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Version vom 2009-02-14
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:08:12 +0200
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1 <?xml version="1.0" encoding="utf-8" ?>
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5 <html>
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7 <head><title></title>
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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12 <meta name="src" content="Einst_Bezie_de_1911.tex" />
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13 <meta name="date" content="2005-02-25 10:08:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Bezie_de_1911.css" />
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15 </head><body >
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17 <!--l. 15--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 16--><p class="noindent">
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21 </p><!--l. 17--><p class="noindent"><span
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22 class="cmr-12">9. </span><span
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23 class="cmbxti-10x-x-120">Eine Beziehung zwischen dem elastischen </span> <br/><span
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24 class="cmbxti-10x-x-120">Verhalten</span>
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25 <span
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26 class="cmbxti-10x-x-120">und der spezifischen W</span><span
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27 class="cmbxti-10x-x-120">ärme bei festen </span> <br/><span
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28 class="cmbxti-10x-x-120">K</span><span
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29 class="cmbxti-10x-x-120">örpern mit</span>
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30 <span
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31 class="cmbxti-10x-x-120">einatomigem Molek</span><span
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32 class="cmbxti-10x-x-120">ül; </span> <br/><span
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33 class="cmbxti-10x-x-120">von A. Einstein.</span></p></div>
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34 <div class="center" >
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36 <!--l. 23--><p class="noindent">
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37 </p><!--l. 24--><p class="noindent">--------</p></div>
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38 <!--l. 27--><p class="indent"> Mein Kollege, Hr. Prof. Zangger, machte mich auf eine <br/>wichtige Bemerkung aufmerksam, die Sutherland<sup ><span
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39 class="cmr-7">1</span></sup>) neulich <br/>publizierte. Dieser stellte sich die Frage, ob die elastischen <br/>Kräfte fester Körper Kräfte derselben Art seien wie diejenigen <br/>Kräfte, welche die Träger der ultraroten Eigenschwingungen <br/>in ihre Ruhelage zurücktreiben, also deren Eigenfrequenzen <br/>bedingen. Er fand, daß diese Frage mit großer Wahrschein-<br/>lichkeit zu bejahen sei auf Grund folgender Tatsache: die <br/>ultraroten Eigenfrequenzen sind von derselben Größenordnung <br/>wie diejenigen Frequenzen, welche man anwenden mußte, um <br/>elastische Transversalschwingungen durch den Körper zu senden, <br/>deren halbe Wellenlänge gleich ist dem Abstand benachbarter
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40 <br/>Moleküle des Körpers.
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41 </p><!--l. 43--><p class="indent"> Bei aller Wichtigkeit der Sutherlandschen Betrachtung <br/>ist es aber klar, daß man auf diesem Wege nicht mehr er-<br/>langen kann als eine rohe Größenordnungsbeziehung, und zwar <br/>insbesondere aus dem Grunde, weil anzunehmen ist, daß die <br/>bekannten ultraroten Eigenschwingungen in der Hauptsache <br/>als Schwingungen der verschieden geladenen Ionen eines Moleküls <br/>gegeneinander, die elastischen Schwingungen aber als Schwin-<br/>gungen der ganzen Moleküle gegeneinander aufzufassen sind. <br/>Es scheint mir deshalb, daß eine genauere Prüfung der Suther-<br/>landschen Idee nur bei Stoffen mit einatomigem Molekül <br/>möglich sei, denen nach der Erfahrung und nach dem theoreti-<br/>schen Bilde optisch nachweisbare Eigenschwingungen von der <br/>bekannten Art nicht zukommenen. Nach der von mir auf die <br/>----------
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42 </p><!--l. 61--><p class="indent"> 1) W. Sutherland, Phil. Mag. (6) <span
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43 class="cmbx-10">20</span>. p. 657. 1910. <pb/>
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44 </p><!--l. 67--><p class="indent">
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46 </p><!--l. 68--><p class="noindent">Planksche Theorie der Strahlung gegründete Theorie der <br/>spezifischen Wärme fester Körper<sup ><span
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47 class="cmr-7">1</span></sup>) ist es aber möglich, die <br/>Eigenfrequenzen der einatomigen Körper, welche Träger der <br/>Wärme sind, aus der Abhängigkeit der spezifischen Wärme <br/>von der Temperatur zu ermitteln. Diese Eigenfrequenzen <br/>kann man benutzen, um die Sutherlandsche Auffassung zu <br/>prüfen, indem man diese Eigenfrequenzen mit jenen vergleicht, <br/>die sich aus der Elastizität ergeben. Eine Art, wie dies ge-<br/>schehen kann, ist im folgenden gegeben, und es sei gleich <br/>hier bemerkt, daß sich beim Silber auf dem angedeuteten <br/>Wege Sutherlands Auffassung von der Wesensgleichheit der
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48 <br/>elastischen und der die Eigenfrequenz bestimmenden Kräfte <br/>befriedigend bestätigte.
