Entropy deficit function

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 ... 

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... 

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. 

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... 

Earlier, 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. Subsequently, we reintroduced heat in conjunction with the concept of entropy. We now complement the earlier presentations with a more meaningful description by introducing a set of units and a method for measuring heat transfers. 

We base our present development on the discussion of Section 1.16 where a distinction was made between the entropy change of a system dS - dQr/T, associated with the reversible transfer of heat across the boundaries, and the entropy change arising from irreversible processes occurring totally within the system. As shown in Section 1.15, the heat transfer is subject to the relation 3Q < TdS whenever the inequality sign applies, one introduces the deficit entropy function 9 so as to render... 

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