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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
author casties
date Fri, 07 Dec 2012 17:05:22 +0100
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1 <?xml version="1.0" encoding="utf-8" ?>
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4 <!--http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd-->
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5 <html>
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7 <head><title></title>
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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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12 <meta name="src" content="Einst_Zural_de_1904.tex" />
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13 <meta name="date" content="2005-03-10 17:39:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Zural_de_1904.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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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-12x-x-120">6. </span><span
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23 class="cmbxti-10x-x-144">Zur allgemeinen molekularen Theorie </span> <br/><span
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24 class="cmbxti-10x-x-144">der W</span><span
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25 class="cmbxti-10x-x-144">ärme;</span>
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26 <span
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27 class="cmbxti-10x-x-144">von A. Einstein.</span></p></div>
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28 <div class="center" >
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30 <!--l. 19--><p class="noindent">
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31 </p><!--l. 20--><p class="noindent">----------</p></div>
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32 <!--l. 23--><p class="indent"> Im folgenden gebe ich einige Ergänzungen zu einer letztes <br/>Jahr von mir publizierten Abhandlung.<sup ><span
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33 class="cmr-8">1</span></sup>)
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34 </p><!--l. 26--><p class="indent"> Wenn ich von ,,allgemeiner molekularer Wärmetheorie&#8220; <br/>spreche, so meine ich damit eine Theorie, welche im wesent-<br/>lichen auf den in <span
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35 class="cmsy-10x-x-120">§ </span>1 der zitierten Abhandlung genannten <br/>Voraussetzungen beruht. Ich setze jene Abhandlung als bekannt <br/>voraus, um unnütze Wiederholungen zu vermeiden, und be-<br/>diene mich der dort gebrauchten Bezeichnungen.
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36 </p><!--l. 34--><p class="indent"> Zuerst wird ein Ausdruck für die Entropie eines Systems <br/>abgeleitet, welcher dem von Boltzmann für ideale Gase ge-<br/>fundenen und von Planck in seiner Theorie der Strahlung <br/>vorausgesetzten vollständig analog ist. Dann wird eine ein-<br/>fache Herleitung des zweiten Hauptsatzes gegeben. Hierauf
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37 <br/>wird die Bedeutung einer universellen Konstanten untersucht, <br/>welche in der allgemeinen molekularen Theorie der Wärme <br/>eine wichtige Rolle spielt. Schließlich folgt eine Anwendung <br/>der Theorie auf die Strahlung schwarzer Körper, wobei sich <br/>zwischen der erwähnten, durch die Größen der Elementar-<br/>quanta der Materie und der Elektrizität bestimmten universellen
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38 <br/>Konstanten und der Größenordnung der Strahlungswellenlängen, <br/>ohne Zuhilfenahme speziellerer Hypothesen, eine höchst inter-<br/>essante Beziehung ergibt.
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39 </p>
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40 <div class="center" >
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42 <!--l. 52--><p class="noindent">
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43 </p><!--l. 53--><p class="noindent"><span
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44 class="cmsy-10x-x-120">§ </span>1. Über den Ausdruck der Entropie.</p></div>
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45 <!--l. 57--><p class="indent"> Für ein System, welches Energie nur in Form von Wärme <br/>aufnehmen kann, oder mit anderen Worten, für ein System, <br/>welches von anderen Systemen nicht adiabatisch beeinflußt <br/>wird, gilt zwischen der absoluten Temperatur <span
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46 class="cmmi-12">T </span>und der
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47 <br/>Energie <span
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48 class="cmmi-12">E</span>, nach <span
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49 class="cmsy-10x-x-120">§ </span>3 und <span
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50 class="cmsy-10x-x-120">§ </span>4, l. c., die Gleichung:</p>
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51 <table width="100%"
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52 class="equation"><tr><td><a
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53 id="x1-2r1"></a>
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55 <center class="math-display" >
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56 <img
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57 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19040x.png" alt=" 1w'(E)- --1--- h = 2w (E) = 4 x T , " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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58 <!--l. 68--><p class="nopar">
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59 </p><!--l. 71--><p class="noindent">----------
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60 </p><!--l. 74--><p class="indent"> 1) A. Einstein, Ann. d. Phys. <span
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61 class="cmbx-12">11. </span>p. 170. 1903. <pb/>
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62 </p><!--l. 79--><p class="indent">
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63
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64 </p><!--l. 80--><p class="noindent">wobei <span
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65 class="cmbxti-10x-x-120">x</span> eine absolute Konstante bedeutet und <span
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66 class="cmmi-12"><img
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67 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-21.png" alt="w" class="12x-x-21" /> </span>(etwas ab-<br/>weichend von der zitierten Abhandlung) durch die Gleichung <br/>definiert sei:</p>
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68 <center class="par-math-display" >
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69 <img
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70 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19041x.png" alt=" E integral +dE
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71 w (E) .dE = d p1 ...d pn . E " class="par-math-display" /></center>
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72 <!--l. 87--><p class="nopar">
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73 </p><!--l. 90--><p class="noindent">Das Integral rechts ist hierbei über alle Werte der den momen-<br/>tanen Zustand des Systems vollkommen und eindeutig definieren-<br/>den Zustandsvariabeln zu erstrecken, denen Werte der Energie <br/>entsprechen, die zwischen <span
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74 class="cmmi-12">E </span>und <span
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75 class="cmmi-12">E </span>+ <span
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76 class="cmmi-12"><img
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77 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /> E</span>
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78 liegen.
