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Unformatted text preview: Chapter 8 Entropy Read BS, Chapter 8 Much as the new property energy arose via consideration of the first law of thermodynamics, we will find it useful to introduce • Entropy : a thermodynamic property which provides a quantitative measure of the disorder of a given thermodynamic state, from consideration of the second law of thermodynamics. The word itself was coined by Clausius, 1 who based it on the combination of ǫν (en) “to put into,” and τρoπ ´ η (trop¯ e), “turn” or “conversion.” The Greek here is a modification of the original Greek of Clausius, who employed ` η τρoπ ` η . An image of the first use of the word is given in Fig. 8.1. In some ways entropy is simply a mathematical convenience and a theoretical construct. However, its resilience is due to the fact that it is useful for engineers to summarize important design concepts for thermal systems such as steam power plants, automobile engines, jet engines, refrigerators, heat pumps, and furnaces. 8.1 Theoretical development Let us motivate the property of entropy by considering Fig. 8.2. Here we perform our analysis on a differential basis. We have a thermal reservoir at T res which delivers a small amount of heat δQ ′ to a reversible cyclic engine, labeled “1.” This engine delivers a small amount of work δW ′ and rejects a small amount of heat δQ to another reservoir at variable T , labeled “2.” This reservoir itself delivers a different small amount of work δW to the surroundings. Let us examine the implications of our temperature definition and the second law of thermodynamics on this scenario. 1 R. Clausius, 1865, “Ueber verschiedene f¨ur die Anwendung bequeme Formen der Hauptgleichungen der mechanischen W¨ armetheorie,” Annalen der Physik und Chemie , 125(7): 353390. 203 204 CHAPTER 8. ENTROPY Figure 8.1: Image capturing the first use of the word entropy, from R. Clausius, 1865. δ Q’ T res reversible cyclic engine δ W’ δ Q T δ W 1 2 combined system boundary Figure 8.2: Sketch of heat engine configuration to motivate the development of entropy. CC BYNCND. 2011, J. M. Powers. 8.1. THEORETICAL DEVELOPMENT 205 We first apply Eq. (7.21) to the configuration of Fig. 8.2: δQ ′ δQ = T res T . (8.1) Thus δQ ′ T res = δQ T . (8.2) Now let us take the combined system, enclosed within the dotted box, to be composed of 1 and 2. The first law in differential form for the combined system is dE = ( δQ ′ ) − ( δW + δW ′ ) . (8.3) Note that we have not yet required the process be cyclic. Also note that δQ is internal and so does not cross the boundary of the combined system and is not present in our first law formulation. Rearrange Eq. (8.3) to get δW + δW ′ = δQ ′ − dE. (8.4) Now use Eq. (8.2) to eliminate δQ ′ in Eq. (8.4): δW + δW ′ = T res δQ T − dE. (8.5) Now let us let this configuration undergo a thermodynamic cycle, indicated by the operation contintegraltext applied to Eq. (8.5): contintegraldisplay...
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 Spring '10
 Powers

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