Positive entropy production

Note that Eq. (6) includes thermodynamic equilibrium (v° = 0) as a special case. However, usually the steady-state condition refers to a stationary nonequilibrium state, with nonzero net flux and positive entropy production. We emphasize the distinction between network stoichiometry and reaction kinetics that is implicit in Eqs. (5) and (6). While kinetic rate functions and the associated parameter values are often not accessible, the stoichiometric matrix is usually (and excluding evolutionary time scales) an invariant property of metabolic reaction networks, that is, its entries are independent of temperature, pH values, and other physiological conditions. 

In other words, under realistic conditions (/ 0), entropy is produced, with the positive entropy production being given by fluxes and forces related to the process j. Equation (5) assumes that all these processes are in series, which is mostly correct. The most obvious contributions are transport resistances, due to the finite conductivities of Li+ and e in electrolyte and electrodes. For usual geometries, these resistances are constant to a good approximation, while for resistances stemming from impeded charge transfer and phase boundaries, the dependence on current can be severe. 

Processes follow certain directions and paths, that must yield positive entropy production. This principle might force chemical reactions to proceed without reaching completion. 

Here, q/7 is the entropy flux and the first term on the right is the entropy production. The simplest constitutive equations satisfying the requirement of the positive entropy production are... 

Open systems do not in general possess thermodynamic potential functions, a fact which often manifests itself with the existence of more than one equilibrium points in the same invariant manifold. Open systems still satisfy the condition of positive entropy production, Eq.(1.5.5). Some interesting implications of the positive entropy production for the steady states of open distributed systems are discussed in Section 2.8. 

From Eq. (1.6.39) it is obvious that the quantity Ajfj is positive at all points outside the equilibrium manifold. This means that in chemical reaction systems with kinetics given by Eq. (1.6.34), each reaction individually leads to positive entropy production, a result beyond the requirements of Postulate 1.5.1. 

A reaction system that is close to equilibrium has a positive entropy production. 

A rough estimation can show that one of the conditions for order under nonequilibrium, a significant amount of negative entropy change, is fulfilled [7]. The next steps in analyzing the nonequilibrium properties are to prove the nonlinearity of the process and to determine whether the distance to equilibrium is supercritical. Therefore we consider similarities to and differences from the irreversible diffusion process, which shows a positive entropy production. 

When a process is completely reversible, the equahty holds, and the lost work is zero. For irreversible processes the inequality holds, and the lost work, that is, the energy that becomes unavailable for work, is positive. The engineering significance of this result is clear The greater the irreversibility of a process, the greater the rate of entropy production and the greater the amount of energy that becomes unavailable for work. Thus, every irreversibility carries with it a price. 

In Equation 2-114, is the rate of entropy production within the control volume symbols with dots refer to the time rate of change of the quantity in question. The second law requires that the rate of entropy production be positive. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

As in the case of the isolated system, the asymmetric part of the transport matrix does not contribute to the scalar product or to the steady-state rate of first entropy production. All of the first entropy produced comes from the reservoirs, as it must since in the steady state the structure of the subsystem and hence its first entropy doesn t change. The rate of first entropy production is of course positive. 

The rate of entropy production is always positive in the present case, since transport processes are irreversible in nature, i.e. always connected with irreversible losses (dissipation) of energy. 

Starting from the second law of thermodynamics, it is possible to derive a principle according to which the change of entropy production in the neighbourhood of a stationary state is always negative if the flows in the system are kept constant and only the forces varied. As already mentioned, the entropy production reaches a minimum value in the stationary state of the system. If it is at a minimum, and a positive fluctuation occurs, the system reverts to the minimum, and a stable state is again reached. 

The second law of thermodynamics asserts that the total entropy 5 of a system may change in time because of exchanges with its environment and internal entropy production which is vanishing at equilibrium and positive out of equilibrium [5]... 

The positivity of the entropy production, dS/dt = J Xi + J2X2 > 0, which is a quadratic form in the thermodynamic forces, implies for the Onsager coefficients... 