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79 </p><!--l. 97--><p class="indent"> Aus Gleichung (1) folgt:
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80 </p>
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81 <center class="par-math-display" >
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82 <img
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83 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19042x.png" alt=" integral d E S = ---- = 2 x log [w (E)]. T " class="par-math-display" /></center>
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84 <!--l. 102--><p class="nopar">
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85 </p><!--l. 105--><p class="noindent">Der Ausdruck stellt also (unter Weglassung der willkürlichen <br/>Integrationskonstanten) die Entropie des Systems dar. Dieser <br/>Ausdruck für die Entropie eines Systems gilt übrigens keines-<br/>wegs nur für Systeme, welche nur rein thermische Zustands-<br/>änderungen erfahren, sondern auch für solche, welche beliebige <br/>adiabatische und isopyknische Zustandsänderungen durch-<br/>laufen.
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86 </p><!--l. 115--><p class="indent"> Der Beweis kann aus der letzten Gleichung von <span
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87 class="cmsy-10x-x-120">§ </span>6, l. c., <br/>geführt werden; ich unterlasse dies, da ich hier keine An-<br/>wendung des Satzes in seiner allgemeinen Bedeutung zu machen <br/>beabsichtige.
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88 </p>
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89 <div class="center" >
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90
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91 <!--l. 122--><p class="noindent">
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92 </p><!--l. 123--><p class="noindent"><span
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93 class="cmsy-10x-x-120">§ </span>2. Herleitung des zweiten Hauptsatzes.</p></div>
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94 <!--l. 126--><p class="indent"> Befindet sich ein System in einer Umgebung von be-<br/>stimmter konstanter Temperatur <span
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95 class="cmmi-12">T</span><sub ><span
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96 class="cmr-8">0</span></sub> und steht es mit dieser <br/>Umgebung in thermischer Wechselwirkung (,,Berührung&#8220;), so <br/>nimmt es ebenfalls erfahrungsgemäß die Temperatur <span
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97 class="cmmi-12">T</span><sub ><span
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98 class="cmr-8">0</span></sub> an <br/>und behält die Temperatur <span
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99 class="cmmi-12">T</span><sub ><span
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100 class="cmr-8">0</span></sub> für alle Zeiten bei.
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101 </p><!--l. 133--><p class="indent"> Nach der molekularen Theorie der Wärme gilt jedoch <br/>dieser Satz nicht streng, sondern nur mit gewisser -- wenn <br/>auch für alle der direkten Untersuchung zugänglichen Systeme <br/>mit sehr großer -- Annäherung. Hat sich vielmehr das be-<br/>trachtete System unendlich lange in der genannten Umgebung <br/>befunden, so ist die Wahrscheinlichkeit <span
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102 class="cmmi-12">W </span>dafür, daß in einem <br/><pb/>
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103
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104 </p><!--l. 143--><p class="indent">
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105
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106 </p><!--l. 144--><p class="noindent">beliebig heransgegriffenen Zeitpunkt der Wert der Energie des <br/>Systems sichz wischen den Grenzen <span
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107 class="cmmi-12">E </span>und <span
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108 class="cmmi-12">E </span>+ 1 befindet (<span
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109 class="cmsy-10x-x-120">§ </span>3, l c.):
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110 </p>
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111 <center class="par-math-display" >
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112 <img
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113 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19043x.png" alt=" - -E-- W = C e 2x T0 w(E) , " class="par-math-display" /></center>
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114 <!--l. 151--><p class="nopar">
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115 </p><!--l. 155--><p class="noindent">wobei <span
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116 class="cmmi-12">C </span>eine Konstante bedeutet. Dieser Wert ist für jedes <span
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117 class="cmmi-12">E </span> <br/>ein von Null verschiedener, hat jedoch für ein bestimmtes <span
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118 class="cmmi-12">E </span> <br/>ein Maximum und nimmt -- wenigstens für alle der direkten <br/>Untersuchung zugänglichen Systeme -- für jedes merklich <br/>größere oder kleinere <span
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119 class="cmmi-12">E </span>einen sehr kleinen Wert an. Wir <br/>nennen das System ,,Wärmereservoir&#8220; und sagen kurz: obiger <br/>Ausdruck stellt die Wahrscheinlichkeit dafür dar, daß die <br/>Energie des betrachteten Wärmereservoirs in der genannten <br/>Umgebung den Wert <span
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120 class="cmmi-12">E </span>hat. Nach dem Ergebnis des vorigen
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121 <br/>Paragraphen kann man auch schreiben:
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122 </p>
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123 <center class="par-math-display" >
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124 <img
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125 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19044x.png" alt=" 1-(S- E-) W = C e 2x T0 , " class="par-math-display" /></center>
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126 <!--l. 171--><p class="nopar">
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127 </p><!--l. 175--><p class="noindent">wobei <span
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128 class="cmmi-12">S </span>die Entropie des Wärmereservoirs bedeutet.
