"> Aus diesen Gleichungen folgen unmittelbar die folgenden |
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268 </p> |
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269 <center class="par-math-display" > |
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270 <img |
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271 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_19129x.png" alt=" sum |
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272 d Eg = dn1{{cn1 T + b1}, } d S = sum dn c lg T + c - R - R lg n1 , g 1 n1 1 V " class="par-math-display" /></center> |
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273 <!--l. 326--><p class="nopar"> |
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274 </p><!--l. 330--><p class="noindent">wobei</p> |
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275 <table width="100%" |
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276 class="equation"><tr><td><a |
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277 id="x1-6r5"></a> |
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278 <center class="math-display" > |
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279 <img |
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280 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191210x.png" alt="dn1 = -1, dn2 = +1, dn3 = +1 " class="math-display" /></center></td><td width="5%">(4a<span |
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281 class="cmsy-10">'</span>)</td></tr></table> |
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282 <!--l. 337--><p class="nopar"> |
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283 <pb/> |
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284 </p><!--l. 344--><p class="indent"> |
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285 |
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286 </p><!--l. 345--><p class="noindent">zu setzen ist. Gleichung (4) nimmt vermöge dieser Gleichungen <br/>für die Variationen und der Gleichung (3) die Form an |
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287 </p> |
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288 <center class="par-math-display" > |
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289 <img |
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290 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191211x.png" alt=" ( ) N-e- A- - RTs + lg a - lg A' r = 0, " class="par-math-display" /></center> |
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291 <!--l. 353--><p class="nopar"> |
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292 </p><!--l. 357--><p class="noindent">wobei zur Abkürzung gesetzt ist</p> |
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293 <table width="100%" |
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294 class="equation"><tr><td><a |
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295 id="x1-7r5"></a> |
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296 <center class="math-display" > |
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297 <img |
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298 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191212x.png" alt=" { } lg a = N-e-+ 1- sum dn1 cn lg T + c1- R - cn - b1- . R T R 1 1 T " class="math-display" /></center></td><td width="5%">(4a<span |
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299 class="cmsy-10">''</span>)</td></tr></table> |
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300 <!--l. 365--><p class="nopar"> |
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301 </p><!--l. 368--><p class="noindent">Die mit <span |
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302 class="cmmi-10"><img |
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303 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /> </span>bezeichnete Größe ist von <span |
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304 class="cmmi-10">T</span><sub ><span |
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305 class="cmmi-7">s</span></sub> unabhängig. |
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306 </p> |
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307 <div class="center" > |
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308 |
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309 <!--l. 371--><p class="noindent"> |
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310 </p><!--l. 372--><p class="noindent"><span |
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311 class="cmsy-10">§ </span>3. Schlußfolgerungen aus der Gleichgewichtsbedingung.</p></div> |
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312 <!--l. 375--><p class="indent"> Wir schreiben nun (4a) in der Form |
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313 </p> |
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314 <table width="100%" |
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315 class="equation"><tr><td><a |
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316 id="x1-8r5"></a> |
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317 <center class="math-display" > |
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318 <img |
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319 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191213x.png" alt=" ' -N-e r = A-a-e-R Ts. A " class="math-display" /></center></td><td width="5%">(4b)</td></tr></table> |
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320 <!--l. 380--><p class="nopar"> |
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321 </p><!--l. 383--><p class="noindent">Da die Beziehung zwischen <span |
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322 class="cmmi-10">T</span><sub ><span |
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323 class="cmmi-7">s</span></sub> und <span |
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324 class="cmmi-10"><img |
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325 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>unabhängig sein muß <br/>von <span |
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326 class="cmmi-10">T</span>, müssen die Größen <span |
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327 class="cmmi-10">A</span><span |
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328 class="cmsy-10">'</span><span |
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329 class="cmmi-10"><img |
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330 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" />/A </span>und <span |
