The entropy function

In Section 3.3 we concluded that an isolated system can be returned to its original state only when all processes that take place within the system are reversible otherwise, in attempting a cyclic process, at least one work reservoir within the isolated system will have done work and some heat reservoir, also within the isolated system, will have absorbed a quantity of heat. We sought a monotonically varying function that describes these results. The reversible Carnot cycle was introduced to investigate the properties of reversible cycles, and the generality of the results has been shown in the preceding sections. We now introduce the entropy function. 

We continue with a reversible heat engine operating in a Carnot cycle, but center our attention on the working substance rather than on the entire system consisting of the heat engine, the work reservoir, and the two heat reservoirs. For such a cycle we can write 

The quantity [(Q2/T2) + (QJTJ] is therefore the value of the line integral around the cycle, and we find that this value is zero. We thus have 

However, the cyclic integral of an exact differential is zero and therefore QJT is an exact differential of some function. The notation dQ, is used to emphasize that the process is reversible. The new function is called the entropy function and is defined in terms of its differential, so 

11 The change in the value of the entropy function of an isolated system 

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Also called the uncertainty or - because of its formal similarity to the entropy function used in statistical mechanics - Shannon entropy. 

The second general principle is based on the properties of the entropy function, and is contained in the aphorism of Clausius... 

The significance of this relationship is that although qx and q2 are not state functions, q/T is a state function, since the sum of the q/T terms in the cycle add to zero. This important observation led to the formulation of the entropy function. 

In the absence of any data, the maximum of the entropy functional is reached for p(r) = m(r). Any substantial departure from the model, observed in the final map, is really contained in the data as, with the new definition, it costs entropy. This paper presents illustrations of model testing in the case of intermetallic and molecular compounds. 

Mixing SCs in a mixture and overlapping many ordered sequences over a retention axis are two faces of the same coin they are both entropy-creating processes and can equally be interpreted by using the entropy function concept (Dondi et al., 1998). Moreover, it is also worthwhile to compare similarities and differences between two processes producing randomness, the above-described Poisson type (Fig. 4.2a and Eq. 4.4), and the Gaussian one that is so often evocated in many fundamental branches of natural sciences (Feller, 1971). The latter refers to the addition of independent... 

Fig. 3.3. Typical results from a density-of-states simulation in which one generates the entropy for aliquid at fixed N and V (i.e., fixed density) (adapted from [29]). The dimensionless entropy. r/ In ( is shown as a function of potential energy U for the 110-particle Lennard-Jones fluid at p = 0.88. Given an input temperature, the entropy function can be reweighted to obtain canonical probabilities. The most probable potential energy U for a given temperature is related to the slope of this curve, d// /dU(U ) = l/k T, and this temperature-energy relationship is shown by the dotted line. Energy and temperature are expressed in Lennard-Jones units...

The final postulated property of the entropy function is that it vanishes in the state for which (dU/dS)v,Nj = 0, i.e. at zero temperature. An immediate implication of this postulate is that S, unlike U has a uniquely defined zero. 

The most important new concept to come from thermodynamics is entropy. Like volume, internal energy and mole number it is an extensive property of a system and together with these, and other variables it defines an elegant self-consistent theory. However, there is one important difference entropy is the only one of the extensive thermodynamic functions that has no obvious physical interpretation. It is only through statistical integration of the mechanical behaviour of microsystems that a property of the average macrosystem, that resembles the entropy function, emerges. 

That we again have a quantity whose sum over a closed cycle is zero suggests that Q/T a thermodynamic property, even though we know that Q is not a thermodynamic property. Acting on this suggestion, we define the entropy function, as Clausius did, by the equatifon... 

Ultimately, we must realize that entropy is essentially a mathematical function. It is a concise function of the variables of experience, such as temperamre, pressure, and composition. Natural processes tend to occur only in certain directions that is, the variables pressure, temperature, and composition change only in certain—but very complicated—ways, which are described most concisely by the change in a single function, the entropy function (AS > 0). 

