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Version vom 2009-02-14
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:08:12 +0200
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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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11 <!-- xhtml,html -->
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12 <meta name="src" content="Einst_Ueber_de_1905.tex" />
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13 <meta name="date" content="2005-03-10 17:33:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Ueber_de_1905.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">5.</span><span
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23 class="cmbxti-10x-x-144">Über die von der molekularkinetischen Theorie </span> <br/><span
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24 class="cmbxti-10x-x-144">der</span>
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25 <span
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26 class="cmbxti-10x-x-144">W</span><span
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27 class="cmbxti-10x-x-144">ärme geforderte Bewegung von in ruhenden</span>
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28 <br/><span
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29 class="cmbxti-10x-x-144">Fl</span><span
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30 class="cmbxti-10x-x-144">üssigkeiten suspendierten Teilchen; </span> <br/><span
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31 class="cmbxti-10x-x-144">von A.</span>
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32 <span
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33 class="cmbxti-10x-x-144">Einstein.</span></p></div>
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34 <div class="center" >
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36 <!--l. 21--><p class="noindent">
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37 </p><!--l. 22--><p class="noindent">--------</p></div>
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38 <!--l. 26--><p class="indent"> In dieser Arbeit soll gezeigt werden, daß nach der molekular-<br/>kinetischen Theorie der Wärme in Flüssigkeiten suspendierte <br/>Körper von mikroskopisch sichtbarer Größe infolge der Mole-<br/>kularbewegung der Wärme Bewegungen von solcher Größe <br/>ausführen müssen, daß diese Bewegungen leicht mit dem <br/>Mikroskop nachgewiesen werden können. Es ist möglich, daß <br/>die hier zu behandelnden Bewegungen mit der sogenannten <br/>,,Brown schen Molekularbewegung&#8220; identisch sind; die mir <br/>erreichbaren Angaben über letztere sind jedoch so ungenau, <br/>daß ich mir hierüber kein Urteil bilden konnte.
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39 </p><!--l. 39--><p class="indent"> Wenn sich die hier zu behandelnde Bewegung samt den <br/>für sie zu erwartenden Gesetzmäßigkeiten wirklich beobachten <br/>läßt, so ist die klassische Thermodynamik schon für mikro-<br/>skopisch unterscheidbare Räume nicht mehr als genau gültig
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40 <br/>anzusehen und es ist dann eine exakte Bestimmung der wahren <br/>Atomgröße möglich. Erwiese sich umgekehrt die Voraussage <br/>dieser Bewegung als unzutreffend, so wäre damit ein schwer-<br/>wiegendes Argument gegen die molekularkinetische Auffassung <br/>der Wärme gegeben.
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41 </p>
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42 <div class="center" >
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44 <!--l. 52--><p class="noindent">
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45 </p><!--l. 53--><p class="noindent"><span
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46 class="cmsy-10x-x-120">§ </span>1. Über den suspendierten Teilchen zuzuschreibenden <br/>osmotischen Druck.</p></div>
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47 <!--l. 58--><p class="indent"> Im Teilvolumen <span
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48 class="cmmi-12">V</span> <sup ><span
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49 class="cmsy-8">*</span></sup> einer Flüssigkeit vom Gesamtvolumen <span
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50 class="cmmi-12">V </span> <br/>seien
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51 <span
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52 class="cmmi-12">z</span>-Gramm - Moleküle eines Nichtelektrolyten gelöst. Ist <br/>das Volumen <span
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53 class="cmmi-12">V</span> <sup ><span
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54 class="cmsy-8">*</span></sup> durch eine für das Lösungsmittel, nicht aber <br/>für die gelöste Substanz durchlässige Wand vom reinen Lösungs-<br/><pb/>
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55 </p><!--l. 66--><p class="indent">
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56
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57 </p><!--l. 67--><p class="noindent">mittel getrennt, so wirkt auf diese Wand der sogenannte os-<br/>motische Druck, welcher bei genügend großen Werten von <span
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58 class="cmmi-12">V</span> <sup ><span
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59 class="cmsy-8">*</span></sup>/<span
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60 class="cmmi-12">z </span> <br/>der Gleichung genügt:
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61 </p>
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62 <center class="par-math-display" >
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63 <img
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64 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19050x.png" alt="p V * = R T z . " class="par-math-display" /></center>
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65 <!--l. 75--><p class="nopar">
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66 </p><!--l. 79--><p class="indent"> Sind hingegen statt der gelösten Substanz in dem Teil-<br/>volumen <span
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67 class="cmmi-12">V</span> <sup ><span
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68 class="cmsy-8">*</span></sup> der Flüssigkeit kleine suspendierte Körper vor-<br/>handen, welche ebenfalls nicht durch die für das Lösungs-<br/>mittel durchlässige Wand hindurchtreten können, so hat man <br/>nach der klassischen Theorie der Thermodynamik -- wenigstens <br/>bei Vernachlässigung der uns hier nicht interessierenden Schwer-<br/>kraft -- nicht zu erwarten, daß auf die Wand eine Kraft <br/>wirke; denn die ,,freie Energie&#8220; des Systems scheint nach der <br/>üblichen Auffassung nicht von der Lage der Wand und der <br/>suspendierten Körper abzuhängen, sondern nur von den Ge-<br/>samtmassen und Qualitäten der suspendierten Substanz, der <br/>Flüssigkeit und der Wand, sowie von Druck und Temperatur. <br/>Es kämen allerdings für die Berechnung der freien Energie <br/>noch Energie und Entropie der Grenzflächen in Betracht
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69 <br/>(Kapillarkräfte); hiervon können wir jedoch absehen, indem <br/>bei den ins Auge zu fassenden Lagenänderungen der Wand <br/>und der suspendierten Körper Änderungen der Größe und <br/>Beschaffenheit der Berührungsflächen nicht eintreten mögen.