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331 class="cmmi-10"><img |
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332 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /> </span>von <span |
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333 class="cmmi-10">T </span>unabhängig <br/>sein. Da diese Größen auch von <span |
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334 class="cmmi-10">T</span><sub ><span |
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335 class="cmmi-7">s</span></sub> |
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336 unabhängig sind, so sind <br/>wir damit zu demjenigen Zusammenhang zwischen <span |
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337 class="cmmi-10"><img |
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338 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>und |
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339 <span |
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340 class="cmmi-10">T</span><sub ><span |
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341 class="cmmi-7">s</span></sub> <br/>gelangt, der der Wienschen Strahlungsformel entspricht. Wir <br/>schließen hieraus: |
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342 </p><!--l. 395--><p class="indent"> <span |
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343 class="cmti-10">Die im </span><span |
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344 class="cmsy-10">§</span><span |
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345 class="cmti-10">1 zugrunde gelegten Annahmen </span><span |
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346 class="cmti-10">über den Verlauf </span> <br/><span |
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347 class="cmti-10">photochemischer</span> |
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348 <span |
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349 class="cmti-10">Vorg</span><span |
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350 class="cmti-10">änge sind mit dem empirisch bekannten </span> <br/><span |
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351 class="cmti-10">Gesetze der W</span><span |
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352 class="cmti-10">ärmestrahlung nur</span> |
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353 <span |
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354 class="cmti-10">vereinbar, insofern die wirkende </span> <br/><span |
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355 class="cmti-10">Strahlung in den G</span><span |
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356 class="cmti-10">ültigkeitsbereich des Wienschen</span> |
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357 <span |
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358 class="cmti-10">Strahlungs- </span> <br/><span |
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359 class="cmti-10">gesetzes f</span><span |
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360 class="cmti-10">ällt; in diesem Falle aber ist Wiens Gesetz eine Kon- </span> <br/><span |
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361 class="cmti-10">sequenz</span> |
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362 <span |
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363 class="cmti-10">unserer Annahmen.</span> |
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364 </p><!--l. 397--><p class="indent"> Schreiben wir Wiens Strahlungsformel unter Einführung <br/>der Planckschen Konstanten in der Form |
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365 </p> |
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366 <center class="par-math-display" > |
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367 <img |
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368 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191214x.png" alt=" h-n- r = 8p-hn3e- xTs , c3 " class="par-math-display" /></center> |
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369 <!--l. 403--><p class="nopar"> |
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370 </p><!--l. 407--><p class="noindent">so sehen wir durch Vergleichung mit (4b), daß die Gleichungen |
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371 </p> |
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372 <table width="100%" |
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373 class="equation"><tr><td><a |
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374 id="x1-9r5"></a> |
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375 <center class="math-display" > |
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376 <img |
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377 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191215x.png" alt="e = h n0, " class="math-display" /></center></td><td width="5%">(5)</td></tr></table> |
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378 <!--l. 413--><p class="nopar"> |
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379 </p> |
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380 <table width="100%" |
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381 class="equation"><tr><td><a |
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382 id="x1-10r6"></a> |
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383 |
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384 <center class="math-display" > |
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385 <img |
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386 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191216x.png" alt=" ' 3 A-a-= 8-phn30- A c " class="math-display" /></center></td><td width="5%">(6)</td></tr></table> |
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387 <!--l. 420--><p class="nopar"> |
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388 </p><!--l. 424--><p class="noindent">erfülit sein müssen. Als wichtigste Konsequenz folgt also (5), <br/><span |
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389 class="cmti-10">da</span><span |
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390 class="cmmi-10"><img |
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391 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-c.png" alt="b" class="cmmi-10x-x-c" align="middle" /> </span><span |
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392 class="cmti-10">ein Gasmolek</span><span |