The second law is more subtle and difficult to comprehend than the first. The full scope of the second law only became clear after an extended period of time in which (as expressed by Gibbs) truth and error were in a confusing state of mixture. In the present chapter, we focus primarily on the work of Carnot (Sidebar 4.1), Thomson (Sidebar 4.2), and Clausius (Sidebar 4.3), which culminated in Clausius clear enunciation of the second law in terms of the entropy function. This in turn led to the masterful reformulation by J. W. Gibbs, which underlies the modem theory of chemical and phase thermodynamics and is introduced in Chapter 5. 

The entropy function S immediately simplifies thermodynamic theory in important respects. From (4.28), we can recognize that S and V are the basic variables of the energy function U,... 

The entropy function S also simplifies the graphical depiction of the Carnot cycle. Consider, for example, the form of the Carnot cycle shown in the PV diagram of Fig. 4.4a. The corresponding ST diagram for the same Carnot cycle is shown in Fig. 4.4b. As can be seen, the ST representation of the Carnot cycle is a simple rectangle whose... 

It was the principal genius of J. W. Gibbs (Sidebar 5.1) to recognize how the Clausius statement could be recast in a form that made reference only to the analytical properties of individual equilibrium states. The essence of the Clausius statement is that an isolated system, in evolving toward a state of thermodynamic equilibrium, undergoes a steady increase in the value of the entropy function. Gibbs recognized that, as a consequence of this increase, the entropy function in the eventual equilibrium state must have the character of a mathematical maximum. As a consequence, this extremal character of the entropy function makes possible an analytical characterization of the second law, expressible entirely in terms of state properties of the individual equilibrium state, without reference to cycles, processes, perpetual motion machines, and the like. 

Gibbs criterion (I) In an isolated equilibrium system, the entropy function has the mathematical character of a maximum with respect to variations that do not alter the energy. 

According to the Gibbs criterion, the entropy function S is a maximum (with respect to certain allowed variations). Recall that for a general function f(x, y,...), the conditions that/be a maximum are ... 

Let us now return to the original problem of maximizing the entropy function (5.7) subject to the constraints (5.6b-d). With Lagrange multipliers Av, Ay, and AN, the constrained function S is... 

The inequality (5.26) merely says that the entropy function was at a maximum before the variation, which is the counterpart of the Clausius statement [cf. (4.48)]... 

Figure 5.1 Schematic plots of (a) constrained entropy Sjj and (b) unconstrained entropy S as functions of a general extensive property X near equilibrium, Xeq. In each case, the negative curvature of the entropy function (constrained or unconstrained) carries it below its equilibrium tangent (dashed line).

As shown in the figure, the curvature of the entropy function always causes it to fall below its tangent planes. A mathematical object having such distinctive global curvature (such as an eggshell or an upside-down bowl) is called convex. Accordingly, we may restate the Gibbs criterion in terms of this intrinsic convexity property of the entropy function S = S(U,V,N) ... 

Figure 5.2 Schematic 3-dimensional depiction of the entropy function S(U, A), showing the tangent plane (planar grid) at an equilibrium state (small circle). According to the curvature (stability) condition, the entropy function always falls below its equilibrium tangent planes, and thus has the form of a convex function. ...

Since both AG and AHX have been dissected into initial-state and transition-state contributions for the tetraethyltin/mercuric chloride reaction, it is possible to achieve a similar separation in terms of the entropy function some values given by Abraham24 are in Table 16. 

Thermodynamics comprises a field of knowledge that is fundamental and applicable to a vast area of human experience. It is a study of the interactions between two or more bodies, the interactions being described in terms of the basic concepts of heat and work. These concepts are deduced from experience, and it is this experience that leads to statements of the first and second laws of thermodynamics. The first law leads to the definition of the energy function, and the second law leads to the definition of the entropy function. With the experimental establishment of these laws, thermodynamics gives an elegant and exact method of studying and determining the properties of natural systems. 

In the development of the second law and the definition of the entropy function, we use the phenomenological approach as we did for the first law. First, the concept of reversible and irreversible processes is developed. The Carnot cycle is used as an example of a reversible heat engine, and the results obtained from the study of the Carnot cycle are generalized and shown to be the same for all reversible heat engines. The relations obtained permit the definition of a thermodynamic temperature scale. Finally, the entropy function is defined and its properties are discussed. 

Having defined the entropy function, we must next determine some of its properties, particularly its change in reversible and irreversible processes taking place in isolated systems. (In each case a simple process is considered first, then a generalization.)... 

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