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49 </p><!--l. 85--><p class="indent"> An eine <span
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50 class="cmti-10">exakte </span>Berechnung der Eigenschwingungsfrequenzen <br/>aus den elastischen Konstanten ist vorläufig nicht zu denken. <br/>Wir bedienen uns vielmehr hier einer rohen, der in der voran-<br/>gehenden Arbeit benutzten ähnlichen Rechenmethode, die aber <br/>wohl im Wesentlichen das Richtige treffen dürfte.
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51 </p><!--l. 92--><p class="indent"> Wir denken uns zunächst die Moleküle der Substanz nach <br/>einem quadratischen Raumgitter angeordnet. Es hat dann <br/>jedes Molekül 26 Nachbarmoleküle, die allerdings nicht gleich <br/>weit von demselben entfernt sind. Wir werden aber so rechnen,
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52 <br/>wie wenn diese 26 Nachbarmoleküle im Ruhestande alle gleich <br/>weit vom betrachteten Molekül entfernt wären.
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53 </p><!--l. 100--><p class="indent"> Wir haben nun irgend eine plausible, möglichst einfache <br/>Darstellung der Molekularkräfte zu wählen. Da führen wir <br/>zuerst die für das folgende fundamentale, in der vorangehenden <br/>Mitteilung für Flüssigkeiten erwiesene Voraussetzung ein, daß
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54 <br/>jedes Molekül nur mit seinen Nachbarmolekülen, nicht aber <br/>mit entfernteren Molekülen in Wechselwirkung stehe. Zwei <br/>Nachbarmoleküle mögen eine Zentralkraft aufeinander ausüben, <br/>welche verschwindet, wenn der Abstand der Moleküle gleich <span
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55 class="cmmi-10">d</span>
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56 <br/>ist. Ist ihr Abstand gleich <span
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57 class="cmmi-10">d </span><span
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58 class="cmsy-10">- </span><img
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59 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" />, so wirke eine Abstoßungs-<br/>kraft von der Größe
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60 <span
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61 class="cmmi-10">a</span><img
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62 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" />.
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63 </p><!--l. 114--><p class="indent"> Nun berechnen wir die Kraft, welche die 26 Nachbar-<br/>moleküle der Verrückung eines Moleküls entgegensetzen. Dabei <br/>denken wir uns die 26 Nachbarmoleküle, statt auf einer Würfel-<br/>----------
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64 </p><!--l. 120--><p class="indent"> 1) A. Einstein, Ann. d. Phys. <span
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65 class="cmbx-10">22. </span>p. 180. 1907. <pb/>
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66 </p><!--l. 126--><p class="indent">
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67
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68 </p><!--l. 127--><p class="noindent">oberfläche, auf einer Kugelfläche von gleich großem räumlichem <br/>Inhalt verteilt, deren Radius gleich <span
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69 class="cmmi-10">d </span>zu wählen ist, so daß <br/>wir haben
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70 </p>
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71 <table width="100%"
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72 class="equation"><tr><td><a
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73 id="x1-2r1"></a>
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74 <center class="math-display" >
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75 <img
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76 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19110x.png" alt="4 3 n- 3 d p = 8 N , " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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77 <!--l. 134--><p class="nopar">
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78 </p><!--l. 138--><p class="noindent">wenn <span
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79 class="cmmi-10">v </span>das Molekularvolumen der Substanz und <span
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80 class="cmmi-10">N </span>die Zahl <br/>der Moleküle in einem Grammolekül bedeutet. Wir denken <br/>uns das im Mittelpunkt der Kugel liegende Molekül in be-<br/>liebiger Richtung um die gegen <span
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81 class="cmmi-10">d </span>kleine Länge <span
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82 class="cmmi-10">x </span>verschoben <br/>und berechnen die der Verschiebung entgegenwirkende Kraft <br/>so, wie wenn die Masse der 26 Moleküle gleichförmig über <br/>die Kugeloberfläche verteilt wäre. Auf dem vom Molekül aus <br/>gezogenen elementar kleinen körperlichen Winkel <span
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83 class="cmmi-10">d<img