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129 </p><!--l. 178--><p class="indent"> Es mögen nun eine Anzahl Wärmereservoirs vorliegen, <br/>welche sich sämtlich in der Umgebung von der Temperatur <span
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130 class="cmmi-12">T</span><sub ><span
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131 class="cmr-8">0</span></sub> <br/>befinden. Die Wahrscheinlichkeit dafür, daß die Energie des <br/>ersten Reservoirs den Wert <span
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132 class="cmmi-12">E</span><sub ><span
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133 class="cmr-8">1</span></sub>, des zweiten den Wert <span
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134 class="cmmi-12">E</span><sub ><span
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135 class="cmr-8">2</span></sub><span
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136 class="cmmi-12">...</span> <br/>des letzten den Wert <span
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137 class="cmmi-12">E</span><sub ><span
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138 class="cmmi-8">l</span></sub> besitzt, ist dann in leicht verständ-<br/>licher Bezeichnung:
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139 </p>
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140 <table width="100%"
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141 class="equation"><tr><td><a
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142 id="x1-3r2"></a>
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143 <center class="math-display" >
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144 <img
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145 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19045x.png" alt=" { sum l sum lE} 12x 1 s--1T0 W = W1 .W2 ...Wl = C1 .C2 ...Cle . " class="math-display" /></center></td><td width="5%">(a)</td></tr></table>
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146 <!--l. 190--><p class="nopar">
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147
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148 </p><!--l. 194--><p class="indent"> Diese Reservoirs mögen nun in Wechselwirkung treten <br/>mit einer Maschine, wobei letztere einen Kreisprozeß durch-<br/>läuft. Bei diesem Vorgange finde weder zwischen Wärme-<br/>reservoirs und Umgebung noch zwischen Maschine und Um-<br/>gebung ein Wärmeaustausch statt. Nach dem betrachteten <br/>Vorgange seien die Energien und Entropien der Systeme:
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149 </p>
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150 <center class="par-math-display" >
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151 <img
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152 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19046x.png" alt="E '1, E '2 ...E l', " class="par-math-display" /></center>
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153 <!--l. 205--><p class="nopar">
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154 </p><!--l. 209--><p class="noindent">bez.</p>
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155 <center class="par-math-display" >
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156 <img
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157 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19047x.png" alt="S1', S2'...S 'l " class="par-math-display" /></center>
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158 <!--l. 216--><p class="nopar"> <pb/>
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159 </p><!--l. 223--><p class="indent">
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160
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161 </p><!--l. 224--><p class="noindent">Dem Gesamtzustande der Wärmereservoirs, welcher durch <br/>diese Werte definiert ist, kommt die Wahrscheinlichkeit zu:</p>
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162 <table width="100%"
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163 class="equation"><tr><td><a
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164 id="x1-4r2"></a>
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165 <center class="math-display" >
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166 <img
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167 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19048x.png" alt=" ( ) sum l sum lE' 12x s'--1T0- W '= C1 .C2 ...Cle 1 . " class="math-display" /></center></td><td width="5%">(b)</td></tr></table>
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168 <!--l. 234--><p class="nopar">
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169 </p><!--l. 238--><p class="indent"> Bei dem Vorgange hat sich weder der Zustand der Um-<br/>gebung noch der Zustand der Maschine geändert, da letztere <br/>einen Kreisprozeß durchlief.