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393 class="cmti-10">ül,</span> |
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394 <span |
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395 class="cmti-10">welches unter Absorption von Strahlung von </span> <br/><pb/> |
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396 </p><!--l. 429--><p class="indent"> |
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397 |
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398 </p><!--l. 430--><p class="noindent"><span |
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399 class="cmti-10">der Frequenz </span><span |
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400 class="cmmi-10"><img |
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401 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span><sub ><span |
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402 class="cmr-7">0</span></sub> <span |
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403 class="cmti-10">zerf</span><span |
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404 class="cmti-10">ällt, bei seinem Zerfall </span>(<span |
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405 class="cmti-10">im Mittel</span>) <span |
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406 class="cmti-10">die </span> <br/><span |
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407 class="cmti-10">Strahlungsenergie </span><span |
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408 class="cmmi-10">h <img |
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409 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /></span><sub ><span |
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410 class="cmr-7">0</span></sub> |
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411 <span |
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412 class="cmti-10">absorbiert. </span>Wir haben die einfachste <br/>Art der Reaktion vorausgesetzt, hätten aber Gleichung (5) auf <br/>demselben Wege wie hier auch für andere unter Lichtabsorption <br/>vor sich gehende Gasreaktionen ableiten können. Ebenso liegt <br/>es auf der Hand, daß die Beziehung in ähnlicher Weise für <br/>verdünnte Lösungen bewiesen werden kann. Sie dürfte wohl <br/>allgemein gültig sein. |
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413 </p><!--l. 437--><p class="indent"> Wir ersetzen ferner mit Hilfe von (6) in (4a<span |
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414 class="cmsy-10">''</span>) die Größe <span |
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415 class="cmmi-10"><img |
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416 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /></span>, <br/>so erhalten wir mit Berücksichtigung von (3), indem wir zur <br/>Abkürzung <span |
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417 class="cmmi-10"><img |
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418 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-11.png" alt="j" class="cmmi-10x-x-11" align="middle" /></span><sub ><span |
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419 class="cmr-7">2</span></sub> <span |
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420 class="cmmi-10"><img |
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421 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-11.png" alt="j" class="cmmi-10x-x-11" align="middle" /></span><sub ><span |
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422 class="cmr-7">3</span></sub> <img |
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423 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191217x.png" alt="/" class="left" align="middle" /> <span |
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424 class="cmmi-10"><img |
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425 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/cmmi10-11.png" alt="j" class="cmmi-10x-x-11" align="middle" /></span><sub ><span |
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426 class="cmr-7">1</span></sub> = <span |
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427 class="cmmi-10">z </span>setzen und Wiens Strahlungsgesetz <br/>anwenden</p> |
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428 <center class="par-math-display" > |
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429 <img |
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430 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Therm_de_1912/fulltext/img/Einst_Therm_de_191218x.png" alt=" sum { } lgz = N-hv0-- N-hv0-+ 1- dn1 cn1 lg T + c1- (cn1 + R) - b1 . R T R Ts R T " class="par-math-display" /></center> |
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431 <!--l. 446--><p class="nopar"> |
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432 </p><!--l. 449--><p class="noindent">Diese Gleichung geht für <span |
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433 class="cmmi-10">T </span>= <span |
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434 class="cmmi-10">T</span><sub ><span |
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435 class="cmmi-7">s</span></sub> in die bekannte Gleichung <br/>für das Dissoziationsgleichgewicht für Gase über, ein Beweis <br/>dafür, daß die vorstehende Theorie mit der thermodynamischen <br/>Theorie der Dissoziation nicht in Widerspruch gerät. |
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436 </p><!--l. 456--><p class="indent"> Prag, Januar 1912. |
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437 </p> |
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438 <div class="center" > |
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439 |
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440 <!--l. 458--><p class="noindent"> |
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441 </p><!--l. 459--><p class="noindent">(Eingegangen 18. Januar 1912.)</p></div> |
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442 <div class="center" > |
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443 |
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444 <!--l. 463--><p class="noindent"> |
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445 </p><!--l. 464--><p class="noindent">----------</p></div> |
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446 |
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447 </body></html> |
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448 |
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449 |
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450 |