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84 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /></span>, dessen <br/>Achse mit der Richtung der Verschiebung <span
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85 class="cmmi-10">x </span>den Winkel <span
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86 class="cmmi-10"><img
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87 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-23.png" alt="h" class="10x-x-23" /> </span> <br/>bilde, liegen dann 26<span
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88 class="cmmi-10">.</span><img
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89 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19111x.png" alt="(dx /4p)" class="left" align="middle" /> Moleküle, welche in Richtung <br/>der Verschiebung <span
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90 class="cmmi-10">x </span>die Kraft
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91 </p>
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92 <center class="par-math-display" >
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93 <img
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94 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19112x.png" alt=" 26 - 4p-dx.a .x cos h .cos h " class="par-math-display" /></center>
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95 <!--l. 158--><p class="nopar">
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96 </p><!--l. 162--><p class="noindent">liefern. Durch Integration bekommen wir für die auf das <br/>verschobene Molekül wirkende Kraft den Wert
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97 </p>
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98 <center class="par-math-display" >
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99 <img
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100 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19113x.png" alt="- 26 ax. 3 " class="par-math-display" /></center>
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101 <!--l. 168--><p class="nopar">
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102 </p><!--l. 172--><p class="indent"> Hieraus ergibt sich, wenn man hinzunimmt, daß <span
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103 class="cmmi-10">M</span><img
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104 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19114x.png" alt="/" class="left" align="middle" /> <span
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105 class="cmmi-10">N </span> <br/>gleich ist der Masse eines Moleküls (<span
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106 class="cmmi-10">M </span>= Molekulargewicht der <br/>Substanz), die Eigenfrequenz <span
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107 class="cmmi-10"><img
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108 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /> </span>und die dieser entsprechende <br/>Vakuumwellenlänge <span
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109 class="cmmi-10"><img
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110 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-15.png" alt="c" class="10x-x-15" /> </span>des Moleküls. Es ist
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111 </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
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116 <center class="math-display" >
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117 <img
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118 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19115x.png" alt=" V~ -------- n = 1-- 26 a. N- 2p 3 M " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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119 <!--l. 184--><p class="nopar">
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120 </p><!--l. 188--><p class="noindent">und
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121 </p>
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122 <table width="100%"
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123 class="equation"><tr><td><a
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124 id="x1-4r3"></a>
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125 <center class="math-display" >
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126 <img
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127 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19116x.png" alt=" V~ -3---M-- c = 2p e -- ---. 26 a N " class="math-display" /></center></td><td width="5%">(2a)</td></tr></table>
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128 <!--l. 195--><p class="nopar">
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129 </p><!--l. 199--><p class="indent"> Wir berechnen nun auf Grund derselben Näherungs-<br/>annahmen den Kompressibilitätskoeffizienten der Substanz. Zu <br/>diesem Zwecke drücken wir die bei einer gleichmäßigen Kom-<br/>pression aufzuwendende Arbeit <span
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130 class="cmmi-10">A </span>auf zwei verschiedene Arten <br/>aus und setzen beide Ausdrücke einander gleich.