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170 </p><!--l. 242--><p class="indent"> Nehmen wir nun an, daß nie unwahrscheinlichere Zu-<br/>stände auf wahrscheinlichere folgen, so ist:
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171 </p>
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172 <center class="par-math-display" >
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173 <img
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174 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_19049x.png" alt=" ' W &gt;= W . " class="par-math-display" /></center>
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175 <!--l. 248--><p class="nopar">
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176 </p><!--l. 252--><p class="indent"> Es ist aber auch nach dem Energieprinzip:
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177 </p>
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178 <center class="par-math-display" >
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179 <img
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180 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190410x.png" alt=" sum l sum l E = E'. 1 1 " class="par-math-display" /></center>
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181 <!--l. 257--><p class="nopar">
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182 </p><!--l. 260--><p class="noindent">Berücksichtigt man dies, so folgt aus Gleichungen (a) und (b):
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183 </p>
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184 <center class="par-math-display" >
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185 <img
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186 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190411x.png" alt=" sum sum
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187 S' &gt;= S . " class="par-math-display" /></center>
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188 <!--l. 267--><p class="nopar">
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189
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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. 271--><p class="noindent">
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194 </p><!--l. 272--><p class="noindent"><span
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195 class="cmsy-10x-x-120">§ </span>3. Über die Bedeutung der Konstanten <span
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196 class="cmbxti-10x-x-120">x</span> in der kinetischen <br/>Atomtheorie.</p></div>
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197 <!--l. 278--><p class="indent"> Es werde ein physikalisches System betrachtet, dessen <br/>momentaner Zustand durch die Werte der Zustandsvariabeln
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198 </p>
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199 <center class="par-math-display" >
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200 <img
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201 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190412x.png" alt="p1 , p2 ...pn " class="par-math-display" /></center>
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202 <!--l. 284--><p class="nopar">
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203 </p><!--l. 288--><p class="noindent">vollständig bestimmt sei.
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204 </p><!--l. 291--><p class="indent"> Wenn das betrachtete System mit einem System von relativ <br/>unendlich großer Energie und der absoluten Temperatur <span
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205 class="cmmi-12">T</span><sub ><span
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206 class="cmr-8">0</span></sub> <br/>in ,,Berührung&#8220; steht, so ist dessen Zustandsverteilung durch <br/>die Gleichung bestimmt:
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207 </p>
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208 <center class="par-math-display" >
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209 <img
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210 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190413x.png" alt=" - 2ExT- d W = C e 0 d p1 ...d pn . " class="par-math-display" /></center>
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211 <!--l. 300--><p class="nopar">
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212 </p><!--l. 303--><p class="noindent">In dieser Gleichung ist <span
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213 class="cmbxti-10x-x-120">x</span> eine universelle Konstante, deren <br/>Bedeutung nun untersucht werden soll.
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214 </p><!--l. 306--><p class="indent"> Unter Zugrundelegung der kinetischen Atomtheorie gelangt <br/>man auf folgendem, aus Boltzmanns Arbeiten über Gas-<br/>theorie geläufigen Wege zu einer Deutung dieser Konstanten.
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215 </p><!--l. 311--><p class="indent"> Es seien die <span
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216 class="cmmi-12">p</span><sub ><span
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217 class="cmmi-8"><img
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218 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die rechtwinkligen Koordinaten <span
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219 class="cmmi-12">x</span><sub ><span
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220 class="cmr-8">1</span></sub> <span
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221 class="cmmi-12">y</span><sub ><span
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222 class="cmr-8">1</span></sub> <span
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223 class="cmmi-12">z</span><sub ><span
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224 class="cmr-8">1</span></sub>, <br/><span
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225 class="cmmi-12">x</span><sub ><span
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226 class="cmr-8">2</span></sub> <span
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227 class="cmmi-12">y</span><sub ><span
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228 class="cmr-8">2</span></sub><span
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229 class="cmmi-12">...</span><span
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230 class="cmmi-12">, x</span><sub ><span
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231 class="cmmi-8">n</span></sub> <span
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232 class="cmmi-12">y</span><sub ><span
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233 class="cmmi-8">n</span></sub> <span
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234 class="cmmi-12">z</span><sub ><span
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235 class="cmmi-8">n</span></sub>
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236 und <span
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237 class="cmmi-12"><img
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238 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /></span><sub ><span
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239 class="cmr-8">1</span></sub> <span
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240 class="cmmi-12"><img
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241 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-11.png" alt="j" class="cmmi-12x-x-11" align="middle" /></span><sub ><span
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242 class="cmr-8">1</span></sub> <span
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243 class="cmmi-12"><img
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244 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-10.png" alt="z" class="cmmi-12x-x-10" align="middle" /></span><sub ><span
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245 class="cmr-8">1</span></sub><span
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246 class="cmmi-12">, <img
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247 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /></span><sub ><span
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248 class="cmr-8">2</span></sub> <span
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249 class="cmmi-12"><img
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250 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-11.png" alt="j" class="cmmi-12x-x-11" align="middle" /></span><sub ><span
