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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
parents 22d6a63640c6
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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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8 <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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11 <!-- xhtml,html -->
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12 <meta name="src" content="Einst_Neueb_de_1906.tex" />
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13 <meta name="date" content="2005-03-02 20:06:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Neueb_de_1906.css" />
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15 </head><body >
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17 <!--l. 12--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 13--><p class="noindent">
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21 </p><!--l. 14--><p class="noindent"><span
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22 class="cmbx-12">3. </span><span
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23 class="cmbxti-10x-x-120">Eine neue Bestimmung der Molek</span><span
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24 class="cmbxti-10x-x-120">ül- </span> <br/><span
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25 class="cmbxti-10x-x-120">dimensionen;</span>
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26 <span
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27 class="cmbxti-10x-x-120">von A. Einstein.</span></p></div>
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28 <div class="center" >
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30 <!--l. 19--><p class="noindent">
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31 </p><!--l. 20--><p class="noindent">--------</p></div>
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32 <!--l. 23--><p class="indent"> Die ältesten Bestimmungen der wahren Größe der Moleküle <br/>hat die kinetische Theorie der Gase ermöglicht, während die <br/>an Flüssigkeiten beobachteten physikalischen Phänomene bis <br/>jetzt zur Bestimmung der Molekülgrößen nicht gedient haben. <br/>Es liegt dies ohne Zweifel an den bisher unüberwindlichen <br/>Schwierigkeiten, welche der Entwickelung einer ins einzelne <br/>gehenden molekularkinetischen Theorie der Flüssigkeiten ent-<br/>gegenstehen. In dieser Arbeit soll nun gezeigt werden, daß <br/>man die Größe der Moleküle des gelösten Stoffs in einer <br/>nicht dissoziierten verdünnten Lösung aus der inneren Reibung <br/>der Lösung und des reinen Lösungsmittels und aus der Diffusion <br/>des gelösten Stoffes im Lösungsmittel ermitteln kann, wenn <br/>das Volumen eines Moleküls des gelösten Stoffs groß ist gegen <br/>das Volumen eines Moleküls des Lösungsmittels. Ein derartiges <br/>gelöstes Molekül wird sich nämlich bezüglich seiner Beweg-<br/>lichkeit im Lösungsmittel und bezüglich seiner Beeinflussung
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33 <br/>der inneren Reibung des letzteren annähernd wie ein im <br/>Lösungsmittel suspendierter fester Körper verhalten, und es <br/>wird erlaubt sein, auf die Bewegung des Lösungsmittels in <br/>unmittelbarer Nähe eines Moleküls die hydrodynamischen
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34 <br/>Gleichungen anzuwenden, in welchen die Flüssigkeit als homogen <br/>betrachtet, eine molekulare Struktur derselben also nicht be-<br/>rücksichtigt wird. Als Form der festen Körper, welche die <br/>gelösten Moleküle darstellen sollen, wählen wir die Kugelform.
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35 </p>
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36 <div class="center" >
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38 <!--l. 55--><p class="noindent">
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39 </p><!--l. 56--><p class="noindent"><span
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40 class="cmsy-10">§ </span>1. Über die Beeinflussung der Bewegung einer Flüssigkeit <br/>durch eine sehr kleine in derselben suspendierte Kugel.</p></div>
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41 <!--l. 61--><p class="indent"> Es liege eine inkompressible homogene Flüssigkeit mit <br/>dem Reibungskoeffizienten
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42 <span
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43 class="cmmi-10">k </span>der Betrachtung zugrunde, deren <br/>Geschwindigkeitskomponenten <span
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44 class="cmmi-10">u</span>, <span
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45 class="cmmi-10">v</span>, <span
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46 class="cmmi-10">w </span>als Funktionen der <br/>Koordinaten <span
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47 class="cmmi-10">x</span>, <span
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48 class="cmmi-10">y</span>, <span
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49 class="cmmi-10">z </span>und der Zeit gegeben seien. Von einem <br/>beliebigen Punkt <span
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50 class="cmmi-10">x</span><sub ><span
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51 class="cmr-7">0</span></sub>, <span
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52 class="cmmi-10">y</span><sub ><span
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53 class="cmr-7">0</span></sub>, <span
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54 class="cmmi-10">z</span><sub ><span
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55 class="cmr-7">0</span></sub> aus denken wir uns die Funk-<br/>tionen <span
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56 class="cmmi-10">u</span>, <span
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57 class="cmmi-10">v</span>, <span
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58 class="cmmi-10">w </span>als Funktionen von
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59 <span
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60 class="cmmi-10">x </span><span
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61 class="cmsy-10">- </span><span
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62 class="cmmi-10">x</span><sub ><span
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63 class="cmr-7">0</span></sub>, <span
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64 class="cmmi-10">y </span><span
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65 class="cmsy-10">- </span><span
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66 class="cmmi-10">y</span><sub ><span
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67 class="cmr-7">0</span></sub>, <span
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68 class="cmmi-10">z </span><span
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69 class="cmsy-10">- </span><span
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70 class="cmmi-10">z</span><sub ><span
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71 class="cmr-7">0</span></sub> nach <br/><pb/>
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72 </p><!--l. 73--><p class="indent">
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73
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74 </p><!--l. 74--><p class="noindent">dem Taylorschen Satze entwickelt und um diesen Punkt ein <br/>so kleines Gebiet <span
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75 class="cmmi-10">G</span>
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76 abgegrenzt, daß innerhalb desselben nur <br/>die linearen Glieder dieser Entwickelung berücksichtigt werden <br/>müssen. Die Bewegung der in <span
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77 class="cmmi-10">G </span>enthaltenen Flüssigkeit kann
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78 <br/>dann bekanntlich als die Superposition dreier Bewegungen auf-<br/>gefaßt werden, nämlich
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79 </p><!--l. 82--><p class="indent"> 1. einer Parallelverschiebung aller Flüssigkeitsteilchen ohne <br/>Änderung von deren relativer Lage,
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80 </p><!--l. 85--><p class="indent"> 2. einer Drehung der Flüssigkeit ohne Änderung der <br/>relativen Lage der Flüssigkeitsteilchen,
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81 </p><!--l. 88--><p class="indent"> 3. einer Dilatationsbewegung in drei aufeinander senk-<br/>rechten Richtungen (den Hauptdilatationsrichtungen).
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82 </p><!--l. 91--><p class="indent"> Wir denken uns nun im Gebiete <span
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83 class="cmmi-10">G </span>einen kugelförmigen starren <br/>Körper, dessen Mittelpunkt im Punkte <span
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84 class="cmmi-10">x</span><sub ><span
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85 class="cmr-7">0</span></sub>, <span
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86 class="cmmi-10">y</span><sub ><span
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87 class="cmr-7">0</span></sub>, <span
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88 class="cmmi-10">z</span><sub ><span
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89 class="cmr-7">0</span></sub> liege und dessen <br/>Dimensionen gegen diejenigen des Gebietes <span
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90 class="cmmi-10">G </span>sehr klein seien. <br/>Wir nehmen ferner an, daß die betrachtete Bewegung eine so <br/>langsame sei, daß die kinetische Energie der Kugel sowie <br/>diejenige der Flüssigkeit vernachlässigt werden können. Es <br/>werde ferner angenommen, daß die Geschwindigkeitskompo-<br/>nenten eines Oberflächenelementes der Kugel mit den ent-<br/>sprechenden Geschwindigkeitskomponenten der unmittelbar be-<br/>nachbarten Flüssigkeitsteilchen übereinstimme, d. h., daß auch <br/>die (kontinuierlich gedachte) Trennungsschicht überall einen <br/>nicht unendlich kleinen Koeffizienten der inneren Reibung <br/>aufweise.
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91 </p><!--l. 107--><p class="indent"> Es ist ohne weiteres klar, daß die Kugel die Teil-<br/>bewegungen 1. und 2. einfach mitmacht, ohne die Bewegung <br/>der benachbarten Flüssigkeit zu modifizieren, da sich bei diesen <br/>Teilbewegungen die Flüssigkeit wie ein starrer Körper bewegt, <br/>und da wir die Wirkungen der Trägheit vernachlässigt haben.
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92 </p><!--l. 114--><p class="indent"> Die Bewegung 3. aber wird durch das Vorhandensein der <br/>Kugel modifiziert, und es wird unsere nächste Aufgabe sein, <br/>den Einfluß der Kugel auf diese Flüssigkeitsbewegung zu unter-<br/>suchen. Beziehen wir die Bewegung 3. auf ein Koordinaten-<br/>system, dessen Achsen den Hauptdilatationsrichtungen parallel <br/>sind, und setzen wir
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93 </p>
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94 <center class="par-math-display" >
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95 <img
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96 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19060x.png" alt="x- x = q, y- y0 = j, z- z0 = z, 0 " class="par-math-display" /></center>
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97 <!--l. 130--><p class="nopar"> <pb/>
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98 </p><!--l. 137--><p class="indent">
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99
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100 </p><!--l. 138--><p class="noindent">so läßt sich jene Bewegung, falls die Kugel nicht vorhanden <br/>ist, durch die Gleichungen darstellen:</p>
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101 <table width="100%"
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102 class="equation"><tr><td><a
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103 id="x1-2r1"></a>
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104 <center class="math-display" >
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105 <img
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106 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19061x.png" alt=" u0 = A q, { v0 = B j, w0 = C z; " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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107 <!--l. 149--><p class="nopar">
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108 </p><!--l. 152--><p class="noindent"><span
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109 class="cmmi-10">A</span>, <span
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110 class="cmmi-10">B</span>, <span
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111 class="cmmi-10">C </span>sind Konstanten, welche wegen der Inkompressibilität <br/>der Flüssigkeit die Bedingung erfüllen:</p>
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112 <table width="100%"
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113 class="equation"><tr><td><a
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114 id="x1-3r2"></a>
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115 <center class="math-display" >
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116 <img
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117 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19062x.png" alt="A + B + C = 0. " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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118 <!--l. 161--><p class="nopar">
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119 </p><!--l. 164--><p class="noindent">Befindet sich nun im Punkte <span
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120 class="cmmi-10">x</span><sub ><span
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121 class="cmr-7">0</span></sub>, <span
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122 class="cmmi-10">y</span><sub ><span
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123 class="cmr-7">0</span></sub>, <span
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124 class="cmmi-10">z</span><sub ><span
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125 class="cmr-7">0</span></sub> die starre Kugel mit <br/>dem Radius <span
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126 class="cmmi-10">P</span>, so ändert sich in der Umgebung derselben die <br/>Flüssigkeitsbewegung. Im folgenden wollen wir der Bequemlich-<br/>keit wegen <span
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127 class="cmmi-10">P </span>als ,,endlich&#8220; bezeichnen, dagegen die Werte <br/>von
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128 <span
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129 class="cmmi-10"><img
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130 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-18.png" alt="q" class="cmmi-10x-x-18" align="middle" />, <img
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131 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-11.png" alt="j" class="cmmi-10x-x-11" align="middle" />, <img
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132 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-10.png" alt="z" class="cmmi-10x-x-10" align="middle" />, </span>für welche die Flüssigkeitsbewegung durch die <br/>Kugel nicht mehr merklich modifiziert wird, als ,,unend-<br/>lich groß&#8220;.
