Deficit function

As an extension of earlier definitions, Q may now be viewed as the excess or deficit function needed to have Q - (AE + W - S(i)/iidni) vanish identically for any process under consideration this is equivalent to the requirement imposed by the First Law that E be a function of state. For the case where only mechanical work is involved we may therefore introduce in the usual manner the total entropy function S such that... 

The above Law has important ramifications. In any nonadiabatic process the work performed, W, differs from Wa- Since now AE — W 0 it is expedient introduce a deficit function Q that restores the equality we write Q — (AE — W) = 0, where Q is called the heat transfer. 

In this section we study among other topics the processes that involve a difference in temperature between a system and the surroundings with which it interacts. This matter seems not to have received sufficient attention in the literature, even though such a difference is needed in any process that involves a transfer of heat. The discussion is much facilitated by the introduction of a deficit function at various stages of the derivation. These deliberations then lead quite naturally to the generation of a variety of useful functions of state and to a study of their extremal properties. 

The use of inequalities is very awkward in any subsequent mathematical manipulations. As a remedy it is apposite to introduce a positive entropy deficit function d9 > 0 that converts Eq. (1.12.2b) into an equality ... 

This equation shows that in an infinitesimal irreversible process the entropy change of the system is related but not equal to the heat transfer, that involves the inverse of the temperature of the reservoir. To, which is a well characterized quantity. Eq. (1.12.5) replaces the standard inequality dS > diQ/T and provides a second interpretation of the deficit function dO specifies the difference between the entropy change dS and the experimentally determined value of diQfTo in an infinitesimal step. It is only for a reversible process that these quantities match and that do drops out in the irreversible case the heat transfer is numerically smaller. 

This completes the construction of the elementary functions of state for a closed system their generalization to open system is deferred to Section 1.20. In the formulations (l.lS.lg), (1.13.2g), (1.13.3g), (1.13.4f) the deficit function was written out in terms of differentials of functions of state and in terms of (To —T) and (Po — -P)- As a first approximation one may expand these differences in a Taylor s series up to second powers. The same applies to the generalized functions of state (1.13. Id), (1.13.2d), (1.13.3d), (1.13.4c). The ramifications of such steps have not been explored, but a version equivalent to the first power expansion is provided in Chapter 6. 

Up to now heat has been treated as a somewhat aetherial quantity, having been introduced as a deficit function that restores the balance between energy changes and work performance in a system. We complement this earlier presentation with a more meaningful description by introducing a set of units and a method for measuring heat transfers. 

We also generalize the definition of heat Q as the deficit function needed so that dQ — (dE — dW — "J i l i dn ) vanishes identically, in order that E remains a function of state. We then view the entropy change as being given by... 

Could you have adopted the view that the term — li drii should have been grouped with the deficit function dQ rather than with dWl What would be the consequences of adopting such a stance ... 

The above law leads to important consequences. While, as claimed earlier, in an adiabatic process, A —Wa = 0, whereas in any nonadiabatic process, the work performed, W, differs from W, so that A W A 0. Inequalities are not particularly pleasant quantities to deal with. We can rectify this situation by introducing a deficit function Q, called the heat transfer, that permits us to restore the equality. Q is so constructed that Q — [A — W] = 0. This rather austere definition provides very little insight on the physical nature of heat, nor on methods for its detection more on that later on. For the moment, suffice it to state that heat transfers attend to all changes in properties of a system that are not accomplished by execution of work as defined in Section 1.5, or by compositional changes described in Section 1.12. As experienced by humans, heating effects normally manifest themselves as increases in the temperature of the system (hotness levels, in conventional language) or as changes in the phase structure of the system. 

So far all variants of the deficit functions have been presented in differential form. To obtain the total deficit function 6, some type of integration is required. By way of illustration in general terms, consider Eq. (1.10.4f) here we must specify how T,P,V,Si,Vi, change in the irreversible process. The simplest procedure involves introducing t as a parameter and then specifying how the above variables evolve with t during the process. In Appendix B we provide a specific example. 

In summary, in the last two sections, we have addressed the problem of handling irreversibilities by introducing a deficit function that converts the commonly encountered inequalities into equalities. What starts out as a book-keeping operation allows us, at any stage of a given... 

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