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251 class="cmr-8">2</span></sub> <span
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252 class="cmmi-12">...</span><span
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253 class="cmmi-12">, <img
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254 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /></span><sub ><span
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255 class="cmmi-8">n</span></sub> <span
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256 class="cmmi-12"><img
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257 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-11.png" alt="j" class="cmmi-12x-x-11" align="middle" /></span><sub ><span
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258 class="cmmi-8">n</span></sub> <span
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259 class="cmmi-12"><img
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260 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-10.png" alt="z" class="cmmi-12x-x-10" align="middle" /></span><sub ><span
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261 class="cmmi-8">n</span></sub> die Geschwindigkeiten <br/><pb/>
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262 </p><!--l. 318--><p class="indent">
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263
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264 </p><!--l. 319--><p class="noindent">der einzelnen (punktförmig gedachten) Atome des Systems. <br/>Diese Zustandsvariabeln können gewählt werden, weil sie der <br/>Bedingung <span
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265 class="cmex-10x-x-120"><img
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266 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmex10-c-50.png" alt=" sum " class="10-120x-x-50" /></span>
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267 <span
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268 class="cmmi-12"><img
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269 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" /> <img
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270 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /></span><sub ><span
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271 class="cmmi-8"><img
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272 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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273 class="cmmi-12">/<img
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274 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-40.png" alt="@" class="12x-x-40" /> p</span><sub ><span
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275 class="cmmi-8"><img
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276 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> = 0 Genüge leisten (l. c., <span
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277 class="cmsy-10x-x-120">§ </span>2). Man <br/>hat dann:
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278 </p>
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279 <center class="par-math-display" >
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280 <img
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281 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190414x.png" alt=" sum n mn E = P (x1 ...zn) + --- (q2n + j2n + zn2), 1 2 " class="par-math-display" /></center>
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282 <!--l. 329--><p class="nopar">
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283 </p><!--l. 333--><p class="noindent">wobei der erste Summand die potentielle Energie, der zweite <br/>die lebendige Kraft des Systems bezeichnet. Sei nun ein <br/>unendlich kleines Gebiet <span
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284 class="cmmi-12">dx</span><sub ><span
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285 class="cmr-8">1</span></sub><span
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286 class="cmmi-12">...</span><span
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287 class="cmmi-12">dz</span><sub ><span
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288 class="cmmi-8">n</span></sub> gegeben. Wir finden <br/>den Mittelwert der Größe
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289 </p>
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290 <center class="par-math-display" >
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291 <img
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292 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190415x.png" alt="mn- (q2n + j2n + z2n), 2 " class="par-math-display" /></center>
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293 <!--l. 341--><p class="nopar">
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294 </p><!--l. 345--><p class="noindent">welcher diesem Gebiete entspricht:</p>
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295 <center class="par-math-display" >
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296 <img
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297 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190416x.png" alt="-- ----------------- Ln = m- (q2n + j2n + z2n) 2 " class="par-math-display" /></center>
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298 <!--l. 353--><p class="nopar"></p>
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299 <center class="math-display" >
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300 <img
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301 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190417x.png" alt=" n sum m (P-x1...zn) integral m 1-2n(q2n+j2n+z2n) e- 4xT0 d x1...d zn --n (q2n + j2n + z2n) e 2xT0 d q1...d zn = -------------------------2-------- sum ------------------------------- - (P-x1...zn) integral --mn2 (q2n+j2n+z2n) e 4xT0 dx1 ...d zn e 2xT0 dq1...d zn " class="math-display" /></center>
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302 <!--l. 363--><p class="nopar">
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303 </p>
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304 <center class="math-display" >
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305 <img
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306 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190418x.png" alt=" + integral oo
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307 2 mnq2n mn qn e4xT0 dqn - oo
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308 = 3 --- integral + oo ------------= 3 x T0. mnq2n e 4xT0 d qn - oo
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309 " class="math-display" /></center>
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310 <!--l. 370--><p class="nopar">
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311 </p><!--l. 373--><p class="noindent">Diese Größe ist also unabhängig von der Wahl des Gebietes <br/>und von der Wahl des Atoms, ist also überhaupt der Mittel-<br/>wert des Atoms bei der absoluten Temperatur <span
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312 class="cmmi-12">T</span><sub ><span
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313 class="cmr-8">0</span></sub>. Die Größe 3 <span
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314 class="cmbxti-10x-x-120">x</span> <br/>ist gleich dem Quotienten aus der mittleren lebendigen Kraft <br/>eines Atoms in die absolute Temperatur.<sup ><span
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315 class="cmr-8">1</span></sup>)
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316 </p><!--l. 378--><p class="indent"> Die Konstante <span
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317 class="cmbxti-10x-x-120">x</span> ist ferner aufs engste verknüpft mit der <br/>Anzahl <span
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318 class="cmmi-12">N </span>der wirklichen Moleküle, welche in einem Molekül <br/>----------
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319 </p><!--l. 382--><p class="indent"> 1) Vgl. L. Boltzmann, Vorl. über Gastheorie <span
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320 class="cmbx-12">2. </span><span
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321 class="cmsy-10x-x-120">§ </span>42. 1898. <pb/>
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322 </p><!--l. 387--><p class="indent">
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323
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324 </p><!--l. 388--><p class="noindent">im Sinne des Chemikers (Äquivalentgewicht bezogen auf 1 g <br/>Wasserstoff als Einheit) enthalten sind.