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133 </p><!--l. 175--><p class="indent"> Zunächst ist wegen der Symmetrie der betrachteten <br/>Flüssigkeitsbewegung klar, daß die Kugel bei der betrachteten <br/>Bewegung weder eine Translation noch eine Drehung aus-<br/>führen kann, und wir erhalten die Grenzbedingungen:
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134 </p>
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135 <center class="par-math-display" >
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136 <img
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137 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19063x.png" alt="u = v = w = 0 f¨ur r = P , " class="par-math-display" /></center>
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138 <!--l. 186--><p class="nopar">
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139 </p><!--l. 190--><p class="noindent">wobei
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140 </p>
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141 <center class="par-math-display" >
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142 <img
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143 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19064x.png" alt="r = V~ q-2-+-j2 +-z2 &gt; 0 " class="par-math-display" /></center>
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144 <!--l. 197--><p class="nopar">
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145 </p><!--l. 201--><p class="noindent">gesetzt ist. Hierbei bedeuten <span
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146 class="cmmi-10">u</span>, <span
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147 class="cmmi-10">v</span>, <span
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148 class="cmmi-10">w </span>die Geschwindigkeits-<br/>komponenten der nun betrachteten (durch die Kugel modifizierten) <br/>Bewegung. Setzt man</p>
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149 <table width="100%"
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150 class="equation"><tr><td><a
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151 id="x1-4r3"></a>
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152 <center class="math-display" >
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153 <img
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154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19065x.png" alt=" u = A q + u1 , { v = B j + v1 , w = C z + w1, " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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155 <!--l. 213--><p class="nopar">
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156 </p><!--l. 217--><p class="noindent">so müßte, da die in Gleichungen (3) dargestellte Bewegung <br/>im Unendlichen in die in Gleichungen (1) dargestellte über-<br/>gehen soll, die Geschwindigkeiten <span
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157 class="cmmi-10">u</span><sub ><span
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158 class="cmr-7">1</span></sub>, <span
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159 class="cmmi-10">v</span><sub ><span
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160 class="cmr-7">1</span></sub>, <span
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161 class="cmmi-10">w</span><sub ><span
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162 class="cmr-7">1</span></sub> im Unendlichen <br/>verschwinden.
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163 </p><!--l. 223--><p class="indent"> Die Funktionen <span
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164 class="cmmi-10">u</span>, <span
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165 class="cmmi-10">v</span>, <span
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166 class="cmmi-10">w </span>haben den Gleichungen der Hydro-<br/>dynamik zu genügen unter Berücksichtigung der inneren Reibung <br/><pb/>
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167 </p><!--l. 229--><p class="indent">
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168
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169 </p><!--l. 230--><p class="noindent">und unter Vernachlässigung der Trägheit. Es gelten also die <br/>Gleichungen<sup ><span
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170 class="cmr-7">1</span></sup>)</p>
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171 <table width="100%"
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172 class="equation"><tr><td><a
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173 id="x1-5r4"></a>
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174 <center class="math-display" >
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175 <img
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176 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19066x.png" alt=" @p- @p- @-p { @q = kD u @j = kD v @ z = k D w, @ u @v @ w -@q + @j-+ @-z = 0, " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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177 <!--l. 247--><p class="nopar">
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178 </p><!--l. 251--><p class="noindent">wobei <img
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179 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /> den Operator
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180 </p>
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181 <center class="par-math-display" >
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182 <img
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183 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19067x.png" alt=" 2 2 2 -@-2 + @--2 +-@-2 @ q @ j @ z " class="par-math-display" /></center>
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184 <!--l. 259--><p class="nopar">
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185 </p><!--l. 263--><p class="noindent">und <span
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186 class="cmmi-10">p </span>den hydrostatischen Druck bedeutet.
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187 </p><!--l. 266--><p class="indent"> Da die Gleichungen (1) Lösungen der Gleichungen (4) und <br/>letztere linear sind, müssen nach (3) auch die Größen <span
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188 class="cmmi-10">u</span><sub ><span
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189 class="cmr-7">1</span></sub>, <span
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190 class="cmmi-10">v</span><sub ><span
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191 class="cmr-7">1</span></sub>, <span
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192 class="cmmi-10">w</span><sub ><span
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193 class="cmr-7">1</span></sub> <br/>den Gleichungen (4) genügen. Ich bestimmte <span
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194 class="cmmi-10">u</span><sub ><span
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195 class="cmr-7">1</span></sub>, <span
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196 class="cmmi-10">v</span><sub ><span
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197 class="cmr-7">1</span></sub>, <span
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198 class="cmmi-10">w</span><sub ><span
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199 class="cmr-7">1</span></sub> und <span
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200 class="cmmi-10">p. </span> <br/>nach einer im <span
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201 class="cmsy-10">§ </span>4 der erwähnten Kirchhoffschen Vorlesung <br/>angegebenen Methode<sup ><span
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202 class="cmr-7">2</span></sup>) und fand:
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203 </p><!--l. 274--><p class="noindent">----------
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204 </p><!--l. 277--><p class="indent"> 1) G. Kirchhoff, Vorlesungen über Mechanik. 26. Vorl.
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205 </p><!--l. 279--><p class="indent"> 2) ,,Aus den Gleichungen (4) folgt <img
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206 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /><span
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207 class="cmmi-10">p </span>= 0<span
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208 class="cmmi-10">. </span>Ist <span
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209 class="cmmi-10">p </span>dieser Bedingung <br/>gemäß angenommen und eine Funktion <span
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210 class="cmmi-10">V </span>bestimmt, die der Gleichung
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211 </p>
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212 <center class="par-math-display" >
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213 <img
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214 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19068x.png" alt=" 1 D V = k p " class="par-math-display" /></center>
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215 <!--l. 287--><p class="nopar">
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216 </p><!--l. 291--><p class="noindent">genügt, so erfüllt man die Gleichungen (4), wenn man
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217 </p>
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218 <center class="par-math-display" >
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219 <img
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220 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_19069x.png" alt=" @-V- ' @ V- ' @-V- ' u = @ q + u , v = @ j + v , w = @ z + w " class="par-math-display" /></center>
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221 <!--l. 300--><p class="nopar">
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222 </p><!--l. 304--><p class="noindent">setzt und <span
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223 class="cmmi-10">u</span><span
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224 class="cmsy-10">'</span><span
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225 class="cmmi-10">, v</span><span
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226 class="cmsy-10">'</span><span
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227 class="cmmi-10">, w</span><span
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228 class="cmsy-10">' </span>so wählt, daß <img
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229 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /> <span
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230 class="cmmi-10">u</span><span
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231 class="cmsy-10">' </span>= 0<span
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232 class="cmmi-10">, </span><img
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233 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /> <span
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234 class="cmmi-10">v</span><span
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235 class="cmsy-10">' </span>= 0 und <br/><img
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236 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /> <span
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237 class="cmmi-10">w</span><span
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238 class="cmsy-10">' </span>= 0 und
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239 </p>
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240 <center class="par-math-display" >
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241 <img
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242 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190610x.png" alt="@u'-+ @v' + @-w' = -1-p @q @j @z k " class="par-math-display" /></center>
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243 <!--l. 314--><p class="nopar">
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244 </p><!--l. 318--><p class="noindent">ist.&#8220;
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245 </p><!--l. 321--><p class="indent"> Setzt man nun
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246 </p>
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247 <center class="par-math-display" >
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248 <img
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249 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190611x.png" alt="p- @2-1r k = 2 c@ q2 " class="par-math-display" /></center>
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250 <!--l. 328--><p class="nopar">
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251 </p><!--l. 332--><p class="noindent">und im Einklang hiermit
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252 </p>
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253 <center class="par-math-display" >
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254 <img
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255 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190612x.png" alt=" @2r- @21r- a( 2 j2 z2) V = c@ q2 + b@q2 + 2 q - 2 - 2 " class="par-math-display" /></center>
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256 <!--l. 342--><p class="nopar">
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257 </p><!--l. 346--><p class="noindent">und
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258 </p>
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259 <center class="par-math-display" >
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260 <img
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261 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190613x.png" alt=" @ 1r u'= - 2c@-q, v' = 0, w' = 0 , " class="par-math-display" /></center>
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262 <!--l. 354--><p class="nopar">
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263 </p><!--l. 358--><p class="noindent">so lassen sich die Konstanten <span
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264 class="cmmi-10">a</span>, <span
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265 class="cmmi-10">b</span>, <span
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266 class="cmmi-10">c </span>so bestimmen, daß für <span
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267 class="cmmi-10"><img
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268 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>= <span
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269 class="cmmi-10">P </span> <br/><span
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270 class="cmmi-10">u </span>= <span
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271 class="cmmi-10">v </span>= <span