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70 </p><!--l. 102--><p class="indent"> Vom Standpunkte der molekularkinetischen Wärmetheorie <br/>aus kommt man aber zu einer anderen Auffassung. Nach <br/>dieser Theorie unterscheidet sich eingelöstes Molekül von einem <br/>suspendierten Körper <span
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71 class="cmti-12">lediglich </span>durch die Größe, und man sieht <br/>nicht ein, warum einer Anzahl suspendierter Körper nicht der-<br/>selbe osmotische Druck entsprechen sollte, wie der nämlichen <br/>Anzahl gelöster Moleküle. Man wird anzunehmen haben, daß <br/>die suspendierten Körper infolge der Molekularbewegung der <br/>Flüssigkeit eine wenn auch sehr langsame ungeordnete Be-<br/>wegung in der Flüssigkeit ausführen; werden sie durch die <br/>Wand verhindert, das Volumen <span
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72 class="cmmi-12">V</span> <sup ><span
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73 class="cmsy-8">*</span></sup> zu verlassen, so werden sie
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74 <br/>auf die Wand Kräfte ausüben, ebenso wie gelöste Moleküle. <br/>Sind also <span
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75 class="cmmi-12">n</span>
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76 suspendierte Körper im Volumen <span
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77 class="cmmi-12">V</span> <sup ><span
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78 class="cmsy-8">*</span></sup>, also <span
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79 class="cmmi-12">n/V</span> <sup ><span
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80 class="cmsy-8">*</span></sup> = <span
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81 class="cmmi-12"><img
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82 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span> <br/>in der Volumeneinheit vorhanden, und sind benachbarte unter <br/>ihnen genügend weit voneinander entfernt, so wird ihnen ein <br/>osmotischer Druck <span
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83 class="cmmi-12">p </span>entsprechen von der Größe:
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84 <pb/>
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85 </p><!--l. 125--><p class="indent">
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86
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87 </p>
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88 <center class="par-math-display" >
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89 <img
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90 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19051x.png" alt=" R T n R T p = -V-*-N--= -N---.n , " class="par-math-display" /></center>
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91 <!--l. 130--><p class="nopar">
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92 </p><!--l. 134--><p class="noindent">wobei <span
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93 class="cmmi-12">N </span>die Anzahl der in einem Gramm-Molekül enthaltenen <br/>wirklichen Moleküle bedeutet. Im nächsten Paragraph soll <br/>gezeigt werden, daß die molekularkinetische Theorie der Wärme <br/>wirklich zu dieser erweiterten Auffassung des osmotischen <br/>Druckes führt.
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94 </p>
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95 <div class="center" >
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96
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97 <!--l. 143--><p class="noindent">
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98 </p><!--l. 144--><p class="noindent"><span
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99 class="cmsy-10x-x-120">§ </span>2. Der osmotische Druck vom Standpunkte der molekular-<br/>kinetischen Theorie der Wärme.<sup ><span
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100 class="cmr-8">1</span></sup>)</p></div>
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101 <!--l. 149--><p class="indent"> Sind <span
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102 class="cmmi-12">p</span><sub ><span
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103 class="cmr-8">1</span></sub> <span
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104 class="cmmi-12">p</span><sub ><span
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105 class="cmr-8">2</span></sub><span
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106 class="cmmi-12">...</span><span
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107 class="cmmi-12">p</span><sub ><span
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108 class="cmmi-8">l</span></sub> Zustandsvariable eines physikalischen <br/>Systems, welche den momentanen Zustand desselben voll-<br/>kommen bestimmen (z. B. die Koordinaten und Geschwindig-<br/>keitskomponenten aller Atome des Systems) und ist das voll-<br/>ständige System der Veränderungsgleichungen dieser Zustands-<br/>variabeln von der Form
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109 </p>
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110 <center class="par-math-display" >
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111 <img
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112 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19052x.png" alt="@ pn @ t = fn (p1 ...pl)(n = 1, 2 ...l) " class="par-math-display" /></center>
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113 <!--l. 161--><p class="nopar">
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114 </p><!--l. 165--><p class="noindent">gegeben, wobei <span
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115 class="cmex-10x-x-120"><img
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116 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmex10-c-50.png" alt=" sum " class="10-120x-x-50" /></span>
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117 <img
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118 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19053x.png" alt="@ fn @-p-- n" class="frac" align="middle" /> = 0<span
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119 class="cmmi-12">, </span>so ist die Entropie des Systems <br/>durch den Ausdruck gegeben:
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120 </p>
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121 <center class="par-math-display" >
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122 <img
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123 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19054x.png" alt=" integral
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124 E- ---E-- S = T + 2 x lg e 2x T d p1... d pl. " class="par-math-display" /></center>
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125 <!--l. 174--><p class="nopar">
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126 </p><!--l. 177--><p class="noindent">Hierbei bedeutet <span
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127 class="cmmi-12">T </span>die absolute Temperatur, <img
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128 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19055x.png" alt="E" class="bar" /> die Energie <br/>des Systems, <span
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129 class="cmmi-12">E </span>die Energie als Funktion der <span
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130 class="cmmi-12">p</span><sub ><span
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131 class="cmmi-8"><img
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132 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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133 class="cmmi-12">. </span>Das Inte-<br/>gral ist über alle mit den Bedingungen des Problems ver-<br/>einbaren Wertekombinationen der <span
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134 class="cmmi-12">p</span><sub ><span
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135 class="cmmi-8"><img
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136 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub><span
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137 class="cmmi-12">. </span>zu erstrecken. <span
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138 class="cmbxti-10x-x-120">x</span> ist mit <br/>der oben erwähnten Konstanten <span
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139 class="cmmi-12">N </span>durch die Relation 2 <span
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140 class="cmbxti-10x-x-120">x</span> <span
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141 class="cmmi-12">N </span>= <span
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142 class="cmmi-12">R </span> <br/>verbunden. Für die freie Energie <span
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143 class="cmmi-12">F </span>erhalten wir daher:
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144 </p>
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145 <center class="par-math-display" >
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146 <img
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147 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19056x.png" alt=" integral
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148 R - EN- R T F = - N--T lg e RT d p1... dpl = - -N---lg B . " class="par-math-display" /></center>
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149 <!--l. 191--><p class="nopar">
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150 </p><!--l. 194--><p class="noindent">----------
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151 </p><!--l. 197--><p class="indent"> 1) In diesem Paragraph sind die Arbeiten des Verfassers über die <br/>Grundlagen der Thermodynamik als bekannt vorausgesetzt (vgl. Ann. d. <br/>Phys. <span
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152 class="cmbx-12">9. </span>p. 417. 1902;
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153 <span
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154 class="cmbx-12">11. </span>p. 170. 1903). Für das Verständnis der <br/>Resultate der vorliegenden Arbeit ist die Kenntnis jener Arbeiten sowie <br/>dieses Paragraphen der vorliegenden Arbeit entbehrlich. <pb/>
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155 </p><!--l. 207--><p class="indent">
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156
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157 </p><!--l. 208--><p class="indent"> Wir denken uns nun eine in dem Volumen <span
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158 class="cmmi-12">V </span>eingeschlossene <br/>Flüssigkeit; in dem Teilvolumen <span
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159 class="cmmi-12">V</span> <sup ><span
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160 class="cmsy-8">*</span></sup> von <span
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161 class="cmmi-12">V </span>mögen sich <span
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162 class="cmmi-12">n </span>ge-<br/>löste Moleküle bez. suspendierte Körper befinden, welche im <br/>Volumen <span
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163 class="cmmi-12">V</span> <sup ><span
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164 class="cmsy-8">*</span></sup> durch eine semipermeabele Wand festgehalten <br/>seien; es werden hierdurch die Integrationsgrenzen des in den <br/>Ausdrücken für <span
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165 class="cmmi-12">S </span>und <span
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166 class="cmmi-12">F </span>auftretenden Integrales <span
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167 class="cmmi-12">B </span>beeinflußt. <br/>Das Gesamtvolumen der gelösten Moleküle bez. suspendierten <br/>Körper sei klein gegen <span
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168 class="cmmi-12">V</span> <sup ><span
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169 class="cmsy-8">*</span></sup>. Dies System werde im Sinne der <br/>erwähnten Theorie durch die Zustandsvariabeln <span
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170 class="cmmi-12">p</span><sub ><span
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171 class="cmr-8">1</span></sub> <span
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172 class="cmmi-12">...</span><span
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173 class="cmmi-12">p</span><sub ><span
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174 class="cmmi-8">l</span></sub> voll-<br/>ständig dargestellt.
