Thermodynamic Inequalities

The inequality between geometric mean and arithmetic mean turns out by entropy considerations [16, 17]. 

We consider n identical heat reservoirs each having the same constant heat capacity C. However, heat reservoirs should be initially at different absolute temperatures Ti, i = 1. n. 

When the heat reservoirs are allowed to come into thermal contact they will reach Anally an equilibrium temperature Tg. From the energy conservation it follows 

In the special case of equal heat capacity, Tg is the geometric mean of the initial temperatures, i.e.. 

if we start from a nonequilibrium state the total entropy must increase, or in the limiting case of initial equilibrium be equal to zero. From Eq. (10.11) it follows that 

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The two main conditions (besides the stability of the a radical towards the solvent to observe such an electron catalysis are a sufficient high rate of addition of the nucleophile and the thermodynamic inequality E°ll >E°12 implying a fast displacement of the latter equilibrium to the direction of the formation of the anion radical of 71. 

The basic thermodynamic inequality contained in Eq. (7) can conveniently be expressed in terms of the dimensionless coupling strength. 

Incidentally, a thermodynamic inequality determines whether a shock is stable, ie, whether it does break into several shocks or becomes a gradually rising wave. A shock is stable if the isentrope satisfies the condition ... 

The thermodynamic inequality may also guide the selection of general constitutive laws governing the diffusion of momentum, energy, and species mass in non-equilibrium chemical reacting mixtures. 

From a multiple scale modeling perspective, the presence of phenomenological parameters in various effective theories provides an opportunity for information passage in which one theory s phenomenological parameters are seen as derived quantities of another. We have already seen that although the linear theory of elasticity is silent on the particular values adopted by the elastic moduli (except for important thermodynamic inequalities), these parameters may be deduced on the basis of microscopic analysis. The advent of reliable models of material behavior makes it possible to directly calculate these parameters, complementing the more traditional approach which is to determine them experimentally. 

The expressions (62.15) and (62.16) constitute the sufficient conditions that ascertain the thermodynamic inequality. ... 

The elastic theory of biaxial nematic liquid crystals was developed as the extension of the above-mentioned elastic theory of uniaxial nematics [38]. Fel has discovered a number of volume and surface elastic constants for biaxial nematic liquid crystals in 32-point crystallographic symmetry groups, as well as a number of the corresponding thermodynamic inequalities between these constants, to provide nonnegative values of the elastic energy. 

The sign of AG can be used to predict the direction in which a reaction moves to reach its equilibrium position. A reaction is always thermodynamically favored when enthalpy decreases and entropy increases. Substituting the inequalities AH < 0 and AS > 0 into equation 6.2 shows that AG is negative when a reaction is thermodynamically favored. When AG is positive, the reaction is unfavorable as written (although the reverse reaction is favorable). Systems at equilibrium have a AG of zero. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

In some cases besides the governing algebraic or differential equations, the mathematical model that describes the physical system under investigation is accompanied with a set of constraints. These are either equality or inequality constraints that must be satisfied when the parameters converge to their best values. The constraints may be simply on the parameter values, e.g., a reaction rate constant must be positive, or on the response variables. The latter are often encountered in thermodynamic problems where the parameters should be such that the calculated thermophysical properties satisfy all constraints imposed by thermodynamic laws. We shall first consider equality constraints and subsequently inequality constraints. 

Membrane separations to most scientists and engineers equates to a separation brought about by the application of a pressure difference across the membrane with the higher pressure on the mixture side. The thermodynamic basis for the separation is the inequality in the chemical potential, /z across the membrane for each component ... 

These bounds originate from the systematic errors (biases) due to the finite sampling in free energy simulations and they differ from other inequalities such as those based on mathematical statements or the second law of thermodynamics. The bounds become tighter with more sampling. It can be shown that, statistically, in a forward calculation AA(M) < AA(N) for sample sizes M and N and M > N. In a reverse calculation, AA(M) > AA(N). In addition, one can show that the inequality (6.27) presents a tighter bound than that of the second law of thermodynamics... 

