Entropy coupling

Negative values for redox couple entropy have also been obtained for the Cu(II)/Cu(I) reduction, in aqueous medium, of the blue copper proteins stellacyanin, plastocyanin and azurin.14 The decrease in molecular disorder has been attributed in this case to the fact that the charge neutralization of the redox site (from + 1 to 0) favours the formation of hydrogen bonds between the solvent (water) and the copper centre.17... 

The error in doing this involves the neglect of coupling entropy and consequently the free energy and enthalpy of interaction become equal. Cvdw calculation is a difficult task, and is usually done by numerical integration of potential functions of the Lennard-Jones type as in equation (48). An approximate analytical method for Cvdw evaluation is described in Ref. 35. 

It is still necessary to consider the role of entropy m irreversible changes. To do this we return to the system considered earlier in section A2.1.4.2. the one composed of two subsystems in themial contact, each coupled with the outside tliroiigh movable adiabatic walls. Earlier this system was described as a function of tliree independent variables, F , and 0 (or 7). Now, instead of the temperature, the entropy S = +. S P will be... 

Since entropy plays the determining role in the elasticity of an ideal elastomer, let us review a couple of ideas about this important thermodynamic variable ... 

In addition to the entropy term we assume that there is an extra local coupling between the fields via and, in addition to the coulombic coupling which is long range, we assume the existence of a short-range non-local coupling via We can choose several functional forms to... 

If the coefficients dy vanish, dy = 28y, we recover the exact Debye-Huckel limiting law and its dependence on the power 3/2 of the ionic densities. This non-analytic behavior is the result of the functional integration which introduces a sophisticated coupling between the ideal entropy and the coulomb interaction. In this case the conditions (33) and (34) are verified and the... 

Structure 12.148 was already supported by results from Stamm and Zollinger s investigation (1957) in which activation entropies were determined. The mechanism was corroborated by further activation entropy studies (Demian, 1972, 1973 Demian et al., 1983) and investigations on the 2/4-ratio of coupling of 1-naphthylamine carried out by Hashida et al. (1975 b). 

Clearly, the sticking coefficient for the direct adsorption process is small since a considerable amount of entropy is lost when the molecule is frozen in on an adsorption site. In fact, adsorption of most molecules occurs via a mobile precursor state. Nevertheless, direct adsorption does occur, but it is usually coupled with the activated dissociation of a highly stable molecule. An example is the dissociative adsorption of CH4, with sticking coefScients of the order 10 -10 . In this case the sticking coefficient not only contains the partition functions but also an exponential... 

The parity matrix commutes with the first entropy matrix, gS = Sg, because there is no coupling between variables of opposite parity at equilibrium, (xiXj)0 = 0 if e,e - = — 1. If variables of the same parity are grouped together, the first entropy matrix is block diagonal. 

A significant question is whether the asymmetric contribution to the transport matrix is zero or nonzero. That is, is there any coupling between the transport of variables of opposite parity The question will recur in the discussion of the rate of entropy production later. The earlier analysis cannot decide the issue, since can be zero or nonzero in the earlier results. But some insight can be gained into the possible behavior of the system from the following analysis. 

The present analysis shows that when a thermodynamic gradient is first applied to a system, there is a transient regime in which dynamic order is induced and in which the dynamic order increases over time. The driving force for this is the dissipation of first entropy (i.e., reduction in the gradient), and what opposes it is the cost of the dynamic order. The second entropy provides a quantitative expression for these processes. In the nonlinear regime, the fluxes couple to the static structure, and structural order can be induced as well. The nature of this combined order is to dissipate first entropy, and in the transient regime the rate of dissipation increases with the evolution of the system over time. 

In these systems the converter is producing waste heat, which has to be released to the ambient connected to an entropy flow caused by the irreversibilities within the converter. The discharging process will be a heat pump process, where the entropy has to be taken from the ambient. Therefore it is obvious that these systems have to be coupled to the ambient conditions. Such a storage is not self-sufficient. These systems are called indirect thermal energy storages. 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

This tells us that the reduced magnetic entropy change will be lower in an antiferromagnetic system than in an uncoupled one, as can be seen below, as the spins are not fully saturated below the conditions of lowest temperature and strongest field. The antiferromagnetic coupling in Gd(III)7 is also a hindrance, with behaviour similar to Gd(III)5, when compared to the relevant Brillouin functions. 

A famous example of the same category in irreversible coupling phenomena is "active transportation" [44], in which K+ ions are transported through a membrane from a diluted side to the other concentrated side against entropy increase, with the expense of another entropy increase induced by H+ transfer through the same membrane in a countercurrent. 

However, often the minimum in Si or Ti which is reached at first is shallow and thermal energy will allow escape into other areas on the Si or Ti surface before return to So occurs (Fig. 3, path e). This is particularly true in the Ti state which has longer lifetimes due to the spin-forbidden nature of both its radiative and non-radiative modes of return to So-The rate of the escape should depend on temperature and is determined in the simplest case by the height and shape of the wall around the minimum, similarly as in ground state reactions (concepts such as activation energy and entropy should be applicable). In cases of intermediate complexity, non-unity transmission coefficients may become important, as discussed above. Finally, in unfavorable cases, vibronic coupling between two or more states has to be considered at all times and simple concepts familiar from ground-state chemistry are not applicable. Pres-... 

If the stability constant measurements are coupled with enthalpy determinations, then entropy values (as well as the corresponding free energy values) may be calculated ... 

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