Entropy sources

THL.2.1. Prigogine, Sources et courants d entropie (Entropy sources and fluxes), Comptes Rendus Acad. Set, 1945. 

The entropy production is due to the contributions of the different irreversible processes taking place in the system. The entropy source is given by the sum [5]... 

One of plausible candidates for the entropy source is a dynamic structural disorder in the HS phase, which should be settled down in the LS phase. The crystallographic data for [Mn(taa)] [11] provide a clue, i.e., the presence of C3 axis in the HS molecule. An Mn(III) ion in the 5E state is a well-known Jahn-Teller ion [19]. Since the C3 site-symmetry cannot lift the orbital degeneracy of the 5E term (Fig. 1(b)), it is likely that the Mn ion is subjected to the E e Jahn-Teller effect, which gives rise to three energetically equivalent deformation structures. The apparent C3 symmetry should be observed in a time-averaged structure over three deformed structures. 

Spin-crossover phase transition of a manganese(IU) complex [Mn(taa)] was studied by variable-temperature laser Raman spectroscopy and it was found that the vibrational contribution in the transition entropy is not dominant in contrast to the cases of ordinary iron spin-crossover systems. The discovery of a dynamic disorder in the HS phase by means of dielectric measurements provided an alternative entropy source to explain the thermally induced spin-crossover transition. This dynamic disorder was attributed to the reorienting distortion dipoles accompanying the E e Jahn-Teller effect in HS manganese(III) ions. 

The entropy source strength < > is the rate of entropy production per unit volume... 

The product of the entropy source strength and the absolute temperature is called the dissipation function 4 ... 

When the T is the environmental temperature, the dissipation function represents the energy dissipated to the environment. The entropy source strength and the dissipation function are not state functions, and they depend on the path between the given states. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

Equation (3.116) shows that the rate of change of the entropy per unit volume of substance is due to the convection entropy flow p.w. the conduction entropy flow j v. and the entropy source strength The conduction entropy flow is... 

From Eqs. (3.118), (3.137), and (3.138), the entropy source strength or the rate of local entropy production per unit volume < > is defined by... 

The rate of entropy production is obtained from the local value of entropy production or entropy source strength d>... 

One of the contributions to the modified heat flow is the motion of the Brownian particles. The entropy source strength is due to heat flow and due to diffusion in velocity space (internal degree of freedom), which is the contribution of the motion of the Brownian particles in the heat bath. 

FIGURE 4 Temperature dependence of the Gibbs free energy (G) and the enthalpy (//), at constant pressure, for two polymorphs, I and 2. Tt is the transition temperature and S is the entropy. Source From Ref. 9. 

To reduce the lost work in industrial process plants, the minimization of entropy production rates in process equipment is suggested as a strategy for future process design and optimization [81]. The method is based on the hypothesis that the state of operation that has a minimum total entropy production is characterized by equipartition of the local entropy production. In this context we need to quantify the entropy sources of the various irreversible unit operations that occur in the industrial system. 

The way in which the separation of the terms of the right hand side of the entropy equation into the divergence of a flux and a source term has been achieved may at first sight seem to be to some extent arbitrary The two groups of terms must, however, satisfy a number of requirements which determine this separation uniquely First, one such requirement is that the entropy source term totai must be zero if the thermodynamic equilibrium conditions are satisfied within the system. Another requirement the source term must satisfy is that it should be invariant under a Galilean transformation (e.g., [147]), since the notations of reversible and irreversible behavior must be invariant under such a transformation. The terms included in the source term satisfy this requirement [32]. 

Delhaye JM (1974) Jump Conditions and Entropy Sources in Two-Phase Systems Local Instant Formulation. Int J Multiphase Flow 1 395-409... 

Delhaye JM (1974) Jump Conditions and Entropy Sources in Two-Phase Systems Local Instant Formulation. Int J Multiphase Flow 1 395-409 Delhaye JM, Achard JL (1977) On the averaging operators introduced in two-phase flow. In Banerjee S, Weaver JR (eds) Transient Two-phase Flow. Proc. CSNI Specialists Meeting, Toronto, 3.-4. august... 

Entropy source for ligand binding and structural changes... 

Whenever La-Hu transitions are observed thermotropically, the La phase is always the lower temperature phase. The fact that the Hu phase, which has two-dimensional symmetry, occurs at higher temperatures than the one-dimensional La phase indicates that the lattice entropy is not the dominant entropy contribution within the system. Instead, the dominant entropy sources are the large number of configurations available to the lipid hydrocarbon chains and (probably to a lesser extent) to the water. The shape of the mean molecular volume available to the lipid hydrocarbon chains changes dramatically at the transition, which suggests that the hydrocarbon entropy increases at the transition. This hypothesis is supported by recent deuterium NMR studies of the chain order parameter profile that indicate that the Hu chains are more disordered than the La chains (19, 20). 

In this framework, the intensity of an entropy source is represented by a quadratic form of thermodynamic forces. The corresponding phenomenological coefficients form a matrix with remarkable properties. These properties, formulated as the Onsager reciprocity theorem, allow to reduce the number of independent quantities and to find relations between various physical effects. 

The following step is the reduction of equation (5.190) to the form (5.182), for which purpose one has to collect terms containing divergence. After simple algebra one obtains the expression for entropy flux and entropy source a, known as dissipative function... 

We next calculate the entropy source 5 or the energy TS dissipated by viscous phenomena (e epressed per unit length of the line in the t/-direction). Using rf to designate the viscosity, we start with the general expression... 

When the chemical reaction reaches equilibrium, affinity vanishes A = —St /za, = 0. The entropy generated per unit time and unit volume is called the rate of volumetric entropy production or the entropy source of density ... 

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