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175 </p><!--l. 221--><p class="indent"> Wäre nun auch das molekulare Bild bis in alle Einzel-<br/>heiten festgelegt, so böte doch die Ausrechnung des Integrales <span
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176 class="cmmi-12">B </span> <br/>solche Schwierigkeiten, daß an eine exakte Berechnung von <span
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177 class="cmmi-12">F </span> <br/>kaum gedacht werden könnte. Wir brauchen jedoch hier nur <br/>zu wissen, wie <span
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178 class="cmmi-12">F </span>von der Größe des Volumens <span
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179 class="cmmi-12">V</span> <sup ><span
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180 class="cmsy-8">*</span></sup> abhängt, <br/>in welchem alle gelösten Moleküle bez. suspendierten Körper <br/>(im folgenden kurz ,,Teilchen&#8220; genannt) enthalten sind.
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181 </p><!--l. 231--><p class="indent"> Wir nennen <span
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182 class="cmmi-12">x</span><sub ><span
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183 class="cmr-8">1</span></sub><span
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184 class="cmmi-12">, y</span><sub ><span
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185 class="cmr-8">1</span></sub><span
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186 class="cmmi-12">, z</span><sub ><span
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187 class="cmr-8">1</span></sub> die rechtwinkligen Koordinaten des <br/>Schwerpunktes des ersten Teilchens, <span
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188 class="cmmi-12">x</span><sub ><span
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189 class="cmr-8">2</span></sub><span
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190 class="cmmi-12">, y</span><sub ><span
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191 class="cmr-8">2</span></sub><span
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192 class="cmmi-12">, z</span><sub ><span
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193 class="cmr-8">2</span></sub> die des zweiten etc., <br/><span
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194 class="cmmi-12">x</span><sub ><span
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195 class="cmmi-8">n</span></sub><span
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196 class="cmmi-12">, y</span><sub ><span
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197 class="cmmi-8">n</span></sub><span
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198 class="cmmi-12">, z</span><sub ><span
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199 class="cmmi-8">n</span></sub><span
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200 class="cmmi-12">, </span>die des letzten Teilchens und geben für die Schwer-<br/>punkte der Teilchen die unendlich kleinen parallelepiped-<br/>förmigen Gebiete <span
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201 class="cmmi-12">dx</span><sub ><span
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202 class="cmr-8">1</span></sub> <span
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203 class="cmmi-12">dy</span><sub ><span
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204 class="cmr-8">1</span></sub> <span
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205 class="cmmi-12">dz</span><sub ><span
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206 class="cmr-8">1</span></sub><span
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207 class="cmmi-12">, dx</span><sub ><span
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208 class="cmr-8">2</span></sub> <span
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209 class="cmmi-12">dy</span><sub ><span
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210 class="cmr-8">2</span></sub> <span
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211 class="cmmi-12">dz</span><sub ><span
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212 class="cmr-8">2</span></sub> <span
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213 class="cmmi-12">...</span> <span
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214 class="cmmi-12">dx</span><sub ><span
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215 class="cmmi-8">n</span></sub> <span
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216 class="cmmi-12">dy</span><sub ><span
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217 class="cmmi-8">n</span></sub> <span
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218 class="cmmi-12">dz</span><sub ><span
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219 class="cmmi-8">n</span></sub><span
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220 class="cmmi-12">,</span>
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221 <br/>welche alle in <span
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222 class="cmmi-12">V</span> <sup ><span
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223 class="cmsy-8">*</span></sup> gelegen seien. Gesucht sei der Wert des <br/>im Ausdruck für <span
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224 class="cmmi-12">F</span>
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225 auftretenden Integrales mit der Beschränkung, <br/>daß die Teilchenschwerpunkte in den ihnen soeben zugewiesenen <br/>Gebieten liegen. Dies Integral läßt sich jedenfalls auf die Form
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226 </p>
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227 <center class="par-math-display" >
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228 <img
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229 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19057x.png" alt="d B = dx1 d y1 ... d zn.J " class="par-math-display" /></center>
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230 <!--l. 248--><p class="nopar">
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231 </p><!--l. 252--><p class="noindent">bringen, wobei <span
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232 class="cmmi-12">J </span>von <span
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233 class="cmmi-12">dx</span><sub ><span
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234 class="cmr-8">1</span></sub> <span
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235 class="cmmi-12">dy</span><sub ><span
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236 class="cmr-8">1</span></sub> etc., sowie von <span
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237 class="cmmi-12">V</span> <sup ><span
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238 class="cmsy-8">*</span></sup>, d. h. von <br/>der Lage der semipermeabeln Wand, unabhängig ist. <span
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239 class="cmmi-12">J </span>ist <br/>aber auch unabhängig von der speziellen Wahl <span
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240 class="cmti-12">der Lagen </span>der <br/>Schwerpunktsgebiete und von dem Werte von <span
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241 class="cmmi-12">V</span> <sup ><span
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242 class="cmsy-8">*</span></sup>, wie sogleich <br/>gezeigt werden soll. Sei nämlich ein zweites System von un-<br/>endlich kleinen Gebieten für die Teilchenschwerpunkte gegeben <br/>und bezeichnet durch
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243 <span
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244 class="cmmi-12">dx</span><span
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245 class="cmsy-10x-x-120">'</span><sub ><span
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246 class="cmr-8">1</span> </sub> <span
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247 class="cmmi-12">dy</span><span
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248 class="cmsy-10x-x-120">'</span><sub ><span
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249 class="cmr-8">1</span></sub> <span
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250 class="cmmi-12">dz</span><span
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251 class="cmsy-10x-x-120">'</span><sub ><span
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252 class="cmr-8">2</span></sub><span
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253 class="cmmi-12">, dx</span><span
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254 class="cmsy-10x-x-120">'</span><sub ><span
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255 class="cmr-8">2</span></sub> <span
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256 class="cmmi-12">dy</span><span
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257 class="cmsy-10x-x-120">'</span><sub ><span
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258 class="cmr-8">2</span></sub> <span
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259 class="cmmi-12">dz</span><span
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260 class="cmsy-10x-x-120">'</span><sub ><span
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261 class="cmr-8">2</span></sub> <span
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262 class="cmmi-12">...</span> <span
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263 class="cmmi-12">dx</span><span
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264 class="cmsy-10x-x-120">'</span><sub ><span
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265 class="cmmi-8">n</span></sub> <span
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266 class="cmmi-12">dy</span><span
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267 class="cmsy-10x-x-120">'</span><sub ><span
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268 class="cmmi-8">n</span></sub> <span
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269 class="cmmi-12">dz</span><span
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270 class="cmsy-10x-x-120">'</span><sub ><span
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271 class="cmmi-8">n</span></sub><span
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272 class="cmmi-12">, </span> <br/>welche Gebiete sich von den ursprünglich gegebenen nur durch <br/>ihre Lage, nicht aber durch ihre Größe unterscheiden mögen <br/>und ebenfalls alle in <span
