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We represent two such paths in Figure 2.12 by dashed lines. Rather, if they can be connected, it must be through irreversible adiabatic paths for which d S ≠ 0. However, states on surface S 2, for example, cannot be connected to states on either S 1 or S 3 by any reversible adiabatic path. Presumably all points on the same surface can be connected by some solution curve (reversible adiabatic process). For our purposes, let us assume S 1 > S 2 > S 3 in Figure 2.12. Thus, the surfaces can be expected to be ordered monotonically, either systematically increasing (or decreasing) as one proceeds in a given direction from surface to surface. If they did, states located at the points of intersection would have multiple values of entropy, and this would violate a fundamental property of state functions. Two points which can be connected by a reversible adiabatic path must lie on the same entropy surface and a solution curve must lie wholly within a solution surface. It is easy to see that dS = 0 for a reversible adiabatic process because δ q rev = 0. Each solution surface contains a set of thermodynamic states for which the entropy is constant. The integration yields a family of solution surfaces, S = S( x 1, … x n) = constant. Since d S is an exact differential, equations for d S = 0 can be integrated. In more complicated systems, the paths may be obtained by point-to-point numerical integrations. gg The solid curves in Figure 2.11a and 2.11b are examples of these. But it is still possible to identify reversible adiabatic paths for which the equality δ q rev = 0 is satisfied.
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As an inexact differential expression, equations of the form, δ q rev = 0, cannot be integrated to yield a general solution for a surface q rev = q rev( x 1, … x n). We have seen that for reversible processes, δ q rev is an inexact Pfaffian differential expression, and d S is an exact one. To do so, we return to the realm of Pfaffian differential expressions. We will now consider the last stage in CarathéAodory's development of the second law - the establishment of Clausius' statement represented by equation (2.41). Bevan Ott, Juliana Boerio-Goates, in Chemical Thermodynamics: Principles and Applications, 2000 2.2g Entropy Changes for Reversible and Irreversible Paths