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272 class="cmmi-10">w </span>= 0 ist. Durch Superposition dreier derartiger Lösungen erhält <br/>man die in den Gleichungen (5) und (5a) angegebene Lösung. <pb/>
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273 </p><!--l. 366--><p class="indent">
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274
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275 </p>
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276 <table width="100%"
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277 class="equation"><tr><td><a
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278 id="x1-6r5"></a>
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279 <center class="math-display" >
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280 <img
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281 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190614x.png" alt=" { } 5 3 @2(1r) @2(1r)- @2(1r) p = - 3 kP A @ q2 + B @ j2 + C @ z2 + konst., 5 3 -q @D- { u = A q - 3 P A r3- @q , v = B j - 53 P 3 B jr3-- @@Dj-, 5 3 z- @D- w = C z - 3 P C r3 - @z , " class="math-display" /></center></td><td width="5%">(5)</td></tr></table>
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282 <!--l. 391--><p class="nopar">
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283 </p><!--l. 395--><p class="noindent">wobei
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284 </p>
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285 <table width="100%"
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286 class="equation"><tr><td><a
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287 id="x1-7r6"></a>
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288 <center class="math-display" >
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289 <img
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290 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190615x.png" alt=" { } D = A 5 P3 @2r2 + 1 P5@2(1r2) {6 @ q 6 @ q } { + B 5 P3-@2r2 + 1 P5@2(1r2) {6 @ j 6 @ j } + C 5P3 @2r2-+ 1P 5@2(1r)2- . 6 @ z 6 @ z " class="math-display" /></center></td><td width="5%">(5a)</td></tr></table>
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291 <!--l. 415--><p class="nopar">
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292 </p><!--l. 418--><p class="noindent">Es ist leicht zu beweisen, daß die Gleichungen (5) Lösungen <br/>der Gleichungen (4) sind. Denn da
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293 </p>
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294
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295 <center class="par-math-display" >
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296 <img
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297 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190616x.png" alt=" 1 2 D q = 0, D r = 0, D r = r " class="par-math-display" /></center>
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298 <!--l. 428--><p class="nopar">
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299 </p><!--l. 432--><p class="noindent">und
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300 </p>
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301 <center class="par-math-display" >
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302 <img
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303 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190617x.png" alt=" ( ) { ( )} D q- = --@- D 1 = 0, r3 @q r " class="par-math-display" /></center>
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304 <!--l. 441--><p class="nopar">
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305 </p><!--l. 445--><p class="noindent">erhält man
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306 </p><!--l. 448--><p class="indent"> <span
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307 class="cmmi-10">k </span><img
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308 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /><span
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309 class="cmmi-10">u </span>= <span
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310 class="cmsy-10">-</span><span
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311 class="cmmi-10">k</span><img
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312 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190618x.png" alt="@@q" class="frac" align="middle" /><img
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313 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190619x.png" alt=" {D D}" class="left" align="middle" /> = <span
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314 class="cmsy-10">-</span><span
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315 class="cmmi-10">k</span><img
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316 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190620x.png" alt="@@q" class="frac" align="middle" /><img
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317 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190621x.png" alt="{ 21 21 } 53 P3A @@qr2 + 53 P3B @@jr2 + ..." class="left" align="middle" /><span
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318 class="cmmi-10">.</span>
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319 </p><!--l. 457--><p class="noindent">Der zuletzt erhaltene Ausdruck ist aber nach der ersten der <br/>Gleichungen (5) mit
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320 <span
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321 class="cmmi-10"><img
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322 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-40.png" alt="@" class="10x-x-40" /> n</span><img
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323 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190622x.png" alt="/" class="left" align="middle" /><span
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324 class="cmmi-10"><img
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325 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-40.png" alt="@" class="10x-x-40" /> <img
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326 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-18.png" alt="q" class="cmmi-10x-x-18" align="middle" /></span> identisch. Auf gleiche Weise zeigt <br/>man, daß die zweite und dritte der Gleichungen (4) erfüllt ist. <br/>Ferner erhält man
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327 </p>
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328 <center class="par-math-display" >
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329 <img
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330 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190623x.png" alt="@u-+ @-v + @w- = (A + B + C) @q @ j @q " class="par-math-display" /></center>
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331 <!--l. 472--><p class="nopar">
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332 </p>
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333 <center class="par-math-display" >
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334 <img
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335 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190624x.png" alt=" 5 { @2(1r) @2(1r) @2(1r)} + 3 P3 A -@q2- + B -@j2- + C -@z2-- - D D. " class="par-math-display" /></center>
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336 <!--l. 485--><p class="nopar"> <pb/>
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337 </p><!--l. 492--><p class="indent">
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338
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339 </p><!--l. 493--><p class="noindent">Da aber nach Gleichung (5a)
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340 </p>
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341 <center class="par-math-display" >
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342 <img
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343 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190625x.png" alt=" { 21 2 1 2 1 } D D = 53 A P3 A @(@rq2)- + B @@(jr2) + C@@(zr2) , " class="par-math-display" /></center>
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344 <!--l. 504--><p class="nopar">
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345 </p><!--l. 508--><p class="noindent">so folgt, daß auch die letzte der Gleichungen (4) erfüllt ist. <br/>Was die Grenzbedingungen betrifft, so gehen zunächst für <br/>unendlich große <span
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346 class="cmmi-10"><img
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347 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>unsere Gleichungen für <span
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348 class="cmmi-10">u</span>, <span
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349 class="cmmi-10">v</span>, <span
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350 class="cmmi-10">w </span>in die
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351 <br/>Gleichungen (1) über. Durch Einsetzen des Wertes von <span
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352 class="cmmi-10">D </span>aus <br/>Gleichung (5a) in die zweite der Gleichungen (5) erhält man:</p>
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353 <table width="100%"
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354 class="equation"><tr><td><a
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355 id="x1-8r6"></a>
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356 <center class="math-display" >
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357 <img
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358 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190626x.png" alt=" 5P3 ( 2 2 2) { u = A q - 2r5 q A q + B j + C z 5 P5 ( 2 2 2) P5 +2 r7 q Aq + B j + C z - r5 A q . " class="math-display" /></center></td><td width="5%">(6)</td></tr></table>
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359 <!--l. 530--><p class="nopar">
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360 </p><!--l. 533--><p class="noindent">Man erkennt, daß <span
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361 class="cmmi-10">u </span>für <span
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362 class="cmmi-10"><img
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363 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>= <span
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364 class="cmmi-10">P </span>verschwindet. Gleiches gilt <br/>aus Symmetriegründen für
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365 <span
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366 class="cmmi-10">v </span>und <span
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367 class="cmmi-10">w</span>. Es ist nun bewiesen, <br/>daß durch die Gleichungen (5) sowohl den Gleichungen (4) als <br/>auch den Grenzbedingungen der Aufgabe Genüge geleistet ist.
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368 </p><!--l. 540--><p class="indent"> Es läßt sich auch beweisen, daß die Gleichungen (5) die <br/>einzige mit den Grenzbedingungen der Aufgabe verträgliche <br/>Lösung der Gleichungen (4) sind. Der Beweis soll hier nur <br/>angedeutet werden. Es mögen in einem endlichen Raume die
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369 <br/>Geschwindigkeitskomponenten <span
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370 class="cmmi-10">u</span>, <span
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371 class="cmmi-10">v</span>, <span
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372 class="cmmi-10">w </span>einer Flüssigkeit den <br/>Gleichungen (4) genügen. Existierte noch eine andere Lösung <br/><span
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373 class="cmmi-10">U</span>, <span
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374 class="cmmi-10">V </span>, <span
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375 class="cmmi-10">W </span>der Gleichungen (4), bei welcher an den Grenzen des <br/>betrachteten Raumes <span
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376 class="cmmi-10">U </span>= <span
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377 class="cmmi-10">u, V </span>= <span
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378 class="cmmi-10">v, W </span>= <span
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379 class="cmmi-10">w </span>ist, so ist (<span
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380 class="cmmi-10">U</span>
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381 -- <span
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382 class="cmmi-10">u</span>, <br/><span
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383 class="cmmi-10">V </span>-- <span
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384 class="cmmi-10">v</span>, <span
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385 class="cmmi-10">W </span>-- <span
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386 class="cmmi-10">w</span>) eine Lösung der Gleichungen (4), bei welcher <br/>die Geschwindigkeitskomponenten an der Grenze des Raumes <br/>verschwinden. Der in dem betrachteten Raume befindlichen <br/>Flüssigkeit wird also keine mechanische Arbeit zugeführt. Da <br/>wir die lebendige Kraft der Flüssigkeit vernachlässigt haben, <br/>so folgt daraus, daß auch die im betrachteten Raume in Wärme <br/>verwandelte Arbeit gleich Null ist. Hieraus folgert man, daß <br/>im ganzen Raume <span
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387 class="cmmi-10">u </span>= <span
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388 class="cmmi-10">u</span><sub ><span
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389 class="cmr-7">1</span></sub><span
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390 class="cmmi-10">, v </span>= <span
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391 class="cmmi-10">v</span><sub ><span
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392 class="cmr-7">1</span></sub> <span
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393 class="cmmi-10">w </span>= <span
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394 class="cmmi-10">w</span><sub ><span
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395 class="cmr-7">1</span></sub> sein muß, falls der <br/>Raum wenigstens zum Teil durch ruhende Wände begrenzt <br/>ist. Durch Grenzübergang kann dies Resultat auch auf den <br/>Fall ausgedehnt werden, daß, wie in
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396 dem oben betrachteten <br/>Falle, der betrachtete Raum unendlich ist. Man kann so
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397 <br/>dartun, daß die oben gefundene Lösung die einzige Lösung <br/>der Aufgabe ist.
