Cyclic path

When we go through a cyclic process and end up where we started, AZ = 0, since we have returned to the same state, and therefore, the sum of changes in Z for the steps in the closed cyclic path must also equal 0. We can write this mathematically as... 

We have already established that an integral of a differential expression associated with a state function is zero over a closed path. Now, we must consider whether the converse of that statement is true. That is, if the integral of a general differential expression over a closed path is found to be zero, this expression is the differential expression of some state function. To answer the question, let us reconsider the example described in Figure (A 1.1) and equations (A 1.13) and (A 1.14) and assume that equation (A 1.17) is true for all closed (cyclic) paths. Then, for a path 1 and a path 3 that connect the same two states, 1 and 2,... 

To have a better appreciation of the utility of these representations let us first consider the laws that govern flow rates and pressure drops in a pipeline network. These are the counterparts to KirchofTs laws for electrical circuits, namely, (i) the algebraic sum of flows at each vertex must be zero (ii) the algebraic sum of pressure drops around any cyclic path must be zero. For a connected network with N vertices and P edges there will be (N — 1) independent equations corresponding to the first law (KirchofTs current... 

In what follows, a one-dimensional picture will mainly be employed to avoid a notational proliferation. We will also consider a single atomic solute. The methods discussed are easily generalized to three dimensions and many particles the x, y, and z coordinates of the cyclic path for each atom are represented by a set of Fourier coefficients. 

The sum of the free-energy change around any cyclic path must be zero. 

The product of equilibrium constants around any cyclic path of reactions must equal 1. 

A third difference between AU and W lies in the values obtained if one uses a cyclic path, as in moving the boulder up the hill and then back down to the... 

The energetics of enzymatic and their corresponding uncatalyzed reference reactions can be understood by the cyclic path that allows for substrate conversion to product by the uncatalyzed and enzymatic routes (Fig. 2). Note that the uncatalyzed reaction is characterized by a transition state that is far less stable than its enzymatic counterpart. Note also that the initial and final conditions are the same for either route, an absolute requirement for any catalyzed process i.e., no effect on the overall equilibrium constant). 

The important point is that the final value of the integral depends only on the two endpoints, i.e., the value of the function z at (x, yi) and (x2, y2), but not the chosen path of integration (as illustrated in Sidebar 1.4). Moreover, in the special case of a cyclic integral (denoted ), where initial and final limits coincide, the integral (1.15) necessarily vanishes for an exact differential, independent of how the cyclic path is chosen. We can therefore state the following integral criterion for exactness ... 

Given the validity of (4.25b) for any Carnot cycle, we can extend the result to a general cyclic path, as shown in the following diagram ... 

If we now integrate both sides of inequality (4.38) over any cyclic path, we obtain... 

Figure 3.4 Schematic representation of a quantum particle solvated in a bath of classical molecules. The cyclic path represents the quantum particle in the field created by the classical solvent molecules.

CSA2h index -> charged partial surface area descriptors (O HDCA index) cube root molecular weight -> molecular weight cycle cyclic path - graph cycle matrices (C)... 

A self-returning path is called a cyclic path (or cycle or circuit), i.e. a cycle is a walk with no repeated vertices (i.e. a path) other than its first and last ones (vj = v ). The number of independent cycles (or rings) in a graph is the -> cyclomatic number. Cyclicity C is the number of all possible cycles in a graph. 

In the general acid-catalyzed dehydration of acetaldehyde hydrate, Eigen (1965) has proposed a one-encounter mechanism (transition state 17), in which both the acidity and the basicity (conjugate base) of the catalysts are important (moderated by solvent). Bell (1966) has further discussed the occurrence of cyclic paths in carbonyl hydration. Reimann and Jencks (1966) have concluded from rate and equilibrium data on the addition of hydroxylamine to an aldehyde, that proton... 

The above is a simple example of the anal3dic summation over an alternating series. One can similarly treat more complex paths, which consist of sums of combinations of cyclic paths. In [1], we derive in detail sums over paths (which start and finish at i) constructed on two elemental t3rpes of graphs -star-graphs and chain-graphs. We quote below the results ... 

Degenerate Mechanisms, "Catastrophic Cyclic Paths, and the Unique Path Assunyjtion... 

If sets 1 and 2 are simultaneously important, they constitute a stable cyclic path in which population can move from to (v2,v ) along one path but return to along a distinct second path without loss of total vibrational population A single temperature distribution results. 

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