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131 </p><!--l. 206--><p class="indent"> Es ist <img
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132 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19117x.png" alt="(a /2)" class="left" align="middle" /> <img
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133 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /><sup ><span
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134 class="cmr-7">2</span></sup> die für die Verkleinerung des Abstandes <br/>zweier benachbarter Moleküle um <img
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135 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /> aufzuwendende Arbeit. <br/><pb/>
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136 </p><!--l. 213--><p class="indent">
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137
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138 </p><!--l. 214--><p class="noindent">Da jedes Molekül 26 benachbarte Moleküle hat, so ist die <br/>zur Verkleinerung seines Abstandes von den Nachbarmolekülen <br/>aufzuwendende Arbeit 26<span
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139 class="cmmi-10">.</span><img
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140 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19118x.png" alt="(a/2)" class="left" align="middle" /> <img
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141 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /><sup ><span
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142 class="cmr-7">2</span></sup> Da es in der Volumen-<br/>einheit <span
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143 class="cmmi-10">N</span><img
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144 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_19119x.png" alt="/" class="left" align="middle" /> <span
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145 class="cmmi-10"><img
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146 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-17.png" alt="n" class="10x-x-17" /> </span>Moleküle gibt und jeder Term <img
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147 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191110x.png" alt="(a/2)" class="left" align="middle" /> <img
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148 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /><sup ><span
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149 class="cmr-7">2</span></sup> zu zwei
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150 <br/>Molekülen gehört, erhält man
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151 </p>
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152 <center class="par-math-display" >
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153 <img
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154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191111x.png" alt=" 26 N- 2 A = 4 . n aD . " class="par-math-display" /></center>
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155 <!--l. 231--><p class="nopar">
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156 </p><!--l. 235--><p class="indent"> Ist <span
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157 class="cmmi-10"><img
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158 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /> </span>andererseits die Kompressibilität, <img
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159 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-2.png" alt="Q" class="10x-x-2" /> die Kontraktion <br/>der Volumeneinheit, so ist <span
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160 class="cmmi-10">A </span>= 1<img
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161 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191112x.png" alt="/" class="left" align="middle" /> 2<span
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162 class="cmmi-10"><img
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163 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" />.</span><img
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164 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-2.png" alt="Q" class="10x-x-2" /><sup ><span
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165 class="cmr-7">2</span></sup><span
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166 class="cmmi-10">, </span>oder, da <img
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167 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-2.png" alt="Q" class="10x-x-2" /> = 3<img
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168 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /><img
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169 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191113x.png" alt="/" class="left" align="middle" /> <span
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170 class="cmmi-10">d </span>ist:
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171 </p>
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172 <center class="par-math-display" >
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173 <img
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174 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191114x.png" alt=" 9 D2 A = - ---2-. 2 x.d " class="par-math-display" /></center>
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175 <!--l. 247--><p class="nopar">
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176 </p><!--l. 251--><p class="indent"> Durch gleichsetzen dieser beiden Werte für <span
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177 class="cmmi-10">A </span>erhält man
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178 </p>
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179 <table width="100%"
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180 class="equation"><tr><td><a
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181 id="x1-5r3"></a>
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182 <center class="math-display" >
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183 <img
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184 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191115x.png" alt=" 18 ---v--- x = 26 a.d2.N . " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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185 <!--l. 257--><p class="nopar">
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186 </p><!--l. 261--><p class="indent"> Durch Eliminieren von <span
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187 class="cmmi-10">a </span>und <span
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188 class="cmmi-10">d </span>aus den Gleichungen (1), <br/>2a) und (3) erhält man
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189 </p>
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190 <center class="par-math-display" >
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191 <img
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192 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191116x.png" alt=" ( )1/3 c = 2 V~ p- -6 -C1-M 1/3r1/6 V~ x = 1,08.103.M 1/3r1/6 V~ x 6 p N /3 " class="par-math-display" /></center>
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193
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194 <!--l. 281--><p class="nopar">
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195 </p><!--l. 285--><p class="indent"> Die Formel setzt natürlich voraus, daß Polymerisation <br/>nicht stattfindet. Im folgenden sind die Eigenwellenlängen <br/>(als Maß für die Eigenfrequenzen) derjenigen Metalle nach <br/>dieser Formel berechnet, für welche Grüneisen<sup ><span
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196 class="cmr-7">1</span></sup>) die kubische
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197 <br/>Kompressibilität angegeben hat. Es ergibt sich<sup ><span
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198 class="cmr-7">2</span></sup>):
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199 </p><!--l. 295--><p class="noindent"><!--tex4ht:inline--></p><div class="tabular"><table class="tabular"
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200 cellspacing="0" cellpadding="0" rules="groups"
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201 frame="border" id="TBL-1-" ><colgroup id="TBL-1-1g"><col
