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1 #+TITLE: Statistical Mechanics
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2 #+AUTHOR: E.T. Jaynes; edited by Dylan Holmes
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3 #+EMAIL: rlm@mit.edu
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4 #+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes
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5 #+SETUPFILE: ../../aurellem/org/setup.org
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6 #+INCLUDE: ../../aurellem/org/level-0.org
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7 #+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"
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8
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9 # "extensions/eqn-number.js"
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10
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11 #+begin_quote
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12 *Note:* The following is a typeset version of
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13 [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]]. I have only made
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14 minor changes, e.g. to correct typographical errors, add references, or format equations. The
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15 content itself is intact. --- Dylan
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16 #+end_quote
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17
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18 * Development of Thermodynamics
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19 Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature
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20 arise from the sensations of warmth and cold associated with our
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21 sense of touch . Yet science has been able to convert this qualitative
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22 sensation into an accurately defined quantitative notion,
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23 which can be applied far beyond the range of our direct experience.
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24 Today an experimentalist will report confidently that his
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25 spin system was at a temperature of 2.51 degrees Kelvin; and a
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26 theoretician will report with almost as much confidence that the
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27 temperature at the center of the sun is about \(2 \times 10^7\) degrees
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28 Kelvin.
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29
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30 The /fact/ that this has proved possible, and the main technical
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31 ideas involved, are assumed already known to the reader;
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32 and we are not concerned here with repeating standard material
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33 already available in a dozen other textbooks . However
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34 thermodynamics, in spite of its great successes, firmly established
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35 for over a century, has also produced a great deal of confusion
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36 and a long list of \ldquo{}paradoxes\rdquo{} centering mostly
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37 around the second law and the nature of irreversibility.
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38 For this reason and others noted below, we want to dwell here at
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39 some length on the /logic/ underlying the development of
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40 thermodynamics . Our aim is to emphasize certain points which,
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41 in the writer's opinion, are essential for clearing up the
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42 confusion and resolving the paradoxes; but which are not
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43 sufficiently ernphasized---and indeed in many cases are
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44 totally ignored---in other textbooks.
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45
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46 This attention to logic
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47 would not be particularly needed if we regarded classical
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48 thermodynamics (or, as it is becoming called increasingly,
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49 /thermostatics/) as a closed subject, in which the fundamentals
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50 are already completely established, and there is
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51 nothing more to be learned about them. A person who believes
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52 this will probably prefer a pure axiomatic approach, in which
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53 the basic laws are simply stated as arbitrary axioms, without
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54 any attempt to present the evidence for them; and one proceeds
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55 directly to working out their consequences.
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56 However, we take the attitude here that thermostatics, for
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57 all its venerable age, is very far from being a closed subject,
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58 we still have a great deal to learn about such matters as the
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59 most general definitions of equilibrium and reversibility, the
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60 exact range of validity of various statements of the second and
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61 third laws, the necessary and sufficient conditions for
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62 applicability of thermodynamics to special cases such as
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63 spin systems, and how thermodynamics can be applied to such
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64 systems as putty or polyethylene, which deform under force,
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65 but retain a \ldquo{}memory\rdquo{} of their past deformations.
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66 Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by
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67 no means rule out the possibility that still more laws of
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68 thermodynamics exist, as yet undiscovered, which would be
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69 useful in such applications.
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70
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71
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72 It is only by careful examination of the logic by which
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73 present thermodynamics was created, asking exactly how much of
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74 it is mathematical theorems, how much is deducible from the laws
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75 of mechanics and electrodynamics, and how much rests only on
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76 empirical evidence, how compelling is present evidence for the
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77 accuracy and range of validity of its laws; in other words,
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78 exactly where are the boundaries of present knowledge, that we
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79 can hope to uncover new things. Clearly, much research is still
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80 needed in this field, and we shall be able to accomplish only a
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81 small part of this program in the present review.
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82
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83
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84 It will develop that there is an astonishingly close analogy
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85 with the logic underlying statistical theory in general, where
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86 again a qualitative feeling that we all have (for the degrees of
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87 plausibility of various unproved and undisproved assertions) must
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88 be convertefi into a precisely defined quantitative concept
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89 (probability). Our later development of probability theory in
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90 Chapter 6,7 will be, to a considerable degree, a paraphrase
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91 of our present review of the logic underlying classical
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92 thermodynamics.
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93
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94 ** The Primitive Thermometer
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95
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96 The earliest stages of our
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97 story are necessarily speculative, since they took place long
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98 before the beginnings of recorded history. But we can hardly
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99 doubt that primitive man learned quickly that objects exposed
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100 to the sun‘s rays or placed near a fire felt different from
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101 those in the shade away from fires; and the same difference was
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102 noted between animal bodies and inanimate objects.
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103
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104
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105 As soon as it was noted that changes in this feeling of
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106 warmth were correlated with other observable changes in the
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107 behavior of objects, such as the boiling and freezing of water,
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108 cooking of meat, melting of fat and wax, etc., the notion of
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109 warmth took its first step away from the purely subjective
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110 toward an objective, physical notion capable of being studied
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111 scientifically.
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112
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113 One of the most striking manifestations of warmth (but far
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114 from the earliest discovered) is the almost universal expansion
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115 of gases, liquids, and solids when heated . This property has
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116 proved to be a convenient one with which to reduce the notion
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117 of warmth to something entirely objective. The invention of the
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118 /thermometer/, in which expansion of a mercury column, or a gas,
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119 or the bending of a bimetallic strip, etc. is read off on a
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120 suitable scale, thereby giving us a /number/ with which to work,
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121 was a necessary prelude to even the crudest study of the physical
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122 nature of heat. To the best of our knowledge, although the
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123 necessary technology to do this had been available for at least
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124 3,000 years, the first person to carry it out in practice was
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125 Galileo, in 1592.
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126
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127 Later on we will give more precise definitions of the term
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128 \ldquo{}thermometer.\rdquo{} But at the present stage we
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129 are not in a position to do so (as Galileo was not), because
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130 the very concepts needed have not yet been developed;
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131 more precise definitions can be
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132 given only after our study has revealed the need for them. In
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133 deed, our final definition can be given only after the full
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134 mathematical formalism of statistical mechanics is at hand.
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135
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136 Once a thermometer has been constructed, and the scale
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137 marked off in a quite arbitrary way (although we will suppose
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138 that the scale is at least monotonic: i.e., greater warmth always
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139 corresponds to a greater number), we are ready to begin scien
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140 tific experiments in thermodynamics. The number read eff from
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141 any such instrument is called the /empirical temperature/, and we
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142 denote it by \(t\). Since the exact calibration of the thermometer
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143 is not specified), any monotonic increasing function
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144 \(t‘ = f(t)\) provides an equally good temperature scale for the
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145 present.
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146
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147
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|
148 ** Thermodynamic Systems
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149
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150 The \ldquo{}thermodynamic systems\rdquo{} which
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151 are the objects of our study may be, physically, almost any
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152 collections of objects. The traditional simplest system with
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153 which to begin a study of thermodynamics is a volume of gas.
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154 We shall, however, be concerned from the start also with such
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155 things as a stretched wire or membrane, an electric cell, a
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156 polarized dielectric, a paramagnetic body in a magnetic field, etc.
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157
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158 The /thermodynamic state/ of such a system is determined by
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159 specifying (i.e., measuring) certain macrcoscopic physical
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160 properties. Now, any real physical system has many millions of such
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161 preperties; in order to have a usable theory we cannot require
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162 that /all/ of them be specified. We see, therefore, that there
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163 must be a clear distinction between the notions of
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164 \ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical
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165 system.\rdquo{}
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166 A given /physical/ system may correspond to many different
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167 /thermodynamic systems/, depending
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168 on which variables we choose to measure or control; and which
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169 we decide to leave unmeasured and/or uncontrolled.
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170
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171
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172 For example, our physical system might consist of a crystal
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173 of sodium chloride. For one set of experiments we work with
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174 temperature, volume, and pressure; and ignore its electrical
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175 properties. For another set of experiments we work with
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176 temperature, electric field, and electric polarization; and
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177 ignore the varying stress and strain. The /physical/ system,
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178 therefore, corresponds to two entirely different /thermodynamic/
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179 systems. Exactly how much freedom, then, do we have in choosing
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180 the variables which shall define the thermodynamic state of our
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181 system? How many must we choose? What [criteria] determine when
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182 we have made an adequate choice? These questions cannot be
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183 answered until we say a little more about what we are trying to
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184 accomplish by a thermodynamic theory. A mere collection of
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185 recorded data about our system, as in the [[http://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics][/Handbook of Physics and
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186 Chemistry/]], is a very useful thing, but it hardly constitutes
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187 a theory. In order to construct anything deserving of such a
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188 name, the primary requirement is that we can recognize some kind
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189 of reproducible connection between the different properties con
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190 sidered, so that information about some of them will enable us
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191 to predict others. And of course, in order that our theory can
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192 be called thermodynamics (and not some other area of physics),
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193 it is necessary that the temperature be one of the quantities
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194 involved in a nontrivial way.
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195
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196 The gist of these remarks is that the notion of
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197 \ldquo{}thermodynamic system\rdquo{} is in part
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198 an anthropomorphic one; it is for us to
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199 say which set of variables shall be used. If two different
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200 choices both lead to useful reproducible connections, it is quite
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201 meaningless to say that one choice is any more \ldquo{}correct\rdquo{}
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202 than the other. Recognition of this fact will prove crucial later in
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203 avoiding certain ancient paradoxes.
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204
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205 At this stage we can determine only empirically which other
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206 physical properties need to be introduced before reproducible
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207 connections appear. Once any such connection is established, we
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208 can analyze it with the hope of being able to (1) reduce it to a
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209 /logical/ connection rather than an empirical one; and (2) extend
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210 it to an hypothesis applying beyond the original data, which
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211 enables us to predict further connections capable of being
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212 tested by experiment. Examples of this will be given presently.
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213
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214
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215 There will remain, however, a few reproducible relations
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216 which to the best of present knowledge, are not reducible to
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217 logical relations within the context of classical thermodynamics
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218 (and. whose demonstration in the wider context of mechanics,
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219 electrodynamics, and quantum theory remains one of probability
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220 rather than logical proof); from the standpoint of thermodynamics
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221 these remain simply statements of empirical fact which must be
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222 accepted as such without any deeper basis, but without which the
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223 development of thermodynamics cannot proceed. Because of this
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224 special status, these relations have become known as the
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225 \ldquo{}laws\rdquo{}
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226 of thermodynamics . The most fundamental one is a qualitative
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227 rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
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228
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229 ** Equilibrium; the Zeroth Law
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230
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231 It is a common experience
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232 that when objects are placed in contact with each other but
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233 isolated from their surroundings, they may undergo observable
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234 changes for a time as a result; one body may become warmer,
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235 another cooler, the pressure of a gas or volume of a liquid may
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236 change; stress or magnetization in a solid may change, etc. But
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237 after a sufficient time, the observable macroscopic properties
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238 settle down to a steady condition, after which no further changes
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239 are seen unless there is a new intervention from the outside.
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240 When this steady condition is reached, the experimentalist says
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241 that the objects have reached a state of /equilibrium/ with each
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242 other. Once again, more precise definitions of this term will
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243 be needed eventually, but they require concepts not yet developed.
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244 In any event, the criterion just stated is almost the only one
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245 used in actual laboratory practice to decide when equilibrium
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246 has been reached.
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247
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248
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249 A particular case of equilibrium is encountered when we
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250 place a thermometer in contact with another body. The reading
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251 \(t\) of the thermometer may vary at first, but eventually it reach es
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ocsenave@0
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252 a steady value. Now the number \(t\) read by a thermometer is always.
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253 by definition, the empirical temperature /of the thermometer/ (more
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254 precisely, of the sensitive element of the thermometer). When
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255 this number is constant in time, we say that the thermometer is
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256 in /thermal equilibrium/ with its surroundings; and we then extend
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257 the notion of temperature, calling the steady value \(t\) also the
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258 /temperature of the surroundings/.
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259
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260 We have repeated these elementary facts, well known to every
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261 child, in order to emphasize this point: Thermodynamics can be
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262 a theory /only/ of states of equilibrium, because the very
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263 procedure by which the temperature of a system is defined by
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264 operational means, already presupposes the attainment of
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265 equilibrium. Strictly speaking, therefore, classical
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266 thermodynamics does not even contain the concept of a
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267 \ldquo{}time-varying temperature.\rdquo{}
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268
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269 Of course, to recognize this limitation on conventional
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270 thermodynamics (best emphasized by calling it instead,
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271 thermostatics) in no way rules out the possibility of
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272 generalizing the notion of temperature to nonequilibrium states.
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273 Indeed, it is clear that one could define any number of
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274 time-dependent quantities all of which reduce, in the special
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275 case of equilibrium, to the temperature as defined above.
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276 Historically, attempts to do this even antedated the discovery
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ocsenave@0
|
277 of the laws of thermodynamics, as is demonstrated by
|
|
ocsenave@0
|
278 \ldquo{}Newton's law of cooling.\rdquo{} Therefore, the
|
|
ocsenave@0
|
279 question is not whether generalization is /possible/, but only
|
|
ocsenave@0
|
280 whether it is in any way /useful/; i.e., does the temperature so
|
|
ocsenave@0
|
281 generalized have any connection with other physical properties
|
|
ocsenave@0
|
282 of our system, so that it could help us to predict other things?
|
|
ocsenave@0
|
283 However, to raise such questions takes us far beyond the
|
|
ocsenave@0
|
284 domain of thermostatics; and the general laws of nonequilibrium
|
|
ocsenave@0
|
285 behavior are so much more complicated that it would be virtually
|
|
ocsenave@0
|
286 hopeless to try to unravel them by empirical means alone. For
|
|
ocsenave@0
|
287 example, even if two different kinds of thermometer are calibrated
|
|
ocsenave@0
|
288 so that they agree with each other in equilibrium situations,
|
|
ocsenave@0
|
289 they will not agree in general about the momentary value a
|
|
ocsenave@0
|
290 \ldquo{}time-varying temperature.\rdquo{} To make any real
|
|
ocsenave@0
|
291 progress in this area, we have to supplement empirical observation by the guidance
|
|
ocsenave@0
|
292 of a rather hiqhly-developed theory. The notion of a
|
|
ocsenave@0
|
293 time-dependent temperature is far from simple conceptually, and we
|
|
ocsenave@0
|
294 will find that nothing very helpful can be said about this until
|
|
ocsenave@0
|
295 the full mathematical apparatus of nonequilibrium statistical
|
|
ocsenave@0
|
296 mechanics has been developed.
|
|
ocsenave@0
|
297
|
|
ocsenave@0
|
298 Suppose now that two bodies have the same temperature; i.e.,
|
|
ocsenave@0
|
299 a given thermometer reads the same steady value when in contact
|
|
ocsenave@0
|
300 with either. In order that the statement, \ldquo{}two bodies have the
|
|
ocsenave@1
|
301 same temperature\rdquo{} shall describe a physical property of the bodies,
|
|
ocsenave@0
|
302 and not merely an accidental circumstance due to our having used
|
|
ocsenave@0
|
303 a particular kind of thermometer, it is necessary that /all/
|
|
ocsenave@0
|
304 thermometers agree in assigning equal temperatures to them if
|
|
ocsenave@0
|
305 /any/ thermometer does . Only experiment is competent to determine
|
|
ocsenave@0
|
306 whether this universality property is true. Unfortunately, the
|
|
ocsenave@0
|
307 writer must confess that he is unable to cite any definite
|
|
ocsenave@0
|
308 experiment in which this point was subjected to a careful test.
|
|
ocsenave@0
|
309 That equality of temperatures has this absolute meaning, has
|
|
ocsenave@0
|
310 evidently been taken for granted so much that (like absolute
|
|
ocsenave@0
|
311 sirnultaneity in pre-relativity physics) most of us are not even
|
|
ocsenave@0
|
312 consciously aware that we make such an assumption in
|
|
ocsenave@0
|
313 thermodynamics. However, for the present we can only take it as a familiar
|
|
ocsenave@0
|
314 empirical fact that this condition does hold, not because we can
|
|
ocsenave@0
|
315 cite positive evidence for it, but because of the absence of
|
|
ocsenave@0
|
316 negative evidence against it; i.e., we think that, if an
|
|
ocsenave@0
|
317 exception had ever been found, this would have created a sensation in
|
|
ocsenave@0
|
318 physics, and we should have heard of it.
|
|
ocsenave@0
|
319
|
|
ocsenave@0
|
320 We now ask: when two bodies are at the same temperature,
|
|
ocsenave@0
|
321 are they then in thermal equilibrium with each other? Again,
|
|
ocsenave@0
|
322 only experiment is competent to answer this; the general
|
|
ocsenave@0
|
323 conclusion, again supported more by absence of negative evidence
|
|
ocsenave@0
|
324 than by specific positive evidence, is that the relation of
|
|
ocsenave@0
|
325 equilibrium has this property:
|
|
ocsenave@0
|
326 #+begin_quote
|
|
ocsenave@0
|
327 /Two bodies in thermal equilibrium
|
|
ocsenave@0
|
328 with a third body, are thermal equilibrium with each other./
|
|
ocsenave@0
|
329 #+end_quote
|
|
ocsenave@0
|
330
|
|
ocsenave@0
|
331 This empirical fact is usually called the \ldquo{}zero'th law of
|
|
ocsenave@0
|
332 thermodynamics.\rdquo{} Since nothing prevents us from regarding a
|
|
ocsenave@0
|
333 thermometer as the \ldquo{}third body\rdquo{} in the above statement,
|
|
ocsenave@0
|
334 it appears that we may also state the zero'th law as:
|
|
ocsenave@0
|
335 #+begin_quote
|
|
ocsenave@0
|
336 /Two bodies are in thermal equilibrium with each other when they are
|
|
ocsenave@0
|
337 at the same temperature./
|
|
ocsenave@0
|
338 #+end_quote
|
|
ocsenave@0
|
339 Although from the preceding discussion it might appear that
|
|
ocsenave@0
|
340 these two statements of the zero'th law are entirely equivalent
|
|
ocsenave@0
|
341 (and we certainly have no empirical evidence against either), it
|
|
ocsenave@0
|
342 is interesting to note that there are theoretical reasons, arising
|
|
ocsenave@0
|
343 from General Relativity, indicating that while the first
|
|
ocsenave@0
|
344 statement may be universally valid, the second is not. When we
|
|
ocsenave@0
|
345 consider equilibrium in a gravitational field, the verification
|
|
ocsenave@0
|
346 that two bodies have equal temperatures may require transport
|
|
ocsenave@0
|
347 of the thermometer through a gravitational potential difference;
|
|
ocsenave@0
|
348 and this introduces a new element into the discussion. We will
|
|
ocsenave@0
|
349 consider this in more detail in a later Chapter, and show that
|
|
ocsenave@0
|
350 according to General Relativity, equilibrium in a large system
|
|
ocsenave@0
|
351 requires, not that the temperature be uniform at all points, but
|
|
ocsenave@0
|
352 rather that a particular function of temperature and gravitational
|
|
ocsenave@0
|
353 potential be constant (the function is \(T\cdot \exp{(\Phi/c^2})\), where
|
|
ocsenave@0
|
354 \(T\) is the Kelvin temperature to be defined later, and \(\Phi\) is the
|
|
ocsenave@0
|
355 gravitational potential).
|
|
ocsenave@0
|
356
|
|
ocsenave@0
|
357 Of course, this effect is so small that ordinary terrestrial
|
|
ocsenave@0
|
358 experiments would need to have a precision many orders of
|
|
ocsenave@0
|
359 magnitude beyond that presently possible, before one could hope even
|
|
ocsenave@0
|
360 to detect it; and needless to say, it has played no role in the
|
|
ocsenave@0
|
361 development of thermodynamics. For present purposes, therefore,
|
|
ocsenave@0
|
362 we need not distinguish between the two above statements of the
|
|
ocsenave@0
|
363 zero'th law, and we take it as a basic empirical fact that a
|
|
ocsenave@0
|
364 uniform temperature at all points of a system is an essential
|
|
ocsenave@0
|
365 condition for equilibrium. It is an important part of our
|
|
ocsenave@0
|
366 ivestigation to determine whether there are other essential
|
|
ocsenave@0
|
367 conditions as well. In fact, as we will find, there are many
|
|
ocsenave@0
|
368 different kinds of equilibrium; and failure to distinguish between
|
|
ocsenave@0
|
369 them can be a prolific source of paradoxes.
|
|
ocsenave@0
|
370
|
|
ocsenave@0
|
371 ** Equation of State
|
|
ocsenave@0
|
372 Another important reproducible connection is found when
|
|
ocsenave@0
|
373 we consider a thermodynamic system defined by
|
|
ocsenave@0
|
374 three parameters; in addition to the temperature we choose a
|
|
ocsenave@0
|
375 \ldquo{}displacement\rdquo{} and a conjugate \ldquo{}force.\rdquo{}
|
|
ocsenave@0
|
376 Subject to some qualifications given below, we find experimentally
|
|
ocsenave@0
|
377 that these parameters are not independent, but are subject to a constraint.
|
|
ocsenave@0
|
378 For example, we cannot vary the equilibrium pressure, volume,
|
|
ocsenave@0
|
379 and temperature of a given mass of gas independently; it is found
|
|
ocsenave@0
|
380 that a given pressure and volume can be realized only at one
|
|
ocsenave@0
|
381 particular temperature, that the gas will assume a given tempera~
|
|
ocsenave@0
|
382 ture and volume only at one particular pressure, etc. Similarly,
|
|
ocsenave@0
|
383 a stretched wire can be made to have arbitrarily assigned tension
|
|
ocsenave@0
|
384 and elongation only if its temperature is suitably chosen, a
|
|
ocsenave@0
|
385 dielectric will assume a state of given temperature and
|
|
ocsenave@0
|
386 polarization at only one value of the electric field, etc.