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273 class="cmmi-12">V</span> <sup ><span
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274 class="cmsy-8">*</span></sup> enthalten seien, so gilt analog:
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275 </p>
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276 <center class="par-math-display" >
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277 <img
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278 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19058x.png" alt="d B'= d x'1d y'1 ... dz'n .J', " class="par-math-display" /></center>
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279 <!--l. 271--><p class="nopar"> <pb/>
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280 </p><!--l. 276--><p class="indent">
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281
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282 </p><!--l. 277--><p class="noindent">wobei
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283 </p>
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284 <center class="par-math-display" >
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285 <img
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286 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_19059x.png" alt="dx1 d y1 ... d zn = d x'1 dy'1 ... dz'n . " class="par-math-display" /></center>
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287 <!--l. 285--><p class="nopar">
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288 </p><!--l. 288--><p class="noindent">Es ist also:
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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_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190510x.png" alt="d B J ---'= --'. dB J " class="par-math-display" /></center>
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293 <!--l. 294--><p class="nopar">
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294 </p><!--l. 298--><p class="indent"> Aus der in den zitierten Arbeiten gegebenen molekularen <br/>Theorie der Wärme läßt sich aber leicht folgern<sup ><span
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295 class="cmr-8">1</span></sup>), daß <span
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296 class="cmmi-12">dB/B </span> <br/>bez. <span
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297 class="cmmi-12">dB</span><span
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298 class="cmsy-10x-x-120">'</span><span
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299 class="cmmi-12">/B </span>gleich ist der Wahrscheinlichkeit dafür, daß sich <br/>in einem beliebig herausgegriffenen Zeitpunkte die Teilchen-<br/>schwerpunkte in den Gebieten (<span
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300 class="cmmi-12">dx</span><sub ><span
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301 class="cmr-8">1</span></sub> <span
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302 class="cmmi-12">...</span><span
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303 class="cmmi-12">dz</span><sub ><span
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304 class="cmmi-8">n</span></sub>) bez. in den Ge-<br/>bieten (<span
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305 class="cmmi-12">dx</span><span
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306 class="cmsy-10x-x-120">'</span><sub ><span
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307 class="cmr-8">1</span></sub> <span
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308 class="cmmi-12">...</span><span
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309 class="cmmi-12">dz</span><span
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310 class="cmsy-10x-x-120">'</span><sub ><span
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311 class="cmmi-8">n</span></sub>) befinden. Sind nun die Bewegungen der <br/>einzelnen Teilchen (mit genügender Annäherung) voneinander <br/>unabhängig, ist die Flüssigkeit homogen und wirken auf die <br/>Teilchen keine Kräfte, so müssen bei gleicher Größe der Ge-<br/>biete die den beiden Gebietssystemen zukommenden Wahr-<br/>scheinlichkeiten einander gleich sein, so daß gilt:
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312 </p>
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313 <center class="par-math-display" >
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314 <img
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315 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190511x.png" alt=" ' d-B-= d-B-. B B " class="par-math-display" /></center>
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316 <!--l. 316--><p class="nopar">
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317 </p><!--l. 319--><p class="noindent">Aus dieser und aus der zuletzt gefundenen Gleichung folgt aber
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318 </p>
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319 <center class="par-math-display" >
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320 <img
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321 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190512x.png" alt="J = J'. " class="par-math-display" /></center>
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322 <!--l. 326--><p class="nopar">
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323 </p><!--l. 330--><p class="indent"> Es ist somit erwiesen, daß <span
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324 class="cmmi-12">J </span>weder von <span
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325 class="cmmi-12">V</span> <sup ><span
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326 class="cmsy-8">*</span></sup> noch von <br/><span
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327 class="cmmi-12">x</span><sub >
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328 <span
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329 class="cmr-8">1</span></sub><span
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330 class="cmmi-12">, y</span><sub ><span
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331 class="cmr-8">1</span></sub><span
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332 class="cmmi-12">...</span><span
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333 class="cmmi-12">z</span><sub ><span
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334 class="cmmi-8">n</span></sub> abhängig ist. Durch Integration erhält man
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335 </p>
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336 <center class="par-math-display" >
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337 <img
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338 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190513x.png" alt=" integral
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339 B = J dx1 ...dzn = J V *n " class="par-math-display" /></center>
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340 <!--l. 337--><p class="nopar">
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341 </p><!--l. 341--><p class="noindent">und daraus
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342 </p>
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343 <center class="par-math-display" >
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344 <img
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345 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190514x.png" alt="F = - R-T-{lgJ + n lg V *} N " class="par-math-display" /></center>
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346 <!--l. 347--><p class="nopar">
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347 </p><!--l. 351--><p class="noindent">und
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348 </p>
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349 <center class="par-math-display" >
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350 <img
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351 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190515x.png" alt=" @-F-- R-T-n-- R-T- p = - @ V * = V * N = N n . " class="par-math-display" /></center>
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352 <!--l. 358--><p class="nopar">
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353 </p><!--l. 362--><p class="indent"> Durch diese Betrachtung ist gezeigt, daß die Existenz <br/>des osmotischen Druckes eine Konsequenz der molekular-<br/>kinetischen Theorie der Wärme ist, und daß nach dieser Theorie <br/>gelöste Moleküle und suspendierte Körper von gleicher Anzahl <br/>sich in bezug auf osmotischen Druck bei großer Verdünnung <br/>vollkommen gleich verhalten.
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354 </p><!--l. 370--><p class="noindent">----------
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355 </p><!--l. 373--><p class="indent"> 1) A. Einstein, Ann. d. Phys. <span
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356 class="cmbx-12">11. </span>p. 170. 1903. <pb/>
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357 </p><!--l. 378--><p class="indent">
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358
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359 </p>
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360 <div class="center" >
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361
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362 <!--l. 379--><p class="noindent">
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363 </p><!--l. 380--><p class="noindent"><span
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364 class="cmsy-10x-x-120">§ </span>3. Theorie der Diffusion kleiner suspendierter Kugeln.</p></div>
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365 <!--l. 384--><p class="indent"> In einer Flüssigkeit seien suspendierte Teilchen regellos <br/>verteilt. Wir wollen den dynamischen Gleichgewichtszustand <br/>derselben untersuchen unter der Voraussetzung, daß auf die <br/>einzelnen Teilchen eine Kraft <span
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366 class="cmmi-12">K </span>wirkt, welche vom Orte, <br/>nicht aber von der Zeit abhängt. Der Einfachheit halber
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367 <br/>werde angenommen, daß die Kraft überall die Richtung der <br/><span
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368 class="cmmi-12">X</span>-Achse habe.