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398 <pb/>
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399 </p><!--l. 570--><p class="indent">
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400
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401 </p><!--l. 571--><p class="indent"> Wir legen nun um den Punkt <span
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402 class="cmmi-10">x</span><sub ><span
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403 class="cmr-7">0</span></sub>, <span
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404 class="cmmi-10">y</span><sub ><span
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405 class="cmr-7">0</span></sub>, <span
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406 class="cmmi-10">z</span><sub ><span
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407 class="cmr-7">0</span></sub> eine Kugel vom <br/>Radius <span
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408 class="cmmi-10">R</span>, wobei <span
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409 class="cmmi-10">R </span>gegen
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410 <span
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411 class="cmmi-10">P </span>unendlich groß sei, und berechnen <br/>die Energie, welche in der innerhalb der Kugel befindlichen <br/>Flüssigkeit (in der Zeiteinheit) in Wärme verwandelt wird. <br/>Diese Energie
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412 <span
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413 class="cmmi-10">W </span>ist gleich der der Flüssigkeit mechanisch <br/>zugeführten Arbeit. Bezeichnet man die Komponenten des <br/>auf die Oberfläche der Kugel vom Radius <span
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414 class="cmmi-10">R </span>ausgeübten <br/>Druckes mit <span
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415 class="cmmi-10">X</span><sub ><span
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416 class="cmmi-7">n</span></sub>, <span
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417 class="cmmi-10">Y</span> <sub ><span
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418 class="cmmi-7">n</span></sub>, <span
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419 class="cmmi-10">Z</span><sub ><span
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420 class="cmmi-7">n</span></sub>, so ist:
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421 </p>
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422 <center class="par-math-display" >
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423 <img
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424 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190627x.png" alt=" integral
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425 W = (Xn u + Ynv + Zn w)ds , " class="par-math-display" /></center>
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426 <!--l. 586--><p class="nopar">
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427 </p><!--l. 590--><p class="noindent">wobei das Integral über die Oberfläche der Kugel vom Radius <span
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428 class="cmmi-10">R </span> <br/>zu erstrecken ist. Hierbei ist:
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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_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190628x.png" alt=" ( q j z) Xn = - Xq - + Xj --+ Xz - , ( r r r) Yn = - Yq q + Yj j-+ Yz z , ( r r r) q j- z Zn = - Zq r + Zj r + Zz r , " class="par-math-display" /></center>
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433 <!--l. 608--><p class="nopar">
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434 </p><!--l. 612--><p class="noindent">wobei
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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_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190629x.png" alt=" @u (@ v @w ) Xq = p- 2k@-q, Yz = Zj = - k @-z + @j- , ( ) Yj = p -2k @v, Zq = Xz = - k @w- + @u- , @j (@ q @ z) @w- @-u @-v Zz = p - 2k@z , Xj = Yq = - k @ j + @ q . " class="par-math-display" /></center>
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439 <!--l. 635--><p class="nopar">
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440 </p><!--l. 638--><p class="noindent">Die Ausdrücke für <span
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441 class="cmmi-10">u</span>, <span
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442 class="cmmi-10">v</span>, <span
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443 class="cmmi-10">w </span>vereinfachen sich, wenn wir be-<br/>achten, daß für <span
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444 class="cmmi-10"><img
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445 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>= <span
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446 class="cmmi-10">R </span>die Glieder mit dem Faktor <span
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447 class="cmmi-10">P</span><sup ><span
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448 class="cmr-7">5</span></sup><img
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449 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190630x.png" alt="/" class="left" align="middle" /> <span
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450 class="cmmi-10"><img
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451 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /></span><sup ><span
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452 class="cmr-7">5</span></sup> <br/>gegenüber denen mit dem Faktor <span
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453 class="cmmi-10">P</span><sup ><span
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454 class="cmr-7">3</span></sup><img
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455 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190631x.png" alt="/" class="left" align="middle" /> <span
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456 class="cmmi-10"><img
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457 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /></span><sup ><span
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458 class="cmr-7">3</span></sup>
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459 verschwinden. Wir <br/>haben zu setzen:
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460 </p>
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461 <table width="100%"
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462 class="equation"><tr><td><a
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463 id="x1-9r7"></a>
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464
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465 <center class="math-display" >
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466 <img
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467 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190632x.png" alt=" 5 3q(Aq2+B-j2+C-z2) u = A q - 2 P r5 , { v = B j - 5P3j-(A-q2+B5j2+C-z2), 2 2 r 2 2 w = C z - 52 P3z-(A-q+Br5j+C-z-). " class="math-display" /></center></td><td width="5%">(6a)</td></tr></table>
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468 <!--l. 658--><p class="nopar">
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469 </p><!--l. 661--><p class="noindent">Für <span
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470 class="cmmi-10">p </span>erhalten wir aus der ersten der Gleichungen (5) durch <br/>die entsprechenden Vernachlässigungen
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471 </p>
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472 <center class="par-math-display" >
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473 <img
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474 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190633x.png" alt=" 2 2 2 p = - 5kP 3 A-q-+-B-j-+-C-z + konst. r5 " class="par-math-display" /></center>
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475 <!--l. 670--><p class="nopar"> <pb/>
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476 </p><!--l. 678--><p class="indent">
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477
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478 </p><!--l. 679--><p class="noindent">Wir erhalten zunächst:</p>
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479 <center class="par-math-display" >
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480 <img
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481 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190634x.png" alt=" A q2 q2(A q2 + B j2 + C z2) Xq = - 2kA + 10kP 3-r5- - 25k P 3---------r7----------, 2( 2 2 2) Xj = +10 kp3A-qj - 25kp3 j--A-q-+-B-j--+-C-z--, r5 ( r7 ) 3A-qz 3 z2-A-q2-+-B-j2-+-C-z2- Xz = +10 kp r5 - 25 kp r7 , " class="par-math-display" /></center>
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482 <!--l. 697--><p class="nopar">
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483 </p><!--l. 701--><p class="noindent">und hieraus</p>
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484 <center class="par-math-display" >
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485 <img
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486 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190635x.png" alt=" ( ) q 3 q2 3 q A q2 + B j2 + C z2 Xn = 2Ak r - 10k p r4-+ 25kp ---------r6---------. " class="par-math-display" /></center>
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487 <!--l. 708--><p class="nopar">
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488 </p><!--l. 711--><p class="noindent">Mit Hilfe der durch zyklische Vertauschung abzuleitenden Aus-<br/>drücke für <span
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489 class="cmmi-10">Y</span> <sub ><span
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490 class="cmmi-7">n</span></sub> und <span
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491 class="cmmi-10">Z</span><sub ><span
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492 class="cmmi-7">n</span></sub>
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493 erhält man unter Vernachlässigung aller <br/>Glieder, die das Verhältnis <span
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494 class="cmmi-10">P</span><img
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495 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190636x.png" alt="/" class="left" align="middle" /> <span
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496 class="cmmi-10"><img
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497 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>in einer höheren als der dritten <br/>Potenz enthalten:
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498 </p><!--l. 719--><p class="indent"> </p><center class="gathered"><table class="gathered"
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499 border="0" cellpadding="0" cellspacing="15"><tr><td><span
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500 class="cmmi-10">X</span><sub ><span
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501 class="cmmi-7">n</span></sub> <span
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502 class="cmmi-10">u </span>+ <span
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503 class="cmmi-10">Y</span> <sub ><span
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504 class="cmmi-7">n</span></sub> <span
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505 class="cmmi-10">v </span>+ <span
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506 class="cmmi-10">Z</span><sub ><span
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507 class="cmmi-7">n</span></sub> <span
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508 class="cmmi-10">w </span>+ <img
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509 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190637x.png" alt="2k- r" class="frac" align="middle" /><img
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510 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190638x.png" alt="(A2q2 + B2j2 + C2z2)" class="left" align="middle" />
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511 <span
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512 class="cmsy-10">-</span>10<span
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513 class="cmmi-10">k</span><img
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514 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190639x.png" alt="P-3 r4" class="frac" align="middle" /><img
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515 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190640x.png" alt="( 22 ) A q + .+ ." class="left" align="middle" /> + 20<span
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516 class="cmmi-10">k</span><img
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517 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190641x.png" alt="P-3 r6" class="frac" align="middle" /><img
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518 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190642x.png" alt="( 2 ) Aq + .+ ." class="left" align="middle" /><sup ><span
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519 class="cmr-7">2</span></sup><span
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520 class="cmmi-10">.</span> </td></tr></table></center>
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521 Integriert man über die Kugel und berücksichtigt, daß <center class="par-math-display" >
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522 <img
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523 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190643x.png" alt=" integral
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524 ds = 4R2p, integral integral integral
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525 2 2 2 4 4 q ds = j ds = z d s = 3 pR , integral 4 integral 4 integral 4 4 6 q ds = j ds = zd s = 5 pR , integral integral integral
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526 j2z2d s = z2q2ds = q2j2d s = 416 pR6, integral ( ) ( ) Aq2 + B j2 + C z22d s = 416-pR6 A2 + B2 + C2 , " class="par-math-display" /></center>
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527 <!--l. 753--><p class="nopar">
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528 </p><!--l. 757--><p class="noindent">so erhält man:</p>
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529 <table width="100%"
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530 class="equation"><tr><td><a
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531 id="x1-10r7"></a>
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532
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533 <center class="math-display" >
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534 <img
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535 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190644x.png" alt="W = 83 pR3 kd2 - 83 pP 3kd2 = 2d2 k(V - P) , " class="math-display" /></center></td><td width="5%">(7)</td></tr></table>
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536 <!--l. 766--><p class="nopar">
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537 </p><!--l. 770--><p class="noindent">wobei</p>
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538 <center class="par-math-display" >
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539 <img
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540 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190645x.png" alt="d = A2 + B2 + C2, 4 pR3 = V 3 " class="par-math-display" /></center>
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541 <!--l. 778--><p class="nopar">
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542 </p><!--l. 782--><p class="noindent">und</p>
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543 <center class="par-math-display" >
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544 <img
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545 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190646x.png" alt="43 p P 3 = P " class="par-math-display" /></center>
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546 <!--l. 787--><p class="nopar">
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547 </p><!--l. 791--><p class="noindent">gesetzt ist. Wäre die suspendierte Kugel nicht vorhanden <br/><img
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548 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190647x.png" alt="(P = 0)" class="left" align="middle" />, so erhielte man für die im Volumen <span
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549 class="cmmi-10">V </span>verzehrte <br/>Energie
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550 </p>
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551 <table width="100%"
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552 class="equation"><tr><td><a
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553 id="x1-11r8"></a>
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554 <center class="math-display" >
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555 <img
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556 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190648x.png" alt="W0 = 2d2k V . " class="math-display" /></center></td><td width="5%">(7a)</td></tr></table>
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557 <!--l. 798--><p class="nopar">
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558 <pb/>
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559 </p><!--l. 805--><p class="indent">
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560
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561 </p><!--l. 806--><p class="noindent">Durch das Vorhandensein der Kugel wird also die verzehrte <br/>Energie um 2<span
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562 class="cmmi-10"><img
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563 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /></span><sup ><span
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564 class="cmr-7">2</span></sup> <span
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565 class="cmmi-10">k </span><img
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566 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmr10-8.png" alt="P" class="10x-x-8" />
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567 verkleinert. Es ist bemerkenswert, daß <br/>der Einfluß der suspendierten Kugel auf die Größe der ver-<br/>zehrten Energie gerade so groß ist, wie er wäre, wenn durch <br/>die Anwesenheit der Kugel die Bewegung der sie umgebenden <br/>Flüssigkeit gar nicht modifiziert würde.