|
|
ocsenave@0
|
387 These simplest nontrivial thermodynamic systems (three
|
|
ocsenave@0
|
388 parameters with one constraint) are said to possess two
|
|
ocsenave@0
|
389 /degrees of freedom/; for the range of possible equilibrium states is defined
|
|
ocsenave@0
|
390 by specifying any two of the variables arbitrarily, whereupon the
|
|
ocsenave@0
|
391 third, and all others we may introduce, are determined.
|
|
ocsenave@0
|
392 Mathematically, this is expressed by the existence of a functional
|
|
ocsenave@4
|
393 relationship of the form[fn:: Edit: The set of solutions to an equation
|
|
ocsenave@0
|
394 like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is
|
|
ocsenave@0
|
395 saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional
|
|
ocsenave@0
|
396 rule\rdquo{}, so the set of physically allowed combinations of /X/,
|
|
ocsenave@0
|
397 /x/, and /t/ in equilibrium states can be
|
|
ocsenave@0
|
398 expressed as the level set of a function.
|
|
ocsenave@0
|
399
|
|
ocsenave@0
|
400 But not every function expresses a constraint relation; for some
|
|
ocsenave@0
|
401 functions, you can specify two of the variables, and the third will
|
|
ocsenave@0
|
402 still be undetermined. (For example, if f=X^2+x^2+t^2-3,
|
|
ocsenave@0
|
403 the level set /f(X,x,t)=0/ is a sphere, and specifying /x=1/, /t=1/
|
|
ocsenave@0
|
404 leaves you with two potential possibilities for /X/ =\pm 1.)
|
|
ocsenave@0
|
405
|
|
ocsenave@1
|
406 A function like /f/ has to possess one more propery in order for its
|
|
ocsenave@1
|
407 level set to express a constraint relationship: it must be monotonic in
|
|
ocsenave@0
|
408 each of its variables /X/, /x/, and /t/.
|
|
ocsenave@0
|
409 #the partial derivatives of /f/ exist for every allowed combination of
|
|
ocsenave@0
|
410 #inputs /x/, /X/, and /t/.
|
|
ocsenave@0
|
411 In other words, the level set has to pass a sort of
|
|
ocsenave@0
|
412 \ldquo{}vertical line test\rdquo{} for each of its variables.]
|
|
ocsenave@0
|
413
|
|
ocsenave@0
|
414 #Edit Here, Jaynes
|
|
ocsenave@0
|
415 #is saying that it is possible to express the collection of allowed combinations \(\langle X,x,t \rangle\) of force, quantity, and temperature as a
|
|
ocsenave@0
|
416 #[[http://en.wikipedia.org/wiki/Level_set][level set]] of some function \(f\). However, not all level sets represent constraint relations; consider \(f(X,x,t)=X^2+x^2+t^2-1\)=0.
|
|
ocsenave@0
|
417 #In order to specify
|
|
ocsenave@0
|
418
|
|
ocsenave@0
|
419 \begin{equation}
|
|
ocsenave@0
|
420 f(X,x,t) = O
|
|
ocsenave@0
|
421 \end{equation}
|
|
ocsenave@0
|
422
|
|
ocsenave@0
|
423 where $X$ is a generalized force (pressure, tension, electric or
|
|
ocsenave@0
|
424 magnetic field, etc.), $x$ is the corresponding generalized
|
|
ocsenave@0
|
425 displacement (volume, elongation, electric or magnetic polarization,
|
|
ocsenave@1
|
426 etc.), and $t$ is the empirical temperature. Equation (1-1) is
|
|
ocsenave@0
|
427 called /the equation of state/.
|
|
ocsenave@0
|
428
|
|
ocsenave@0
|
429 At the risk of belaboring it, we emphasize once again that
|
|
ocsenave@0
|
430 all of this applies only for a system in equilibrium; for
|
|
ocsenave@0
|
431 otherwise not only.the temperature, but also some or all of the other
|
|
ocsenave@0
|
432 variables may not be definable. For example, no unique pressure
|
|
ocsenave@0
|
433 can be assigned to a gas which has just suffered a sudden change
|
|
ocsenave@0
|
434 in volume, until the generated sound waves have died out.
|
|
ocsenave@0
|
435
|
|
ocsenave@0
|
436 Independently of its functional form, the mere fact of the
|
|
ocsenave@0
|
437 /existence/ of an equation of state has certain experimental
|
|
ocsenave@0
|
438 consequences. For example, suppose that in experiments on oxygen
|
|
ocsenave@0
|
439 gas, in which we control the temperature and pressure
|
|
ocsenave@0
|
440 independently, we have found that the isothermal compressibility $K$
|
|
ocsenave@0
|
441 varies with temperature, and the thermal expansion coefficient
|
|
ocsenave@0
|
442 \alpha varies with pressure $P$, so that within the accuracy of the data,
|
|
ocsenave@0
|
443
|
|
ocsenave@0
|
444 \begin{equation}
|
|
ocsenave@0
|
445 \frac{\partial K}{\partial t} = - \frac{\partial \alpha}{\partial P}
|
|
ocsenave@0
|
446 \end{equation}
|
|
ocsenave@0
|
447
|
|
ocsenave@0
|
448 Is this a particular property of oxygen; or is there reason to
|
|
ocsenave@0
|
449 believe that it holds also for other substances? Does it depend
|
|
ocsenave@0
|
450 on our particular choice of a temperature scale?
|
|
ocsenave@0
|
451
|
|
ocsenave@0
|
452 In this case, the answer is found at once; for the definitions of $K$,
|
|
ocsenave@0
|
453 \alpha are
|
|
ocsenave@0
|
454
|
|
ocsenave@0
|
455 \begin{equation}
|
|
ocsenave@0
|
456 K = -\frac{1}{V}\frac{\partial V}{\partial P},\qquad
|
|
ocsenave@0
|
457 \alpha=\frac{1}{V}\frac{\partial V}{\partial t}
|
|
ocsenave@0
|
458 \end{equation}
|
|
ocsenave@0
|
459
|
|
ocsenave@0
|
460 which is simply a mathematical expression of the fact that the
|
|
ocsenave@0
|
461 volume $V$ is a definite function of $P$ and $t$; i.e., it depends
|
|
ocsenave@0
|
462 only
|
|
ocsenave@0
|
463 on their present values, and not how those values were attained.
|
|
ocsenave@0
|
464 In particular, $V$ does not depend on the direction in the \((P, t)\)
|
|
ocsenave@0
|
465 plane through which the present values were approached; or, as we
|
|
ocsenave@0
|
466 usually say it, \(dV\) is an /exact differential/.
|
|
ocsenave@0
|
467
|
|
ocsenave@1
|
468 Therefore, although at first glance the relation (1-2) appears
|
|
ocsenave@0
|
469 nontrivial and far from obvious, a trivial mathematical analysis
|
|
ocsenave@0
|
470 convinces us that it must hold regardless of our particular
|
|
ocsenave@0
|
471 temperature scale, and that it is true not only of oxygen; it must
|
|
ocsenave@0
|
472 hold for any substance, or mixture of substances, which possesses a
|
|
ocsenave@0
|
473 definite, reproducible equation of state \(f(P,V,t)=0\).
|
|
ocsenave@0
|
474
|
|
ocsenave@0
|
475 But this understanding also enables us to predict situations in which
|
|
ocsenave@1
|
476 (1-2) will /not/ hold. Equation (1-2), as we have just learned, expresses
|
|
ocsenave@0
|
477 the fact that an equation of state exists involving only the three
|
|
ocsenave@0
|
478 variables \((P,V,t)\). Now suppose we try to apply it to a liquid such
|
|
ocsenave@0
|
479 as nitrobenzene. The nitrobenzene molecule has a large electric dipole
|
|
ocsenave@0
|
480 moment; and so application of an electric field (as in the
|
|
ocsenave@0
|
481 [[http://en.wikipedia.org/wiki/Kerr_effect][electro-optical Kerr cell]]) causes an alignment of molecules which, as
|
|
ocsenave@0
|
482 accurate measurements will verify, changes the pressure at a given
|
|
ocsenave@0
|
483 temperature and volume. Therefore, there can no longer exist any
|
|
ocsenave@0
|
484 unique equation of state involving \((P, V, t)\) only; with
|
|
ocsenave@0
|
485 sufficiently accurate measurements, nitrobenzene must be regarded as a
|
|
ocsenave@0
|
486 thermodynamic system with at least three degrees of freedom, and the
|
|
ocsenave@0
|
487 general equation of state must have at least a complicated a form as
|
|
ocsenave@0
|
488 \(f(P,V,t,E) = 0\).
|
|
ocsenave@0
|
489
|
|
ocsenave@0
|
490 But if we introduce a varying electric field $E$ into the discussion,
|
|
ocsenave@0
|
491 the resulting varying electric polarization $M$ also becomes a new
|
|
ocsenave@0
|
492 thermodynamic variable capable of being measured. Experimentally, it
|
|
ocsenave@0
|
493 is easiest to control temperature, pressure, and electric field
|
|
ocsenave@0
|
494 independently, and of course we find that both the volume and
|
|
ocsenave@0
|
495 polarization are then determined; i.e., there must exist functional
|
|
ocsenave@0
|
496 relations of the form \(V = V(P,t,E)\), \(M = M(P,t,E)\), or in more
|
|
ocsenave@0
|
497 symmetrical form
|
|
ocsenave@0
|
498
|
|
ocsenave@0
|
499 \begin{equation}
|
|
ocsenave@0
|
500 f(V,P,t,E) = 0 \qquad g(M,P,t,E)=0.
|
|
ocsenave@0
|
501 \end{equation}
|
|
ocsenave@0
|
502
|
|
ocsenave@0
|
503 In other words, if we regard nitrobenzene as a thermodynamic system of
|
|
ocsenave@0
|
504 three degrees of freedom (i.e., having specified three parameters
|
|
ocsenave@0
|
505 arbitrarily, all others are then determined), it must possess two
|
|
ocsenave@0
|
506 independent equations of state.
|
|
ocsenave@0
|
507
|
|
ocsenave@0
|
508 Similarly, a thermodynamic system with four degrees of freedom,
|
|
ocsenave@0
|
509 defined by the termperature and three pairs of conjugate forces and
|
|
ocsenave@0
|
510 displacements, will have three independent equations of state, etc.
|
|
ocsenave@0
|
511
|
|
ocsenave@0
|
512 Now, returning to our original question, if nitrobenzene possesses
|
|
ocsenave@0
|
513 this extra electrical degree of freedom, under what circumstances do
|
|
ocsenave@0
|
514 we exprect to find a reproducible equation of state involving
|
|
ocsenave@0
|
515 \((p,V,t)\) only? Evidently, if $E$ is held constant, then the first
|
|
ocsenave@0
|
516 of equations (1-5) becomes such an equation of state, involving $E$ as
|
|
ocsenave@0
|
517 a fixed parameter; we would find many different equations of state of
|
|
ocsenave@0
|
518 the form \(f(P,V,t) = 0\) with a different function $f$ for each
|
|
ocsenave@0
|
519 different value of the electric field. Likewise, if \(M\) is held
|
|
ocsenave@0
|
520 constant, we can eliminate \(E\) between equations (1-5) and find a
|
|
ocsenave@0
|
521 relation \(h(P,V,t,M)=0\), which is an equation of state for
|
|
ocsenave@0
|
522 \((P,V,t)\) containing \(M\) as a fixed parameter.
|
|
ocsenave@0
|
523
|
|
ocsenave@0
|
524 More generally, if an electrical constraint is imposed on the system
|
|
ocsenave@0
|
525 (for example, by connecting an external charged capacitor to the
|
|
ocsenave@0
|
526 electrodes) so that \(M\) is determined by \(E\); i.e., there is a
|
|
ocsenave@0
|
527 functional relation of the form
|
|
ocsenave@0
|
528
|
|
ocsenave@0
|
529 \begin{equation}
|
|
ocsenave@0
|
530 g(M,E) = \text{const.}
|
|
ocsenave@0
|
531 \end{equation}
|
|
ocsenave@0
|
532
|
|
ocsenave@0
|
533 then (1-5) and (1-6) constitute three simultaneous equations, from
|
|
ocsenave@0
|
534 which both \(E\) and \(M\) may be eliminated mathematically, leading
|
|
ocsenave@0
|
535 to a relation of the form \(h(P,V,t;q)=0\), which is an equation of
|
|
ocsenave@0
|
536 state for \((P,V,t)\) involving the fixed parameter \(q\).
|
|
ocsenave@0
|
537
|
|
ocsenave@0
|
538 We see, then, that as long as a fixed constraint of the form (1-6) is
|
|
ocsenave@0
|
539 imposed on the electrical degree of freedom, we can still observe a
|
|
ocsenave@0
|
540 reproducible equation of state for nitrobenzene, considered as a
|
|
ocsenave@0
|
541 thermodynamic system of only two degrees of freedom. If, however, this
|
|
ocsenave@0
|
542 electrical constraint is removed, so that as we vary $P$ and $t$, the
|
|
ocsenave@0
|
543 values of $E$ and $M$ vary in an uncontrolled way over a
|
|
ocsenave@0
|
544 /two-dimensional/ region of the \((E, M)\) plane, then we will find no
|
|
ocsenave@0
|
545 definite equation of state involving only \((P,V,t)\).
|
|
ocsenave@0
|
546
|
|
ocsenave@0
|
547 This may be stated more colloqually as follows: even though a system
|
|
ocsenave@0
|
548 has three degrees of freedom, we can still consider only the variables
|
|
ocsenave@0
|
549 belonging to two of them, and we will find a definite equation of
|
|
ocsenave@0
|
550 state, /provided/ that in the course of the experiments, the unused
|
|
ocsenave@0
|
551 degree of freedom is not \ldquo{}tampered with\rdquo{} in an
|
|
ocsenave@0
|
552 uncontrolled way.
|
|
ocsenave@0
|
553
|
|
ocsenave@0
|
554 We have already emphasized that any physical system corresponds to
|
|
ocsenave@0
|
555 many different thermodynamic systems, depending on which variables we
|
|
ocsenave@0
|
556 choose to control and measure. In fact, it is easy to see that any
|
|
ocsenave@0
|
557 physical system has, for all practical purposes, an /arbitrarily
|
|
ocsenave@0
|
558 large/ number of degrees of freedom. In the case of nitrobenzene, for
|
|
ocsenave@0
|
559 example, we may impose any variety of nonuniform electric fields on
|
|
ocsenave@1
|
560 our sample. Suppose we place $(n+1)$ different electrodes, labelled
|
|
ocsenave@1
|
561 \(\{e_0,e_1, e_2 \ldots e_n\}\) in contact with the liquid in various
|
|
ocsenave@1
|
562 positions. Regarding \(e_0\) as the \ldquo{}ground\rdquo{}, maintained
|
|
ocsenave@1
|
563 at zero potential, we can then impose $n$ different potentials
|
|
ocsenave@1
|
564 \(\{v_1, \ldots, v_n\}\) on the other electrodes independently, and we
|
|
ocsenave@1
|
565 can also measure the $n$ different conjugate displacements, as the
|
|
ocsenave@1
|
566 charges \(\{q_1,\ldots, q_n\}\) accumulated on electrodes
|
|
ocsenave@1
|
567 \(\{e_1,\ldots e_n\}\). Together with the pressure (understood as the
|
|
ocsenave@1
|
568 pressure measured at one given position), volume, and temperature, our
|
|
ocsenave@1
|
569 sample of nitrobenzene is now a thermodynamic system of $(n+1)$
|
|
ocsenave@1
|
570 degrees of freedom. This number may be as large as we please, limited
|
|
ocsenave@1
|
571 only by our patience in constructing the apparatus needed to control
|
|
ocsenave@1
|
572 or measure all these quantities.
|
|
ocsenave@1
|
573
|
|
ocsenave@1
|
574 We leave it as an exercise for the reader (Problem 1) to find the most
|
|
ocsenave@1
|
575 general condition on the variables \(\{v_1, q_1, v_2, q_2, \ldots
|
|
ocsenave@1
|
576 v_n,q_n\}\) which will ensure that a definite equation of state
|
|
ocsenave@1
|
577 $f(P,V,t)=0$ is observed in spite of all these new degrees of
|
|
ocsenave@1
|
578 freedom. The simplest special case of this relation is, evidently, to
|
|
ocsenave@1
|
579 ground all electrodes, thereby inposing the conditions $v_1 = v_2 =
|
|
ocsenave@1
|
580 \ldots = v_n = 0$. Equally well (if we regard nitrobenzene as having
|
|
ocsenave@1
|
581 negligible electrical conductivity) we may open-circuit all
|
|
ocsenave@1
|
582 electrodes, thereby imposing the conditions \(q_i = \text{const.}\) In
|
|
ocsenave@1
|
583 the latter case, in addition to an equation of state of the form
|
|
ocsenave@1
|
584 \(f(P,V,t)=0\), which contains these constants as fixed parameters,
|
|
ocsenave@1
|
585 there are \(n\) additional equations of state of the form $v_i =
|
|
ocsenave@1
|
586 v_i(P,t)$. But if we choose to ignore these voltages, there will be no
|
|
ocsenave@1
|
587 contradiction in considering our nitrobenzene to be a thermodynamic
|
|
ocsenave@1
|
588 system of two degrees of freedom, involving only the variables
|
|
ocsenave@1
|
589 \(P,V,t\).
|
|
ocsenave@1
|
590
|
|
ocsenave@1
|
591 Similarly, if our system of interest is a crystal, we may impose on it
|
|
ocsenave@1
|
592 a wide variety of nonuniform stress fields; each component of the
|
|
ocsenave@1
|
593 stress tensor $T_{ij}$ may bary with position. We might expand each of
|
|
ocsenave@1
|
594 these functions in a complete orthonormal set of functions
|
|
ocsenave@1
|
595 \(\phi_k(x,y,z)\):
|
|
ocsenave@1
|
596
|
|
ocsenave@1
|
597 \begin{equation}
|
|
ocsenave@1
|
598 T_{ij}(x,y,z) = \sum_k a_{ijk} \phi_k(x,y,z)
|
|
ocsenave@1
|
599 \end{equation}
|
|
ocsenave@1
|
600
|
|
ocsenave@1
|
601 and with a sufficiently complicated system of levers which in various
|
|
ocsenave@1
|
602 ways squeeze and twist the crystal, we might vary each of the first
|
|
ocsenave@1
|
603 1,000 expansion coefficients $a_{ijk}$ independently, and measure the
|
|
ocsenave@1
|
604 conjugate displacements $q_{ijk}$. Our crystal is then a thermodynamic
|
|
ocsenave@1
|
605 system of over 1,000 degrees of freedom.
|
|
ocsenave@1
|
606
|
|
ocsenave@1
|
607 The notion of \ldquo{}numbers of degrees of freedom\rdquo{} is
|
|
ocsenave@1
|
608 therefore not a /physical property/ of any system; it is entirely
|
|
ocsenave@1
|
609 anthropomorphic, since any physical system may be regarded as a
|
|
ocsenave@1
|
610 thermodynamic system with any number of degrees of freedom we please.
|
|
ocsenave@1
|
611
|
|
ocsenave@1
|
612 If new thermodynamic variables are always introduced in pairs,
|
|
ocsenave@1
|
613 consisting of a \ldquo{}force\rdquo{} and conjugate
|
|
ocsenave@1
|
614 \ldquo{}displacement\rdquo{}, then a thermodynamic system of $n$
|
|
ocsenave@1
|
615 degrees of freedom must possess $(n-1)$ independent equations of
|
|
ocsenave@1
|
616 state, so that specifying $n$ quantities suffices to determine all
|
|
ocsenave@1
|
617 others.
|
|
ocsenave@1
|
618
|
|
ocsenave@1
|
619 This raises an interesting question; whether the scheme of classifying
|
|
ocsenave@1
|
620 thermodynamic variables in conjugate pairs is the most general
|
|
ocsenave@1
|
621 one. Why, for example, is it not natural to introduce three related
|
|
ocsenave@1
|
622 variables at a time? To the best of the writer's knowledge, this is an
|
|
ocsenave@1
|
623 open question; there seems to be no fundamental reason why variables
|
|
ocsenave@1
|
624 /must/ always be introduced in conjugate pairs, but there seems to be
|
|
ocsenave@1
|
625 no known case in which a different scheme suggests itself as more
|
|
ocsenave@1
|
626 appropriate.
|
|
ocsenave@1
|
627
|
|
ocsenave@1
|
628 ** Heat
|
|
ocsenave@1
|
629 We are now in a position to consider the results and interpretation of
|
|
ocsenave@1
|
630 a number of elementary experiments involving
|
|
ocsenave@2
|
631 thermal interaction, which can be carried out as soon as a primitive
|
|
ocsenave@2
|
632 thermometer is at hand. In fact these experiments, which we summarize
|
|
ocsenave@2
|
633 so quickly, required a very long time for their first performance, and
|
|
ocsenave@2
|
634 the essential conclusions of this Section were first arrived at only
|
|
ocsenave@2
|
635 about 1760---more than 160 years after Galileo's invention of the
|
|
ocsenave@4
|
636 thermometer---[[http://web.lemoyne.edu/~giunta/blackheat.html][by Joseph Black]], who was Professor of Chemistry at
|
|
ocsenave@2
|
637 Glasgow University. Black's analysis of calorimetric experiments
|
|
ocsenave@2
|
638 initiated by G. D. Fahrenheit before 1736 led to the first recognition
|
|
ocsenave@2
|
639 of the distinction between temperature and heat, and prepared the way
|
|
ocsenave@2
|
640 for the work of his better-known pupil, James Watt.