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325 </p><!--l. 391--><p class="indent"> Liege nämlich eine solche Quantität eines idealen Gases <br/>vor, so ist bekanntlich, wenn Gramm und Zentimeter als Ein-<br/>heiten benutzt werden
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326 </p>
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327 <center class="par-math-display" >
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328 <img
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329 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190419x.png" alt=" 7 p v = R T, wobei R = 8,31 .10 . " class="par-math-display" /></center>
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330 <!--l. 398--><p class="nopar">
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331 </p><!--l. 401--><p class="noindent">Nach der kinetischen Gastheorie ist aber:
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332 </p>
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333 <center class="par-math-display" >
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334 <img
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335 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190420x.png" alt="p v = 2N L-, 3 " class="par-math-display" /></center>
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336 <!--l. 407--><p class="nopar">
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337 </p><!--l. 411--><p class="noindent">wobei <span class="overline"><span
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338 class="cmmi-12">L</span></span> den Mittelwert der lebendigen Kraft der Schwerpunkts-<br/>bewegung eines Moleküles bedeutet. Berücksichtigt man <br/>noch, daß
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339 </p>
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340 <center class="par-math-display" >
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341 <img
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342 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190421x.png" alt="L-= L-, n " class="par-math-display" /></center>
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343 <!--l. 418--><p class="nopar">
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344 </p><!--l. 420--><p class="noindent">so erhält man:
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345 </p>
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346 <center class="par-math-display" >
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347 <img
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348 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190422x.png" alt="N.2 x = R . " class="par-math-display" /></center>
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349 <!--l. 425--><p class="nopar">
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350 </p><!--l. 428--><p class="noindent">Die Konstante 2 <span
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351 class="cmbxti-10x-x-120">x</span> ist also gleich dem Quotienten der Kon-<br/>stanten <span
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352 class="cmmi-12">R </span>in Anzahl der in einem Äquivalent enthaltenen <br/>Moleküle.
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353 </p><!--l. 431--><p class="indent"> Setzt man mit O. E. Meyer <span
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354 class="cmmi-12">N </span>= 6<span
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355 class="cmmi-12">, </span>4 <span
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356 class="cmmi-12">. </span>10<sup ><span
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357 class="cmr-8">23</span></sup><span
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358 class="cmmi-12">, </span>so erhält <br/>man <span
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359 class="cmbxti-10x-x-120">x</span> = 6<span
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360 class="cmmi-12">, </span>5 <span
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361 class="cmmi-12">. </span>10<sup ><span
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362 class="cmsy-8">-</span><span
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363 class="cmr-8">17</span></sup><span
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364 class="cmmi-12">.</span>
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365 </p>
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366 <div class="center" >
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367
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368 <!--l. 435--><p class="noindent">
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369 </p><!--l. 436--><p class="noindent"><span
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370 class="cmsy-10x-x-120">§ </span>4. Allgemeine Bedeutung der Konstanten <span
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371 class="cmbxti-10x-x-120">x</span>.</p></div>
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372 <!--l. 440--><p class="indent"> Ein gegebenes System berühre ein System von relativ <br/>unendlich großer Energie und der Temperatur <span
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373 class="cmmi-12">T. </span>Die Wahr-<br/>scheinlichkeit <span
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374 class="cmmi-12">dW </span>dafür, daß der Wert seiner Energie in einem <br/>beliebig herausgegriffenen Zeitpunkte zwischen <span
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375 class="cmmi-12">E </span>und <span
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376 class="cmmi-12">E </span>+ <span
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377 class="cmmi-12">dE</span>
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378
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379 <br/>liegt, ist:
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380 </p>
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381 <center class="par-math-display" >
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382 <img
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383 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190423x.png" alt=" -E- d W = C e- 2xT wE dE . " class="par-math-display" /></center>
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384 <!--l. 450--><p class="nopar">
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385 </p><!--l. 453--><p class="noindent">Für den Mittelwert <img
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386 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190424x.png" alt="E" class="bar" /> von <span
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387 class="cmmi-12">E </span>erhält man:
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388 </p>
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389 <center class="par-math-display" >
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390 <img
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391 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190425x.png" alt=" oo integral
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392 E = C E e-2ExT w E dE .
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393 0 " class="par-math-display" /></center>
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394 <!--l. 458--><p class="nopar">
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395 </p><!--l. 461--><p class="noindent">Da ferner
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396 </p>
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397 <center class="par-math-display" >
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398 <img
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399 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190426x.png" alt=" integral oo
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400 1 = C e- 2ExT w E d E ,
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401 0 " class="par-math-display" /></center>
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402 <!--l. 467--><p class="nopar"> <pb/>
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403 </p><!--l. 474--><p class="indent">
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404
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405 </p><!--l. 475--><p class="noindent">so ist
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406 </p>
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407 <center class="par-math-display" >
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408 <img
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409 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190427x.png" alt=" integral oo E (E - E) e- 2xT w(E) d E = 0.