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369 </p><!--l. 393--><p class="indent"> Es sei <span
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370 class="cmmi-12"><img
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371 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>die Anzahl der suspendierten Teilchen pro <br/>Volumeneinheit, so ist im Falle des thermodynamischen Gleich-<br/>gewichtes <span
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372 class="cmmi-12"><img
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373 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>eine solche Funktion von <span
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374 class="cmmi-12">x</span>, daß für eine beliebige <br/>virtuelle Verrückung <span
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375 class="cmmi-12"><img
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376 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /> x </span>der suspendierten Substanz die Variation <br/>der freien Energie verschwindet. Man hat also:
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377 </p>
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378 <center class="par-math-display" >
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379 <img
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380 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190516x.png" alt="d F = d E - T dS = 0. " class="par-math-display" /></center>
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381 <!--l. 403--><p class="nopar">
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382 </p><!--l. 406--><p class="noindent">Es werde angenommen, daß die Flüssigkeit senkrecht zur <br/><span
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383 class="cmmi-12">X</span>-Achse den Querschnitt 1 habe und durch die Ebenen <span
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384 class="cmmi-12">x </span>= 0 <br/>und <span
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385 class="cmmi-12">x </span>= <span
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386 class="cmmi-12">l </span>begrenzt sei. Man hat dann:
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387 </p>
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388 <center class="par-math-display" >
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389 <img
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390 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190517x.png" alt=" integral l d E = - K n dx dx
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391 0 " class="par-math-display" /></center>
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392 <!--l. 417--><p class="nopar">
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393 </p><!--l. 421--><p class="noindent">und
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394 </p>
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395 <center class="par-math-display" >
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396 <img
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397 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190518x.png" alt=" integral l n @ d x R integral l@ n d S = R --------dx = - --- --- dx dx . N @ x N @ x 0 0 " class="par-math-display" /></center>
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398 <!--l. 431--><p class="nopar">
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399 </p><!--l. 434--><p class="noindent">Die gesuchte Gleichgewichtsbedingung ist also:
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400 </p>
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401 <table width="100%"
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402 class="equation"><tr><td><a
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403 id="x1-2r1"></a>
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404
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405 <center class="math-display" >
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406 <img
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407 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190519x.png" alt="- K n + R-T- @-n = 0 N @ x " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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408 <!--l. 440--><p class="nopar">
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409 </p><!--l. 444--><p class="noindent">oder
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410 </p>
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411 <center class="par-math-display" >
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412 <img
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413 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190520x.png" alt="K n - @-p = 0 . @ x " class="par-math-display" /></center>
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414 <!--l. 450--><p class="nopar">
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415 </p><!--l. 453--><p class="noindent">Die letzte Gleichung sagt aus, daß der Kraft <span
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416 class="cmmi-12">K </span>durch osmo-<br/>tische Druckkräfte das Gleichgewicht geleistet wird.
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417 </p><!--l. 458--><p class="indent"> Die Gleichung (1) benutzen wir, um den Diffusionskoeffi-<br/>zienten der suspendierten Substanz zu ermitteln. Wir können <br/>den eben betrachteten dynamischen Gleichgewichtszustand als <br/><pb/>
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418 </p><!--l. 464--><p class="indent">
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419
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420 </p><!--l. 465--><p class="noindent">die Superposition zweier in umgekehrtem Sinne verlaufender <br/>Prozesse auffassen, nämlich
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421 </p><!--l. 468--><p class="indent"> 1. einer Bewegung der suspendierten Substanz unter der <br/>Wirkung der auf jedes einzelne suspendierte Teilchen wirken-<br/>den Kraft <span
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422 class="cmmi-12">K</span>,
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423 </p><!--l. 472--><p class="indent"> 2. eines Diffusionsvorganges, welcher als Folge der un-<br/>geordneten Bewegungen der Teilchen infolge der Molekular-<br/>bewegung der Wärme aufzufassen ist.
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424 </p><!--l. 476--><p class="indent"> Haben die suspendierten Teilchen Kugelform (Kugelradius <span
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425 class="cmmi-12">P</span>) <br/>und besitzt die Flüssigkeit den Reibungskoeffizienten <span
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426 class="cmmi-12">k</span>, so <br/>erteilt die Kraft <span
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427 class="cmmi-12">K </span>dem einzelnen Teilchen die Geschwindigkeit<sup ><span
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428 class="cmr-8">1</span></sup>)
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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_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190521x.png" alt="--K---- 6 pk P , " class="par-math-display" /></center>
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433 <!--l. 484--><p class="nopar">
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434 </p><!--l. 488--><p class="noindent">und es treten durch die Querschnittseinheit pro Zeiteinheit
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435 </p>
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436 <center class="par-math-display" >
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437 <img
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438 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190522x.png" alt="-n-K--- 6p k P " class="par-math-display" /></center>
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439 <!--l. 494--><p class="nopar">
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440 </p><!--l. 497--><p class="noindent">Teilchen hindurch.
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441 </p><!--l. 500--><p class="indent"> Bezeichnet ferner <span
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442 class="cmmi-12">D </span>den Diffusionskoeffizienten der sus-<br/>pendierten Substanz und <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_Ueber_de_1905/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>die Masse eines Teilchens, so treten <br/>pro Zeiteinheit infolge der Diffusion
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445 </p>
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446 <center class="par-math-display" >
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447 <img
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448 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190523x.png" alt=" @ (m n) - D ------- Gramm @ x " class="par-math-display" /></center>
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449 <!--l. 508--><p class="nopar">
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450 </p><!--l. 512--><p class="noindent">oder
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451 </p>
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452 <center class="par-math-display" >
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453 <img
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454 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190524x.png" alt="- D @ n @ x " class="par-math-display" /></center>
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455 <!--l. 518--><p class="nopar">
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456 </p><!--l. 521--><p class="noindent">Teilchen durch die Querschnittseinheit. Da dynamisches Gleich-<br/>gewicht herrschen
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457 soll, so muß sein:
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458 </p>
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459 <table width="100%"
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460 class="equation"><tr><td><a
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461 id="x1-3r2"></a>
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462 <center class="math-display" >
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463 <img
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464 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190525x.png" alt=" n K @ n -------- D --- = 0 . 6p k P @ x " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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465 <!--l. 529--><p class="nopar">
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466 </p><!--l. 533--><p class="indent"> Aus den beiden für das dynamische Gleichgewicht ge-<br/>fundenen Bedingungen (1) und (2) kann man den Diffusions-<br/>koeffizienten berechnen. Man erhält:
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467 </p>
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468 <center class="par-math-display" >
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469 <img
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470 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190526x.png" alt=" R-T- --1---- D = N 6p kP . " class="par-math-display" /></center>
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471 <!--l. 540--><p class="nopar">
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472 </p><!--l. 543--><p class="noindent">Der Diffusionskoeffizient der suspendierten Substanz hängt also <br/>----------
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473 </p><!--l. 547--><p class="indent"> 1) Vgl. z. B. G. Kirchhoff, Vorlesungen über Mechanik, 26. Vor-<br/>lesung <span
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474 class="cmsy-10x-x-120">§ </span>4.