|
|
ocsenave@1
|
641
|
|
ocsenave@2
|
642 We first observe that if two bodies at different temperatures are
|
|
ocsenave@2
|
643 separated by walls of various materials, they sometimes maintain their
|
|
ocsenave@2
|
644 temperature difference for a long time, and sometimes reach thermal
|
|
ocsenave@2
|
645 equilibrium very quickly. The differences in behavior observed must be
|
|
ocsenave@2
|
646 ascribed to the different properties of the separating walls, since
|
|
ocsenave@2
|
647 nothing else is changed. Materials such as wood, asbestos, porous
|
|
ocsenave@2
|
648 ceramics (and most of all, modern porous plastics like styrofoam), are
|
|
ocsenave@2
|
649 able to sustain a temperature difference for a long time; a wall of an
|
|
ocsenave@2
|
650 imaginary material with this property idealized to the point where a
|
|
ocsenave@2
|
651 temperature difference is maintained indefinitely is called an
|
|
ocsenave@2
|
652 /adiabatic wall/. A very close approximation to a perfect adiabatic
|
|
ocsenave@2
|
653 wall is realized by the Dewar flask (thermos bottle), of which the
|
|
ocsenave@2
|
654 walls consist of two layers of glass separated by a vacuum, with the
|
|
ocsenave@2
|
655 surfaces silvered like a mirror. In such a container, as we all know,
|
|
ocsenave@2
|
656 liquids may be maintained hot or cold for days.
|
|
ocsenave@1
|
657
|
|
ocsenave@2
|
658 On the other hand, a thin wall of copper or silver is hardly able to
|
|
ocsenave@2
|
659 sustain any temperature difference at all; two bodies separated by
|
|
ocsenave@2
|
660 such a partition come to thermal equilibrium very quickly. Such a wall
|
|
ocsenave@2
|
661 is called /diathermic/. It is found in general that the best
|
|
ocsenave@2
|
662 diathermic materials are the metals and good electrical conductors,
|
|
ocsenave@2
|
663 while electrical insulators make fairly good adiabatic walls. There
|
|
ocsenave@2
|
664 are good theoretical reasons for this rule; a particular case of it is
|
|
ocsenave@2
|
665 given by the [[http://en.wikipedia.org/wiki/Wiedemann_franz_law][Wiedemann-Franz law]] of solid-state theory.
|
|
ocsenave@2
|
666
|
|
ocsenave@2
|
667 Since a body surrounded by an adiabatic wall is able to maintain its
|
|
ocsenave@2
|
668 temperature independently of the temperature of its surroundings, an
|
|
ocsenave@2
|
669 adiabatic wall provides a means of thermally /isolating/ a system from
|
|
ocsenave@2
|
670 the rest of the universe; it is to be expected, therefore, that the
|
|
ocsenave@2
|
671 laws of thermal interaction between two systems will assume the
|
|
ocsenave@2
|
672 simplest form if they are enclosed in a common adiabatic container,
|
|
ocsenave@2
|
673 and that the best way of carrying out experiments on thermal
|
|
ocsenave@2
|
674 peroperties of substances is to so enclose them. Such an apparatus, in
|
|
ocsenave@2
|
675 which systems are made to interact inside an adiabatic container
|
|
ocsenave@2
|
676 supplied with a thermometer, is called a /calorimeter/.
|
|
ocsenave@2
|
677
|
|
ocsenave@2
|
678 Let us imagine that we have a calorimeter in which there is initially
|
|
ocsenave@2
|
679 a volume $V_W$ of water at a temperature $t_1$, and suspended above it
|
|
ocsenave@2
|
680 a volume $V_I$ of some other substance (say, iron) at temperature
|
|
ocsenave@2
|
681 $t_2$. When we drop the iron into the water, they interact thermally
|
|
ocsenave@2
|
682 (and the exact nature of this interaction is one of the things we hope
|
|
ocsenave@2
|
683 to learn now), the temperature of both changing until they are in
|
|
ocsenave@2
|
684 thermal equilibrium at a final temperature $t_0$.
|
|
ocsenave@2
|
685
|
|
ocsenave@2
|
686 Now we repeat the experiment with different initial temperatures
|
|
ocsenave@2
|
687 $t_1^\prime$ and $t_2^\prime$, so that a new equilibrium is reached at
|
|
ocsenave@2
|
688 temperature $t_0^\prime$. It is found that, if the temperature
|
|
ocsenave@2
|
689 differences are sufficiently small (and in practice this is not a
|
|
ocsenave@2
|
690 serious limitation if we use a mercury thermometer calibrated with
|
|
ocsenave@2
|
691 uniformly spaced degree marks on a capillary of uniform bore), then
|
|
ocsenave@2
|
692 whatever the values of $t_1^\prime$, $t_2^\prime$, $t_1$, $t_2$, the
|
|
ocsenave@2
|
693 final temperatures $t_0^\prime$, $t_0$ will adjust themselves so that
|
|
ocsenave@2
|
694 the following relation holds:
|
|
ocsenave@2
|
695
|
|
ocsenave@2
|
696 \begin{equation}
|
|
ocsenave@2
|
697 \frac{t_2 - t_0}{t_0 - t_1} = \frac{t_2^\prime -
|
|
ocsenave@2
|
698 t_0^\prime}{t_0^\prime - t_1^\prime}
|
|
ocsenave@2
|
699 \end{equation}
|
|
ocsenave@2
|
700
|
|
ocsenave@2
|
701 in other words, the /ratio/ of the temperature changes of the iron and
|
|
ocsenave@2
|
702 water is independent of the initial temperatures used.
|
|
ocsenave@2
|
703
|
|
ocsenave@2
|
704 We now vary the amounts of iron and water used in the calorimeter. It
|
|
ocsenave@2
|
705 is found that the ratio (1-8), although always independent of the
|
|
ocsenave@2
|
706 starting temperatures, does depend on the relative amounts of iron and
|
|
ocsenave@2
|
707 water. It is, in fact, proportional to the mass $M_W$ of water and
|
|
ocsenave@2
|
708 inversely proportional to the mass $M_I$ of iron, so that
|
|
ocsenave@2
|
709
|
|
ocsenave@2
|
710 \begin{equation}
|
|
ocsenave@2
|
711 \frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_I M_I}
|
|
ocsenave@2
|
712 \end{equation}
|
|
ocsenave@2
|
713
|
|
ocsenave@2
|
714 where $K_I$ is a constant.
|
|
ocsenave@2
|
715
|
|
ocsenave@2
|
716 We next repeat the above experiments using a different material in
|
|
ocsenave@2
|
717 place of the iron (say, copper). We find again a relation
|
|
ocsenave@2
|
718
|
|
ocsenave@2
|
719 \begin{equation}
|
|
ocsenave@2
|
720 \frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_C \cdot M_C}
|
|
ocsenave@2
|
721 \end{equation}
|
|
ocsenave@2
|
722
|
|
ocsenave@2
|
723 where $M_C$ is the mass of copper; but the constant $K_C$ is different
|
|
ocsenave@2
|
724 from the previous $K_I$. In fact, we see that the constant $K_I$ is a
|
|
ocsenave@2
|
725 new physical property of the substance iron, while $K_C$ is a physical
|
|
ocsenave@2
|
726 property of copper. The number $K$ is called the /specific heat/ of a
|
|
ocsenave@2
|
727 substance, and it is seen that according to this definition, the
|
|
ocsenave@2
|
728 specific heat of water is unity.
|
|
ocsenave@2
|
729
|
|
ocsenave@2
|
730 We now have enough experimental facts to begin speculating about their
|
|
ocsenave@2
|
731 interpretation, as was first done in the 18th century. First, note
|
|
ocsenave@2
|
732 that equation (1-9) can be put into a neater form that is symmetrical
|
|
ocsenave@2
|
733 between the two substances. We write $\Delta t_I = t_0 - t_2$, $\Delta
|
|
ocsenave@2
|
734 t_W = t_0 - t_1$ for the temperature changes of iron and water
|
|
ocsenave@2
|
735 respectively, and define $K_W \equiv 1$ for water. Equation (1-9) then
|
|
ocsenave@2
|
736 becomes
|
|
ocsenave@2
|
737
|
|
ocsenave@2
|
738 \begin{equation}
|
|
ocsenave@2
|
739 K_W M_W \Delta t_W + K_I M_I \Delta t_I = 0
|
|
ocsenave@2
|
740 \end{equation}
|
|
ocsenave@2
|
741
|
|
ocsenave@2
|
742 The form of this equation suggests a new experiment; we go back into
|
|
ocsenave@2
|
743 the laboratory, and find $n$ substances for which the specific heats
|
|
ocsenave@2
|
744 \(\{K_1,\ldots K_n\}\) have been measured previously. Taking masses
|
|
ocsenave@2
|
745 \(\{M_1, \ldots, M_n\}\) of these substances, we heat them to $n$
|
|
ocsenave@2
|
746 different temperatures \(\{t_1,\ldots, t_n\}\) and throw them all into
|
|
ocsenave@2
|
747 the calorimeter at once. After they have all come to thermal
|
|
ocsenave@2
|
748 equilibrium at temperature $t_0$, we find the differences $\Delta t_j
|
|
ocsenave@2
|
749 = t_0 - t_j$. Just as we suspected, it turns out that regardless of
|
|
ocsenave@2
|
750 the $K$'s, $M$'s, and $t$'s chosen, the relation
|
|
ocsenave@2
|
751 \begin{equation}
|
|
ocsenave@2
|
752 \sum_{j=0}^n K_j M_j \Delta t_j = 0
|
|
ocsenave@2
|
753 \end{equation}
|
|
ocsenave@2
|
754 is always satisfied. This sort of process is an old story in
|
|
ocsenave@2
|
755 scientific investigations; although the great theoretician Boltzmann
|
|
ocsenave@3
|
756 is said to have remarked: \ldquo{}Elegance is for tailors\rdquo{}, it
|
|
ocsenave@2
|
757 remains true that the attempt to reduce equations to the most
|
|
ocsenave@2
|
758 symmetrical form has often suggested important generalizations of
|
|
ocsenave@2
|
759 physical laws, and is a great aid to memory. Witness Maxwell's
|
|
ocsenave@2
|
760 \ldquo{}displacement current\rdquo{}, which was needed to fill in a
|
|
ocsenave@2
|
761 gap and restore the symmetry of the electromagnetic equations; as soon
|
|
ocsenave@2
|
762 as it was put in, the equations predicted the existence of
|
|
ocsenave@2
|
763 electromagnetic waves. In the present case, the search for a rather
|
|
ocsenave@2
|
764 rudimentary form of \ldquo{}elegance\rdquo{} has also been fruitful,
|
|
ocsenave@2
|
765 for we recognize that (1-12) has the standard form of a /conservation
|
|
ocsenave@2
|
766 law/; it defines a new quantity which is conserved in thermal
|
|
ocsenave@2
|
767 interactions of the type just studied.
|
|
ocsenave@2
|
768
|
|
ocsenave@2
|
769 The similarity of (1-12) to conservation laws in general may be seen
|
|
ocsenave@3
|
770 as follows. Let $A$ be some quantity that is conserved; the \(i\)th
|
|
ocsenave@2
|
771 system has an amount of it $A_i$. Now when the systems interact such
|
|
ocsenave@2
|
772 that some $A$ is transferred between them, the amount of $A$ in the
|
|
ocsenave@3
|
773 \(i\)th system is changed by a net amount \(\Delta A_i = (A_i)_{final} -
|
|
ocsenave@2
|
774 (A_i)_{initial}\); and the fact that there is no net change in the
|
|
ocsenave@2
|
775 total amount of $A$ is expressed by the equation \(\sum_i \Delta
|
|
ocsenave@3
|
776 A_i = 0\). Thus, the law of conservation of matter in a chemical
|
|
ocsenave@2
|
777 reaction is expressed by \(\sum_i \Delta M_i = 0\), where $M_i$ is the
|
|
ocsenave@3
|
778 mass of the \(i\)th chemical component.
|
|
ocsenave@2
|
779
|
|
ocsenave@3
|
780 What is this new conserved quantity? Mathematically, it can be defined
|
|
ocsenave@3
|
781 as $Q_i = K_i\cdot M_i \cdot t_i$; whereupon (1-12) becomes
|
|
ocsenave@2
|
782
|
|
ocsenave@2
|
783 \begin{equation}
|
|
ocsenave@2
|
784 \sum_i \Delta Q_i = 0
|
|
ocsenave@2
|
785 \end{equation}
|
|
ocsenave@2
|
786
|
|
ocsenave@2
|
787 and at this point we can correct a slight quantitative inaccuracy. As
|
|
ocsenave@2
|
788 noted, the above relations hold accurately only when the temperature
|
|
ocsenave@2
|
789 differences are sufficiently small; i.e., they are really only
|
|
ocsenave@2
|
790 differential laws. On sufficiently accurate measurements one find that
|
|
ocsenave@2
|
791 the specific heats $K_i$ depend on temperature; if we then adopt the
|
|
ocsenave@2
|
792 integral definition of $\Delta Q_i$,
|
|
ocsenave@2
|
793 \begin{equation}
|
|
ocsenave@2
|
794 \Delta Q_i = \int_{t_{i}}^{t_0} K_i(t) M_i dt
|
|
ocsenave@2
|
795 \end{equation}
|
|
ocsenave@2
|
796
|
|
ocsenave@2
|
797 the conservation law (1-13) will be found to hold in calorimetric
|
|
ocsenave@2
|
798 experiments with liquids and solids, to any accuracy now feasible. And
|
|
ocsenave@2
|
799 of course, from the manner in which the $K_i(t)$ are defined, this
|
|
ocsenave@2
|
800 relation will hold however our thermometers are calibrated.
|
|
ocsenave@2
|
801
|
|
ocsenave@2
|
802 Evidently, the stage is now set for a \ldquo{}new\rdquo{} physical
|
|
ocsenave@2
|
803 theory to account for these facts. In the 17th century, both Francis
|
|
ocsenave@2
|
804 Bacon and Isaac Newton had expressed their opinions that heat was a
|
|
ocsenave@2
|
805 form of motion; but they had no supporting factual evidence. By the
|
|
ocsenave@2
|
806 latter part of the 18th century, one had definite factual evidence
|
|
ocsenave@2
|
807 which seemed to make this view untenable; by the calorimetric
|
|
ocsenave@2
|
808 \ldquo{}mixing\rdquo{} experiments just described, Joseph Black had
|
|
ocsenave@2
|
809 recognized the distinction between temperature $t$ as a measure of
|
|
ocsenave@2
|
810 \ldquo{}hotness\rdquo{}, and heat $Q$ as a measure of /quantity/ of
|
|
ocsenave@2
|
811 something, and introduced the notion of heat capacity. He also
|
|
ocsenave@2
|
812 recognized the latent heats of freezing and vaporization. To account
|
|
ocsenave@2
|
813 for the conservation laws thus discovered, the theory then suggested
|
|
ocsenave@2
|
814 itself, naturally and almost inevitably, that heat was /fluid/,
|
|
ocsenave@2
|
815 indestructable and uncreatable, which had no appreciable weight and
|
|
ocsenave@2
|
816 was attracted differently by different kinds of matter. In 1787,
|
|
ocsenave@2
|
817 Lavoisier invented the name \ldquo{}caloric\rdquo{} for this fluid.
|
|
ocsenave@2
|
818
|
|
ocsenave@2
|
819 Looking down today from our position of superior knowledge (i.e.,
|
|
ocsenave@2
|
820 hindsight) we perhaps need to be reminded that the caloric theory was
|
|
ocsenave@2
|
821 a perfectly respectable scientific theory, fully deserving of serious
|
|
ocsenave@2
|
822 consideration; for it accounted quantitatively for a large body of
|
|
ocsenave@2
|
823 experimental fact, and made new predictions capable of being tested by
|
|
ocsenave@2
|
824 experiment.
|
|
ocsenave@2
|
825
|
|
ocsenave@2
|
826 One of these predictions was the possibility of accounting for the
|
|
ocsenave@2
|
827 thermal expansion of bodies when heated; perhaps the increase in
|
|
ocsenave@2
|
828 volume was just a measure of the volume of caloric fluid
|
|
ocsenave@2
|
829 absorbed. This view met with some disappointment as a result of
|
|
ocsenave@2
|
830 experiments which showed that different materials, on absorbing the
|
|
ocsenave@2
|
831 same quantity of heat, expanded by different amounts. Of course, this
|
|
ocsenave@2
|
832 in itself was not enough to overthrow the caloric theory, because one
|
|
ocsenave@2
|
833 could suppose that the caloric fluid was compressible, and was held
|
|
ocsenave@2
|
834 under different pressure in different media.
|
|
ocsenave@2
|
835
|
|
ocsenave@2
|
836 Another difficulty that seemed increasingly serious by the end of the
|
|
ocsenave@2
|
837 18th century was the failure of all attempts to weigh this fluid. Many
|
|
ocsenave@2
|
838 careful experiments were carried out, by Boyle, Fordyce, Rumford and
|
|
ocsenave@2
|
839 others (and continued by Landolt almost into the 20th century), with
|
|
ocsenave@2
|
840 balances capable of detecting a change of weight of one part in a
|
|
ocsenave@2
|
841 million; and no change could be detected on the melting of ice,
|
|
ocsenave@2
|
842 heating of substances, or carrying out of chemical reactions. But even
|
|
ocsenave@2
|
843 this is not really a conclusive argument against the caloric theory,
|
|
ocsenave@2
|
844 since there is no /a priori/ reason why the fluid should be dense
|
|
ocsenave@2
|
845 enough to weigh with balances (of course, we know today from
|
|
ocsenave@2
|
846 Einstein's $E=mc^2$ that small changes in weight should indeed exist
|
|
ocsenave@2
|
847 in these experiments; but to measure them would require balances about
|
|
ocsenave@2
|
848 10^7 times more sensitive than were available).
|
|
ocsenave@2
|
849
|
|
ocsenave@2
|
850 Since the caloric theory derives entirely from the empirical
|
|
ocsenave@2
|
851 conservation law (1-33), it can be refuted conclusively only by
|
|
ocsenave@2
|
852 exhibiting new experimental facts revealing situations in which (1-13)
|
|
ocsenave@2
|
853 is /not/ valid. The first such case was [[http://www.chemteam.info/Chem-History/Rumford-1798.html][found by Count Rumford (1798)]],
|
|
ocsenave@2
|
854 who was in charge of boring cannon in the Munich arsenal, and noted
|
|
ocsenave@2
|
855 that the cannon and chips became hot as a result of the cutting. He
|
|
ocsenave@2
|
856 found that heat could be produced indefinitely, as long as the boring
|
|
ocsenave@2
|
857 was continued, without any compensating cooling of any other part of
|
|
ocsenave@2
|
858 the system. Here, then, was a clear case in which caloric was /not/
|
|
ocsenave@2
|
859 conserved, as in (1-13); but could be created at will. Rumford wrote
|
|
ocsenave@2
|
860 that he could not conceive of anything that could be produced
|
|
ocsenave@2
|
861 indefinitely by the expenditure of work, \ldquo{}except it be /motion/\rdquo{}.
|
|
ocsenave@2
|
862
|
|
ocsenave@2
|
863 But even this was not enough to cause abandonment of the caloric
|
|
ocsenave@2
|
864 theory; for while Rumford's observations accomplished the negative
|
|
ocsenave@2
|
865 purpose of showing that the conservation law (1-13) is not universally
|
|
ocsenave@2
|
866 valid, they failed to accomplish the positive one of showing what
|
|
ocsenave@2
|
867 specific law should replace it (although he produced a good hint, not
|
|
ocsenave@2
|
868 sufficiently appreciated at the time, in his crude measurements of the
|
|
ocsenave@2
|
869 rate of heat production due to the work of one horse). Within the
|
|
ocsenave@2
|
870 range of the original calorimetric experiments, (1-13) was still
|
|
ocsenave@2
|
871 valid, and a theory successful in a restricted domain is better than
|
|
ocsenave@2
|
872 no theory at all; so Rumford's work had very little impact on the
|
|
ocsenave@2
|
873 actual development of thermodynamics.
|
|
ocsenave@2
|
874
|
|
ocsenave@2
|
875 (This situation is a recurrent one in science, and today physics offers
|
|
ocsenave@2
|
876 another good example. It is recognized by all that our present quantum
|
|
ocsenave@2
|
877 field theory is unsatisfactory on logical, conceptual, and
|
|
ocsenave@2
|
878 mathematical grounds; yet it also contains some important truth, and
|
|
ocsenave@2
|
879 no responsible person has suggested that it be abandoned. Once again,
|
|
ocsenave@2
|
880 a semi-satisfactory theory is better than none at all, and we will
|
|
ocsenave@2
|
881 continue to teach it and to use it until we have something better to
|
|
ocsenave@2
|
882 put in its place.)
|
|
ocsenave@2
|
883
|
|
ocsenave@2
|
884 # what is "the specific heat of a gas at constant pressure/volume"?
|
|
ocsenave@2
|
885 # changed t for temperature below from capital T to lowercase t.
