This equation does not require that AS be positive for any irreversible process. Q can be sufficiently negative (i.e., heat is removed from the system) to more than compensate for the entropy increase due to irreversibilities. All that Eq (1.10) requires is that in such processes, the entropy change is more positive than the heat addition divided by the temperature. That is, both sides of the inequality in Eq (1.10) may be negative, but the right side is more negative than the left side.

The Second Law can also be written for a differential portion of a process:

V T jrev V T jirr

These equations are the Second-Law analogs of Eq (1.5) for the First Law.

When the process involves a complete cycle, as in Fig. 1.15, the system's entropy returns to its initial value so that AS(cycle) = (pSS = 0 . For the cycle, Eqs (1.9) and (1.10) become:

where the equality applies if all steps in the cycle are reversible and the inequality describes a cycle with irreversibilities. The latter is called the Inequality of Clausius, and can be regarded as another form of the Second Law. Equation (1.12) is the Second-Law analog of Eq (1.7) for the First Law.

The preceding discussion of the Second Law has dealt exclusively with the system, without regard to the entropy changes in the surroundings. Considering the system and surroundings (the "universe"), the total entropy change is:

where the equality applies to reversible processes. Contrary to the energy analog given by Eq (1.8), entropy is not conserved in processes with irreversibilities. The logical consequence of this fact, that the universe is destined to degrade to a uniform mass of indistinguishable dust, has troubled philosophers and cosmologists for a century. However, experience, both scientific and practical, has empirically demonstrated, without exception, the correctness of Eq (1.13).

A practically important special case of Eq (1.13) is provided by changes in an isolated system. Such a system is completely cut off from its surroundings, so that no matter what transpires in the system, ASsl]lr = 0. Equation (1.13) still applies, and takes the form:

The appropriate interpretation of Eq (1.14) is that the entropy of an isolated system seeks its maximum value, which occurs at the state of equilibrium. This equation is often referred to as the maximum entropy principle. Two examples of its application follow. Problems 3.19 and 3.20 utilize this principle for similar situations.

Example: The direction of heat flow

The First Law cannot determine the direction of the heat flows between systems 1 and 2 in Fig. 1.16. We suppose that the reservoirs are infinite in size, so that exchange of a finite quantity of heat does not affect their temperatures, which are taken to be Ti > T2. We know from experience that heat will flow from system 1 to system 2, or; in terms of the directions indicated by the arrows in the drawing, that Q1 must be negative and Q2 must be positive. The Second-Law proof of this intuitive conclusion starts by noting that because T1 and T2 are not infinitesimally close, the process must be irreversible, so

The entropy changes of the individual reservoirs, on the other hand, are given by the reversible formula Eq (1.19). The reason is that no irreversibilities occur within the reservoirs; changes in S of these bodies depend only on the quantity of heat transferred, and not on its origin or destination. Therefore, the above inequality becomes:

q1 qi

Getting Started With Solar

Getting Started With Solar

Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.

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