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568 </p>
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569 <div class="center" >
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570
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571 <!--l. 815--><p class="noindent">
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572 </p><!--l. 816--><p class="noindent"><span
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573 class="cmsy-10">§ </span>2. Berechnung des Reibungskoeffizienten einer Flüssigkeit, in <br/>welcher sehr viele kleine Kugeln in regelloser Verteilung sus-<br/>pendiert sind.</p></div>
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574 <!--l. 821--><p class="indent"> Wir haben im vorstehenden den Fall betrachtet, daß in <br/>einem Gebiete <span
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575 class="cmmi-10">G </span>von der oben definierten Größenordnung eine <br/>relativ zu diesem Gebiete sehr kleine Kugel suspendiert ist <br/>und untersucht, wie dieselbe die Flüssigkeitsbewegung beein-<br/>flußt. Wir wollen nun annehmen, daß in dem Gebiete <span
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576 class="cmmi-10">G </span> <br/>unendlich viele Kugeln von gleichem, und zwar so kleinem <br/>Radius regellos verteilt sind, daß das Volumen aller Kugeln <br/>zusammen sehr klein sei gegen das Gebiet <span
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577 class="cmmi-10">G</span>. Die Zahl der <br/>auf die Volumeneinheit entfallenden Kugeln sei <span
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578 class="cmmi-10">n</span>, wobei <span
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579 class="cmmi-10">n </span> <br/>allenthalben in der Flüssigkeit bis auf Vernachlässigbares kon-<br/>stant sei.
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580 </p><!--l. 835--><p class="indent"> Wir gehen nun wieder aus von einer Bewegung einer <br/>homogenen Flüssigkeit ohne suspendierte Kugeln und betrachten <br/>wieder die allgemeinste Dilatationsbewegung. Sind keine <br/>Kugeln vorhanden, so können wir bei passender Wahl des <br/>Koordinatensystems die Geschwindigkeitskomponenten <span
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581 class="cmmi-10">u</span><sub ><span
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582 class="cmr-7">0</span></sub>, <span
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583 class="cmmi-10">v</span><sub ><span
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584 class="cmr-7">0</span></sub>, <span
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585 class="cmmi-10">w</span><sub ><span
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586 class="cmr-7">0</span></sub> <br/>in dem beliebigen Punkte <span
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587 class="cmmi-10">x</span>, <span
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588 class="cmmi-10">y</span>, <span
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589 class="cmmi-10">z </span>des Gebietes <span
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590 class="cmmi-10">G </span>darstellen <br/>durch die Gleichungen:
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591 </p>
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592 <center class="par-math-display" >
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593 <img
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594 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190649x.png" alt="u0 = A x, v0 = B y, w0 = C z, " class="par-math-display" /></center>
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595 <!--l. 851--><p class="nopar">
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596 </p><!--l. 855--><p class="noindent">wobei
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597 </p>
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598 <center class="par-math-display" >
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599 <img
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600 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190650x.png" alt="A + B + C = 0. " class="par-math-display" /></center>
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601 <!--l. 862--><p class="nopar">
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602 </p><!--l. 865--><p class="noindent">Eine im Punkte <span
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603 class="cmmi-10">x</span><sub ><span
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604 class="cmmi-7">v</span></sub>, <span
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605 class="cmmi-10">y</span><sub ><span
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606 class="cmmi-7">v</span></sub>, <span
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607 class="cmmi-10">z</span><sub ><span
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608 class="cmmi-7">v</span></sub> suspendierte Kugel beeinflußt nun <br/>diese Bewegung in der aus Gleichung (6) ersichtlichen Weise. <br/>Da wir den mittleren Abstand benachbarter Kugeln als sehr <br/>groß gegen deren Radius wählen, und folglich die von allen
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609 <br/><pb/>
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610 </p><!--l. 875--><p class="indent">
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611
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612 </p><!--l. 876--><p class="noindent">suspendierten Kugeln zusammen herrührenden zusätzlichen <br/>Geschwindigkeitskomponenten gegen <span
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613 class="cmmi-10">u</span><sub ><span
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614 class="cmr-7">0</span></sub>, <span
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615 class="cmmi-10">v</span><sub ><span
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616 class="cmr-7">0</span></sub>, <span
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617 class="cmmi-10">w</span><sub ><span
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618 class="cmr-7">0</span></sub> sehr klein sind, <br/>so erhalten wir für die Geschwindigkeitskomponenten <span
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619 class="cmmi-10">u</span>,
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620 <span
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621 class="cmmi-10">v</span>, <span
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622 class="cmmi-10">w </span> <br/>in der Flüssigkeit unter Berücksichtigung der suspendierten <br/>Kugeln und unter Vernachlässigung von Gliedern höherer Ord-<br/>nungen:</p>
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623 <table width="100%"
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624 class="equation"><tr><td><a
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625 id="x1-12r8"></a>
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626 <center class="math-display" >
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627 <img
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628 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190651x.png" alt=" sum { 5P-3qv(A-q2v+-Bj2v+-Cz2v) u = Ax - 2 r2v r3v 5q Aq2+ Bj2+C z2 5 } - 52Pr4v v(--v--r3v-v----v)-+ Pr4v Aqrvv ,
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629
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630 sum { 5 P3jv(Aq2v+B-j2v-+Cz2v) lv = By - 2 r2v r3v { 5j A q2+ Bj2+C z2 5 } - 52Pr4v -v(--v--r3v-v----v)-+ Pr4v Bjrvv ,
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631
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632 sum { 5P-3zv(A-q2v+-Bj2v+-Cz2v) w = Cz - 2 r2v r3v 5 z Aq2+ Bj2+C z2 5 } - 52Pr4v v(--v--r3v-v---v)-+ Pr4v Czrvv , " class="math-display" /></center></td><td width="5%">(8)</td></tr></table>
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633 <!--l. 921--><p class="nopar">
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634 </p><!--l. 924--><p class="noindent">wobei die Summation über alle Kugeln des Gebietes <span
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635 class="cmmi-10">G </span>zu <br/>erstrecken ist und
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636 </p>
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637 <center class="par-math-display" >
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638 <img
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639 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190652x.png" alt="qv = x - xv, V~ -2----2----2 jv = y - yv, rv = qv + jv + zv , zv = z - zv, " class="par-math-display" /></center>
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640 <!--l. 936--><p class="nopar">
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641
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642 </p><!--l. 940--><p class="noindent">gesetzt ist. <span
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643 class="cmmi-10">x</span><sub ><span
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644 class="cmmi-7">v</span></sub>, <span
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645 class="cmmi-10">y</span><sub ><span
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646 class="cmmi-7">v</span></sub>, <span
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647 class="cmmi-10">z</span><sub ><span
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648 class="cmmi-7">v</span></sub> sind die Koordinaten der Kugelmittel-<br/>punkte. Aus den Gleichungen (7) und (7a) schließen wir ferner, <br/>daß die Anwesenheit jeder der Kugeln bis auf unendlich <br/>Kleines höherer Ordnung eine Verringerung der Wärme-<br/>produktion pro Zeiteinheit um 2<span
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649 class="cmmi-10"><img
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650 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /></span><sup ><span
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651 class="cmr-7">2</span></sup> <span
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652 class="cmmi-10">k </span><img
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653 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmr10-8.png" alt="P" class="10x-x-8" /> zum Gefolge hat und <br/>daß im Gebiete <span
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654 class="cmmi-10">G </span>die pro Volumeneinheit in Wärme ver-<br/>wandelte Energie den Wert hat:
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655 </p>
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656 <center class="par-math-display" >
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657 <img
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658 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190653x.png" alt=" 2 2 W = 2d k - 2n d kP , " class="par-math-display" /></center>
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659 <!--l. 954--><p class="nopar">
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660 </p><!--l. 958--><p class="noindent">oder
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661 </p>
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662 <table width="100%"
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663 class="equation"><tr><td><a
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664 id="x1-13r9"></a>
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665 <center class="math-display" >
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666 <img
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667 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190654x.png" alt="W = 2d2k (1 - f) , " class="math-display" /></center></td><td width="5%">(7b)</td></tr></table>
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668 <!--l. 964--><p class="nopar">
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669 <pb/>
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670 </p><!--l. 971--><p class="indent">
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671
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672 </p><!--l. 972--><p class="noindent">wobei <span
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673 class="cmmi-10"><img
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674 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span>den von den Kugeln eingenommenen Bruchteil des <br/>Volumens bedeutet.