|
|
ocsenave@3
|
886 Another failure of the conservation law (1-13) was [[http://web.lemoyne.edu/~giunta/mayer.html][noted in 1842]] by
|
|
ocsenave@2
|
887 R. Mayer, a German physician, who pointed out that the data already
|
|
ocsenave@2
|
888 available showed that the specific heat of a gas at constant pressure,
|
|
ocsenave@2
|
889 C_p, was greater than at constant volume $C_v$. He surmised that the
|
|
ocsenave@2
|
890 difference was due to the work done in expansion of the gas against
|
|
ocsenave@2
|
891 atmospheric pressure, when measuring $C_p$. Supposing that the
|
|
ocsenave@2
|
892 difference $\Delta Q = (C_p - C_v)\Delta t$ calories, in the heat
|
|
ocsenave@2
|
893 required to raise the temperature by $\Delta t$ was actually a
|
|
ocsenave@2
|
894 measure of amount of energy, he could estimate from the amount
|
|
ocsenave@2
|
895 $P\Delta V$ ergs of work done the amount of mechanical energy (number
|
|
ocsenave@2
|
896 of ergs) corresponding to a calorie of heat; but again his work had
|
|
ocsenave@2
|
897 very little impact on the development of thermodynamics, because he
|
|
ocsenave@2
|
898 merely offered this notion as an interpretation of the data without
|
|
ocsenave@2
|
899 performing or suggesting any new experiments to check his hypothesis
|
|
ocsenave@2
|
900 further.
|
|
ocsenave@2
|
901
|
|
ocsenave@2
|
902 Up to the point, then, one has the experimental fact that a
|
|
ocsenave@2
|
903 conservation law (1-13) exists whenever purely thermal interactions
|
|
ocsenave@2
|
904 were involved; but in processes involving mechanical work, the
|
|
ocsenave@2
|
905 conservation law broke down.
|
|
ocsenave@2
|
906
|
|
ocsenave@2
|
907 ** The First Law
|
|
ocsenave@3
|
908 Corresponding to the partially valid law of \ldquo{}conservation of
|
|
ocsenave@3
|
909 heat\rdquo{}, there had long been known another partially valid
|
|
ocsenave@3
|
910 conservation law in mechanics. The principle of conservation of
|
|
ocsenave@3
|
911 mechanical energy had been given by Leibnitz in 1693 in noting that,
|
|
ocsenave@3
|
912 according to the laws of Newtonian mechanics, one could define
|
|
ocsenave@3
|
913 potential and kinetic energy so that in mechanical processes they were
|
|
ocsenave@3
|
914 interconverted into each other, the total energy remaining
|
|
ocsenave@3
|
915 constant. But this too was not universally valid---the mechanical
|
|
ocsenave@3
|
916 energy was conserved only in the absence of frictional forces. In
|
|
ocsenave@3
|
917 processes involving friction, the mechanical energy seemed to
|
|
ocsenave@3
|
918 disappear.
|
|
ocsenave@3
|
919
|
|
ocsenave@3
|
920 So we had a law of conservation of heat, which broke down whenever
|
|
ocsenave@3
|
921 mechanical work was done; and a law of conservation of mechanical
|
|
ocsenave@3
|
922 energy, which broke down when frictional forces were present. If, as
|
|
ocsenave@3
|
923 Mayer had suggested, heat was itself a form of energy, then one had
|
|
ocsenave@3
|
924 the possibility of accounting for both of these failures in a new law
|
|
ocsenave@3
|
925 of conservation of /total/ (mechanical + heat) energy. On one hand,
|
|
ocsenave@3
|
926 the difference $C_p-C_v$ of heat capacities of gases would be
|
|
ocsenave@3
|
927 accounted for by the mechanical work done in expansion; on the other
|
|
ocsenave@3
|
928 hand, the disappearance of mechanical energy would be accounted for by
|
|
ocsenave@3
|
929 the heat produced by friction.
|
|
ocsenave@3
|
930
|
|
ocsenave@3
|
931 But to establish this requires more than just suggesting the idea and
|
|
ocsenave@3
|
932 illustrating its application in one or two cases --- if this is really
|
|
ocsenave@3
|
933 a new conservation law adequate to replace the two old ones, it must
|
|
ocsenave@3
|
934 be shown to be valid for /all/ substances and /all/ kinds of
|
|
ocsenave@3
|
935 interaction. For example, if one calorie of heat corresponded to $E$
|
|
ocsenave@3
|
936 ergs of mechanical energy in the gas experiments, but to a different
|
|
ocsenave@3
|
937 amoun $E^\prime$ in heat produced by friction, then there would be no
|
|
ocsenave@3
|
938 universal conservation law. This \ldquo{}first law\rdquo{} of
|
|
ocsenave@3
|
939 thermodynamics must therefore take the form:
|
|
ocsenave@3
|
940 #+begin_quote
|
|
ocsenave@3
|
941 There exists a /universal/ mechanical equivalent of heat, so that the
|
|
ocsenave@3
|
942 total (mechanical energy) + (heat energy) remeains constant in all
|
|
ocsenave@3
|
943 physical processes.
|
|
ocsenave@3
|
944 #+end_quote
|
|
ocsenave@3
|
945
|
|
ocsenave@3
|
946 It was James Prescott Joule who provided the [[http://www.chemteam.info/Chem-History/Joule-Heat-1845.html][first experimental data]]
|
|
ocsenave@3
|
947 indicating this universality, and providing the first accurate
|
|
ocsenave@3
|
948 numerical value of this mechanical equivalent. The calorie had been
|
|
ocsenave@3
|
949 defined as the amount of heat required to raise the temperature of one
|
|
ocsenave@3
|
950 gram of water by one degree Centigrade (more precisely, to raise it
|
|
ocsenave@3
|
951 from 14.5 to 15.5$^\circ C$). Joule measured the heating of a number
|
|
ocsenave@3
|
952 of different liquids due to mechanical stirring and electrical
|
|
ocsenave@3
|
953 heating, and established that, within the experimental accuracy (about
|
|
ocsenave@3
|
954 one percent) a /calorie/ of heat always corresponded to the same
|
|
ocsenave@3
|
955 amount of energy. Modern measurements give this numerical value as: 1
|
|
ocsenave@3
|
956 calorie = 4.184 \times 10^7 ergs = 4.184 joules.
|
|
ocsenave@3
|
957 # capitalize Joules? I think the convention is to spell them out in lowercase.
|
|
ocsenave@3
|
958
|
|
ocsenave@3
|
959 The circumstances of this important work are worth noting. Joule was
|
|
ocsenave@3
|
960 in frail health as a child, and was educated by private tutors,
|
|
ocsenave@3
|
961 including the chemist, John Dalton, who had formulated the atomic
|
|
ocsenave@3
|
962 hypothesis in the early nineteenth century. In 1839, when Joule was
|
|
ocsenave@3
|
963 nineteen, his father (a wealthy brewer) built a private laboratory for
|
|
ocsenave@3
|
964 him in Manchester, England; and the good use he made of it is shown by
|
|
ocsenave@3
|
965 the fact that, within a few months of the opening of this laboratory
|
|
ocsenave@3
|
966 (1840), he had completed his first important piece of work, at the
|
|
ocsenave@3
|
967 age of twenty. This was his establishment of the law of \ldquo{}Joule
|
|
ocsenave@3
|
968 heating,\rdquo{} $P=I^2 R$, due to the electric current in a
|
|
ocsenave@3
|
969 resistor. He then used this effect to determine the universality and
|
|
ocsenave@3
|
970 numerical value of the mechanical equivalent of heat, reported
|
|
ocsenave@3
|
971 in 1843. His mechanical stirring experiments reported in 1849 yielded
|
|
ocsenave@3
|
972 the value 1 calorie = 4.154 \times 10^7 ergs, amount 0.7% too low;
|
|
ocsenave@3
|
973 this determination was not improved upon for several decades.
|
|
ocsenave@3
|
974
|
|
ocsenave@3
|
975 The first law of thermodynamics may then be stated mathematically as
|
|
ocsenave@3
|
976 follows:
|
|
ocsenave@3
|
977
|
|
ocsenave@3
|
978 #+begin_quote
|
|
ocsenave@3
|
979 There exists a state function (i.e., a definite function of the
|
|
ocsenave@3
|
980 thermodynamic state) $U$, representing the total energy of any system,
|
|
ocsenave@3
|
981 such that in any process in which we change from one equilibrium to
|
|
ocsenave@3
|
982 another, the net change in $U$ is given by the difference of the heat
|
|
ocsenave@3
|
983 $Q$ supplied to the system, and the mechanical work $W$ done by the
|
|
ocsenave@3
|
984 system.
|
|
ocsenave@3
|
985 #+end_quote
|
|
ocsenave@3
|
986 On an infinitesimal change of state, this becomes
|
|
ocsenave@3
|
987
|
|
ocsenave@3
|
988 \begin{equation}
|
|
ocsenave@3
|
989 dU = dQ - dW.
|
|
ocsenave@3
|
990 \end{equation}
|
|
ocsenave@3
|
991
|
|
ocsenave@3
|
992 For a system of two degrees of freedom, defined by pressure $P$,
|
|
ocsenave@3
|
993 volume $V$, and temperature $t$, we have $dW = PdV$. Then if we regard
|
|
ocsenave@3
|
994 $U$ as a function $U(V,t)$ of volume and temperature, the fact that
|
|
ocsenave@3
|
995 $U$ is a state function means that $dU$ must be an exact differential;
|
|
ocsenave@3
|
996 i.e., the integral
|
|
ocsenave@3
|
997
|
|
ocsenave@3
|
998 \begin{equation}
|
|
ocsenave@3
|
999 \int_1^2 dU = U(V_2,t_2) - U(V_1,t_1)
|
|
ocsenave@3
|
1000 \end{equation}
|
|
ocsenave@3
|
1001 between any two thermodynamic states must be independent of the
|
|
ocsenave@3
|
1002 path. Equivalently, the integral $\oint dU$ over any closed cyclic
|
|
ocsenave@3
|
1003 path (for example, integrate from state 1 to state 2 along path A,
|
|
ocsenave@3
|
1004 then back to state 1 by a different path B) must be zero. From (1-15),
|
|
ocsenave@3
|
1005 this gives for any cyclic integral,
|
|
ocsenave@3
|
1006
|
|
ocsenave@3
|
1007 \begin{equation}
|
|
ocsenave@3
|
1008 \oint dQ = \oint P dV
|
|
ocsenave@3
|
1009 \end{equation}
|
|
ocsenave@3
|
1010
|
|
ocsenave@3
|
1011 another form of the first law, which states that in any process in
|
|
ocsenave@3
|
1012 which the system ends in the same thermodynamic state as the initial
|
|
ocsenave@3
|
1013 one, the total heat absorbed by the system must be equal to the total
|
|
ocsenave@3
|
1014 work done.
|
|
ocsenave@3
|
1015
|
|
ocsenave@3
|
1016 Although the equations (1-15)-(1-17) are rather trivial
|
|
ocsenave@3
|
1017 mathematically, it is important to avoid later conclusions that we
|
|
ocsenave@3
|
1018 understand their exact meaning. In the first place, we have to
|
|
ocsenave@3
|
1019 understand that we are now measuring heat energy and mechanical energy
|
|
ocsenave@3
|
1020 in the same units; i.e. if we measured $Q$ in calories and $W$ in
|
|
ocsenave@3
|
1021 ergs, then (1-15) would of course not be correct. It does
|
|
ocsenave@3
|
1022 not matter whether we apply Joule's mechanical equivalent of heat
|
|
ocsenave@3
|
1023 to express $Q$ in ergs, or whether we apply it in the opposite way
|
|
ocsenave@3
|
1024 to express $U$ and $W$ in calories; each procedure will be useful in
|
|
ocsenave@3
|
1025 various problems. We can develop the general equations of
|
|
ocsenave@3
|
1026 thermodynamics
|
|
ocsenave@3
|
1027 without committing ourselves to any particular units,
|
|
ocsenave@3
|
1028 but of course all terms in a given equation must be expressed
|
|
ocsenave@3
|
1029 in the same units.
|
|
ocsenave@3
|
1030
|
|
ocsenave@3
|
1031 Secondly, we have already stressed that the theory being
|
|
ocsenave@3
|
1032 developed must, strictly speaking, be a theory only of
|
|
ocsenave@3
|
1033 equilibrium states, since otherwise we have no operational definition
|
|
ocsenave@4
|
1034 of temperature When we integrate over any \ldquo{}path\rdquo{} in the $(V-t)$
|
|
ocsenave@3
|
1035 plane, therefore, it must be understood that the path of
|
|
ocsenave@3
|
1036 integration is, strictly speaking, just a /locus of equilibrium
|
|
ocsenave@3
|
1037 states/; nonequilibrium states cannot be represented by points
|
|
ocsenave@3
|
1038 in the $(V-t)$ plane.
|
|
ocsenave@3
|
1039
|
|
ocsenave@3
|
1040 But then, what is the relation between path of equilibrium
|
|
ocsenave@3
|
1041 states appearing in our equations, and the sequence of conditions
|
|
ocsenave@3
|
1042 produced experimentally when we change the state of a system in
|
|
ocsenave@3
|
1043 the laboratory? With any change of state (heating, compression,
|
|
ocsenave@3
|
1044 etc.) proceeding at a finite rate we do not have equilibrium in
|
|
ocsenave@3
|
1045 termediate states; and so there is no corresponding \ldquo{}path\rdquo{} in
|
|
ocsenave@3
|
1046 the $(V-t)$ plane ; only the initial and final equilibrium states
|
|
ocsenave@3
|
1047 correspond to definite points. But if we carry out the change
|
|
ocsenave@3
|
1048 of state more and more slowly, the physical states produced are
|
|
ocsenave@3
|
1049 nearer and nearer to equilibrium state. Therefore, we interpret
|
|
ocsenave@3
|
1050 a path of integration in the $(V-t)$ plane, not as representing
|
|
ocsenave@3
|
1051 the intermediate states of any real experiment carried out at
|
|
ocsenave@3
|
1052 a finite rate, but as the /limit/ of this sequence of states, in
|
|
ocsenave@3
|
1053 the limit where the change of state takes place arbitrarily
|
|
ocsenave@3
|
1054 slowly.
|
|
ocsenave@3
|
1055
|
|
ocsenave@3
|
1056 An arbitrarily slow process, so that we remain arbitrarily
|
|
ocsenave@3
|
1057 near to equilibrium at all times, has another important property.
|
|
ocsenave@3
|
1058 If heat is flowing at an arbitrarily small rate, the temperature
|
|
ocsenave@3
|
1059 difference producing it must be arbitrarily small, and therefore
|
|
ocsenave@3
|
1060 an arbitrarily small temperature change would be able to reverse
|
|
ocsenave@3
|
1061 the direction of heat flow. If the Volume is changing very
|
|
ocsenave@3
|
1062 slowly, the pressure difference responsible for it must be very
|
|
ocsenave@3
|
1063 small; so a small change in pressure would be able to reverse
|
|
ocsenave@3
|
1064 the direction of motion. In other words, a process carried out
|
|
ocsenave@3
|
1065 arbitrarily slowly is /reversible/; if a system is arbitrarily
|
|
ocsenave@3
|
1066 close to equilibrium, then an arbitrarily small change in its
|
|
ocsenave@3
|
1067 environment can reverse the direction of the process.
|
|
ocsenave@3
|
1068 Recognizing this, we can then say that the paths of integra
|
|
ocsenave@3
|
1069 tion in our equations are to be interpreted physically as
|
|
ocsenave@4
|
1070 /reversible paths/ In practice, some systems (such as gases)
|
|
ocsenave@3
|
1071 come to equilibrium so rapidly that rather fast changes of
|
|
ocsenave@3
|
1072 state (on the time scale of our own perceptions) may be quite
|
|
ocsenave@3
|
1073 good approximations to reversible changes; thus the change of
|
|
ocsenave@3
|
1074 state of water vapor in a steam engine may be considered
|
|
ocsenave@3
|
1075 reversible to a useful engineering approximation.
|
|
ocsenave@3
|
1076
|
|
ocsenave@3
|
1077
|
|
ocsenave@3
|
1078 ** Intensive and Extensive Parameters
|
|
ocsenave@3
|
1079
|
|
ocsenave@3
|
1080 The literature of thermodynamics has long recognized a distinction between two
|
|
ocsenave@3
|
1081 kinds of quantities that may be used to define the thermodynamic
|
|
ocsenave@3
|
1082 state. If we imagine a given system as composed of smaller
|
|
ocsenave@3
|
1083 subsystems, we usually find that some of the thermodynamic variables
|
|
ocsenave@3
|
1084 have the same values in each subsystem, while others are additive,
|
|
ocsenave@3
|
1085 the total amount being the sum of the values of each subsystem.
|
|
ocsenave@3
|
1086 These are called /intensive/ and /extensive/ variables, respectively.
|
|
ocsenave@3
|
1087 According to this definition, evidently, the mass of a system is
|
|
ocsenave@3
|
1088 always an extensive quantity, and at equilibrium the temperature
|
|
ocsenave@3
|
1089 is an intensive ‘quantity. Likewise, the energy will be extensive
|
|
ocsenave@3
|
1090 provided that the interaction energy between the subsystems can
|
|
ocsenave@3
|
1091 be neglected.
|
|
ocsenave@3
|
1092
|
|
ocsenave@3
|
1093 It is important to note, however, that in general the terms
|
|
ocsenave@3
|
1094 \ldquo{}intensive\rdquo{} and \ldquo{}extensive\rdquo{}
|
|
ocsenave@3
|
1095 so defined cannot be regarded as
|
|
ocsenave@3
|
1096 establishing a real physical distinction between the variables.
|
|
ocsenave@3
|
1097 This distinction is, like the notion of number of degrees of
|
|
ocsenave@3
|
1098 freedom, in part an anthropomorphic one, because it may depend
|
|
ocsenave@3
|
1099 on the particular kind of subdivision we choose to imagine. For
|
|
ocsenave@3
|
1100 example, a volume of air may be imagined to consist of a number
|
|
ocsenave@3
|
1101 of smaller contiguous volume elements. With this subdivision,
|
|
ocsenave@3
|
1102 the pressure is the same in all subsystems, and is therefore in
|
|
ocsenave@4
|
1103 tensive; while the volume is additive and therefore extensive
|
|
ocsenave@3
|
1104 But we may equally well regard the volume of air as composed of
|
|
ocsenave@3
|
1105 its constituent nitrogen and oxygen subsystems (or we could re
|
|
ocsenave@3
|
1106 gard pure hydrogen as composed of two subsystems, in which the
|
|
ocsenave@3
|
1107 molecules have odd and even rotational quantum numbers
|
|
ocsenave@4
|
1108 respectively, etc.) With this kind of subdivision the volume is the
|
|
ocsenave@3
|
1109 same in all subsystems, while the pressure is the sum of the
|
|
ocsenave@3
|
1110 partial pressures of its constituents; and it appears that the
|
|
ocsenave@3
|
1111 roles of \ldquo{}intensive\rdquo{} and \ldquo{}extensive\rdquo{}
|
|
ocsenave@3
|
1112 have been interchanged. Note that this ambiguity cannot be removed by requiring
|
|
ocsenave@3
|
1113 that we consider only spatial subdivisions, such that each sub
|
|
ocsenave@4
|
1114 system has the same local composi tion For, consider a s tressed
|
|
ocsenave@3
|
1115 elastic solid, such as a stretched rubber band. If we imagine
|
|
ocsenave@3
|
1116 the rubber band as divided, conceptually, into small subsystems
|
|
ocsenave@3
|
1117 by passing planes through it normal to its axis, then the tension
|
|
ocsenave@3
|
1118 is the same in all subsystems, while the elongation is additive.
|
|
ocsenave@3
|
1119 But if the dividing planes are parallel to the axis, the elonga
|
|
ocsenave@3
|
1120 tion is the same in all subsystems, while the tension is
|
|
ocsenave@3
|
1121 additive; once again, the roles of \ldquo{}extensive\rdquo{} and
|
|
ocsenave@3
|
1122 \ldquo{}intensive\rdquo{} are
|
|
ocsenave@3
|
1123 interchanged merely by imagining a different kind of subdivision.
|
|
ocsenave@3
|
1124 In spite of the fundamental ambiguity of the usual definitions,
|
|
ocsenave@3
|
1125 the notions of extensive and intensive variables are useful,
|
|
ocsenave@3
|
1126 and in practice we seem to have no difficulty in deciding
|
|
ocsenave@3
|
1127 which quantities should be considered intensive. Perhaps the
|
|
ocsenave@3
|
1128 distinction is better characterized, not by considering
|
|
ocsenave@3
|
1129 subdivisions at all, but by adopting a different definition, in which
|
|
ocsenave@3
|
1130 we recognize that some quantities have the nature of a \ldquo{}force\rdquo{}
|
|
ocsenave@3
|
1131 or \ldquo{}potential\rdquo{}, or some other local physical property, and are
|
|
ocsenave@3
|
1132 therefore called intensive, while others have the nature of a
|
|
ocsenave@3
|
1133 \ldquo{}displacement\rdquo{} or a \ldquo{}quantity\rdquo{} of
|
|
ocsenave@3
|
1134 something (i.e. are proportional to the size of the system),
|
|
ocsenave@3
|
1135 and are therefore called extensive. Admittedly, this definition is somewhat vague, in a
|
|
ocsenave@3
|
1136 way that can also lead to ambiguities ; in any event, let us agree
|
|
ocsenave@3
|
1137 to class pressure, stress tensor, mass density, energy density,
|
|
ocsenave@3
|
1138 particle density, temperature, chemical potential, angular
|
|
ocsenave@3
|
1139 velocity, as intensive, while volume, mass, energy, particle
|
|
ocsenave@3
|
1140 numbers, strain, entropy, angular momentum, will be considered
|
|
ocsenave@3
|
1141 extensive.
|
|
ocsenave@3
|
1142
|
|
ocsenave@3
|
1143 ** The Kelvin Temperature Scale
|
|
ocsenave@3
|
1144 The form of the first law,
|
|
ocsenave@3
|
1145 $dU = dQ - dW$, expresses the net energy increment of a system as
|
|
ocsenave@3
|
1146 the heat energy supplied to it, minus the work done by it. In
|
|
ocsenave@3
|
1147 the simplest systems of two degrees of freedom, defined by
|
|
ocsenave@3
|
1148 pressure and volume as the thermodynamic variables, the work done
|
|
ocsenave@3
|
1149 in an infinitesimal reversible change of state can be separated
|
|
ocsenave@3
|
1150 into a product $dW = PdV$ of an intensive and an extensive quantity.