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410 0 " class="par-math-display" /></center>
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411 <!--l. 481--><p class="nopar">
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412 </p><!--l. 484--><p class="noindent">Differenziert man diese Gleichung nach <span
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413 class="cmmi-12">T</span>, so erhält man:
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414 </p>
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415 <center class="par-math-display" >
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416 <img
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417 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190428x.png" alt=" integral oo ( ) 2 d-E 2 - 2ExT 2 x T d T + E E - E e wE dE = 0. 0
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418 " class="par-math-display" /></center>
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419 <!--l. 492--><p class="nopar">
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420 </p><!--l. 495--><p class="noindent">Diese Gleichung besagt, daß der Mittelwert der Klammer ver-<br/>schwindet, also:</p>
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421 <center class="par-math-display" >
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422 <img
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423 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190429x.png" alt=" 2 dE 2 2 x T ----= E - E E . dT " class="par-math-display" /></center>
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424 <!--l. 502--><p class="nopar">
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425 </p><!--l. 505--><p class="noindent">Im allgemeinen unterscheidet sich der Momentanwert <span
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426 class="cmmi-12">E </span>der <br/>Energie von <img
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427 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190430x.png" alt="E" class="bar" />
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428 um eine gewisse Größe, welche wir ,,Energie-<br/>schwankung&#8220; nennen; wir setzen:
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429 </p>
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430 <center class="par-math-display" >
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431 <img
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432 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190431x.png" alt="E = E + e. " class="par-math-display" /></center>
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433 <!--l. 513--><p class="nopar">
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434 </p><!--l. 516--><p class="noindent">Man erhält dann</p>
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435 <center class="par-math-display" >
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436 <img
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437 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190432x.png" alt=" 2 -- 2 d E E - E E = e2 = 2 x T ----. d T " class="par-math-display" /></center>
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438 <!--l. 523--><p class="nopar">
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439 </p><!--l. 526--><p class="noindent">Die Größe <span class="overline"><span
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440 class="cmmi-12"><img
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441 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-22.png" alt="e" class="12x-x-22" /></span><sup ><span
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442 class="cmr-8">2</span></sup></span> ist ein Maß für die thermische Stabilität des <br/>Systems; je größer <span class="overline"><span
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443 class="cmmi-12"><img
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444 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-22.png" alt="e" class="12x-x-22" /></span><sup ><span
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445 class="cmr-8">2</span></sup></span>,
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446 desto kleiner diese Stabilität.
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447 </p><!--l. 532--><p class="indent"> Die absolute Konstante <span
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448 class="cmbxti-10x-x-120">x</span> bestimmt also die thermische <br/>Stabilität der Systeme. Die zuletzt gefundene Beziehung ist <br/>darum interessant, weil in derselben keine Größe mehr vor-<br/>kommt, welche an die der Theorie zugrunde liegenden An-<br/>nahmen erinnert.
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449 </p><!--l. 536--><p class="indent"> Durch wiederholtes Differenzieren kann man ohne Schwierig-<br/>keit die Größen
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450 <span class="overline"><span
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451 class="cmmi-12"><img
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452 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-22.png" alt="e" class="12x-x-22" /></span><sup ><span
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453 class="cmr-8">3</span></sup></span>, <span class="overline"><span
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454 class="cmmi-12"><img
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455 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-22.png" alt="e" class="12x-x-22" /></span><sup ><span
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456 class="cmr-8">4</span></sup></span> etc. berechnen.
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457 </p>
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458 <div class="center" >
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459
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460 <!--l. 542--><p class="noindent">
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461 </p><!--l. 543--><p class="noindent"><span
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462 class="cmsy-10x-x-120">§ </span>5. Anwendung auf die Strahlung.</p></div>
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463 <!--l. 547--><p class="indent"> Die zuletzt gefundene Gleichung würde eine exakte Be-<br/>stimmung der universellen Konstanten <span
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464 class="cmbxti-10x-x-120">x</span> zulassen, wenn es <br/>möglich wäre, den Mittelwert des Quadrates der Energie-<br/>schwankung eines Systems zu bestimmen; dies ist jedoch bei <br/>dem gegenwärtigen Stande unseres Wissens nicht der Fall.
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465 <br/><pb/>
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466 </p><!--l. 553--><p class="indent">
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467
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468 </p><!--l. 554--><p class="noindent">Wir können überhaupt nur bei einer einzigen Art physi-<br/>kalischer Systeme aus der Erfahrung vermuten, daß ihnen eine <br/>Energieschwankung zukomme; es ist dies der mit Temperatur-<br/>strahlung erfüllte leere Raum.