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475 <pb/>
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476 </p><!--l. 552--><p class="indent">
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477
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478 </p><!--l. 553--><p class="noindent">außer von universellen Konstanten und der absoluten Tem-<br/>peratur nur vom Reibungskoeffizienten der Flüssigkeit und von <br/>der Größe der suspendierten Teilchen ab.
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479 </p>
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480 <div class="center" >
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481
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482 <!--l. 559--><p class="noindent">
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483 </p><!--l. 560--><p class="noindent"><span
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484 class="cmsy-10x-x-120">§ </span>4. Über die ungeordnete Bewegung von in einer Flüssigkeit
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485 <br/>suspendierten Teilchen und deren Beziehung zur Diffusion.</p></div>
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486 <!--l. 567--><p class="indent"> Wir gehen nun dazu über, die ungeordneten Bewegungen <br/>genauer zu untersuchen, welche, von der Molekularbewegung <br/>der Wärme hervorgerufen, Anlaß zu der im letzten Para-<br/>graphen untersuchten Diffusion geben.
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487 </p><!--l. 573--><p class="indent"> Es muß offenbar angenommen werden, daß jedes einzelne <br/>Teilchen eine Bewegung ausführe, welche unabhängig ist von <br/>der Bewegung aller anderen Teilchen; es werden auch die <br/>Bewegungen eines und desselben Teilchens in verschiedenen <br/>Zeitintervallen als voneinander unabhängige Vorgänge aufzu-<br/>fassen sein, solange wir diese Zeitintervalle nicht zu klein ge-<br/>wählt denken.
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488 </p><!--l. 582--><p class="indent"> Wir führen ein Zeitintervall <span
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489 class="cmmi-12"><img
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490 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>in die Betrachtung ein, <br/>welches sehr klein sei gegen die beobachtbaren Zeitintervalle, <br/>aber doch so groß, daß die in zwei aufeinanderfolgenden Zeit-<br/>intervallen <span
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491 class="cmmi-12"><img
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492 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>von einem Teilchen ausgeführten Bewegungen als <br/>voneinander unabhängige Ereignisse aufzufassen sind.
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493 </p><!--l. 589--><p class="indent"> Seien nun in einer Flüssigkeit im ganzen <span
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494 class="cmmi-12">n </span>suspendierte <br/>Teilchen vorhanden. In einem Zeitintervall <span
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495 class="cmmi-12"><img
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496 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>werden sich die <br/><span
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497 class="cmmi-12">X</span>-Koordinaten der einzelnen Teilchen um <img
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498 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />
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499 vergrößern, wobei <br/><img
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500 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> für jedes Teilchen einen anderen (positiven oder negativen)
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501 <br/>Wert hat. Es wird für <img
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502 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> ein gewisses Häufigkeitsgesetz gelten; <br/>die Anzahl <span
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503 class="cmmi-12">d n</span>
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504 der Teilchen, welche in dem Zeitintervall <span
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505 class="cmmi-12"><img
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506 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span> <br/>eine Verschiebung erfahren, welche zwischen <img
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507 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> und <img
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508 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> + <span
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509 class="cmmi-12">d </span><img
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510 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> <br/>liegt, wird durch eine Gleichung von der Form
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511 </p>
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512 <center class="par-math-display" >
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513 <img
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514 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190527x.png" alt="dn = n f (D) d D " class="par-math-display" /></center>
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515 <!--l. 604--><p class="nopar">
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516 </p><!--l. 608--><p class="noindent">ausdrückbar sein, wobei
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517 </p>
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518 <center class="par-math-display" >
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519 <img
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520 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190528x.png" alt=" integral +o o
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521
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522 f (D) d D = 1 - oo
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523
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524 " class="par-math-display" /></center>
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525 <!--l. 615--><p class="nopar">
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526 </p><!--l. 619--><p class="noindent">und <span
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527 class="cmmi-12"><img
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528 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>nur für sehr kleine Werte von <img
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529 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> von Null verschieden <br/>ist und die Bedingung
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530 </p>
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531 <center class="par-math-display" >
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532 <img
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533 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190529x.png" alt="f (D) = f(- D ) " class="par-math-display" /></center>
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534 <!--l. 625--><p class="nopar">
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535 </p><!--l. 629--><p class="noindent">erfüllt. <pb/>
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536 </p><!--l. 633--><p class="indent">
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537
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538 </p><!--l. 634--><p class="indent"> Wir untersuchen nun, wie der Diffusionskoeffizient von <span
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539 class="cmmi-12"><img
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540 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span> <br/>abhängt, wobei wir uns wieder auf den Fall beschränken, daß <br/>die Anzahl <span
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541 class="cmmi-12"><img
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542 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>der Teilchen pro Volumeneinheit nur von <span
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543 class="cmmi-12">x </span>und <span
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544 class="cmmi-12">t </span> <br/>abhängt.