|
|
ocsenave@3
|
1151 Furthermore, we know that the pressure $P$ is not only the
|
|
ocsenave@3
|
1152 intensive factor of the work; it is also the \ldquo{}potential\rdquo{}
|
|
ocsenave@3
|
1153 which governs mechanical equilibrium (in this case, equilibrium with respect
|
|
ocsenave@4
|
1154 to exchange of volume) between two systems; ie., if they are
|
|
ocsenave@3
|
1155 separated by a flexible but impermeable membrane, the two systems
|
|
ocsenave@3
|
1156 will exchange volume $dV_1 = -dV_2$ in a direction determined by the
|
|
ocsenave@3
|
1157 pressure difference, until the pressures are equalized. The
|
|
ocsenave@3
|
1158 energy exchanged in this way between the systems is a product
|
|
ocsenave@3
|
1159 of the form
|
|
ocsenave@3
|
1160 #+begin_quote
|
|
ocsenave@3
|
1161 (/intensity/ of something) \times (/quantity/ of something exchanged)
|
|
ocsenave@3
|
1162 #+end_quote
|
|
ocsenave@3
|
1163
|
|
ocsenave@3
|
1164 Now if heat is merely a particular form of energy that can
|
|
ocsenave@3
|
1165 also be exchanged between systems, the question arises whether
|
|
ocsenave@3
|
1166 the quantity of heat energy $dQ$ exchanged in an infinitesimal
|
|
ocsenave@3
|
1167 reversible change of state can also be written as a product of one
|
|
ocsenave@3
|
1168 factor which measures the \ldquo{}intensity\rdquo{} of the heat,
|
|
ocsenave@3
|
1169 times another that represents the \ldquo{}quantity\rdquo{}
|
|
ocsenave@3
|
1170 of something exchanged between
|
|
ocsenave@3
|
1171 the systems, such that the intensity factor governs the
|
|
ocsenave@3
|
1172 conditions of thermal equilibrium and the direction of heat exchange,
|
|
ocsenave@3
|
1173 in the same way that pressure does for volume exchange.
|
|
ocsenave@3
|
1174
|
|
ocsenave@3
|
1175
|
|
ocsenave@3
|
1176 But we already know that the /temperature/ is the quantity
|
|
ocsenave@3
|
1177 that governs the heat flow (i.e., heat flows from the hotter to
|
|
ocsenave@4
|
1178 the cooler body until the temperatures are equalized) So the
|
|
ocsenave@3
|
1179 intensive factor in $dQ$ must be essentially the temperature. But
|
|
ocsenave@3
|
1180 our temperature scale is at present still arbitrary, and we can
|
|
ocsenave@3
|
1181 hardly expect that such a factorization will be possible for all
|
|
ocsenave@3
|
1182 calibrations of our thermometers.
|
|
ocsenave@3
|
1183
|
|
ocsenave@3
|
1184 The same thing is evidently true of pressure; if instead of
|
|
ocsenave@3
|
1185 the pressure $P$ as ordinarily defined, we worked with any mono
|
|
ocsenave@3
|
1186 tonic increasing function $P_1 = P_1 (P)$ we would find that $P_1$ is
|
|
ocsenave@3
|
1187 just as good as $P$ for determining the direction of volume
|
|
ocsenave@3
|
1188 exchange and the condition of mechanical equilibrium; but the work
|
|
ocsenave@3
|
1189 done would not be given by $PdV$; in general, it could not even
|
|
ocsenave@3
|
1190 be expressed in the form $P_1 \cdot dF(V)$, where $F(V)$ is some function
|
|
ocsenave@3
|
1191 of V.
|
|
ocsenave@3
|
1192
|
|
ocsenave@3
|
1193
|
|
ocsenave@3
|
1194 Therefore we ask: out of all the monotonic functions $t_1(t)$
|
|
ocsenave@3
|
1195 corresponding to different empirical temperature scales, is
|
|
ocsenave@3
|
1196 there one (which we denote as $T(t)$) which forms a \ldquo{}natural\rdquo{}
|
|
ocsenave@3
|
1197 intensity factor for heat, such that in a reversible change
|
|
ocsenave@3
|
1198 $dQ = TdS$, where $S(U,V)$ is a new function of the thermodynamic
|
|
ocsenave@3
|
1199 state? If so, then the temperature scale $T$ will have a great
|
|
ocsenave@3
|
1200 theoretical advantage, in that the laws of thermodynamics will
|
|
ocsenave@3
|
1201 take an especially simple form in terms of this particular scale,
|
|
ocsenave@3
|
1202 and the new quantity $S$, which we call the /entropy/, will be a
|
|
ocsenave@3
|
1203 kind of \ldquo{}volume\rdquo{} factor for heat.
|
|
ocsenave@3
|
1204
|
|
ocsenave@3
|
1205 We recall that $dQ = dU + PdV$ is not an exact differential;
|
|
ocsenave@3
|
1206 i.e., on a change from one equilibrium state to another the
|
|
ocsenave@3
|
1207 integral
|
|
ocsenave@3
|
1208
|
|
ocsenave@3
|
1209 \[\int_1^2 dQ\]
|
|
ocsenave@3
|
1210
|
|
ocsenave@3
|
1211 cannot be set equal to the difference $Q_2 - Q_1$ of values of any
|
|
ocsenave@3
|
1212 state function $Q(U,V)$, since the integral has different values
|
|
ocsenave@3
|
1213 for different paths connecting the same initial and final states.
|
|
ocsenave@3
|
1214 Thus there is no \ldquo{}heat function\rdquo{} $Q(U,V)$, and the notion of
|
|
ocsenave@3
|
1215 \ldquo{}amount of heat\rdquo{} $Q$ stored in a body has no meaning
|
|
ocsenave@3
|
1216 (nor does the \ldquo{}amount of work\rdquo{} $W$;
|
|
ocsenave@3
|
1217 only the total energy is a well-defined quantity).
|
|
ocsenave@3
|
1218 But we want the entropy $S(U,V)$ to be a definite quantity,
|
|
ocsenave@3
|
1219 like the energy or volume, and so $dS$ must be an exact differential.
|
|
ocsenave@3
|
1220 On an infinitesimal reversible change from one equilibrium state
|
|
ocsenave@4
|
1221 to another, the first law requires that it satisfy[fn:: Edit: The first
|
|
ocsenave@3
|
1222 equality comes from our requirement that $dQ = T\,dS$. The second
|
|
ocsenave@3
|
1223 equality comes from the fact that $dU = dQ - dW$ (the first law) and
|
|
ocsenave@3
|
1224 that $dW = PdV$ in the case where the state has two degrees of
|
|
ocsenave@3
|
1225 freedom, pressure and volume.]
|
|
ocsenave@3
|
1226
|
|
ocsenave@3
|
1227 \begin{equation}
|
|
ocsenave@3
|
1228 dS(U,V) = \frac{dQ}{T} = \frac{dU}{T} + \frac{P}{T}dV
|
|
ocsenave@3
|
1229 \end{equation}
|
|
ocsenave@3
|
1230
|
|
ocsenave@3
|
1231 Thus $(1/T)$ must be an /integrating factor/ which converts $dQ$ into
|
|
ocsenave@4
|
1232 an exact differential [[fn::Edit: A differential $M(x,y)dx +
|
|
ocsenave@3
|
1233 N(x,y)dy$ is called /exact/ if there is a scalar function
|
|
ocsenave@3
|
1234 $\Phi(x,y)$ such that $M = \frac{\partial \Phi}{\partial x}$ and
|
|
ocsenave@3
|
1235 $N=\frac{\partial \Phi}{\partial y}$. If there is, \Phi is called the
|
|
ocsenave@3
|
1236 /potential function/ of the differential, Conceptually, this means
|
|
ocsenave@3
|
1237 that M(x,y)dx + N(x,y) dy is the derivative of a scalar potential and
|
|
ocsenave@3
|
1238 so consequently corresponds to a conservative field.
|
|
ocsenave@3
|
1239
|
|
ocsenave@3
|
1240 Even if there is no such potential function
|
|
ocsenave@3
|
1241 \Phi for the given differential, it is possible to coerce an
|
|
ocsenave@3
|
1242 inexact differential into an exact one by multiplying by an unknown
|
|
ocsenave@3
|
1243 function $\mu(x,y)$ (called an /integrating factor/) and requiring the
|
|
ocsenave@3
|
1244 resulting differential $\mu M\, dx + \mu N\, dy$ to be exact.
|
|
ocsenave@3
|
1245
|
|
ocsenave@3
|
1246 To complete the analogy, here we have the differential $dQ =
|
|
ocsenave@3
|
1247 dU + PdV$ (by the first law) which is not exact---conceptually, there
|
|
ocsenave@3
|
1248 is no scalar potential nor conserved quantity corresponding to
|
|
ocsenave@3
|
1249 $dQ$. We have introduced a new differential $dS = \frac{1}{T}dQ$, and we
|
|
ocsenave@3
|
1250 are searching for the temperature scale $T(U,V)$ which makes $dS$
|
|
ocsenave@3
|
1251 exact (i.e. which makes $S$ correspond to a conserved quantity). This means
|
|
ocsenave@3
|
1252 that $\frac{1}{T}$ is playing the role of the integrating factor
|
|
ocsenave@3
|
1253 \ldquo{}\mu\rdquo{} for the differential $dQ$.]]
|
|
ocsenave@3
|
1254
|
|
ocsenave@3
|
1255 Now the question of the existence and properties of
|
|
ocsenave@3
|
1256 integrating factors is a purely mathematical one, which can be
|
|
ocsenave@3
|
1257 investigated independently of the properties of any particular
|
|
ocsenave@3
|
1258 substance. Let us denote this integrating factor for the moment
|
|
ocsenave@3
|
1259 by $w(U,V) = T^{-1}$; then the first law becomes
|
|
ocsenave@3
|
1260
|
|
ocsenave@3
|
1261 \begin{equation}
|
|
ocsenave@3
|
1262 dS(U,V) = w dU + w P dV
|
|
ocsenave@3
|
1263 \end{equation}
|
|
ocsenave@3
|
1264
|
|
ocsenave@3
|
1265 from which the derivatives are
|
|
ocsenave@3
|
1266
|
|
ocsenave@3
|
1267 \begin{equation}
|
|
ocsenave@3
|
1268 \left(\frac{\partial S}{\partial U}\right)_V = w, \qquad
|
|
ocsenave@3
|
1269 \left(\frac{\partial S}{\partial V}\right)_U = wP.
|
|
ocsenave@3
|
1270 \end{equation}
|
|
ocsenave@3
|
1271
|
|
ocsenave@3
|
1272 The condition that $dS$ be exact is that the cross-derivatives be
|
|
ocsenave@3
|
1273 equal, as in (1-4):
|
|
ocsenave@3
|
1274
|
|
ocsenave@3
|
1275 \begin{equation}
|
|
ocsenave@3
|
1276 \frac{\partial^2 S}{\partial U \partial V} = \frac{\partial^2
|
|
ocsenave@3
|
1277 S}{\partial V \partial U},
|
|
ocsenave@3
|
1278 \end{equation}
|
|
ocsenave@3
|
1279
|
|
ocsenave@3
|
1280 or
|
|
ocsenave@3
|
1281
|
|
ocsenave@3
|
1282 \begin{equation}
|
|
ocsenave@3
|
1283 \left(\frac{\partial w}{\partial V}\right)_U = \left(\frac{\partial
|
|
ocsenave@3
|
1284 P}{\partial U}\right)_V + P\cdot \left(\frac{\partial w}{\partial U}\right)_V.
|
|
ocsenave@3
|
1285 \end{equation}
|
|
ocsenave@3
|
1286
|
|
ocsenave@3
|
1287 Any function $w(U,V)$ satisfying this differential equation is an
|
|
ocsenave@3
|
1288 integrating factor for $dQ$.
|
|
ocsenave@3
|
1289
|
|
ocsenave@3
|
1290 But if $w(U,V)$ is one such integrating factor, which leads
|
|
ocsenave@3
|
1291 to the new state function $S(U,V)$, it is evident that
|
|
ocsenave@3
|
1292 $w_1(U,V) \equiv w \cdot f(S)$ is an equally good integrating factor, where
|
|
ocsenave@3
|
1293 $f(S)$ is an arbitrary function. Use of $w_1$ will lead to a
|
|
ocsenave@3
|
1294 different state function
|
|
ocsenave@3
|
1295
|
|
ocsenave@3
|
1296 #what's with the variable collision?
|
|
ocsenave@3
|
1297 \begin{equation}
|
|
ocsenave@3
|
1298 S_1(U,V) = \int^S f(S) dS
|
|
ocsenave@3
|
1299 \end{equation}
|
|
ocsenave@3
|
1300
|
|
ocsenave@3
|
1301 The mere conversion of into an exact differential is, therefore,
|
|
ocsenave@3
|
1302 not enough to determine any unique entropy function $S(U,V)$.
|
|
ocsenave@3
|
1303 However, the derivative
|
|
ocsenave@3
|
1304
|
|
ocsenave@3
|
1305 \begin{equation}
|
|
ocsenave@3
|
1306 \left(\frac{dU}{dV}\right)_S = -P
|
|
ocsenave@3
|
1307 \end{equation}
|
|
ocsenave@3
|
1308
|
|
ocsenave@3
|
1309 is evidently uniquely determined; so also, therefore, is the
|
|
ocsenave@3
|
1310 family of lines of constant entropy, called /adiabats/, in the
|
|
ocsenave@3
|
1311 $(U-V)$ plane. But, as (1-24) shows, the numerical value of $S$ on
|
|
ocsenave@3
|
1312 each adiabat is still completely undetermined.
|
|
ocsenave@3
|
1313
|
|
ocsenave@3
|
1314 In order to fix the relative values of $S$ on different
|
|
ocsenave@3
|
1315 adiabats we need to add the condition, not yet put into the equations,
|
|
ocsenave@3
|
1316 that the integrating factor $w(U,V) = T^{-1}$ is to define a new
|
|
ocsenave@4
|
1317 temperature scale In other words, we now ask: out of the
|
|
ocsenave@3
|
1318 infinite number of different integrating factors allowed by
|
|
ocsenave@3
|
1319 the differential equation (1-23), is it possible to find one
|
|
ocsenave@3
|
1320 which is a function only of the empirical temperature $t$? If
|
|
ocsenave@3
|
1321 $w=w(t)$, we can write
|
|
ocsenave@3
|
1322
|
|
ocsenave@3
|
1323 \begin{equation}
|
|
ocsenave@3
|
1324 \left(\frac{\partial w}{\partial V}\right)_U = \frac{dw}{dt}\left(\frac{\partial
|
|
ocsenave@3
|
1325 t}{\partial V}\right)_U
|
|
ocsenave@3
|
1326 \end{equation}
|
|
ocsenave@3
|
1327 \begin{equation}
|
|
ocsenave@3
|
1328 \left(\frac{\partial w}{\partial U}\right)_V = \frac{dw}{dt}\left(\frac{\partial
|
|
ocsenave@3
|
1329 t}{\partial U}\right)_V
|
|
ocsenave@3
|
1330 \end{equation}
|
|
ocsenave@3
|
1331
|
|
ocsenave@3
|
1332
|
|
ocsenave@3
|
1333 and (1-23) becomes
|
|
ocsenave@3
|
1334 \begin{equation}
|
|
ocsenave@3
|
1335 \frac{d}{dt}\log{w} = \frac{\left(\frac{\partial P}{\partial
|
|
ocsenave@3
|
1336 U}\right)_V}{\left(\frac{\partial t}{\partial V}\right)_U-P\left(\frac{\partial t}{\partial U}\right)_V}
|
|
ocsenave@3
|
1337 \end{equation}
|
|
ocsenave@3
|
1338
|
|
ocsenave@3
|
1339
|
|
ocsenave@3
|
1340 which shows that $w$ will be determined to within a multiplicative
|
|
ocsenave@3
|
1341 factor.
|
|
ocsenave@3
|
1342
|
|
ocsenave@3
|
1343 Is the temperature scale thus defined independent of the
|
|
ocsenave@3
|
1344 empirical scale from which we started? To answer this, let
|
|
ocsenave@3
|
1345 $t_1 = t_1(t)$ be any monotonic function which defines a different
|
|
ocsenave@3
|
1346 empirical temperature scale. In place of (1-28), we then have
|
|
ocsenave@3
|
1347
|
|
ocsenave@3
|
1348 \begin{equation}
|
|
ocsenave@3
|
1349 \frac{d}{dt_1}\log{w} \quad=\quad \frac{\left(\frac{\partial P}{\partial
|
|
ocsenave@3
|
1350 U}\right)_V}{\left(\frac{\partial t_1}{\partial V}\right)_U-P\left(\frac{\partial t_1}{\partial U}\right)_V}
|
|
ocsenave@3
|
1351 \quad = \quad
|
|
ocsenave@3
|
1352 \frac{\left(\frac{\partial P}{\partial
|
|
ocsenave@3
|
1353 U}\right)_V}{\frac{dt_1}{dt}\left[ \left(\frac{\partial t}{\partial
|
|
ocsenave@3
|
1354 V}\right)_U-P\left(\frac{\partial t}{\partial U}\right)_V\right]},
|
|
ocsenave@3
|
1355 \end{equation}
|
|
ocsenave@3
|
1356 or
|
|
ocsenave@3
|
1357 \begin{equation}
|
|
ocsenave@3
|
1358 \frac{d}{dt_1}\log{w_1} = \frac{dt}{dt_1}\frac{d}{dt}\log{w}
|
|
ocsenave@3
|
1359 \end{equation}
|
|
ocsenave@3
|
1360
|
|
ocsenave@3
|
1361 which reduces to $d \log{w_1} = d \log{w}$, or
|
|
ocsenave@3
|
1362 \begin{equation}
|
|
ocsenave@3
|
1363 w_1 = C\cdot w
|
|
ocsenave@3
|
1364 \end{equation}
|
|
ocsenave@3
|
1365
|
|
ocsenave@3
|
1366 Therefore, integrating factors derived from whatever empirical
|
|
ocsenave@3
|
1367 temperature scale can differ among themselves only by a
|
|
ocsenave@3
|
1368 multiplicative factor. For any given substance, therefore, except
|
|
ocsenave@3
|
1369 for this factor (which corresponds just to our freedom to choose
|
|
ocsenave@3
|
1370 the size of the units in which we measure temperature), there is
|
|
ocsenave@3
|
1371 only /one/ temperature scale $T(t) = 1/w$ with the property that
|
|
ocsenave@3
|
1372 $dS = dQ/T$ is an exact differential.
|
|
ocsenave@3
|
1373
|
|
ocsenave@3
|
1374 To find a feasible way of realizing this temperature scale
|
|
ocsenave@3
|
1375 experimentally, multiply numerator and denominator of the right
|
|
ocsenave@3
|
1376 hand side of (1-28) by the heat capacity at constant volume,
|
|
ocsenave@3
|
1377 $C_V^\prime = (\partial U/\partial t) V$, the prime denoting that
|
|
ocsenave@3
|
1378 it is in terms of the empirical temperature scale $t$.
|
|
ocsenave@3
|
1379 Integrating between any two states denoted 1 and 2, we have
|
|
ocsenave@3
|
1380
|
|
ocsenave@3
|
1381 \begin{equation}
|
|
ocsenave@3
|
1382 \frac{T_1}{T_2} = \exp\left\{\int_{t_1}^{t_2}
|
|
ocsenave@3
|
1383 \frac{\left(\frac{\partial P}{\partial t}\right)_V dt}{P - C_V^\prime
|
|
ocsenave@3
|
1384 \left(\frac{\partial t}{\partial V}\right)_U} \right\}
|
|
ocsenave@3
|
1385 \end{equation}
|
|
ocsenave@3
|
1386
|
|
ocsenave@3
|
1387 If the quantities on the right-hand side have been determined
|
|
ocsenave@3
|
1388 experimentally, then a numerical integration yields the ratio
|
|
ocsenave@3
|
1389 of Kelvin temperatures of the two states.
|
|
ocsenave@3
|
1390
|
|
ocsenave@3
|
1391 This process is particularly simple if we choose for our
|
|
ocsenave@3
|
1392 system a volume of gas with the property found in Joule's famous
|
|
ocsenave@3
|
1393 expansion experiment; when the gas expands freely into a vacuum
|
|
ocsenave@3
|
1394 (i.e., without doing work, or $U = \text{const.}$), there is no change in
|
|
ocsenave@3
|
1395 temperature. Real gases when sufficiently far from their condensation
|
|
ocsenave@3
|
1396 points are found to obey this rule very accurately.
|
|
ocsenave@3
|
1397 But then
|
|
ocsenave@3
|
1398
|
|
ocsenave@3
|
1399 \begin{equation}
|
|
ocsenave@3
|
1400 \left(\frac{dt}{dV}\right)_U = 0
|
|
ocsenave@3
|
1401 \end{equation}
|
|
ocsenave@3
|
1402
|
|
ocsenave@3
|
1403 and on a change of state in which we heat this gas at constant
|
|
ocsenave@3
|
1404 volume, (1-31) collapses to
|
|
ocsenave@3
|
1405
|
|
ocsenave@3
|
1406 \begin{equation}
|
|
ocsenave@3
|
1407 \frac{T_1}{T_2} = \exp\left\{\int_{t_1}^{t_2}
|
|
ocsenave@3
|
1408 \frac{1}{P}\left(\frac{\partial P}{\partial t}\right)_V dt\right\} = \frac{P_2}{P_1}.
