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Extra info for A First Course in the Mathematical Foundations of Thermodynamics

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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.

21) for every state (V, (J) in ~. 21) (called the Carnot - Clapeyron relation). 21), we assumed without mention that the latent heat is positive. ) We are now in a position to state the following result. 1. Let ff'be a homogeneous fluid body. 23) which hold for every cycle P for ff'. 24) for every standard parameterization (V, if) of each cycle P for ff'. Thus, the laws of thermodynamics imply for homogeneous flUid bodies that the heating divided by temperature has zero integral on cycles. It is interesting to compare this result with the assertion underlying the Caloric Theory, namely, that the heating itself has zero integral on cycles.

Granted the validity of the italicized assertion above, our argument in support of the Second Law proceeds as follows. 1) is false in a particular instance. 3) 15 3. The Second Law; Entropy t WU:%? [emitted] 82 _____ 2 W(<<:%l 8t -----,Tr---- + W(<<:%i [emitted] Figure 8. and, moreover, H+ (C so,) - H+ (C so2 ) = H+ = (C so,) - W(C so,) + W(C S02)- H+ (C S02 ) H- (CsoJ- H- (C so,). 4) Consider now C S02 and (C so) r' the reversal of C so,. The action of (C so) rand C S02 can be depicted schematically as in Figure 8.