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675 </p><!--l. 975--><p class="indent"> Gleichung (7b) erweckt den Anschein, als ob der Reibungs-<br/>koeffizient der von uns betrachteten inhomogenen Mischung <br/>von Flüssigkeit und suspendierten Kugeln (im folgenden kurz <br/>,,Mischung&#8220; genannt) kleiner sei als der Reibungskoeffizient <span
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676 class="cmmi-10">k </span> <br/>der Flüssigkeit. Dies ist jedoch nicht der Fall, da <span
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677 class="cmmi-10">A</span>, <span
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678 class="cmmi-10">B</span>, <span
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679 class="cmmi-10">C </span> <br/>nicht die Werte der Hauptdilatationen der in Gleichungen (8) <br/>dargestellten Flüssigkeitsbewegung sind; wir wollen die Haupt-<br/>dilatationen der Mischung <span
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680 class="cmmi-10">A</span><sup ><span
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681 class="cmmi-7">x</span></sup>, <span
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682 class="cmmi-10">B</span><sup ><span
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683 class="cmmi-7">x</span></sup>, <span
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684 class="cmmi-10">C</span><sup ><span
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685 class="cmmi-7">x</span></sup> nennen. Aus Symmetrie-<br/>gründen folgt, daß die Hauptdilatationsrichtungen der Mischung <br/>den Richtungen der Hauptdilatationen <span
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686 class="cmmi-10">A</span>, <span
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687 class="cmmi-10">B</span>, <span
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688 class="cmmi-10">C</span>, also den Ko-<br/>ordinatenrichtungen parallel sind. Schreiben wir die Glei-<br/>chungen (8) in der Form:
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689 </p>
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690 <center class="par-math-display" >
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691 <img
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692 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190655x.png" alt="u = A x + sum u , v = B y + sum vv, w = C z + sum wv, v " class="par-math-display" /></center>
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693 <!--l. 999--><p class="nopar">
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694 </p><!--l. 1003--><p class="noindent">so erhalten wir:
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695 </p>
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696 <center class="par-math-display" >
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697 <img
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698 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190656x.png" alt=" x (@-u) sum (@-uv) sum (@-uv) A = @ x = A + @x = A - @x0 . x=0 x=0 x=0 " class="par-math-display" /></center>
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699 <!--l. 1013--><p class="nopar">
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700 </p><!--l. 1016--><p class="noindent">Schließen wir die unmittelbaren Umgebungen der einzelnen <br/>Kugeln von der Betrachtung aus, so können wir die zweiten <br/>und dritten Glieder der Ausdrücke von <span
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701 class="cmmi-10">u</span>,
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702 <span
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703 class="cmmi-10">v</span>, <span
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704 class="cmmi-10">w </span>weglassen und <br/>erhalten für <span
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705 class="cmmi-10">x </span>= <span
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706 class="cmmi-10">y </span>= <span
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707 class="cmmi-10">z </span>= 0:</p>
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708 <table width="100%"
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709 class="equation"><tr><td><a
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710 id="x1-14r9"></a>
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711 <center class="math-display" >
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712 <img
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713 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190657x.png" alt=" 5 P3xv(Ax2v+B-y2v+C-z2v) uv = - 2 r2v r3v , { 5P-3yv(A-x2v+B-yv2+Cz2v) vv = - 2 r2v r3v , w = - 5P-3zv(A-x2v+B-y2v+-Cz2v), v 2 rv2 r3v
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714 " class="math-display" /></center></td><td width="5%">(9)</td></tr></table>
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715 <!--l. 1034--><p class="nopar">
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716 </p><!--l. 1038--><p class="noindent">wobei
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717 </p>
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718 <center class="par-math-display" >
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719 <img
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720 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190658x.png" alt=" V~ -2---2----2 rv = x v + yv + zv &gt; 0 " class="par-math-display" /></center>
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721 <!--l. 1045--><p class="nopar">
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722 </p><!--l. 1049--><p class="noindent">gesetzt ist. Die Summierung erstrecken wir über das Volumen <br/>einer Kugel <span
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723 class="cmmi-10">K </span>von sehr großem Radius <span
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724 class="cmmi-10">R</span>, deren Mittelpunkt <br/>im Koordinatenursprung liegt. Betrachten wir ferner die <br/><pb/>
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725 </p><!--l. 1056--><p class="indent">
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726
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727 </p><!--l. 1057--><p class="noindent"><span
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728 class="cmti-10">regellos </span>verteilten Kugeln als <span
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729 class="cmti-10">gleichm</span><span
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730 class="cmti-10">ä</span><span
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731 class="cmmi-10"><img
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732 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-c.png" alt="b" class="cmmi-10x-x-c" align="middle" /></span><span
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733 class="cmti-10">ig </span>verteilt und setzen <br/>an Stelle der Summe ein Integral, so erhalten wir:
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734 </p>
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735 <center class="par-math-display" >
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736 <img
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737 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190659x.png" alt=" integral
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738 @uv- A* = A - n @xv dxv dyvdzv, K integral
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739 = A - n uvxv-ds, rv " class="par-math-display" /></center>
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740 <!--l. 1071--><p class="nopar">
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741 </p><!--l. 1075--><p class="noindent">wobei das letzte Integral über die Oberfläche der Kugel <span
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742 class="cmmi-10">K </span> <br/>zu erstrecken ist. Wir finden unter Berücksichtigung von (9):
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743 </p>
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744 <center class="par-math-display" >
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745 <img
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746 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190660x.png" alt=" integral
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747 A* = A - 5 P36-n x20(A x20 + B y20 + Cz20)d s, 2 R( ) = A - n 43 P 3p A = A (1 - f) . " class="par-math-display" /></center>
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748 <!--l. 1088--><p class="nopar">
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749 </p><!--l. 1091--><p class="noindent">Analog ist
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750 </p>
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751 <center class="par-math-display" >
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752 <img
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753 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190661x.png" alt="B* = B (1 - f) , C* = C (1 - f) . " class="par-math-display" /></center>
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754 <!--l. 1101--><p class="nopar">
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755 </p><!--l. 1104--><p class="noindent">Setzen wir
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756 </p>
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757 <center class="par-math-display" >
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758 <img
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759 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190662x.png" alt="d* = A*2 + B*2 + C*2, " class="par-math-display" /></center>
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760 <!--l. 1111--><p class="nopar">
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761 </p><!--l. 1115--><p class="noindent">so ist bis auf unendlich Kleines höherer Ordnung:
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762 </p>
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763 <center class="par-math-display" >
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764 <img
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765 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190663x.png" alt="d*2 = d2(1 - 2f). " class="par-math-display" /></center>
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766 <!--l. 1122--><p class="nopar">
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767 </p><!--l. 1125--><p class="noindent">Wir haben für die Wärmeentwickelung pro Zeit- und Volumen-<br/>einheit gefunden:
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768 </p>
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769 <center class="par-math-display" >
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770 <img
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771 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190664x.png" alt="W * = 2d2k(1 - f). " class="par-math-display" /></center>
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772 <!--l. 1133--><p class="nopar">
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773 </p><!--l. 1136--><p class="noindent">Bezeichnen wir mit <span
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774 class="cmmi-10">k</span><sup ><sup ><span
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775 class="cmsy-5">*</span></sup>
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776 </sup> den Reibungskoeffizienten des Gemisches, <br/>so ist:
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777 </p>
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778 <center class="par-math-display" >
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779 <img
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780 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190665x.png" alt="W * = 2 d*2 k*. " class="par-math-display" /></center>
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781 <!--l. 1143--><p class="nopar">
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782 </p><!--l. 1146--><p class="noindent">Aus den drei letzten Gleichungen erhält man unter Vernach-<br/>lässigung von unendlich Kleinem höherer Ordnung:
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783 </p>
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784 <center class="par-math-display" >
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785 <img
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786 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190666x.png" alt="k* = k(1 + f). " class="par-math-display" /></center>
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787 <!--l. 1155--><p class="nopar">
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788 </p><!--l. 1158--><p class="noindent">Wir erhalten also das Resultat:
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789 </p><!--l. 1161--><p class="indent"> Werden in einer Flüssigkeit sehr kleine starre Kugeln <br/>suspendiert, so wächst dadurch der Koeffizient der inneren <br/>Reibung um einen Bruchteil, der gleich ist dem Gesamt-<br/><pb/>
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790 </p><!--l. 1168--><p class="indent">
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791
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792 </p><!--l. 1169--><p class="noindent">volumen der in der Volumeneinheit suspendierten Kugeln, <br/>vorausgesetzt, daß dieses Gesamtvolumen sehr klein ist.
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793 </p>
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794 <div class="center" >
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795
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796 <!--l. 1173--><p class="noindent">
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797 </p><!--l. 1174--><p class="noindent"><span
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798 class="cmsy-10">§ </span>3. Über das Volumen einer gelösten Substanz von im Vergleich <br/>zum Lösungsmittel großem Molekularvolumen.</p></div>
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799 <!--l. 1179--><p class="indent"> Es liege eine verdünnte Lösung vor eines Stoffes, welcher <br/>in der Lösung nicht dissoziiert. Ein Molekül des gelösten <br/>Stoffes sei groß gegenüber einem Molekül des Lösungsmittels <br/>und werde als starre Kugel vom Radius <span
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800 class="cmmi-10">P </span>aufgefaßt. Wir
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801 <br/>können dann das in <span
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802 class="cmsy-10">§ </span>2 gewonnene Resultat anwenden. Be-<br/>deutet <span
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803 class="cmmi-10">k</span><sup ><sup ><span
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804 class="cmsy-5">*</span></sup>
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805 </sup> den Reibungskoeffizienten der Lösung, <span
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806 class="cmmi-10">k </span>denjenigen <br/>des reinen Lösungsmittels, so ist:
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807 </p>
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808 <center class="par-math-display" >
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809 <img
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810 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190667x.png" alt="k*- k = 1 + f, " class="par-math-display" /></center>
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811 <!--l. 1193--><p class="nopar">
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812 </p><!--l. 1197--><p class="noindent">wobei <span
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813 class="cmmi-10"><img
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814 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span>das Gesamtvolumen der in Lösung befindlichen Mole-<br/>küle pro Volumeinheit ist.
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815 </p><!--l. 1201--><p class="indent"> Wir wollen <span
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816 class="cmmi-10"><img
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817 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span>für eine 1 proz. wässerige Zuckerlösung be-<br/>rechnen. Nach Beobachtungen von Burkhard (Tabellen von <br/>Landolt und Börnstein) ist bei einer 1proz. wässerigen <br/>Zuckerlösung <span
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818 class="cmmi-10">k</span><sup ><span
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819 class="cmsy-7">*</span></sup><img
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820 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190668x.png" alt="/" class="left" align="middle" /> <span
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821 class="cmmi-10">k </span>= 1<span
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822 class="cmmi-10">,</span>0245 (bei 20<sup ><span
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823 class="cmr-7">0</span></sup> C.), also <span
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824 class="cmmi-10"><img
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825 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span>= 0<span
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826 class="cmmi-10">,</span>0245 für
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827 <br/>(beinahe genau) 0,01 g Zucker. Ein Gramm in Wasser gelöster <br/>Zucker hat also auf den Reibungskoeffizienten denselben Einfluß <br/>wie kleine suspendierte starre Kugeln vom Gesamtvolumen <br/>2,45 cm<sup ><span
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828 class="cmr-7">3</span></sup>.
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829 </p><!--l. 1213--><p class="indent"> Es ist nun daran zu erinnern, daß 1 g festen Zuckers <br/>das Volumen 0,61 cm<sup ><span
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830 class="cmr-7">3</span></sup>
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831 besitzt. Dasselbe Volumen findet man <br/>auch für das spezifische Volumen <span
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832 class="cmmi-10">s </span>des in Lösung befindlichen <br/>Zuckers, wenn man die Zuckerlösung als eine <span
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833 class="cmti-10">Mischung </span>von
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834 <br/>Wasser und Zucker in gelöster Form auffaßt. Die Dichte <br/>einer 1 proz. wässerigen Zuckerlösung (bezogen auf Wasser von <br/>derselben Temperatur) bei 17<span
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835 class="cmmi-10">,</span>5<sup ><span
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836 class="cmr-7">0</span></sup> ist nämlich 1,00388. Man hat <br/>also (unter Vernachlässigung des Dichteunterschiedes von <br/>Wasser von 4<sup ><span
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837 class="cmr-7">0</span></sup> und Wasser von 17<span
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838 class="cmmi-10">,</span>5<sup ><span
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839 class="cmr-7">0</span></sup>:
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840 </p>
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841 <center class="par-math-display" >
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842 <img
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843 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190669x.png" alt="--1----= 0,99 + 0,01s; 1,00388 " class="par-math-display" /></center>
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844 <!--l. 1229--><p class="nopar">
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845 </p>
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846 <center class="par-math-display" >
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847 <img
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848 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190670x.png" alt="also s = 0,61.