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202 id="TBL-1-1" /></colgroup><colgroup id="TBL-1-2g"><col
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203 id="TBL-1-2" /></colgroup><colgroup id="TBL-1-3g"><col
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204 id="TBL-1-3" /></colgroup><colgroup id="TBL-1-4g"><col
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205 id="TBL-1-4" /></colgroup><tr
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206 class="hline"><td><hr /></td><td><hr /></td><td><hr /></td><td><hr /></td></tr><tr
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207 class="hline"><td><hr /></td><td><hr /></td><td><hr /></td><td><hr /></td></tr><tr
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208 valign="baseline" id="TBL-1-1-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-1-1"
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209 class="td11"> Stoff </td><td align="center" style="white-space:nowrap; text-align:center;" id="TBL-1-1-2"
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210 class="td11"> <span
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211 class="cmmi-10"><img
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212 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-15.png" alt="c" class="10x-x-15" />.</span>10<sup ><span
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213 class="cmr-7">4</span></sup> </td><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-1-3"
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214 class="td11"> Stoff </td><td align="right" style="white-space:nowrap; text-align:right;" id="TBL-1-1-4"
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215 class="td11"> <span
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216 class="cmmi-10"><img
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217 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-15.png" alt="c" class="10x-x-15" />.</span>10<sup ><span
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218 class="cmr-7">4</span></sup> </td>
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219 </tr><tr
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220 class="hline"><td><hr /></td><td><hr /></td><td><hr /></td><td><hr /></td></tr><tr
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221 class="hline"><td><hr /></td><td><hr /></td><td><hr /></td><td><hr /></td></tr><tr
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222 valign="baseline" id="TBL-1-2-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-2-1"
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223 class="td11"> </td><td align="center" style="white-space:nowrap; text-align:center;" id="TBL-1-2-2"
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224 class="td11"> </td><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-2-3"
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225 class="td11"> </td><td align="right" style="white-space:nowrap; text-align:right;" id="TBL-1-2-4"
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226 class="td11"> </td>
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227 </tr><tr
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228 valign="baseline" id="TBL-1-3-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-3-1"
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229 class="td11">Aluminium . . . </td><td align="center" style="white-space:nowrap; text-align:center;" id="TBL-1-3-2"
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230 class="td11"> 45 </td><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-3-3"
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231 class="td11">Palladium . . . . </td><td align="right" style="white-space:nowrap; text-align:right;" id="TBL-1-3-4"
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232 class="td11"> 58 </td>
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233 </tr><tr
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234 valign="baseline" id="TBL-1-4-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-4-1"
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235 class="td11">Kupfer . . . . . </td><td align="center" style="white-space:nowrap; text-align:center;" id="TBL-1-4-2"
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236 class="td11"> 53 </td><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-4-3"
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237 class="td11">Platin . . . . . </td><td align="right" style="white-space:nowrap; text-align:right;" id="TBL-1-4-4"
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238 class="td11"> 66 </td>
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239 </tr><tr
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240 valign="baseline" id="TBL-1-5-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-5-1"
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241 class="td11">Silber . . . . . </td><td align="center" style="white-space:nowrap; text-align:center;" id="TBL-1-5-2"
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242 class="td11"> 73 </td><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-5-3"
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243 class="td11">Kadmium . . . . </td><td align="right" style="white-space:nowrap; text-align:right;" id="TBL-1-5-4"
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244 class="td11"> 115 </td>
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245 </tr><tr
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246 valign="baseline" id="TBL-1-6-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-6-1"
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247 class="td11">Gold . . . . . . </td><td align="center" style="white-space:nowrap; text-align:center;" id="TBL-1-6-2"
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248 class="td11"> 79 </td><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-6-3"
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249 class="td11">Zinn . . . . . . </td><td align="right" style="white-space:nowrap; text-align:right;" id="TBL-1-6-4"
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250 class="td11"> 102 </td>
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251 </tr><tr
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252 valign="baseline" id="TBL-1-7-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-7-1"
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253 class="td11">Nickel . . . . . </td><td align="center" style="white-space:nowrap; text-align:center;" id="TBL-1-7-2"
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254 class="td11"> 45 </td><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-7-3"
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255 class="td11">Blei . . . . . . </td><td align="right" style="white-space:nowrap; text-align:right;" id="TBL-1-7-4"
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256 class="td11"> 135 </td>
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257 </tr><tr
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258 valign="baseline" id="TBL-1-8-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-8-1"
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259 class="td11">Eisen . . . . . </td><td align="center" style="white-space:nowrap; text-align:center;" id="TBL-1-8-2"
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260 class="td11"> 46 </td><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-8-3"
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261 class="td11">Wismut . . . . . </td><td align="right" style="white-space:nowrap; text-align:right;" id="TBL-1-8-4"
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262 class="td11"> 168 </td>
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263 </tr><tr
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264 valign="baseline" id="TBL-1-9-"><td align="left" style="white-space:nowrap; text-align:left;" id="TBL-1-9-1"
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265 class="td11"> </td>
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266 </tr></table></div>
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267 <!--l. 311--><p class="indent"> Nach der aus der Planckschen Strahlungstheorie ab-<br/>geleiteten Theorie der spezifischen Wärme soll letztere gegen <br/>----------
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268 </p><!--l. 316--><p class="indent"> 1) E. Grüneisen, Ann. d. Phys. 25. p. 848. 1908.
