Read e-book online A First Course in the Mathematical Foundations of PDF

By David R. Owen (auth.)

Research long ago thirty years at the foundations of thermodynamics has led not just to a greater knowing of the early advancements of the topic but additionally to formulations of the 1st and moment legislation that let either a rigorous research of the implications of those legislation and a considerable broadening of the category of platforms to which the legislation can fruitfully be utilized. in addition, modem formulations of the legislation of thermodynamics have now accomplished logically parallel types at a degree obtainable to less than­ graduate scholars in technological know-how and engineering who've accomplished the traditional calculus series and who desire to comprehend the position which arithmetic can play in clinical inquiry. My objective in scripting this publication is to make a number of the modem boost­ ments in thermodyamics on hand to readers with the history and orientation simply pointed out and to offer this fabric within the type of a textual content compatible for a one-semester junior-level path. such a lot of this presentation is taken from notes that I assembled whereas educating one of these direction on events. i discovered that, apart from a short evaluate of line integrals and precise differentials in dimensions and a quick dialogue of infima and suprema of units of genuine numbers, juniors (and even a few mature sophomores) had enough mathematical history to address the subject material. some of the scholars whom I taught had very restricted adventure with formal and rigorous mathematical exposition.

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Products of Systems and Preservation of the First Law Recall that Carnot's heuristic argument in support of his version of the Second Law involved combining two homogeneous fluid bodies performing Carnot cycles to form a third system for which the heat and work in the combined cycle could be obtained simply by adding the corresponding quantities for the two Carnot cycles. " The next definition shows that, from a mathematical point of view, it is easy to provide such a description by first collecting pairs of process generators and states from two given systems with perfect accessibility to form a "product" system and then taking a work action and a heat action for the product system to be the sums of the corresponding actions for the given systems.

J=l, ... j is identically zero, always positive, or always negative. For each cycle I? ) is the "largest" temperature encountered on I? ) is the "smallest" temperature on I? at which heat is emitted (Figure 14). 1. Let 01 < O2 be positive numbers, ~ a homogeneous fluid body which obeys the First and Second Laws, and I? ) 81 +() :s;l- n JH I? 4) with equality holding if and only if the cycle I? is a Carnot path on which heat is emitted only at temperature 01 and absorbed only at O2 , This theorem implies that Carnot heat engines are the most efficient simple cycles which emit heat only at or above a given temperature and absorb heat only at or below a second, higher temperature.

Henceforth in this chapter, we assume given a hotness manifold vii. 3. The Accumulation Function for an Ideal Gas We specify an ideal gas here by giving not only the state space X ~ ++ and the functions 1', f:.. (V, 0) = V' A(V, 0) = V ~w = ~ ++ o(V, 0) = 0(0), but also by giving an empirical temperature scale CPw with range CPw = ~ ++. 46 IV. A Modem Treatment of the Second Law If (V, (J) is a state of <'#, then we call qJi I( (J) the hotness level of the state (V, (J) (Figure 19). One can think of a device which determines the empirical temperature scale qJ'§: take the ideal gas at a standard pressure Po and, keeping its pressure constant at that value, place it in contact with an object at hotness level L and measure the volume Vof the gas; the number qJ'§(L) is then defined by the relation qJ'§(L) = ~ V.

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