The thennodynainics of complexation between hard cations and hard (O, N donor) hgands often are characterized by positive values of both the enthalpy and entropy changes. A positive AH value indicates that the products are more stable than the reactants, i.e., destabilizes the reaction, while a positive entropy favors it. If TAS > AH°, AG° will be negative and thus log(3 positive, i.e., the reaction occurs spontaneously. Such reactions are termed entropy driven since the favorable entropy overcomes the unfavorable enthalpy. 

The enthalpy value of Eq. (3.23) is very small as might be expected if two Cd-N bonds in Cd(NH3) 2 are replaced by two Cd-N bonds in Cd(en). The favorable equilibrium constants for reactions [Eqs. (3.22) and (3.23)] are due to the positive entropy change. Note that in reaction, Eq. (3.23), two reactant molecules form three product molecules so chelation increases the net disorder (i.e., increase the degrees of freedom) of the system, which contributes a positive AS° change. In reaction Eq. (3.23), the AH is more negative but, again, it is the large, positive entropy that causes the chelation to be so favored. 

This positive entropy change means that there is greater disorder in the product (HjO gas) than the reactant (HjO liquid). In terms of just entropy, the increase in entropy drives the reaction to the right, toward a condition of higher entropy. 

In irreversible thermod3mamics, the second law of thermodynamics dictates that entropy of an isolated system can only increase. From the second law of thermodynamics, entropy production in a system must be positive. When this is applied to diffusion, it means that binary diffusivities as well as eigenvalues of diffusion matrix are real and positive if the phase is stable. This section shows the derivation (De Groot and Mazur, 1962). 

Nonequilibrium Steady State (NESS). The system is driven by external forces (either time dependent or nonconservative) in a stationary nonequilibrium state, where its properties do not change with time. The steady state is an irreversible nonequilibrium process that cannot be described by the Boltzmann-Gibbs distribution, where the average heat that is dissipated by the system (equal to the entropy production of the bath) is positive. 

Under those conditions P behaves as a Lagrangian in mechanics. Furthermore, as P is a nonnegative function for any positive value of the concentrations X,, by a theorem due to Lyapounov, the asymptotic stability of nonequilibrium steady states is ensured (theorem of minimum entropy production.1-23 These steady states are thus characterized by a minimum level of the dissipation in the linear domain of nonequilibrium thermodynamics the systems tend to states approaching equilibrium as much as their constraints permit. Although entropy may be lower than at equilibrium, the equilibrium type of order still prevails. The steady states belong to what has been called the thermodynamic branch, as it contains the equilibrium state as a particular case. 

Reactions with a large, positive entropy change also favor product formation (large Kp). For example, a reaction with a net increase in the number of moles of gas-phase species has a very positive A S°, from the translational entropy gain associated with the additional species. If AS° > 0, high temperatures increase Kp and drive the reaction toward completion (toward the products). If A5° < 0, Kp will increase as the temperature goes down. 

Here Dt is a positive proportionality constant ( diffusion constant for Et), Jfz is z-ward flow induced by the gradient, and superscript e denotes eigenmodt character of the associated force or flow. The proportionality (13.25) corresponds to Fick s first law of diffusion when Et is dominated by mass transport or to Fourier s heat theorem when Et is dominated by heat transport, but it applies here more deeply to the metric eigenvalues that control all transport phenomena. In the near-equilibrium limit (13.25), the local entropy production rate (13.24) is evaluated as... 

Since the entropy production is positive, the transport coefficients Lik must satisfy the relation TAA-Lhh>LhA-TAh [S.R. de Groot, P. Mazur (1962)]. This restricts the range for the charges of transport to aA-Oh< 1, see Eq. (8.56) ff. We should also add that whereas the Ly are phenomenological coefficients appropriate for the description of the experiments on transport, the ly relate directly to the SE s (Eqn. (8.28)) and can be derived from lattice dynamics based theoretical calculations. 

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