|
|
ocsenave@3
|
1409 \end{equation}
|
|
ocsenave@3
|
1410
|
|
ocsenave@3
|
1411 Therefore, with a constant-volume ideal gas thermometer, (or more
|
|
ocsenave@3
|
1412 generally, a thermometer using any substance obeying (1-32) and
|
|
ocsenave@3
|
1413 held at constant volume), the measured pressure is directly
|
|
ocsenave@3
|
1414 proportional to the Kelvin temperature.
|
|
ocsenave@3
|
1415
|
|
ocsenave@3
|
1416 For an imperfect gas, if we have measured $(\partial t /\partial
|
|
ocsenave@3
|
1417 V)_U$ and $C_V^\prime$, Eq. (1-31) determines the necessary
|
|
ocsenave@3
|
1418 corrections to (1-33). However, an alternative form of (1-31), in
|
|
ocsenave@3
|
1419 which the roles of pressure and volume are interchanged, proves to be
|
|
ocsenave@3
|
1420 more convenient for experimental determinations. To derive it, introduce the
|
|
ocsenave@3
|
1421 enthalpy function
|
|
ocsenave@3
|
1422
|
|
ocsenave@3
|
1423 \begin{equation}H = U + PV\end{equation}
|
|
ocsenave@3
|
1424
|
|
ocsenave@3
|
1425 with the property
|
|
ocsenave@3
|
1426
|
|
ocsenave@3
|
1427 \begin{equation}
|
|
ocsenave@3
|
1428 dH = dQ + VdP
|
|
ocsenave@3
|
1429 \end{equation}
|
|
ocsenave@3
|
1430
|
|
ocsenave@3
|
1431 Equation (1-19) then becomes
|
|
ocsenave@3
|
1432
|
|
ocsenave@3
|
1433 \begin{equation}
|
|
ocsenave@3
|
1434 dS = \frac{dH}{T} - \frac{V}{T}dP.
|
|
ocsenave@3
|
1435 \end{equation}
|
|
ocsenave@3
|
1436
|
|
ocsenave@3
|
1437 Repeating the steps (1-20) to (1-31) of the above derivation
|
|
ocsenave@3
|
1438 starting from (1-36) instead of from (1-19), we arrive at
|
|
ocsenave@3
|
1439
|
|
ocsenave@3
|
1440 \begin{equation}
|
|
ocsenave@3
|
1441 \frac{T_2}{T_1} = \exp\left\{\int_{t_1}^{t_2}
|
|
ocsenave@3
|
1442 \frac{\left(\frac{dV}{dt}\right)_P dt}{V + C_P^\prime
|
|
ocsenave@3
|
1443 \left(\frac{\partial t}{\partial P}\right)_H}\right\}
|
|
ocsenave@3
|
1444 \end{equation}
|
|
ocsenave@3
|
1445
|
|
ocsenave@3
|
1446 or
|
|
ocsenave@3
|
1447
|
|
ocsenave@3
|
1448 \begin{equation}
|
|
ocsenave@3
|
1449 \frac{T_2}{T_1} = \exp\left\{\frac{\alpha^\prime
|
|
ocsenave@3
|
1450 dt}{1+\left(C_P^\prime \cdot \mu^\prime / V\right)}\right\}
|
|
ocsenave@3
|
1451 \end{equation}
|
|
ocsenave@3
|
1452
|
|
ocsenave@3
|
1453 where
|
|
ocsenave@3
|
1454 \begin{equation}
|
|
ocsenave@3
|
1455 \alpha^\prime \equiv \frac{1}{V}\left(\frac{\partial V}{\partial t}\right)_P
|
|
ocsenave@3
|
1456 \end{equation}
|
|
ocsenave@3
|
1457 is the thermal expansion coefficient,
|
|
ocsenave@3
|
1458 \begin{equation}
|
|
ocsenave@3
|
1459 C_P^\prime \equiv \left(\frac{\partial H}{\partial t}\right)_P
|
|
ocsenave@3
|
1460 \end{equation}
|
|
ocsenave@3
|
1461 is the heat capacity at constant pressure, and
|
|
ocsenave@3
|
1462 \begin{equation}
|
|
ocsenave@3
|
1463 \mu^\prime \equiv \left(\frac{dt}{dP}\right)_H
|
|
ocsenave@3
|
1464 \end{equation}
|
|
ocsenave@3
|
1465
|
|
ocsenave@3
|
1466 is the coefficient measured in the Joule-Thompson porous plug
|
|
ocsenave@3
|
1467 experiment, the primes denoting again that all are to be measured
|
|
ocsenave@3
|
1468 in terms of the empirical temperature scale $t$.
|
|
ocsenave@3
|
1469 Since $\alpha^\prime$, $C_P^\prime$, $\mu^\prime$ are all
|
|
ocsenave@3
|
1470 easily measured in the laboratory, Eq. (1-38) provides a
|
|
ocsenave@3
|
1471 feasible way of realizing the Kelvin temperature scale experimentally,
|
|
ocsenave@3
|
1472 taking account of the imperfections of real gases.
|
|
ocsenave@3
|
1473 For an account of the work of Roebuck and others based on this
|
|
ocsenave@3
|
1474 relation, see [[http://books.google.com/books?id=KKJKAAAAMAAJ][Zemansky (1943)]]; pp. 252-255.
|
|
ocsenave@3
|
1475
|
|
ocsenave@3
|
1476 Note that if $\mu^\prime = O$ and we heat the gas at constant
|
|
ocsenave@3
|
1477 pressure, (1-38) reduces to
|
|
ocsenave@3
|
1478
|
|
ocsenave@3
|
1479 \begin{equation}
|
|
ocsenave@3
|
1480 \frac{T_2}{T_1} = \exp\left\{ \int_{t_1}^{t_2}
|
|
ocsenave@3
|
1481 \frac{1}{V}\left(\frac{\partial V}{\partial t}\right)_P dt \right\} = \frac{V_2}{V_1}
|
|
ocsenave@3
|
1482 \end{equation}
|
|
ocsenave@3
|
1483
|
|
ocsenave@3
|
1484 so that, with a constant-pressure gas thermometer using a gas for
|
|
ocsenave@3
|
1485 which the Joule-Thomson coefficient is zero, the Kelvin temperature is
|
|
ocsenave@3
|
1486 proportional to the measured volume.
|
|
ocsenave@3
|
1487
|
|
ocsenave@3
|
1488 Now consider another empirical fact, [[http://en.wikipedia.org/wiki/Boyle%27s_law][Boyle's law]]. For gases
|
|
ocsenave@3
|
1489 sufficiently far from their condensation points---which is also
|
|
ocsenave@3
|
1490 the condition under which (1-32) is satisfied---Boyle found that
|
|
ocsenave@3
|
1491 the product $PV$ is a constant at any fixed temperature. This
|
|
ocsenave@3
|
1492 product is, of course proportional to the number of moles $n$
|
|
ocsenave@3
|
1493 present, and so Boyle's equation of state takes the form
|
|
ocsenave@3
|
1494
|
|
ocsenave@3
|
1495 \begin{equation}PV = n \cdot f(t)\end{equation}
|
|
ocsenave@3
|
1496
|
|
ocsenave@3
|
1497 where f(t) is a function that depends on the particular empirical
|
|
ocsenave@3
|
1498 temperature scale used. But from (1-33) we must then have
|
|
ocsenave@3
|
1499 $f(t) = RT$, where $R$ is a constant, the universal gas constant whose
|
|
ocsenave@4
|
1500 numerical value (1.986 calories per mole per degree K), depends
|
|
ocsenave@3
|
1501 on the size of the units in which we choose to measure the Kelvin
|
|
ocsenave@3
|
1502 temperature $T$. In terms of the Kelvin temperature, the ideal gas
|
|
ocsenave@3
|
1503 equation of state is therefore simply
|
|
ocsenave@3
|
1504
|
|
ocsenave@3
|
1505 \begin{equation}
|
|
ocsenave@3
|
1506 PV = nRT
|
|
ocsenave@3
|
1507 \end{equation}
|
|
ocsenave@3
|
1508
|
|
ocsenave@3
|
1509
|
|
ocsenave@3
|
1510 The relations (1-32) and (1-44) were found empirically, but
|
|
ocsenave@3
|
1511 with the development of thermodynamics one could show that they
|
|
ocsenave@3
|
1512 are not logically independent. In fact, all the material needed
|
|
ocsenave@3
|
1513 for this demonstration is now at hand, and we leave it as an
|
|
ocsenave@3
|
1514 exercise for the reader to prove that Joule‘s relation (1-32) is
|
|
ocsenave@3
|
1515 a logical consequence of Boyle's equation of state (1-44) and the
|
|
ocsenave@3
|
1516 first law.
|
|
ocsenave@3
|
1517
|
|
ocsenave@3
|
1518
|
|
ocsenave@3
|
1519 Historically, the advantages of the gas thermometer were
|
|
ocsenave@3
|
1520 discovered empirically before the Kelvin temperature scale was
|
|
ocsenave@3
|
1521 defined; and the temperature scale \theta defined by
|
|
ocsenave@3
|
1522
|
|
ocsenave@3
|
1523 \begin{equation}
|
|
ocsenave@3
|
1524 \theta = \lim_{P\rightarrow 0}\left(\frac{PV}{nR}\right)
|
|
ocsenave@3
|
1525 \end{equation}
|
|
ocsenave@3
|
1526
|
|
ocsenave@3
|
1527 was found to be convenient, easily reproducible, and independent
|
|
ocsenave@3
|
1528 of the properties of any particular gas. It was called the
|
|
ocsenave@3
|
1529 /absolute/ temperature scale; and from the foregoing it is clear
|
|
ocsenave@3
|
1530 that with the same choice of the numerical constant $R$, the
|
|
ocsenave@3
|
1531 absolute and Kelvin scales are identical.
|
|
ocsenave@3
|
1532
|
|
ocsenave@3
|
1533
|
|
ocsenave@3
|
1534 For many years the unit of our temperature scale was the
|
|
ocsenave@3
|
1535 Centigrade degree, so defined that the difference $T_b - T_f$ of
|
|
ocsenave@3
|
1536 boiling and freezing points of water was exactly 100 degrees.
|
|
ocsenave@3
|
1537 However, improvements in experimental techniques have made another
|
|
ocsenave@3
|
1538 method more reproducible; and the degree was redefined by the
|
|
ocsenave@3
|
1539 Tenth General Conference of Weights and Measures in 1954, by
|
|
ocsenave@3
|
1540 the condition that the triple point of water is at 273.l6^\circ K,
|
|
ocsenave@3
|
1541 this number being exact by definition. The freezing point, 0^\circ C,
|
|
ocsenave@3
|
1542 is then 273.15^\circ K. This new degree is called the Celsius degree.
|
|
ocsenave@3
|
1543 For further details, see the U.S. National Bureau of Standards
|
|
ocsenave@3
|
1544 Technical News Bulletin, October l963.
|
|
ocsenave@3
|
1545
|
|
ocsenave@3
|
1546
|
|
ocsenave@3
|
1547 The appearance of such a strange and arbitrary-looking
|
|
ocsenave@3
|
1548 number as 273.16 in the /definition/ of a unit is the result of
|
|
ocsenave@3
|
1549 the historical development, and is the means by which much
|
|
ocsenave@3
|
1550 greater confusion is avoided. Whenever improved techniques make
|
|
ocsenave@3
|
1551 possible a new and more precise (i.e., more reproducible)
|
|
ocsenave@3
|
1552 definition of a physical unit, its numerical value is of course chosen
|
|
ocsenave@3
|
1553 so as to be well inside the limits of error with which the old
|
|
ocsenave@3
|
1554 unit could be defined. Thus the old Centigrade and new Celsius
|
|
ocsenave@3
|
1555 scales are the same, within the accuracy with which the
|
|
ocsenave@3
|
1556 Centigrade scale could be realized; so the same notation, ^\circ C, is used
|
|
ocsenave@4
|
1557 for both Only in this way can old measurements retain their
|
|
ocsenave@3
|
1558 value and accuracy, without need of corrections every time a
|
|
ocsenave@3
|
1559 unit is redefined.
|
|
ocsenave@3
|
1560
|
|
ocsenave@3
|
1561 #capitalize Joules?
|
|
ocsenave@3
|
1562 Exactly the same thing has happened in the definition of
|
|
ocsenave@3
|
1563 the calorie; for a century, beginning with the work of Joule,
|
|
ocsenave@3
|
1564 more and more precise experiments were performed to determine
|
|
ocsenave@4
|
1565 the mechanical equivalent of heat more and more accurately But
|
|
ocsenave@3
|
1566 eventually mechanical and electrical measurements of energy be
|
|
ocsenave@3
|
1567 came far more reproducible than calorimetric measurements; so
|
|
ocsenave@3
|
1568 recently the calorie was redefined to be 4.1840 Joules, this
|
|
ocsenave@3
|
1569 number now being exact by definition. Further details are given
|
|
ocsenave@3
|
1570 in the aforementioned Bureau of Standards Bulletin.
|
|
ocsenave@3
|
1571
|
|
ocsenave@3
|
1572
|
|
ocsenave@3
|
1573 The derivations of this section have shown that, for any
|
|
ocsenave@3
|
1574 particular substance, there is (except for choice of units) only
|
|
ocsenave@3
|
1575 one temperature scale $T$ with the property that $dQ = TdS$ where
|
|
ocsenave@3
|
1576 $dS$ is the exact differential of some state function $S$. But this
|
|
ocsenave@3
|
1577 in itself provides no reason to suppose that the /same/ Kelvin
|
|
ocsenave@3
|
1578 scale will result for all substances; i.e., if we determine a
|
|
ocsenave@3
|
1579 \ldquo{}helium Kelvin temperature\rdquo{} and a
|
|
ocsenave@3
|
1580 \ldquo{}carbon dioxide Kelvin temperature\rdquo{} by the measurements
|
|
ocsenave@3
|
1581 indicated in (1-38), and choose the units so that they agree numerically at one point, will they then
|
|
ocsenave@3
|
1582 agree at other points? Thus far we have given no reason to
|
|
ocsenave@3
|
1583 expect that the Kelvin scale is /universal/, other than the empirical
|
|
ocsenave@3
|
1584 fact that the limit (1-45) is found to be the same for all gases.
|
|
ocsenave@3
|
1585 In section 2.0 we will see that this universality is a conse
|
|
ocsenave@3
|
1586 quence of the second law of thermodynamics (i.e., if we ever
|
|
ocsenave@3
|
1587 find two substances for which the Kelvin scale as defined above
|
|
ocsenave@3
|
1588 is different, then we can take advantage of this to make a
|
|
ocsenave@3
|
1589 perpetual motion machine of the second kind).
|
|
ocsenave@3
|
1590
|
|
ocsenave@3
|
1591
|
|
ocsenave@3
|
1592 Usually, the second law is introduced before discussing
|
|
ocsenave@3
|
1593 entropy or the Kelvin temperature scale. We have chosen this
|
|
ocsenave@3
|
1594 unusual order so as to demonstrate that the concepts of entropy
|
|
ocsenave@3
|
1595 and Kelvin temperature are logically independent of the second
|
|
ocsenave@3
|
1596 law; they can be defined theoretically, and the experimental
|
|
ocsenave@3
|
1597 procedures for their measurement can be developed, without any
|
|
ocsenave@3
|
1598 appeal to the second law. From the standpoint of logic, there
|
|
ocsenave@3
|
1599 fore, the second law serves /only/ to establish that the Kelvin
|
|
ocsenave@3
|
1600 temperature scale is the same for all substances.
|
|
ocsenave@3
|
1601
|
|
ocsenave@3
|
1602
|
|
ocsenave@4
|
1603 ** Entropy of an Ideal Boltzmann Gas
|
|
ocsenave@3
|
1604
|
|
ocsenave@3
|
1605 At the present stage we are far from understanding the physical
|
|
ocsenave@4
|
1606 meaning of the function $S$ defined by (1-19); but we can investigate
|
|
ocsenave@4
|
1607 its mathematical form and numerical values. Let us do this for a
|
|
ocsenave@4
|
1608 system
|
|
ocsenave@4
|
1609 consisting of $n$ moles of a substance which obeys the ideal gas
|
|
ocsenave@3
|
1610 equation of state
|
|
ocsenave@4
|
1611
|
|
ocsenave@4
|
1612 \begin{equation}PV = nRT\end{equation}
|
|
ocsenave@4
|
1613
|
|
ocsenave@4
|
1614 and for which the heat capacity at constant volume
|
|
ocsenave@4
|
1615 $C_V$ is a constant. The difference in entropy between any two states (1)
|
|
ocsenave@3
|
1616 and (2) is from (1-19),
|
|
ocsenave@4
|
1617
|
|
ocsenave@4
|
1618 \begin{equation}
|
|
ocsenave@4
|
1619 S_2 - S_1 = \int_1^2 \frac{dQ}{T} = \int_1^2
|
|
ocsenave@4
|
1620 \left[\left(\frac{\partial S}{\partial V}\right)+\left(\frac{\partial S}{\partial T}\right)_V dT\right]
|
|
ocsenave@4
|
1621 \end{equation}
|
|
ocsenave@3
|
1622
|
|
ocsenave@3
|
1623 where we integrate over any reversible path connecting the two
|
|
ocsenave@4
|
1624 states. From the manner in which $S$ was defined, this integral
|
|
ocsenave@3
|
1625 must be the same whatever path we choose. Consider, then, a
|
|
ocsenave@4
|
1626 path consisting of a reversible expansion at constant
|
|
ocsenave@4
|
1627 temperature to a state 3 which has the initial temperature $T_1$, and the
|
|
ocsenave@4
|
1628 the final volume $V_2$; followed by heating at constant volume to the
|
|
ocsenave@4
|
1629 final temperature $T_2$.
|
|
ocsenave@4
|
1630 Then (1-47) becomes
|
|
ocsenave@4
|
1631
|
|
ocsenave@4
|
1632 \begin{equation}
|
|
ocsenave@4
|
1633 S_2 - S_1 = \int_1^3 \left(\frac{\partial S}{\partial V}\right)_T dV +
|
|
ocsenave@4
|
1634 \int_3^2 \left(\frac{\partial S}{\partial T}\right)_V dT
|
|
ocsenave@4
|
1635 \end{equation}
|
|
ocsenave@4
|
1636
|
|
ocsenave@4
|
1637 To evaluate the integral over $(1\rightarrow 3)$, note that since $dU
|
|
ocsenave@4
|
1638 = TdS - PdV$, the Helmholtz free energy function $F \equiv U -TS$ has
|
|
ocsenave@4
|
1639 the property $dF = -SdT - PdV$; and of course $dF$ is an exact
|
|
ocsenave@4
|
1640 differential since $F$ is a definite state function. The condition
|
|
ocsenave@4
|
1641 that $dF$ be exact is, analogous to (1-22),
|
|
ocsenave@4
|
1642
|
|
ocsenave@4
|
1643 \begin{equation}
|
|
ocsenave@4
|
1644 \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial
|
|
ocsenave@4
|
1645 P}{\partial T}\right)_V
|
|
ocsenave@4
|
1646 \end{equation}
|
|
ocsenave@4
|
1647
|
|
ocsenave@3
|
1648 which is one of the Maxwell relations, discussed further in
|
|
ocsenave@4
|
1649 Chapter 2. But [the value of this expression] is determined by the equation of state
|
|
ocsenave@4
|
1650 (1-46):
|
|
ocsenave@4
|
1651
|
|
ocsenave@4
|
1652 \begin{equation}
|
|
ocsenave@4
|
1653 \left(\frac{\partial S}{\partial V}\right)_T = \frac{nR}{V}
|
|
ocsenave@4
|
1654 \end{equation}
|
|
ocsenave@4
|
1655
|
|
ocsenave@4
|
1656 Likewise, along the path $(3\rightarrow 2)$, we have
|
|
ocsenave@4
|
1657
|
|
ocsenave@4
|
1658 \begin{equation}
|
|
ocsenave@4
|
1659 \left(\frac{\partial S}{\partial T}\right)_V = \frac{n C_V}{T}
|
|
ocsenave@4
|
1660 \end{equation}
|
|
ocsenave@4
|
1661
|
|
ocsenave@4
|
1662 where $C_V$ is the molar heat capacity at constant volume.
|
|
ocsenave@4
|
1663 Collecting these results, we have
|
|
ocsenave@4
|
1664
|
|
ocsenave@4
|
1665 \begin{equation}
|
|
ocsenave@4
|
1666 S_2 - S_1 = \int_1^3 \frac{nR}{V} dV + \int_2^3 \frac{n C_V}{T} dT =
|
|
ocsenave@4
|
1667 nR\log{(V_2/V_1)} + nC_V \log{(T_2/T_1)}
|
|
ocsenave@4
|
1668 \end{equation}
|
|
ocsenave@4
|
1669
|
|
ocsenave@4
|
1670 since $C_V$ was assumed independent of $T$. Thus the entropy function
|
|
ocsenave@3
|
1671 must have the form
|
|
ocsenave@4
|
1672
|
|
ocsenave@4
|
1673 \begin{equation}
|
|
ocsenave@4
|
1674 S(n,V,T) = nR \log{V} + n C_V \log{T} + (\text{const.})
|
|
ocsenave@4
|
1675 \end{equation}
|
|
ocsenave@3
|
1676
|
|
ocsenave@3
|
1677 From the derivation, the additive constant must be independent
|
|
ocsenave@3
|
1678 of V and T; but it can still depend on n. We indicate this by
|
|
ocsenave@3
|
1679 writing
|
|
ocsenave@4
|
1680
|
|
ocsenave@4
|
1681 \begin{equation}
|
|
ocsenave@4
|
1682 S(n,V,T) = n\left[R \log{V} + C_V \log{T}\right] + f(n)
|
|
ocsenave@4
|
1683 \end{equation}
|
|
ocsenave@4
|
1684
|
|
ocsenave@4
|
1685 where $f(n)$ is a function not determined by the definition (1-47).