Note that the true free energy difference is A A = AAlDO, and the other limit gives the average work, W) = AA. The second law of thermodynamics indicates that AAoq = A A < (W) = AA. In fact, due to the monotonic behavior of the Boltzmann factor in (6.84), one has the general inequality... 

The combination of the Clausius inequality (eq. 1.30) and the first law of thermodynamics for a system at constant volume thus gives... 

We often want to perform thermodynamic studies isothermally because, that way, we need no subsequent corrections for inequalities in temperature isothermal measurements generally simplify our calculations. 

Figure 3.12) and is the substance of linear programming. Let us take a simple example in n = 2 dimensions and assume that we are looking for the minimum of a linear function /(x), where x is the vector [x,x2]T, with the equality constraint brx = q (b and q being known constants) and the inequality constraints x1 0,x2>0.ln geochemistry or thermodynamics, the equality constraints are typically conservation equations, while phase or end-member... 

Charge transfer reactions on semiconductor electrodes proceed under the condition of anodic and cathodic polarization in which the Fermi level epfsc) is different either from the Fermi level Eputicox) of redox electron transfer reactions or from the equivalent Fermi level ep,ioN) of ion transfer reactions. For redox electron transfer reactions, thermodynamic requirement for the anodic and cathodic reactions to proceed is given by the following inequalities ... 

Under the condition of photoexcitation, the quasi-Fermi level, instead of the original Fermi level, determines the possibility of redox electron transfer reactions. The thermodynamic requirement is then given, for the transfer of cathodic electrons to proceed from the conduction band to oxidant particles, by the inequality of Eqn. 10-7 ... 

Carnot efficiency is one of the cornerstones of thermodynamics. This concept was derived by Carnot from the impossibility of a perpetuum mobile of the second kind [ 1]. It was used by Clausius to define the most basic state function of thermodynamics, namely the entropy [2]. The Carnot cycle deals with the extraction, during one full cycle, of an amount of work W from an amount of heat Q, flowing from a hot reservoir (temperature Ti) into a cold reservoir (temperature T2 < T ). The efficiency r] for doing so obeys the following inequality ... 

The only thermodynamic condition then required for this reaction to be driven from left to right is that the standard molar free enthalpy of the reaction be negative, that is, the following inequality must hold E pq < E ac, where E ac represents the standard redox potential of the A/C couple. 

For certain types of the series the explicit inequalities involving "mirror reflections" of the discriminant were possible (Passare and Tsikh, 2004). The situation is clearer for series depending on fewer variables. For instance, applying Birkeland approach, we can reduce to two the number of parameters in the case of cubic equation. The "thermodynamic branch" corresponds to the Birkeland series (60) for p — 0 and q—1. The discriminant for cubic equation in Birkeland form is... 

The simplest one-constant limitation concept cannot be applied to all systems. There is another very simple case based on exclusion of "fast equilibria" A Ay. In this limit, the ratio of reaction constants Kij — kij/kji is bounded, 0equilibrium constant", even if there is no relevant thermodynamics.) Ray (1983) discussed that case systematically for some real examples. Of course, it is possible to create the theory for that case very similarly to the theory presented above. This should be done, but it is worth to mention now that the limitation concept can be applied to any modular structure of reaction network. Let for the reaction network if the set of elementary reactions is partitioned on some modules — U j. We can consider the related multiscale ensemble of reaction constants let the ratio of any two-rate constants inside each module be bounded (and separated from zero, of course), but the ratios between modules form a well-separated ensemble. This can be formalized by multiplication of rate constants of each module on a timescale coefficient fc,. If we assume that In fc, are uniformly and independently distributed on a real line (or fc, are independently and log-uniformly distributed on a sufficiently large interval) then we come to the problem of modular limitation. The problem is quite general describe the typical behavior of multiscale ensembles for systems with given modular structure each module has its own timescale and these time scales are well separated. 

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