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849 " class="par-math-display" /></center>
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850 <!--l. 1240--><p class="nopar">
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851 </p><!--l. 1244--><p class="indent"> Während also die Zuckerlösung, was ihre Dichte anbelangt, <br/>sich wie eine Mischung von Wasser und festem Zucker ver-<br/><pb/>
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852 </p><!--l. 1250--><p class="indent">
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853
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854 </p><!--l. 1251--><p class="noindent">hält, ist der Einfluß auf die innere Reibung viermal größer, <br/>als er aus der Suspendierung der gleichen Zuckermenge re-<br/>sultieren würde. Es scheint mir dies Resultat im Sinne der <br/>Molekulartheorie kaum anders gedeutet werden zu können, als
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855 <br/>indem man annimmt, daß das in Lösung befindliche Zucker-<br/>molekül die Beweglichkeit des unmittelbar angrenzenden <br/>Wassers hemme, so daß ein Quantum Wasser, dessen Volumen <br/>ungefähr das Dreifache des Volums des Zuckermoleküls ist,
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856 <br/>an das Zuckermolekül gekettet ist.
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857 </p><!--l. 1263--><p class="indent"> Wir können also sagen, daß ein gelöstes Zuckermolekül <br/>(bez. das Molekül samt dem durch dasselbe festgehaltene <br/>Wasser) in hydrodynamischer Beziehung sich verhält wie eine <br/>Kugel vom Volumen 2,45.342/<span
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858 class="cmmi-10">N </span>cm<sup ><span
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859 class="cmr-7">3</span></sup>, wobei 342 das Molekular-<br/>gewicht des Zuckers und <span
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860 class="cmmi-10">N </span>die Anzahl der wirklichen Mole-<br/>küle in einem Grammolekül ist.
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861 </p>
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862 <div class="center" >
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863
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864 <!--l. 1271--><p class="noindent">
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865 </p><!--l. 1272--><p class="noindent"><span
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866 class="cmsy-10">§ </span>4. Über die Diffusion eines nicht dissoziierten Stoffes in <br/>flüssiger Lösung.</p></div>
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867 <!--l. 1276--><p class="indent"> Es liege eine Lösung vor, wie sie in <span
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868 class="cmsy-10">§ </span>3 betrachtet wurde. <br/>Wirkt auf das Molekül, welches wir als eine Kugel vom Radius <span
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869 class="cmmi-10">P </span> <br/>betrachten, eine Kraft <span
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870 class="cmmi-10">K</span>, so bewegt sich das Molekül mit einer <br/>Geschwindigkeit <span
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871 class="cmmi-10">w</span>, welche durch <span
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872 class="cmmi-10">P </span>und den Reibungskoeffi-<br/>zienten <span
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873 class="cmmi-10">k </span>des Lösungsmittels bestimmt ist. Es besteht nämlich <br/>die Gleichung<sup ><span
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874 class="cmr-7">1</span></sup>):
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875 </p>
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876 <table width="100%"
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877 class="equation"><tr><td><a
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878 id="x1-15r10"></a>
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879 <center class="math-display" >
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880 <img
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881 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190671x.png" alt=" K w = 6pk-P-. " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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882 <!--l. 1287--><p class="nopar">
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883 </p><!--l. 1291--><p class="indent"> Diese Beziehung benutzen wir zur Berechnung des Diffu-<br/>sionskoeffizienten einer nicht dissoziierten Lösung. Bedeutet <span
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884 class="cmmi-10">p </span> <br/>den osmotischen Druck der gelösten Substanz, welcher bei der <br/>betrachteten verdünnten Lösung als die einzige bewegende <br/>Kraft anzusehen sei, so ist die auf die gelöste Substanz pro <br/>Volumeneinheit der Lösung in Richtung der <span
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885 class="cmmi-10">X</span>-Achse ausgeübte <br/>Kraft = <span
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886 class="cmsy-10">-</span><span
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887 class="cmmi-10"><img
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888 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-40.png" alt="@" class="10x-x-40" />p</span><img
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889 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190672x.png" alt="/" class="left" align="middle" /> <span
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890 class="cmmi-10"><img
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891 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-40.png" alt="@" class="10x-x-40" /> x</span>. Befinden sich <span
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892 class="cmmi-10"><img
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893 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /></span>; Gramm in der Volumen-<br/>einheit und ist <span
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894 class="cmmi-10">m </span>das Molekulargewicht des gelösten Stoffes, <br/><span
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895 class="cmmi-10">N </span>die Anzahl wirklicher Moleküle in einem
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896 Grammolekül, so <br/>ist <img
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897 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190673x.png" alt="(r/m)" class="left" align="middle" /> <span
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898 class="cmmi-10">N </span>die Anzahl der (wirklichen) Moleküle in der Vo-<br/>----------
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899 </p><!--l. 1309--><p class="indent"> 1) G. Kirchhoff, Vorlesungen über Mechanik. 26. Vorl., Gl. (22). <pb/>
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900 </p><!--l. 1315--><p class="indent">
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901
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902 </p><!--l. 1316--><p class="noindent">lumeneinheit und die auf ein Molekül infolge des Konzentrations-<br/>gefälles wirkende Kraft:
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903 </p>
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904 <table width="100%"
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905 class="equation"><tr><td><a
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906 id="x1-16r10"></a>
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907 <center class="math-display" >
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908 <img
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909 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190674x.png" alt=" -m--@-p K = - r N @ x . " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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910 <!--l. 1323--><p class="nopar">
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911 </p><!--l. 1326--><p class="noindent">Ist die Lösung genügend verdünnt, so ist der osmotische <br/>Druck durch die Gleichung gegeben:
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912 </p>
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913 <table width="100%"
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914 class="equation"><tr><td><a
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915 id="x1-17r10"></a>
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916 <center class="math-display" >
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917 <img
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918 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190675x.png" alt=" R p = --r T, m " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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919 <!--l. 1333--><p class="nopar">
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920 </p><!--l. 1337--><p class="noindent">wobei <span
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921 class="cmmi-10">T </span>die absolute Temperatur und <span
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922 class="cmmi-10">R </span>= 8<span
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923 class="cmmi-10">,</span>31<span
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924 class="cmmi-10">.</span>10<sup ><span
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925 class="cmr-7">7</span></sup> ist. Aus <br/>den Gleichungen (1), (2) und (3) erhalten wir für die Ge-<br/>schwindigkeit der Wanderung der gelösten Substanz:
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926 </p>
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927 <center class="par-math-display" >
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928 <img
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929 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190676x.png" alt=" R-T- -1--1 @r- w = - 6p k N P r @x . " class="par-math-display" /></center>
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930
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931 <!--l. 1348--><p class="nopar">
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932 </p><!--l. 1352--><p class="indent"> Die pro Zeiteinheit durch die Einheit des Querschnittes <br/>in Richtung der <span
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933 class="cmmi-10">X </span>- Achse hindurchtretende Stoffmenge ist <br/>endlich:
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934 </p>
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935 <table width="100%"
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936 class="equation"><tr><td><a
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937 id="x1-18r10"></a>
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938 <center class="math-display" >
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939 <img
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940 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190677x.png" alt=" -R-T -1--@r- wr = - 6 pk . N P @x . " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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941 <!--l. 1360--><p class="nopar">
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942 </p><!--l. 1363--><p class="noindent">Wir erhalten also für den Diffusionskoeffizienten <span
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943 class="cmmi-10">D</span>:
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944 </p>
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945 <center class="par-math-display" >
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946 <img
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947 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190678x.png" alt="D = R-T-. -1--. 6pk N P " class="par-math-display" /></center>
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948 <!--l. 1371--><p class="nopar">
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949 </p><!--l. 1374--><p class="noindent">Man kann also aus dem Diffusionskoeffizienten und dem <br/>Koeffizienten der inneren Reibung des Lösungsmittels das Pro-<br/>dukt aus der Anzahl <span
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950 class="cmmi-10">N </span>der wirklichen Moleküle in einem <br/>Grammolekül und dem hydrodynamisch wirksamen Molekular-<br/>radius <span
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951 class="cmmi-10">P</span>
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952 berechnen.
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953 </p><!--l. 1382--><p class="indent"> In dieser Ableitung ist der osmotische Druck wie eine <br/>auf die einzelnen Moleküle wirkende Kraft behandelt worden, <br/>was offenbar der Auffassung der kinetischen Molekulartheorie <br/>nicht entspricht, da gemäß letzterer in dem vorliegenden Falle <br/>der osmotische Druck nur als eine scheinbare Kraft aufzu-<br/>fassen ist. Diese Schwierigkeit verschwindet jedoch, wenn man <br/>bedenkt, daß den (scheinbaren) osmotischen Kräften, welche <br/>den Konzentrationsverschiedenheiten der Lösung entsprechen, <br/>durch ihnen numerisch gleiche, entgegengesetzt gerichtete, auf <br/>die einzelnen Moleküle wirkende Kräfte das (dynamische) Gleich-<br/><pb/>
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954 </p><!--l. 1397--><p class="indent">
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955
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956 </p><!--l. 1398--><p class="noindent">gewicht geleistet werden kann, wie auf thermodynamischem <br/>Wege leicht eingesehen werden kann.