|
|
ocsenave@4
|
1686 The form of $f(n)$ is, however, restricted by the condition that
|
|
ocsenave@4
|
1687 the entropy be an extensive quantity; i.e., two identical systems
|
|
ocsenave@4
|
1688 placed together should have twice the entropy of a single system; or
|
|
ocsenave@4
|
1689 more generally,
|
|
ocsenave@4
|
1690
|
|
ocsenave@4
|
1691 \begin{equation}
|
|
ocsenave@4
|
1692 S(qn, qV, T) = q\cdot S(n,v,T),\qquad 0<q<\infty
|
|
ocsenave@4
|
1693 \end{equation}
|
|
ocsenave@4
|
1694
|
|
ocsenave@4
|
1695 Substituting (1-54) into (1-55), we find that $f(n)$ must satisfy
|
|
ocsenave@4
|
1696 the functional equation
|
|
ocsenave@4
|
1697
|
|
ocsenave@4
|
1698 \begin{equation}
|
|
ocsenave@4
|
1699 f(q\cdot n) = q\cdot f(n) - R\cdot n\cdot q\log{q}\end{equation}
|
|
ocsenave@4
|
1700
|
|
ocsenave@4
|
1701
|
|
ocsenave@4
|
1702 To solve this, one can differentiate with respect to $q$ and set
|
|
ocsenave@4
|
1703 $q = 1$; we then obtain the differential equation
|
|
ocsenave@4
|
1704
|
|
ocsenave@4
|
1705 \begin{equation}
|
|
ocsenave@4
|
1706 n\cdot f^\prime(n) - f(n) + R\cdot n = 0
|
|
ocsenave@4
|
1707 \end{equation}
|
|
ocsenave@4
|
1708 # xy' - y + rx = 0
|
|
ocsenave@4
|
1709 which is readily solved; alternatively, just set $n = 1$ in (1-56)
|
|
ocsenave@4
|
1710 and replace $q$ by $n$ By either procedure we find
|
|
ocsenave@4
|
1711
|
|
ocsenave@4
|
1712 \begin{equation}
|
|
ocsenave@4
|
1713 f(n) = n\cdot f(1) - R\cdot n \log{n} (1-58)
|
|
ocsenave@4
|
1714 \end{equation}
|
|
ocsenave@4
|
1715
|
|
ocsenave@4
|
1716 As a check, it is easily verified that this is the solution of (1-56)
|
|
ocsenave@4
|
1717 and (1-57). We then have finally,
|
|
ocsenave@4
|
1718
|
|
ocsenave@4
|
1719 \begin{equation}
|
|
ocsenave@4
|
1720 S(n,V,t) = n\left[C_v\cdot\log{t} + R\cdot \log{\left(\frac{V}{n}\right)} +
|
|
ocsenave@4
|
1721 A\right]
|
|
ocsenave@4
|
1722 \end{equation}
|
|
ocsenave@4
|
1723
|
|
ocsenave@4
|
1724 where $A\equiv f(1)$ is still an arbitrary constant, not determined
|
|
ocsenave@4
|
1725 by the definition (1-19), or by the condition (1-55) that $S$ be
|
|
ocsenave@4
|
1726 extensive. However, $A$ is not without physical meaning; we will
|
|
ocsenave@4
|
1727 see in the next Section that the vapor pressure of this
|
|
ocsenave@4
|
1728 substance (and more generally, its chemical potential) depends on
|
|
ocsenave@4
|
1729 $A$. Later, it will appear that the numerical value of $A$ involves
|
|
ocsenave@3
|
1730 Planck's constant, and its theoretical determination therefore
|
|
ocsenave@4
|
1731 requires quantum statistics.
|
|
ocsenave@3
|
1732
|
|
ocsenave@4
|
1733 #edit: "is constant"
|
|
ocsenave@4
|
1734 We conclude from this that, in any region where experimentally
|
|
ocsenave@4
|
1735 $C_V$ is constant, and the ideal gas equation of state is
|
|
ocsenave@4
|
1736 obeyed, the entropy must have the form (1-59) The fact that
|
|
ocsenave@3
|
1737 classical statistical mechanics does not lead to this result,
|
|
ocsenave@4
|
1738 the term $n\cdot R \cdot \log{(1/n)}$ being missing (Gibbs paradox),
|
|
ocsenave@4
|
1739 was historically one of the earliest clues indicating the need for the
|
|
ocsenave@3
|
1740 quantum theory.
|
|
ocsenave@4
|
1741
|
|
ocsenave@4
|
1742 In the case of a liquid, the volume does not change
|
|
ocsenave@4
|
1743 appreciably on heating, and so $dS = n\cdot C_V\cdot dT/T$, and if
|
|
ocsenave@4
|
1744 $C_V$ is independent of temperature, we would have in place of (1-59),
|
|
ocsenave@4
|
1745
|
|
ocsenave@4
|
1746 \begin{equation}
|
|
ocsenave@4
|
1747 S = n\left[C_V\ln{T}+A_\ell\right]
|
|
ocsenave@4
|
1748 \end{equation}
|
|
ocsenave@4
|
1749
|
|
ocsenave@4
|
1750 where $A_\ell$ is an integration constant, which also has physical
|
|
ocsenave@3
|
1751 meaning in connection with conditions of equilibrium between
|
|
ocsenave@3
|
1752 two different phases.
|
|
ocsenave@4
|
1753
|
|
ocsenave@4
|
1754 ** The Second Law: Definition
|
|
ocsenave@4
|
1755
|
|
ocsenave@4
|
1756 Probably no proposition in physics has been the subject of more deep
|
|
ocsenave@4
|
1757 and sustained confusion
|
|
ocsenave@4
|
1758 than the second law of thermodynamics It is not in the province
|
|
ocsenave@3
|
1759 of macroscopic thermodynamics to explain the underlying reason
|
|
ocsenave@3
|
1760 for the second law; but at this stage we should at least be able
|
|
ocsenave@4
|
1761 to /state/ this law in clear and experimentally meaningful terms.
|
|
ocsenave@3
|
1762 However, examination of some current textbooks reveals that,
|
|
ocsenave@3
|
1763 after more than a century, different authors still disagree as
|
|
ocsenave@3
|
1764 to the proper statement of the second law, its physical meaning,
|
|
ocsenave@3
|
1765 and its exact range of validity.
|
|
ocsenave@4
|
1766
|
|
ocsenave@3
|
1767 Later on in this book it will be one of our major objectives
|
|
ocsenave@4
|
1768 to show, from several different viewpoints, how much clearer and
|
|
ocsenave@3
|
1769 simpler these problems now appear in the light of recent develop
|
|
ocsenave@4
|
1770 ments in statistical mechanics For the present, however, our
|
|
ocsenave@3
|
1771 aim is only to prepare the way for this by pointing out exactly
|
|
ocsenave@3
|
1772 what it is that is to be proved later. As a start on this at
|
|
ocsenave@3
|
1773 tempt, we note that the second law conveys a certain piece of
|
|
ocsenave@4
|
1774 informations about the /direction/ in which processes take place.
|
|
ocsenave@3
|
1775 In application it enables us to predict such things as the final
|
|
ocsenave@3
|
1776 equilibrium state of a system, in situations where the first law
|
|
ocsenave@3
|
1777 alone is insufficient to do this.
|
|
ocsenave@4
|
1778
|
|
ocsenave@4
|
1779
|
|
ocsenave@3
|
1780 A concrete example will be helpful. We have a vessel
|
|
ocsenave@4
|
1781 equipped with a piston, containing $N$ moles of carbon dioxide.
|
|
ocsenave@3
|
1782
|
|
ocsenave@4
|
1783 #changed V_f to V_1
|
|
ocsenave@4
|
1784 The system is initially at thermal equilibrium at temperature $T_0$,
|
|
ocsenave@4
|
1785 volume $V_0$ and pressure $P_O$; and under these conditions it contains
|
|
ocsenave@4
|
1786 $n$ moles of CO_2 in the vapor phase and $N-n$ moles in the liquid
|
|
ocsenave@4
|
1787 phase The system is now thermally insulated from its
|
|
ocsenave@4
|
1788 surroundings, and the piston is moved rapidly (i.e., so that $n$ does not
|
|
ocsenave@3
|
1789 change appreciably during the motion) so that the system has a
|
|
ocsenave@4
|
1790 new volume $V_1$; and immediately after the motion, a new pressure
|
|
ocsenave@4
|
1791 $P_1$ The piston is now held fixed in its new position, and the
|
|
ocsenave@3
|
1792 system allowed to come once more to equilibrium. During this
|
|
ocsenave@4
|
1793 process, will the CO_2 tend to evaporate further, or condense further?
|
|
ocsenave@4
|
1794 What will be the final equilibrium temperature $T_{eq}$,
|
|
ocsenave@4
|
1795 the final pressure $P_eq$, and final value of $n_{eq}$?
|
|
ocsenave@4
|
1796
|
|
ocsenave@4
|
1797 It is clear that the first law alone is incapable of answering
|
|
ocsenave@3
|
1798 these questions; for if the only requirement is conservation of
|
|
ocsenave@4
|
1799 energy, then the CO_2 might condense, giving up its heat of
|
|
ocsenave@4
|
1800 vaporization and raising the temperature of the system; or it might
|
|
ocsenave@3
|
1801 evaporate further, lowering the temperature. Indeed, all values
|
|
ocsenave@4
|
1802 of $n_{eq}$ in $O \leq n_{eq} \leq N$ would be possible without any
|
|
ocsenave@4
|
1803 violation of the first law. In practice, however, this process will be found
|
|
ocsenave@4
|
1804 to go in only one direction and the system will reach a definite
|
|
ocsenave@3
|
1805 final equilibrium state with a temperature, pressure, and vapor
|
|
ocsenave@3
|
1806 density predictable from the second law.
|
|
ocsenave@4
|
1807
|
|
ocsenave@4
|
1808
|
|
ocsenave@3
|
1809 Now there are dozens of possible verbal statements of the
|
|
ocsenave@3
|
1810 second law; and from one standpoint, any statement which conveys
|
|
ocsenave@4
|
1811 the same information has equal right to be called \ldquo{}the second
|
|
ocsenave@4
|
1812 law.\rdquo{} However, not all of them are equally direct statements of
|
|
ocsenave@3
|
1813 experimental fact, or equally convenient for applications, or
|
|
ocsenave@3
|
1814 equally general; and it is on these grounds that we ought to
|
|
ocsenave@4
|
1815 choose among them.
|
|
ocsenave@4
|
1816
|
|
ocsenave@4
|
1817 Some of the mos t popular statements of the second law
|
|
ocsenave@4
|
1818 belong to the class of the well-known \ldquo{}impossibility\rdquo{}
|
|
ocsenave@4
|
1819 assertions; i.e., it is impossible to transfer heat from a lower to a higher
|
|
ocsenave@3
|
1820 temperature without leaving compensating changes in the rest of
|
|
ocsenave@4
|
1821 the universe, it is impossible to convert heat into useful work
|
|
ocsenave@3
|
1822 without leaving compensating changes, it is impossible to make
|
|
ocsenave@3
|
1823 a perpetual motion machine of the second kind, etc.
|
|
ocsenave@3
|
1824
|
|
ocsenave@3
|
1825 Suoh formulations have one clear logical merit; they are
|
|
ocsenave@3
|
1826 stated in such a way that, if the assertion should be false, a
|
|
ocsenave@4
|
1827 single experiment would suffice to demonstrate that fact
|
|
ocsenave@4
|
1828 conclusively. It is good to have our principles stated in such a
|
|
ocsenave@3
|
1829 clear, unequivocal way.
|
|
ocsenave@4
|
1830
|
|
ocsenave@4
|
1831 However, impossibility statements also have some
|
|
ocsenave@4
|
1832 disadvantages In the first place, /they are not, and by their very
|
|
ocsenave@4
|
1833 nature cannot be, statements of eiperimental fact/. Indeed, we
|
|
ocsenave@3
|
1834 can put it more strongly; we have no record of anyone having
|
|
ocsenave@3
|
1835 seriously tried to do any of the various things which have been
|
|
ocsenave@3
|
1836 asserted to be impossible, except for one case which actually
|
|
ocsenave@4
|
1837 succeeded. In the experimental realization of negative spin
|
|
ocsenave@4
|
1838 temperatures, one can transfer heat from a lower to a higher
|
|
ocsenave@3
|
1839 temperature without external changes; and so one of the common
|
|
ocsenave@3
|
1840 impossibility statements is now known to be false [for a clear
|
|
ocsenave@4
|
1841 discussion of this, see the [[../sources/Ramsey.pdf][article of N. F. Ramsey (1956)]];
|
|
ocsenave@3
|
1842 experimental details of calorimetry with negative temperature
|
|
ocsenave@4
|
1843 spin systems are given by Abragam and Proctor (1958)]
|
|
ocsenave@4
|
1844
|
|
ocsenave@4
|
1845
|
|
ocsenave@3
|
1846 Finally, impossibility statements are of very little use in
|
|
ocsenave@4
|
1847 /applications/ of thermodynamics; the assertion that a certain kind
|
|
ocsenave@4
|
1848 of machine cannot be built, or that a certain laboratory feat
|
|
ocsenave@3
|
1849 cannot be performed, does not tell me very directly whether my
|
|
ocsenave@3
|
1850 carbon dioxide will condense or evaporate. For applications,
|
|
ocsenave@3
|
1851 such assertions must first be converted into a more explicit
|
|
ocsenave@3
|
1852 mathematical form.
|
|
ocsenave@4
|
1853
|
|
ocsenave@4
|
1854
|
|
ocsenave@3
|
1855 For these reasons, it appears that a different kind of
|
|
ocsenave@3
|
1856 statement of the second law will be, not necessarily more
|
|
ocsenave@4
|
1857 \ldquo{}correct\rdquo{}, but more useful in practice. Now both Clausius (1875)
|
|
ocsenave@3
|
1858 and Planck (1897) have laid great stress on their conclusion
|
|
ocsenave@3
|
1859 that the most general statement, and also the most immediately
|
|
ocsenave@3
|
1860 useful in applications, is simply the existence of a state
|
|
ocsenave@3
|
1861 function, called the entropy, which tends to increase. More
|
|
ocsenave@3
|
1862 precisely: in an adiabatic change of state, the entropy of
|
|
ocsenave@3
|
1863 a system may increase or may remain constant, but does not
|
|
ocsenave@3
|
1864 decrease. In a process involving heat flow to or from the
|
|
ocsenave@3
|
1865 system, the total entropy of all bodies involved may increase
|
|
ocsenave@4
|
1866 or may remain constant; but does not decrease; let us call this
|
|
ocsenave@4
|
1867 the \ldquo{}weak form\rdquo{} of the second law.
|
|
ocsenave@3
|
1868
|
|
ocsenave@3
|
1869 The weak form of the second law is capable of answering the
|
|
ocsenave@4
|
1870 first question posed above; thus the carbon dioxide will
|
|
ocsenave@4
|
1871 evaporate further if, and only if, this leads to an increase in the
|
|
ocsenave@4
|
1872 total entropy of the system This alone, however, is not enough
|
|
ocsenave@4
|
1873 to answer the second question; to predict the exact final
|
|
ocsenave@4
|
1874 equilibrium state, we need one more fact.
|
|
ocsenave@4
|
1875
|
|
ocsenave@3
|
1876 The strong form of the second law is obtained by adding the
|
|
ocsenave@4
|
1877 further assertion that the entropy not only \ldquo{}tends\rdquo{} to increase;
|
|
ocsenave@4
|
1878 in fact it /will/ increase, /to the maximum value permitted by the
|
|
ocsenave@4
|
1879 constraints imposed[fn::Note, however, that the second law has
|
|
ocsenave@4
|
1880 nothing to say about how rapidly this approach to equilibrium takes place.]/. In the case of the carbon dioxide, these
|
|
ocsenave@4
|
1881 constraints are: fixed total energy (first law), fixed total
|
|
ocsenave@4
|
1882 amount of carbon dioxide, and fixed position of the piston. The
|
|
ocsenave@3
|
1883 final equilibrium state is the one which has the maximum entropy
|
|
ocsenave@4
|
1884 compatible with these constraints, and it can be predicted
|
|
ocsenave@4
|
1885 quantitatively from the strong form of the second law if we know,
|
|
ocsenave@3
|
1886 from experiment or theory, the thermodynamic properties of carbon
|
|
ocsenave@4
|
1887 dioxide (ie, heat capacity, equation of state, heat of vapor
|
|
ocsenave@4
|
1888 ization)
|
|
ocsenave@3
|
1889
|
|
ocsenave@4
|
1890 To illustrate this, we set up the problem in a crude
|
|
ocsenave@4
|
1891 approximation which supposes that (l) in the range of conditions
|
|
ocsenave@4
|
1892 of interest, the molar heat capacity $C_v$ of the vapor, and $C_\ell$ of
|
|
ocsenave@4
|
1893 the liquid, and the molar heat of vaporization $L$, are all con
|
|
ocsenave@4
|
1894 stants, and the heat capacities of cylinder and piston are
|
|
ocsenave@4
|
1895 negligible; (2) the liquid volume is always a small fraction of the
|
|
ocsenave@4
|
1896 total $V$, so that changes in vapor volume may be neglected; (3) the
|
|
ocsenave@4
|
1897 vapor obeys the ideal gas equation of state $PV = nRT$. The
|
|
ocsenave@4
|
1898 internal energy functions of liquid and vapor then have the form
|
|
ocsenave@4
|
1899
|
|
ocsenave@4
|
1900 \begin{equation}
|
|
ocsenave@4
|
1901 U_\ell = (N-n)\left[C_\ell\cdot T + A\right]
|
|
ocsenave@4
|
1902 \end{equation}
|
|
ocsenave@4
|
1903 \begin{equation}
|
|
ocsenave@4
|
1904 U_v = n\left[C_v\cdot T + A + L\right]
|
|
ocsenave@4
|
1905 \end{equation}
|
|
ocsenave@4
|
1906
|
|
ocsenave@4
|
1907 where $A$ is a constant which plays no role in the problem. The
|
|
ocsenave@4
|
1908 appearance of $L$ in (1-62) recognizes that the zero from which we
|
|
ocsenave@3
|
1909 measure energy of the vapor is higher than that of the liquid by
|
|
ocsenave@4
|
1910 the energy $L$ necessary to form the vapor. On evaporation of $dn$
|
|
ocsenave@4
|
1911 moles of liquid, the total energy increment is $dU = dU_\ell + dU_v =
|
|
ocsenave@4
|
1912 0$; or
|
|
ocsenave@4
|
1913
|
|
ocsenave@4
|
1914 \begin{equation}
|
|
ocsenave@4
|
1915 \left[n\cdot C_v + (N-n)C_\ell\right] dT + \left[(C_v-C_\ell)T + L\right]dn = 0
|
|
ocsenave@4
|
1916 \end{equation}
|
|
ocsenave@4
|
1917
|
|
ocsenave@3
|
1918 which is the constraint imposed by the first law. As we found
|
|
ocsenave@4
|
1919 previously (1-59), (1-60) the entropies of vapor and liquid are
|
|
ocsenave@3
|
1920 given by
|
|
ocsenave@4
|
1921
|
|
ocsenave@4
|
1922 \begin{equation}
|
|
ocsenave@4
|
1923 S_v = n\left[C_v\cdot\ln{T} + R\cdot \ln{\left(V/n\right)} + A_v\right]
|
|
ocsenave@4
|
1924 \end{equation}
|
|
ocsenave@4
|
1925 \begin{equation}
|
|
ocsenave@4
|
1926 S_\ell = (N-n)\left[C_\ell\cdot \ln{T}+A_\ell\right]
|
|
ocsenave@4
|
1927 \end{equation}
|
|
ocsenave@4
|
1928
|
|
ocsenave@4
|
1929 where $A_v$, $A_\ell$ are the constants of integration discussed in the
|
|
ocsenave@3
|
1930 last Section.
|
|
ocsenave@4
|
1931
|
|
ocsenave@4
|
1932
|
|
ocsenave@3
|
1933 We leave it as an exercise for the reader to complete the
|
|
ocsenave@4
|
1934 derivation from this point, and show that the total entropy
|
|
ocsenave@4
|
1935 $S = S_\ell + S_v$ is maximized subject to the constraint (1-63), when
|
|
ocsenave@4
|
1936 the values $n_{eq}$, $T_{eq}$ are related by
|
|
ocsenave@4
|
1937
|
|
ocsenave@4
|
1938 \begin{equation}
|
|
ocsenave@4
|
1939 \frac{n_{eq}}{V}= B\cdot T_{eq}^a\cdot \exp{\left(-\frac{L}{RT_{eq}}\right)}
|
|
ocsenave@4
|
1940 \end{equation}
|
|
ocsenave@4
|
1941
|
|
ocsenave@4
|
1942 where $B\equiv \exp{(-1-a-\frac{A_\ell-A_v}{R})}$ and $a\equiv
|
|
ocsenave@4
|
1943 (C_v-C_\ell)/R$ are constants.
|
|
ocsenave@4
|
1944
|
|
ocsenave@4
|
1945
|
|
ocsenave@3
|
1946 Equation (1-66) is recognized as an approximate form of the Vapor
|
|
ocsenave@4
|
1947 pressure formula
|
|
ocsenave@3
|
1948 We note that AQ, AV, which appeared first as integration
|
|
ocsenave@4
|
1949 constants for the entropy with no parti cular physical meaning,
|
|
ocsenave@3
|
1950 now play a role in determining the vapor pressure.