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545 </p><!--l. 640--><p class="indent"> Es sei <span
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546 class="cmmi-12"><img
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547 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /> </span>= <span
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548 class="cmmi-12">f</span>(<span
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549 class="cmmi-12">x, t</span>) die Anzahl der Teilchen pro Volumen-<br/>einheit, wir berechnen die Verteilung der Teilchen zur Zeit <br/><span
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550 class="cmmi-12">t </span>+ <span
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551 class="cmmi-12"><img
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552 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>aus deren Verteilung zur Zeit <span
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553 class="cmmi-12">t. </span>Aus der Definition <br/>der Funktion <span
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554 class="cmmi-12"><img
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555 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>(<img
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556 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />) ergibt sich leicht die Anzahl der Teilchen, <br/>welche sich zur Zeit <span
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557 class="cmmi-12">t </span>+ <span
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558 class="cmmi-12"><img
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559 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>zwischen zwei zur <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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560 class="cmmi-12">X</span>-Achse senk-<br/>rechten Ebenen mit den Abszissen <span
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561 class="cmmi-12">x </span>und <span
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562 class="cmmi-12">x </span>+ <span
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563 class="cmmi-12">dx </span>befinden. <br/>Man erhält:
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564 </p>
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565 <center class="par-math-display" >
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566 <img
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567 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190530x.png" alt=" D=+ oo
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568 integral
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569 f (x, t + t) dx = d x . f (x + D) f(D) d D . D= - oo
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570 " class="par-math-display" /></center>
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571 <!--l. 655--><p class="nopar">
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572 </p><!--l. 658--><p class="noindent">Nun können wir aber, da <span
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573 class="cmmi-12"><img
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574 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-1c.png" alt="t" class="12x-x-1c" /> </span>sehr klein ist, setzen:
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575 </p>
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576 <center class="par-math-display" >
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577 <img
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578 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190531x.png" alt="f(x, t + t ) = f (x, t) + t@-f-. @ t " class="par-math-display" /></center>
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579 <!--l. 666--><p class="nopar">
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580 </p><!--l. 669--><p class="noindent">Ferner entwickeln wir <span
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581 class="cmmi-12">f</span>(<span
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582 class="cmmi-12">x </span>+ <img
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583 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /><span
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584 class="cmmi-12">, t</span>) nach Potenzen von <img
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585 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />:
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586 </p>
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587 <center class="par-math-display" >
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588 <img
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589 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190532x.png" alt=" @ f (x,t) D2 @2f (x, t) f (x + D, t) = f(x, t) + D --------+ ---------2--- ... in inf. @ x 2! @ x " class="par-math-display" /></center>
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590 <!--l. 679--><p class="nopar">
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591 </p><!--l. 682--><p class="noindent">Diese Entwicklung können wir unter dem Integral vornehmen, <br/>da zu letzterem nur sehr kleine Werte von <img
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592 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> etwas beitragen. <br/>Wir erhalten:
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593 </p>
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594 <center class="par-math-display" >
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595 <img
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596 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190533x.png" alt=" integral +o o integral +o o
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597 @ f @ f f + ----.t = f . f (D) d D + ---- D f (D) dD @ t - oo @ x- oo
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598 + oo
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599 @2f integral D 2 + ---2 ----f (D) d D ... @ x- oo 2 " class="par-math-display" /></center>
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600 <!--l. 705--><p class="nopar">
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601 </p><!--l. 708--><p class="noindent">Auf der rechten Seite verschwindet wegen <span
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602 class="cmmi-12"><img
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603 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>(<span
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604 class="cmmi-12">x</span>) = <span
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605 class="cmmi-12"><img
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606 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /> </span>(<span
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607 class="cmsy-10x-x-120">-</span><span
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608 class="cmmi-12">x</span>) das <br/>zweite, vierte etc. Glied, während von dem ersten, dritten, <br/>fünften etc. Gliede jedes folgende gegen das vorhergehende <br/>sehr klein ist. Wir erhalten aus dieser Gleichung, indem wir
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609 <br/>berücksichtigen, daß
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610 </p>
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611 <center class="par-math-display" >
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612 <img
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613 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190534x.png" alt=" integral + oo
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614
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615 f (D) dD = 1 , - oo
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616 " class="par-math-display" /></center>
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617 <!--l. 720--><p class="nopar"> <pb/>
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618 </p><!--l. 727--><p class="indent">
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619
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620 </p><!--l. 728--><p class="noindent">und indem wir
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621 </p>
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622 <center class="par-math-display" >
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623 <img
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624 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190535x.png" alt=" integral + oo 2 1- D-- f (D) dD = D t 2 - oo
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625 " class="par-math-display" /></center>
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626 <!--l. 736--><p class="nopar">
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627 </p><!--l. 740--><p class="noindent">setzen und nur das erste und dritte Glied der rechten Seite <br/>berücksichtigen:</p>
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628 <table width="100%"
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629 class="equation"><tr><td><a
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630 id="x1-4r1"></a>
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631 <center class="math-display" >
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632 <img
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633 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190536x.png" alt=" 2 @ f-= D @-f- . @ t @ x2 " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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634 <!--l. 748--><p class="nopar">
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635 </p><!--l. 752--><p class="indent"> Dies ist die bekannte Differentialgleichung der Diffusion, <br/>und man erkennt, daß
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636 <span
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637 class="cmmi-12">D </span>der Diffusionskoeffizient ist.
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638 </p><!--l. 756--><p class="indent"> An diese Entwicklung läßt sich noch eine wichtige Über-<br/>legung anknüpfen. Wir haben angenommen, daß die einzelnen <br/>Teilchen alle auf dasselbe Koordinatensystem bezogen seien. <br/>Dies ist jedoch nicht nötig, da die Bewegungen der einzelnen <br/>Teilchen voneinander unabhängig sind. Wir wollen nun die
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639 <br/>Bewegung jedes Teilchens auf ein Koordinatensystem beziehen, <br/>dessen Ursprung mit der Lage des Schwerpunktes des be-<br/>treffenden Teilchens zur Zeit <span
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640 class="cmmi-12">t </span>= 0 zusammenfällt, mit dem <br/>Unterschiede, daß jetzt <span
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641 class="cmmi-12">f </span>(<span
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642 class="cmmi-12">x, t</span>) <span
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643 class="cmmi-12">dx </span>die Anzahl der Teilchen be-<br/>deutet, deren <span
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644 class="cmmi-12">X</span>-Koordinaten von der Zeit <span
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645 class="cmmi-12">t </span>= 0 bis zur Zeit <br/><span
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646 class="cmmi-12">t </span>= <span
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647 class="cmmi-12">t </span>um eine Größe <span
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648 class="cmti-12">gewachsen </span>ist, welche zwischen <span
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649 class="cmmi-12">x </span>und <br/><span
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650 class="cmmi-12">x </span>+ <span
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651 class="cmmi-12">dx </span>liegt. Auch in diesem Falle ändert sich also die <br/>Funktion <span
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652 class="cmmi-12">f </span>gemäß Gleichung (1). Ferner muß offenbar für <br/><span
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653 class="cmmi-12">x </span><span
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654 class="msam-10x-x-120"><img
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655 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/msam10-c-3f.png" alt="&gt;&lt;" class="10-120x-x-3f" /> </span>0 und <span
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656 class="cmmi-12">t </span>= 0
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657 </p>
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658 <center class="par-math-display" >
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659 <img
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660 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190537x.png" alt=" integral + oo
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661
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662 f(x,t) = 0 und f(x,t)d x = n - oo
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663 " class="par-math-display" /></center>
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664 <!--l. 779--><p class="nopar">
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665 </p><!--l. 783--><p class="noindent">sein. Das Problem, welches mit dem Problem der Diffusion <br/>von einem Punkte aus (unter Vernachlässigung der Wechsel-<br/>wirkung der diffundierenden Teilchen) übereinstimmt, ist nun <br/>mathematisch vollkommen bestimmt; seine Lösung ist:
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666 </p>
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667 <center class="par-math-display" >
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668 <img
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669 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190538x.png" alt=" x2- ---n----e--4Dt f(x,t) = V~ ------ V~ t . 4 pD " class="par-math-display" /></center>
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670 <!--l. 793--><p class="nopar">
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671 </p><!--l. 797--><p class="indent"> Die Häufigkeitsverteilung der in einer beliebigen Zeit <span
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672 class="cmmi-12">t </span> <br/>erfolgten Lagenänderungen ist also dieselbe wie die der zu-<br/><pb/>
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673 </p><!--l. 802--><p class="indent">
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674
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675 </p><!--l. 803--><p class="noindent">fälligen Fehler, was zu vermuten war. Von Bedeutung aber <br/>ist, wie die Konstante im Exponenten mit dem Diffusions-<br/>koeffizienten zusammenhängt. Wir berechnen nun mit Hilfe <br/>dieser Gleichung die Verrückung <span
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676 class="cmmi-12"><img
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677 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /></span><sub ><span
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678 class="cmmi-8">x</span></sub> in Richtung der <span
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679 class="cmmi-12">X</span>-Achse,
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680 <br/>welche ein Teilchen im Mittel erfäbrt, oder -- genauer aus-<br/>gedrückt -- die Wurzel aus dem arithmetischen Mittel der <br/>Quadrate der Verrückungen in Richtung der <span
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681 class="cmmi-12">X</span>-Achse; es ist:
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682 </p>
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683 <center class="par-math-display" >
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684 <img
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685 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190539x.png" alt=" V~ --- V~ ----- cx = x2 = 2 D t . " class="par-math-display" /></center>
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686 <!--l. 816--><p class="nopar">
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687 </p><!--l. 820--><p class="indent"> Die mittlere Verschiebung ist also proportional der Qua-<br/>dratwurzel aus der Zeit. Man kann leicht zeigen, daß die <br/>Wurzel aus dem Mittelwert der Quadrate der <span
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688 class="cmti-12">Gesamtverschic- </span> <br/><span
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689 class="cmti-12">bungen </span>der Teilchen den Wert <span
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690 class="cmmi-12"><img
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691 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /></span><sub ><span
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692 class="cmmi-8">x</span></sub><img
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693 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190540x.png" alt=" V~ 3" class="sqrt" />
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694 besitzt.