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469 </p><!--l. 561--><p class="indent"> Ist nämlich ein mit Temperaturstrahlung erfüllter Raum <br/>von Lineardimensionen, welche sehr groß gegen die Wellen-<br/>länge ist, der das Energiemaximum der Strahlung bei der be-<br/>treffenden Temperatur zukommt, so wird offenbar der Betrag <br/>der Energieschwankung im Mittel im Vergleich zur mittleren
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470 <br/>Strahlungsenergie dieses Raumes sehr klein sein. Wenn da-<br/>gegen der Strahlungsraum von der Größenordnung jener <br/>Wellenlänge ist, so wird die Energieschwankung von derselben <br/>Größenordnung sein, wie die Energie der Strahlung des <br/>Strahlungsraumes.
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471 </p><!--l. 573--><p class="indent"> Es ist allerdings einzuwenden, daß wir nicht behaupten <br/>können, daß ein Strahlungs<span
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472 class="cmti-12">raum </span>als ein <span
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473 class="cmti-12">System </span>von der von <br/>uns vorausgesetzten Art zu betrachten sei, auch dann nicht, <br/>wenn die Anwendbarkeit der allgemeinen molekularen Theorie <br/>zugestanden wird. Vielleicht müßte man zum Beispiel die
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474 <br/>Grenzen des Raumes als mit den elektromagnetischen Zu-<br/>ständen desselben veränderlich annehmen. Diese Umstände <br/>kommen indessen hier, wo es sich nur um Größenordnungen <br/>handelt, nicht in Betracht.
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475 </p><!--l. 584--><p class="indent"> Setzen wir also in der im vorigen Paragraphen gefundenen <br/>Gleichung</p>
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476 <center class="par-math-display" >
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477 <img
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478 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190433x.png" alt="-2 -2- e = E , " class="par-math-display" /></center>
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479 <!--l. 590--><p class="nopar">
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480 </p><!--l. 594--><p class="noindent">und nach dem Stefan-Boltzmannschen Gesetze
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481 </p>
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482 <center class="par-math-display" >
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483 <img
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484 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190434x.png" alt="-- E = c vT 4, " class="par-math-display" /></center>
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485 <!--l. 600--><p class="nopar">
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486 </p><!--l. 604--><p class="noindent">wobei <span
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487 class="cmmi-12">v </span>das Volumen in cm<sup ><span
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488 class="cmr-8">3</span></sup> und <span
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489 class="cmmi-12">c </span>die Konstante dieses Ge-<br/>setzes bedeutet, so müssen wir für <img
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490 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190435x.png" alt="3 V~ -- v" class="root" align="middle" /> einen Wert von der <br/>Größenordnung der Wellenlänge maximaler Strahlungsenergie <br/>erhalten, welche der betreffenden Temperatur entspricht.
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491 </p><!--l. 610--><p class="indent"> Man erhält:
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492 </p>
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493 <center class="par-math-display" >
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494 <img
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495 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190436x.png" alt=" V~ -x- 2 3 -- 3 V~ v-= ----c-= 0,42-, T T " class="par-math-display" /></center>
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496 <!--l. 615--><p class="nopar">
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497
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498 </p><!--l. 620--><p class="noindent">wobei für <span
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499 class="cmbxti-10x-x-120">x</span> der aus der kinetischen Gastheorie gefundene <br/>Wert und für <span
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500 class="cmmi-12">c </span>der Wert 7,06 . 10<sup ><span
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501 class="cmsy-8">-</span><span
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502 class="cmr-8">15</span></sup> gesetzt ist. <pb/>
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503 </p><!--l. 625--><p class="indent">
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504
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505 </p><!--l. 626--><p class="indent"> Ist <span
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506 class="cmmi-12"><img
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507 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /></span><sub ><span
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508 class="cmmi-8">m</span></sub> die Wellenlänge des Energiemaximums der Strahlung, <br/>so liefert die Erfahrung:</p>
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509 <center class="par-math-display" >
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510 <img
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511 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/Einst_Zural_de_190437x.png" alt="cm = 0,293-. T " class="par-math-display" /></center>
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512 <!--l. 632--><p class="nopar">
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513 </p><!--l. 634--><p class="indent"> Man sieht, daß sowohl die Art der Abhängigkeit von der <br/>Temperatur als auch die Größenordnung von <span
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514 class="cmmi-12"><img
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515 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zural_de_1904/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /></span><sub ><span
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516 class="cmmi-8">m</span></sub> mittels der <br/>allgemeinen molekularen Theorie der Wärme richtig bestimmt <br/>werden kann, und ich glaube, daß diese Übereinstimmung bei <br/>der großen Allgemeinheit unserer Voraussetzungen nicht dem <br/>Zufall zugeschrieben werden darf.
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517 </p><!--l. 642--><p class="indent"> Bern, den 27. März 1904. </p>
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518 <div class="center" >
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519
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520 <!--l. 643--><p class="noindent">
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521 </p><!--l. 644--><p class="noindent">(Eingegangen 29. März 1904.)</p></div>
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522 <div class="center" >
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523
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524 <!--l. 648--><p class="noindent">
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525 </p><!--l. 649--><p class="noindent">----------</p></div>
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526
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527 </body></html>
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530