|
|
ocsenave@4
|
1951
|
|
ocsenave@4
|
1952 ** The Second Law: Discussion
|
|
ocsenave@4
|
1953
|
|
ocsenave@4
|
1954 We have emphasized the distinction between the weak and strong forms
|
|
ocsenave@4
|
1955 of the second law
|
|
ocsenave@4
|
1956 because (with the exception of Boltzmann's original unsuccessful
|
|
ocsenave@4
|
1957 argument based on the H-theorem), most attempts to deduce the
|
|
ocsenave@4
|
1958 second law from statistical mechanics have considered only the
|
|
ocsenave@3
|
1959 weak form; whereas it is evidently the strong form that leads
|
|
ocsenave@3
|
1960 to definite quantitative predictions, and is therefore needed
|
|
ocsenave@4
|
1961 for most applications. As we will see later, a demonstration of
|
|
ocsenave@4
|
1962 the weak form is today almost trivial---given the Hamiltonian form
|
|
ocsenave@4
|
1963 of the equations of motion, the weak form is a necessary
|
|
ocsenave@4
|
1964 condition for any experiment to be reproducible. But demonstration
|
|
ocsenave@4
|
1965 of the strong form is decidedly nontrivial; and we recognize from
|
|
ocsenave@4
|
1966 the start that the job of statistical mechanics is not complete
|
|
ocsenave@4
|
1967 until that demonstration is accomplished.
|
|
ocsenave@2
|
1968
|
|
ocsenave@2
|
1969
|
|
ocsenave@4
|
1970 As we have noted, there are many different forms of the
|
|
ocsenave@4
|
1971 seoond law, that have been favored by various authors. With
|
|
ocsenave@4
|
1972 regard to the entropy statement of the second law, we note the
|
|
ocsenave@4
|
1973 following. In the first place, it is a direct statement of
|
|
ocsenave@4
|
1974 experimental fact, verified in many thousands of quantitative
|
|
ocsenave@4
|
1975 measurements, /which have actually been performed/. This is worth a
|
|
ocsenave@4
|
1976 great deal in an age when theoretical physics tends to draw
|
|
ocsenave@4
|
1977 sweeping conclusions from the assumed outcomes of
|
|
ocsenave@4
|
1978 \ldquo{}thought-experiments.\rdqquo{} Secondly, it has stood the test
|
|
ocsenave@4
|
1979 of time; it is the entropy statement which remained valid in the case
|
|
ocsenave@4
|
1980 of negative spin temperatures, where some others failed. Thirdly, it
|
|
ocsenave@4
|
1981 is very easy to apply in practice, the weak form leading
|
|
ocsenave@4
|
1982 immediately to useful predictions as to which processes will go and
|
|
ocsenave@4
|
1983 which will not; the strong form giving quantitative predictions
|
|
ocsenave@4
|
1984 of the equilibrium state. At the present time, therefore, we
|
|
ocsenave@4
|
1985 cannot understand what motivates the unceasing attempts of many
|
|
ocsenave@4
|
1986 textbook authors to state the second law in new and more
|
|
ocsenave@4
|
1987 complicated ways.
|
|
ocsenave@4
|
1988
|
|
ocsenave@4
|
1989 One of the most persistent of these attempts involves the
|
|
ocsenave@4
|
1990 use of [[http://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Principle_of_Carath.C3.A9odory][Carath\eacute{}odory's principle]]. This states that, in the
|
|
ocsenave@4
|
1991 neighborhood of any thermodynamic state there are other states which
|
|
ocsenave@4
|
1992 cannot be reached by an adiabatic process. After some mathematical
|
|
ocsenave@4
|
1993 analysis
|
|
ocsenave@4
|
1994 [Margenau and Murphy (1943), pp. 26-31; or Wannier (1966),
|
|
ocsenave@4
|
1995 pp. 126-132]
|
|
ocsenave@4
|
1996 one infers the existence of a state function (entropy) which tends
|
|
ocsenave@4
|
1997 to increase; or at least, cannot decrease. From a /mathematical/
|
|
ocsenave@4
|
1998 standpoint there can be no objection at all to this; the analysis
|
|
ocsenave@4
|
1999 is quite rigorous. But from a /physical/ standpoint it is subject
|
|
ocsenave@4
|
2000 to the same objection that its premise is an impossibility
|
|
ocsenave@4
|
2001 statement, and therefore not an experimental fact.
|
|
ocsenave@4
|
2002 Indeed, the conclusion of Carath\eacute{}odory's
|
|
ocsenave@4
|
2003 argument is a far more direct statement of observed fact than its
|
|
ocsenave@4
|
2004 premise; and so it would seem more logical to use the argument
|
|
ocsenave@4
|
2005 backwards. Thus, from the experimental fact that the entropy
|
|
ocsenave@4
|
2006 tends to increase, we would infer that there must exist
|
|
ocsenave@4
|
2007 neighboring states inaccessible in an adiabatic process; but the
|
|
ocsenave@4
|
2008 result is then trivial. In a similar way, other impossibility
|
|
ocsenave@4
|
2009 statements follow trivially from the entropy statement of the
|
|
ocsenave@4
|
2010 second law.
|
|
ocsenave@4
|
2011
|
|
ocsenave@4
|
2012
|
|
ocsenave@4
|
2013 Finally, we note that all statements of the second law are
|
|
ocsenave@4
|
2014 subject to a very important qualification, not always sufficiently
|
|
ocsenave@4
|
2015 emphasized. As we stress repeatedly, conventional thermodynamics
|
|
ocsenave@4
|
2016 is a theory only of states of thermal equilibrium; such concepts
|
|
ocsenave@4
|
2017 as temperature and entropy are not even defined for others.
|
|
ocsenave@4
|
2018 Therefore, all the above statements of the second law must be under
|
|
ocsenave@4
|
2019 stood as describing only the /net result/ of processes /which begin
|
|
ocsenave@4
|
2020 and end in states of complete thermal equilibrium/. Classical
|
|
ocsenave@4
|
2021 thermodynamics has nothing to say about processes that do not
|
|
ocsenave@4
|
2022 meet this condition, or about intermediate states of processes
|
|
ocsenave@4
|
2023 that do. Again, it is nuclear magnetic resonance (NMR)
|
|
ocsenave@4
|
2024 experiments which provide the most striking evidence showing how
|
|
ocsenave@4
|
2025 essential this qualification is; the spin-echo experiment
|
|
ocsenave@4
|
2026 (Hahn, 1950) is, as we will see in detail later, a gross violation of
|
|
ocsenave@4
|
2027 any statement of the second law that fails to include it.
|
|
ocsenave@4
|
2028
|
|
ocsenave@4
|
2029
|
|
ocsenave@4
|
2030 This situation has some interesting consequences, in that
|
|
ocsenave@4
|
2031 impossibility statements may be misleading if we try to read too
|
|
ocsenave@4
|
2032 much into them. From classical thermodynamics alone, we cannot
|
|
ocsenave@4
|
2033 logically infer the impossibility of a \ldquo{}perpetual motion machine\rdquo{}
|
|
ocsenave@4
|
2034 of the second kind (i.e., a machine which converts heat energy
|
|
ocsenave@4
|
2035 into useful work without requiring any low temperature heat sink,
|
|
ocsenave@4
|
2036 as does the Carnot engine); we can infer only that such a machine
|
|
ocsenave@4
|
2037 cannot operate between equilibrium states. More specifically, if
|
|
ocsenave@4
|
2038 the machine operates by carrying out some cyclic process, then
|
|
ocsenave@4
|
2039 the states of (machine + environment) at the beginning and end
|
|
ocsenave@4
|
2040 of a cycle cannot be states of complete thermal equilibrium, as
|
|
ocsenave@4
|
2041 in the reversible Carnot engine. But no real machine operates
|
|
ocsenave@4
|
2042 between equilibrium states anyway. Without some further analysis
|
|
ocsenave@4
|
2043 involving statistical mechanics, we cannot be at all certain that
|
|
ocsenave@4
|
2044 a sufficiently clever inventor could not find a way to convert
|
|
ocsenave@4
|
2045 heat energy into useful work on a commercially profitable scale;
|
|
ocsenave@4
|
2046 the energy is there, and the only question is whether we could
|
|
ocsenave@4
|
2047 persuade it to \ldquo{}organize\rdquo{} itself enough to perform useful work
|
|
ocsenave@4
|
2048 against pistons, magnets, gravitational or electric fields,
|
|
ocsenave@4
|
2049 chemical activation energy hills, etc.
|
|
ocsenave@4
|
2050
|
|
ocsenave@4
|
2051
|
|
ocsenave@4
|
2052 It was Maxwell himself who first ([[../sources/Maxwell-Heat.pdf][1871]])[fn::See also, the [[http://openlibrary.org/books/OL7243600M/Theory_of_heat][Open Library
|
|
ocsenave@4
|
2053 page]], where you can read and download Maxwell's book in a variety of formats.] suggested such
|
|
ocsenave@4
|
2054 possibilities, in his invention of the \ldquo{}Maxwell Demon\rdquo{},
|
|
ocsenave@4
|
2055 an imaginary being (or mechanism) which can regulate valves so as to allow
|
|
ocsenave@4
|
2056 fast molecules to pass through a partition in one direction only,
|
|
ocsenave@4
|
2057 thus heating up one side at the expense of the other. We could
|
|
ocsenave@4
|
2058 then allow the heat to flow back from the hot side to the cold
|
|
ocsenave@4
|
2059 through a conventional Carnot engine, generating useful work; and
|
|
ocsenave@4
|
2060 the whole arrangement would constitute a perpetual motion machine
|
|
ocsenave@4
|
2061 of the second kind.
|
|
ocsenave@4
|
2062
|
|
ocsenave@4
|
2063 #http://naca.larc.nasa.gov/search.jsp?R=19760010893&qs=Ns%3DLoaded-Date|1%26N%3D4294709597
|
|
ocsenave@4
|
2064
|
|
ocsenave@4
|
2065 Maxwell did not regard such a device as impossible in principle;
|
|
ocsenave@4
|
2066 only very difficult technically. Later authors ([[../sources/Szilard.pdf][Szilard, 1929]];
|
|
ocsenave@4
|
2067 Brillouin, 1951, 1956)
|
|
ocsenave@4
|
2068 have argued, on the basis of quantum
|
|
ocsenave@4
|
2069 theory or connections between entropy and information, that it
|
|
ocsenave@4
|
2070 fundamentally impossible. However, all these arguments seem
|
|
ocsenave@4
|
2071 to contain just enough in the way of questionable assumptions or
|
|
ocsenave@4
|
2072 loopholes in the logic, as to leave the critical reader not quite
|
|
ocsenave@4
|
2073 convinced. This is particularly so when we recall the lessons
|
|
ocsenave@4
|
2074 of history; clever experimenters have, over and over again, made
|
|
ocsenave@4
|
2075 fools of theorists who were too quick to assert that something
|
|
ocsenave@4
|
2076 cannot be done.
|
|
ocsenave@4
|
2077
|
|
ocsenave@4
|
2078
|
|
ocsenave@4
|
2079 A recent example worth recalling concerns the Overhauser
|
|
ocsenave@4
|
2080 effect in magnetic resonance (enhancement of the polarization
|
|
ocsenave@4
|
2081 of one set of spins by irradiation of another set coupled to them).
|
|
ocsenave@4
|
2082 When this effect was first proposed, several well-known
|
|
ocsenave@4
|
2083 authorities on thermodynamics and statistical mechanics ridiculed the
|
|
ocsenave@4
|
2084 suggestion and asserted that the effect could not possibly exist,
|
|
ocsenave@4
|
2085 because it violated the second law of thermodynamics. This
|
|
ocsenave@4
|
2086 incident is a valuable reminder of how little we really understand
|
|
ocsenave@4
|
2087 the second law, or how to apply it in new situations.
|
|
ocsenave@4
|
2088
|
|
ocsenave@4
|
2089 In this connection, there is a fascinating little gadget
|
|
ocsenave@4
|
2090 known as the Hilsch tube or Vortex tube, in which a jet of
|
|
ocsenave@4
|
2091 compressed air is injected into a pipe at right angles to its
|
|
ocsenave@4
|
2092 axis, but off center so that it sets up a rapid rotational
|
|
ocsenave@4
|
2093 motion of the gas. In some manner, this causes a separation of
|
|
ocsenave@4
|
2094 the fast and slow molecules, cold air collecting along the axis
|
|
ocsenave@4
|
2095 of the tube, and hot air at the walls. On one side of the jet,
|
|
ocsenave@4
|
2096 a diaphragm with a small hole at the center allows only the cold
|
|
ocsenave@4
|
2097 air to escape, the other side is left open so that the hot air
|
|
ocsenave@4
|
2098 can escape. The result is that when compressed air at room
|
|
ocsenave@4
|
2099 temperature is injected, one can obtain air from the hot side
|
|
ocsenave@4
|
2100 at +100^\circ F from the cold side at -70^\circ F, in sufficient quantities
|
|
ocsenave@4
|
2101 to be used for quick-freezing small objects, or for cooling
|
|
ocsenave@4
|
2102 photomultiplier tubes [for construction drawings and experi
|
|
ocsenave@4
|
2103 mental data, see Stong (1960); for a partial thermodynamic
|
|
ocsenave@4
|
2104 analysis, see Hilsch (19-47)].
|
|
ocsenave@4
|
2105
|
|
ocsenave@4
|
2106 Of course, the air could also be cooled by adiabatic expansion
|
|
ocsenave@4
|
2107 (i.e., by doing work against a piston); and it appears that
|
|
ocsenave@4
|
2108 the amount of cooling achieved in vortex tubes is comparable to,
|
|
ocsenave@4
|
2109 but somewhat less than, what could be obtained this way for the
|
|
ocsenave@4
|
2110 same pressure drop. However, the operation of the vortex tube
|
|
ocsenave@4
|
2111 is manifestly not simple adiabatic since no work is
|
|
ocsenave@4
|
2112 done; rather, part of the gas is heated up, at the cost of cooling
|
|
ocsenave@4
|
2113 the rest; i.e., fast and slow molecules are separated spatially.
|
|
ocsenave@4
|
2114 There is, apparently, no violation of the laws of thermodynamics,
|
|
ocsenave@4
|
2115 since work must be supplied to compress the air; nevertheless,
|
|
ocsenave@4
|
2116 the device resembles the Maxwell Demon so much as to make one
|
|
ocsenave@4
|
2117 uncomfortable.. This is so particularly because of our
|
|
ocsenave@4
|
2118 embarrassing inability to explain in detail (i.e., in molecular terms)
|
|
ocsenave@4
|
2119 how such asimple device works. If we did understand it, would
|
|
ocsenave@4
|
2120 we be able to see still more exciting possibilities? No one
|
|
ocsenave@4
|
2121 knows.
|
|
ocsenave@4
|
2122
|
|
ocsenave@4
|
2123
|
|
ocsenave@4
|
2124 It is interesting to note in passing that such considerations
|
|
ocsenave@4
|
2125 were very much in Planck's mind also; in his [[http://books.google.com/books?id=kOjy3FQqXPQC&printsec=frontcover][/Treatise on Thermodynamics/]] (Planck, 1897; 116), he begins his discussion
|
|
ocsenave@4
|
2126 of the second law in these words (translation of A. Ogg):
|
|
ocsenave@4
|
2127 #+begin_quote
|
|
ocsenave@4
|
2128 \ldquo{}We
|
|
ocsenave@4
|
2129 $\ldots$ put forward the following proposition $\ldots$ :
|
|
ocsenave@4
|
2130 /it is impossible to construct an engine which will work a complete cycle,
|
|
ocsenave@4
|
2131 and produce no effect except the raising of a weight and the cooling of a heat-reservoir./ Such an engine could be used simultaneously
|
|
ocsenave@4
|
2132 as a motor and a refrigerator without any waste of energy or
|
|
ocsenave@4
|
2133 material, and would in any case be the most profitable engine
|
|
ocsenave@4
|
2134 ever made. It would, it is true, not be equivalent to perpetual
|
|
ocsenave@4
|
2135 motion, for it does not produce work from nothing, but from the
|
|
ocsenave@4
|
2136 heat which it draws from the reservoir. It would not, therefore,
|
|
ocsenave@4
|
2137 like perpetual motion, contradict the principle of energy, but
|
|
ocsenave@4
|
2138 would nevertheless possess for man the essential advantage of
|
|
ocsenave@4
|
2139 perpetual motion, the supply of work without cost; for the in
|
|
ocsenave@4
|
2140 exhaustible supply of heat in the earth, in the atmosphere, and
|
|
ocsenave@4
|
2141 in the sea, would, like the oxygen of the atmosphere, be at
|
|
ocsenave@4
|
2142 everybody ‘s immediate disposal. For this reason we take the
|
|
ocsenave@4
|
2143 above proposition as our starting point. Since we are to deduce
|
|
ocsenave@4
|
2144 the second law from it, we expect, at the same time, to make a
|
|
ocsenave@4
|
2145 most serviceable application of any natural phenomenon which may
|
|
ocsenave@4
|
2146 be discovered to deviate from the second law.\rdquo{}
|
|
ocsenave@4
|
2147 #+end_quote
|
|
ocsenave@4
|
2148 The ammonia maser ([[../sources/Townes-Maser.pdf][Townes, 1954]]) is another example of an
|
|
ocsenave@4
|
2149 experimental device which, at first glance, violates the second
|
|
ocsenave@4
|
2150 law by providing \ldquo{}useful work\rdquo{} in the form of coherent microwave
|
|
ocsenave@4
|
2151 radiation at the expense of thermal energy. The ammonia molecule
|
|
ocsenave@4
|
2152 has two energy levels separated by 24.8 GHz, with a large electric
|
|
ocsenave@4
|
2153 dipole moment matrix element connecting them. We cannot obtain
|
|
ocsenave@4
|
2154 radiation from ordinary ammonia gas because the lower state
|
|
ocsenave@4
|
2155 populations are slightly greater than the upper, as given by
|
|
ocsenave@4
|
2156 the usual Boltzmann factors. However, if we release ammonia gas
|
|
ocsenave@4
|
2157 slowly from a tank into a vacuum so that a well-collimated jet
|
|
ocsenave@4
|
2158 of gas is produced, we can separate the upper state molecules
|
|
ocsenave@4
|
2159 from the lower. In an electric field, there is a quadratic
|
|
ocsenave@4
|
2160 Stark effect, the levels \ldquo{}repelling\rdquo{} each other according to
|
|
ocsenave@4
|
2161 the well-known rule of second-order perturbation theory. Thus,
|
|
ocsenave@4
|
2162 the thermally excited upper-state molecules have their energy
|
|
ocsenave@4
|
2163 raised further by a strong field; and vice versa for the lower
|
|
ocsenave@4
|
2164 state molecules. If the field is inhomogeneous, the result is
|
|
ocsenave@4
|
2165 that upper-state molecules experience a force drawing them into
|
|
ocsenave@4
|
2166 regions of weak field; and lower-state molecules are deflected
|
|
ocsenave@4
|
2167 toward strong field regions. The effect is so large that, in a
|
|
ocsenave@4
|
2168 path length of about 15 cm, one can achieve an almost complete
|
|
ocsenave@4
|
2169 spatial separation. The upper-state molecules then pass through
|
|
ocsenave@4
|
2170 a small hole into a microwave cavity, where they give up their
|
|
ocsenave@4
|
2171 energy in the form of coherent radiation.
|
|
ocsenave@4
|
2172
|
|
ocsenave@4
|
2173
|
|
ocsenave@4
|
2174 Again, we have something very similar to a Maxwell Demon;
|
|
ocsenave@4
|
2175 for without performing any work (since no current flows to the
|
|
ocsenave@4
|
2176 electrodes producing the deflecting field) we have separated
|
|
ocsenave@4
|
2177 the high-energy molecules from the low-energy ones, and obtained
|
|
ocsenave@4
|
2178 useful work from the former. This, too, was held to be
|
|
ocsenave@4
|
2179 impossible by some theorists before the experiment succeeded!
|
|
ocsenave@4
|
2180
|
|
ocsenave@4
|
2181 Later in this course, when we have learned how to formulate
|
|
ocsenave@4
|
2182 a general theory of irreversible processes, we will see that the
|
|
ocsenave@4
|
2183 second law can be extended to a new principle that tells us which
|
|
ocsenave@4
|
2184 nonequilibrium states can be reached, reproducibly, from others;
|
|
ocsenave@4
|
2185 and this will of course have a direct bearing on the question of
|
|
ocsenave@4
|
2186 perpetual motion machines of the second kind. However, the full
|
|
ocsenave@4
|
2187 implications of this generalized second law have not yet been
|
|
ocsenave@4
|
2188 worked out; our understanding has advanced just to the point
|
|
ocsenave@4
|
2189 where confident, dogmatic statements on either side now seem
|
|
ocsenave@4
|
2190 imprudent. For the present, therefore, we leave it as an open
|
|
ocsenave@4
|
2191 question whether such machines can or cannot be made.
|
|
ocsenave@4
|
2192
|
|
ocsenave@2
|
2193
|
|
ocsenave@2
|
2194 * COMMENT Appendix
|
|
ocsenave@1
|
2195
|
|
ocsenave@1
|
2196 | Generalized Force | Generalized Displacement |
|
|
ocsenave@1
|
2197 |--------------------+--------------------------|
|
|
ocsenave@1
|
2198 | force | displacement |
|
|
ocsenave@1
|
2199 | pressure | volume |
|
|
ocsenave@1
|
2200 | electric potential | charge |
|