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269 </p><!--l. 318--><p class="indent"> 2) Die Temperaturabhängigkeit der kubischen Kompressibilität ist <br/>hierbei vernachlässigt. <pb/>
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270 </p><!--l. 325--><p class="indent">
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271
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272 </p><!--l. 326--><p class="noindent">den Nullwert der absoluten Temperatur abfallen nach folgen-<br/>dem Gesetz:
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273 </p>
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274 <center class="par-math-display" >
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275 <img
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276 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191117x.png" alt=" - -a (a )2 e T -- C = 3 R (---a--T--)2 - T- e - 1 " class="par-math-display" /></center>
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277 <!--l. 335--><p class="nopar">
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278 </p><!--l. 339--><p class="noindent">wobei <span
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279 class="cmmi-10">C </span>die auf das Grammolekel bezogene spezifische Wärme <br/>bedeutet, und
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280 </p>
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281 <center class="par-math-display" >
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282 <img
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283 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/Einst_Bezie_de_191118x.png" alt="h-n = a = h.e- k k.c " class="par-math-display" /></center>
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284 <!--l. 346--><p class="nopar">
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285 </p><!--l. 350--><p class="noindent">gesetzt ist. Hierbei sind <span
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286 class="cmmi-10">h </span>und <span
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287 class="cmmi-10"><img
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288 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /> </span>die Konstanten der Planck-<br/>schen Strahlungsformel. Man kann daher aus dem Verlauf <br/>der spezifischen Wärme <span
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289 class="cmmi-10"><img
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290 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-15.png" alt="c" class="10x-x-15" /> </span>ein zweites Mal bestimmen. Der <br/>einzige, der oben angeführten Stoffe, dessen spezifische Wärme <br/>bei tiefen Temperaturen hinreichend genau bestimmt ist, ist <br/>das Silber. Für dieses fand Nernst<sup ><span
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291 class="cmr-7">1</span></sup>) <span
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292 class="cmmi-10">a </span>= 162, woraus sich <br/><span
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293 class="cmmi-10"><img
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294 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-15.png" alt="c" class="10x-x-15" />.</span>10<sup ><span
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295 class="cmr-7">4</span></sup> = 90 ergibt, während wir aus den elastischen Kon-<br/>stanten <span
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296 class="cmmi-10"><img
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297 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bezie_de_1911/fulltext/img/cmmi10-15.png" alt="c" class="10x-x-15" />.</span>10<sup ><span
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298 class="cmr-7">4</span></sup> = 73 berechnet haben. Diese nahe Überein-<br/>stimmung ist wahrhaft überraschend. Eine noch exaktere <br/>Prüfung der Sutherlandschen Auffassung wird sich wohl <br/>nur dadurch erzielen lassen, daß man die molekulare Theorie <br/>der festen Körper vervollkommnet.
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299 </p><!--l. 366--><p class="noindent">----------
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300 </p><!--l. 369--><p class="indent"> 1) Vgl. W. Nernst, Bulletin des Seances de la Société franç. de <br/>Phys. 1910. 1 fase.
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301 </p>
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302 <div class="center" >
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303
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304 <!--l. 372--><p class="noindent">
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305 </p><!--l. 373--><p class="noindent">(Eingegangen 30. November 1910.)</p></div>
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306 <div class="center" >
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307
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308 <!--l. 376--><p class="noindent">
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309 </p><!--l. 377--><p class="noindent">----------</p></div>
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310
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311 </body></html>
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