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695 </p>
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696 <div class="center" >
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697
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698 <!--l. 827--><p class="noindent">
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699 </p><!--l. 828--><p class="noindent"><span
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700 class="cmsy-10x-x-120">§ </span>5. Formel für die mittlere Verschiebung suspendierter Teilchen. <br/>Eine neue Methode zur Bestimmung der wahren Größe der Atome.</p></div>
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701 <!--l. 835--><p class="indent"> In <span
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702 class="cmsy-10x-x-120">§ </span>3 haben wir für den Diffusionskoeffizienten <span
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703 class="cmmi-12">D </span>eines <br/>in einer Flüssigkeit in Form von kleinen Kugeln vom Radius <span
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704 class="cmmi-12">P </span> <br/>suspendierten Stoffes den Wert gefunden:
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705 </p>
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706 <center class="par-math-display" >
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707 <img
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708 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190541x.png" alt=" R T 1 D = -----------. N 6p k P " class="par-math-display" /></center>
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709 <!--l. 843--><p class="nopar">
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710 </p><!--l. 846--><p class="noindent">Ferner fanden wir in <span
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711 class="cmsy-10x-x-120">§ </span>4 für den Mittelwert der Verschie-<br/>bungen der Teilchen in Richtung der <span
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712 class="cmmi-12">X</span>-Achse in der Zeit <span
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713 class="cmmi-12">t</span>:
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714 </p>
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715 <center class="par-math-display" >
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716 <img
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717 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190542x.png" alt=" V~ ----- cx = 2D t . " class="par-math-display" /></center>
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718 <!--l. 853--><p class="nopar">
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719 </p><!--l. 856--><p class="noindent">Durch Eliminieren von <span
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720 class="cmmi-12">D </span>erhalten wir:
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721 </p>
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722
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723 <center class="par-math-display" >
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724 <img
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725 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190543x.png" alt=" V~ ------------ V~ - R T 1 cx = t. -----------. N 3p k P " class="par-math-display" /></center>
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726 <!--l. 863--><p class="nopar">
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727 </p><!--l. 866--><p class="noindent">Diese Gleichung läßt erkennen, wie <span
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728 class="cmmi-12"><img
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729 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /></span><sub ><span
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730 class="cmmi-8">x</span></sub> von <span
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731 class="cmmi-12">T, k </span>und <span
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732 class="cmmi-12">P </span>ab-<br/>hängen muß.
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733 </p><!--l. 869--><p class="indent"> Wir wollen berechnen, wie groß <span
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734 class="cmmi-12"><img
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735 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/cmmi12-15.png" alt="c" class="12x-x-15" /></span><sub ><span
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736 class="cmmi-8">x</span></sub> für eine Sekunde ist, <br/>wenn <span
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737 class="cmmi-12">N </span>gemäß den Resultaten der kinetischen Gastheorie <br/>6.10<sup ><span
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738 class="cmr-8">23</span></sup> gesetzt wird; es sei als Flüssigkeit Wasser von 17<sup ><span
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739 class="cmr-8">0</span></sup> C<span
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740 class="cmmi-12">. </span> <br/>gewählt (<span
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741 class="cmmi-12">k </span>= 1<span
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742 class="cmmi-12">, </span>35 <span
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743 class="cmmi-12">. </span>10<sup ><span
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744 class="cmsy-8">-</span><span
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745 class="cmr-8">2</span></sup>) und der Teilchendurchmesser sei
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746 <br/>0,001 mm. Man erhält:
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747 </p>
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748 <center class="par-math-display" >
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749 <img
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750 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190544x.png" alt="cx = 8.10- 5 cm = 0,8 Mikron. " class="par-math-display" /></center>
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751 <!--l. 880--><p class="nopar">
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752 </p><!--l. 883--><p class="noindent">Die mittlere Verschiebung in 1 Min. wäre also ca. 6 Mikron. <pb/>
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753 </p><!--l. 888--><p class="indent">
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754
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755 </p><!--l. 889--><p class="indent"> Umgekehrt läßt sich die gefundene Beziehung zur Be-<br/>stimmung von <span
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756 class="cmmi-12">N</span>
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757 benutzen. Man erhält:
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758 </p>
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759 <center class="par-math-display" >
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760 <img
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761 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1905/fulltext/img/Einst_Ueber_de_190545x.png" alt=" t R T N = --2-.------- . cx 3 p kP " class="par-math-display" /></center>
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762 <!--l. 896--><p class="nopar">
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763 </p><!--l. 900--><p class="indent"> Möge es bald einem Forscher gelingen, die hier auf-<br/>geworfene, für die Theorie der Wärme wichtige Frage zu ent-<br/>scheiden!
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764 </p><!--l. 904--><p class="indent"> Bern, Mai 1905. </p>
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765 <div class="center" >
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766
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767 <!--l. 906--><p class="noindent">
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768 </p><!--l. 907--><p class="noindent">(Eingegangen 11. Mai 1905.)</p></div>
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769 <div class="center" >
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770
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771 <!--l. 912--><p class="noindent">
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772 </p><!--l. 913--><p class="noindent">----------</p></div>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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773
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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774 </body></html>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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775
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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776
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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777