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957 </p><!--l. 1401--><p class="indent"> Der auf die Masseneinheit wirkenden osmotischen Kraft <br/><span
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958 class="cmsy-10">-</span><img
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959 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190679x.png" alt="1 r" class="frac" align="middle" /> <img
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960 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190680x.png" alt="-@p @ x" class="frac" align="middle" /> kann durch die (an den einzelnen gelösten Molekülen <br/>angreifende) Kraft <span
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961 class="cmsy-10">-</span><span
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962 class="cmmi-10">P</span><sub ><span
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963 class="cmmi-7">x</span></sub> das Gleichgewicht geleistet werden, <br/>wenn
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964 </p>
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965 <center class="par-math-display" >
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966 <img
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967 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190681x.png" alt=" 1 @p - - ---- Px = 0. r @x " class="par-math-display" /></center>
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968 <!--l. 1411--><p class="nopar">
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969 </p><!--l. 1415--><p class="indent"> Denkt man sich also an der gelösten Substanz (pro Massen-<br/>einheit) die zwei sich gegenseitig aufhebenden Kräftesysteme <span
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970 class="cmmi-10">P</span><sub ><span
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971 class="cmmi-7">x</span></sub> <br/>und <span
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972 class="cmsy-10">-</span><span
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973 class="cmmi-10">P</span><sub ><span
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974 class="cmmi-7">x</span></sub> angreifend, so leistet <span
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975 class="cmsy-10">-</span><span
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976 class="cmmi-10">P</span><sub ><span
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977 class="cmmi-7">x</span></sub> dem osmotischen Drucke <br/>das Gleichgewicht und es bleibt nur die dem osmotischen <br/>Drucke numerisch gleiche Kraft <span
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978 class="cmmi-10">P</span><sub ><span
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979 class="cmmi-7">x</span></sub> als Bewegungsursache übrig. <br/>Damit ist die erwähnte Schwierigkeit beseitigt.<sup ><span
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980 class="cmr-7">1</span></sup>)
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981 </p>
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982 <div class="center" >
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983
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984 <!--l. 1424--><p class="noindent">
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985 </p><!--l. 1425--><p class="noindent"><span
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986 class="cmsy-10">§ </span>5. Bestimmung der Moleküldimensionen mit Hilfe der <br/>erlangten Relationen.</p></div>
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987 <!--l. 1429--><p class="indent"> Wir haben in <span
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988 class="cmsy-10">§ </span>3 gefunden:
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989 </p>
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990 <center class="par-math-display" >
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991 <img
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992 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190682x.png" alt=" * k- = 1 + f = 1 + n .13 p P 3, k
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993 " class="par-math-display" /></center>
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994 <!--l. 1436--><p class="nopar">
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995 </p><!--l. 1440--><p class="noindent">wobei <span
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996 class="cmmi-10">n </span>die Anzahl der gelösten Moleküle pro Volumeneinheit <br/>und <span
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997 class="cmmi-10">P </span>den hydrodynamisch wirksamen Molekülradius bedeutet. <br/>Berücksichtigt man, daß
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998 </p>
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999 <center class="par-math-display" >
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1000 <img
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1001 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190683x.png" alt="n- r- N = m , " class="par-math-display" /></center>
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1002 <!--l. 1449--><p class="nopar">
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1003 </p><!--l. 1453--><p class="noindent">wobei <span
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1004 class="cmmi-10"><img
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1005 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>die in der Volumeneinheit befindliche Masse des ge-<br/>lösten Stoffes und <span
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1006 class="cmmi-10">m</span>
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1007 dessen Molekulargewicht bedeutet, so <br/>erhält man:
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1008 </p>
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1009 <center class="par-math-display" >
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1010 <img
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1011 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190684x.png" alt=" ( * ) N P 3 = -3-m- k-- 1 . 4p r k " class="par-math-display" /></center>
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1012
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1013 <!--l. 1462--><p class="nopar">
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1014 </p><!--l. 1465--><p class="noindent">Andererseits wurde in <span
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1015 class="cmsy-10">§ </span>4 gefunden:
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1016 </p>
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1017 <center class="par-math-display" >
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1018 <img
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1019 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190685x.png" alt=" R T 1 N P = 6pk-D-. " class="par-math-display" /></center>
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1020 <!--l. 1472--><p class="nopar">
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1021 </p><!--l. 1475--><p class="noindent">Diese beiden Gleichungen setzen uns in den Stand, die Größen <br/><span
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1022 class="cmmi-10">P </span>und <span
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1023 class="cmmi-10">N </span>einzeln zu berechnen, von welchen sich <span
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1024 class="cmmi-10">N </span>als un-<br/>----------
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1025 </p><!--l. 1481--><p class="indent"> 1) Eine ausführliche Darlegung dieses Gedankenganges findet sich <br/>in Ann. d. Phys. <span
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1026 class="cmbx-10">17. </span>p. 549. 1905. <pb/>
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1027 </p><!--l. 1488--><p class="indent">
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1028
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1029 </p><!--l. 1489--><p class="noindent">abhängig von der Natur des Lösungsmittels, der gelösten Sub-<br/>stanz und der Temperatur herausstellen muß, wenn unsere <br/>Theorie den Tatsachen entspricht.
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1030 </p><!--l. 1494--><p class="indent"> Wir wollen die Rechnung für wässerige Zuckerlösung <br/>durchführen. Nach den oben mitgeteilten Angaben über die <br/>innere Reibung der Zuckerlösung folgt zunächst für 2<sup ><span
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1031 class="cmr-7">0</span></sup>
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1032 C.:
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1033 </p>
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1034 <center class="par-math-display" >
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1035 <img
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1036 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190686x.png" alt=" 3 N P = 200. " class="par-math-display" /></center>
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1037 <!--l. 1503--><p class="nopar">
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1038 </p><!--l. 1507--><p class="indent"> Nach Versuchen von Graham (berechnet von Stefan) ist <br/>der Diffusionskoeffizient von Zucker in Wasser bei 9<span
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1039 class="cmmi-10">,</span>5<sup ><span
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1040 class="cmr-7">0</span></sup> C. <br/>0,384, wenn der Tag als Zeiteinheit gewählt wird. Die Zähig-<br/>keit des Wassers bei 9<span
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1041 class="cmmi-10">,</span>5<sup ><span
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1042 class="cmr-7">0</span></sup> ist 0,0135. Wir wollen diese Daten <br/>in unsere Formel für den Diffusionskoeffizienten einsetzen, <br/>trotzdem sie an 10 proz. Lösungen gewonnen sind und eine <br/>genaue Gültigkeit unserer Formel bei so hohen Konzentrationen <br/>nicht zu erwarten ist. Wir erhalten
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1043 </p>
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1044 <center class="par-math-display" >
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1045 <img
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1046 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190687x.png" alt="N P = 2,08 .1016. " class="par-math-display" /></center>
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1047 <!--l. 1521--><p class="nopar">
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1048 </p><!--l. 1525--><p class="indent"> Aus den für <span
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1049 class="cmmi-10">NP</span><sup ><span
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1050 class="cmr-7">3</span></sup> und <span
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1051 class="cmmi-10">NP </span>gefundenen Werten folgt, wenn <br/>wir die Verschiedenheit von <span
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1052 class="cmmi-10">P </span>bei 9<span
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1053 class="cmmi-10">,</span>5<sup ><span
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1054 class="cmr-7">0</span></sup> und 20<sup ><span
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1055 class="cmr-7">0</span></sup> vernach-<br/>lässigen,
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1056 </p>
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1057 <center class="par-math-display" >
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1058 <img
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1059 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190688x.png" alt="P = 9,9 .10-8cm, N = 2,1 . 1023. " class="par-math-display" /></center>
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1060 <!--l. 1536--><p class="nopar">
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1061 </p><!--l. 1540--><p class="indent"> Der für <span
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1062 class="cmmi-10">N </span>gefundene Wert stimmt der Größenordnung <br/>nach mit den durch andere Methoden gefundenen Werten für <br/>diese Größe befriedigend überein.
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1063 </p><!--l. 1545--><p class="indent"> Bern, den 30. April 1905.
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1064 </p>
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1065 <div class="center" >
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1066
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1067 <!--l. 1548--><p class="noindent">
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1068 </p><!--l. 1549--><p class="noindent">(Eingegangen 19. August 1905.)</p></div>
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1069 <div class="center" >
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1070
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1071 <!--l. 1552--><p class="noindent">
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1072 </p><!--l. 1553--><p class="noindent">---------</p></div>
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1073
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1074 <div class="center" >
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1075
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1076 <!--l. 1556--><p class="noindent">
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1077 </p><!--l. 1557--><p class="noindent"><span
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1078 class="cmbx-10">Nachtrag.</span></p></div>
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1079 <!--l. 1560--><p class="indent"> In der neuen Auflage der physikalisch-chemischen Tabellen <br/>von Landolt und Börnstein finden sich weit brauchbarere <br/>Angaben zur Berechnung der Größe des Zuckermoleküls und <br/>der Anzahl <span
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1080 class="cmmi-10">N </span>der wirklichen Moleküle in einem Gramm-<br/>molekül.
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1081 </p><!--l. 1566--><p class="indent"> Thovert fand (Tab. p. 372) für den Diffusionskoeffizienten <br/>von Zucker in Wasser bei 18<span
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1082 class="cmmi-10">,</span>5<sup ><span
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1083 class="cmr-7">0</span></sup> C. und der Konzentration <br/><pb/>
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1084 </p><!--l. 1572--><p class="indent">
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1085
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1086 </p><!--l. 1573--><p class="noindent">0,005 Mol./Liter den Wert 0,33 cm<sup ><span
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1087 class="cmr-7">2</span></sup>/Tage. Aus einer Tabelle <br/>mit Beobachtungsresultaten von Hosking (Tab. p. 81) findet <br/>man ferner durch Interpolation, daß bei verdünnter Zucker-<br/>lösung einer Zunahme des Zuckergehaltes um 1 Proz. bei <br/>18<span
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1088 class="cmmi-10">,</span>5<sup ><span
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1089 class="cmr-7">0</span></sup> C. eine Zunahme des Viskositätskoeffizienten um 0,000 25 <br/>entspricht.
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1090 </p><!--l. 1581--><p class="indent"> Unter Zugrundelegung dieser Angaben findet man
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1091 </p>
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1092 <center class="par-math-display" >
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1093 <img
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1094 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190689x.png" alt=" - 6 P = 0,78 . 10 mm " class="par-math-display" /></center>
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1095 <!--l. 1587--><p class="nopar">
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1096 </p><!--l. 1591--><p class="noindent">und
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1097 </p>
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1098 <center class="par-math-display" >
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1099 <img
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1100 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Neueb_de_1906/fulltext/img/Einst_Neueb_de_190690x.png" alt=" 23 N = 4,15 . 10 . " class="par-math-display" /></center>
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1101 <!--l. 1598--><p class="nopar">
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1102 </p><!--l. 1602--><p class="indent"> Bern, Januar 1906.
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1103 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1104 <div class="center" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1105
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1106 <!--l. 1605--><p class="noindent">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1107 </p><!--l. 1606--><p class="noindent">---------</p></div>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1108
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1109 </body></html>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1110
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1111
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1112