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Unformatted text preview: © M. S. Shell 2009 1/9 last modified 10/6/2009 The fundamental equation ChE210A Previously, we saw that the entropy is really a tool to help us find the most likely macroscopic state of a system, i.e., the macroscopic conditions that have the most number of microstates. We also saw that the conditions for equilibrium between two bodies are intimately linked to derivatives of the entropy: thermal equilibrium: gG ¡ g¢ ¡ £ gG ¤ g¢ ¤ or ¥ ¦ ¡ £ ¥ ¦ ¤ mechanical equilibrium: gG ¡ g§ ¡ £ gG ¤ g§ ¤ or ¨ ¡ ¦ ¡ £ ¨ ¤ ¦ ¤ chemical equilibrium: gG ¡ g© ¡ £ gG ¤ g© ¤ or ª ¡ ¦ ¡ £ ª ¤ ¦ ¤ How do we know that the derivatives of the entropy are equal to the quantities 1/« , ¬/« , and ®/« , and not some other quantities or functions? It is precisely because we use equilibrium to measure these functions in reality . When we measure temperature, for example, we allow a thermometer to come to thermal equilibrium with another body with which it exchanges energy. We then use changes in the properties of the thermometer to indicate where we have reached thermal equilibrium. We construct a scale called temperature that depends on these properties. The fact that the derivatives of the entropy actually involve inverse temperatures is really just an after effect of historical convention. We could have just as easily developed thermometers that measure inverse temperatures. We know the ¯,°, and ±derivatives of the entropy correspond to 1/« , ¬/« , and ®/« , respectively, because these are the quantities that become equal be tween two bodies at equilibrium, and because any measurement of «,¬, or ® necessarily involves bringing a measurement device into equilibrium with the system of interest. The fact that all of the derivatives involve inverse tempera tures is merely a feature of convention. Differential and integrated versions of the fundamental equations A convenient way to summarize the entropy function ²³¯,°,±´ is to use the fundamental equation : µ² £ 1 « µ¯ ¶ ¬ « µ° ® « µ± © M. S. Shell 2009 2/9 last modified 10/6/2009 We call this particular form of the fundamental equation the entropy version . An equivalent energy version can be made by rearranging the differentials: gG ¡ ¢g£ ¤ ¥g¦ § ¨g© Here, this version of the fundamental equation describes the function Gª£,¦,©« , and it sum marizes the partial derivatives as so: ¬ gG g£ ®,¯ ¡ ¢ª£,¦,©« ¬ gG g¦ °,¯ ¡ ¤¥ª£,¦,©« ¬ gG g© °,® ¡ ¨ª£,¦,©« All that has been done in going from the entropy to energy version of the fundamental equa tion is to switch out the independent variable from G to £ . It is as if we transformed the equation ± ¡ ²ª³,´« by solving for ³ as a function of the values ± and ´ , ala ³ ¡ µª±,´« . Here, we have also explicitly indicate the dependence of the derivatives on the independent variables £,¦,© ....
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 Fall '08
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 Thermodynamics, Energy, fundamental equation, M. S. Shell

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