Entropy production and stability

In chapter 12 we considered fluctuations in an isolated system in which /, Uand N k are constant and we obtained conditions for the stability of the equilibrium state. These conditions, in fact, have a more general validity in that they remain valid when other types of boundary condition are imposed on the system. For example, instead of constant U and V, we may consider systems maintained at constant T and V, constant p and S or constant T and p. The main reason for the general validity of the stability conditions is that all these conditions are a direct consequence of the fact that for all spontaneous processes diS 0. As we have seen in Chapter 5, when each of these three pairs of variables is held constant, one of the thermodynamic potentials F, H or G is minimized. In each case we have shown that 

Through these relations, the change of the thermodynamic potentials AF, AG or AH due to a fluctuation can be related to the entropy production Aj S. The system is stable to all fluctuations that result in Aj 5 0, because they do not correspond to the spont eous evolution of a system due to irreversible processes. From the above relations it is clear how one could also characterize stability of the equilibrium state by stating that the system is stable to fluctuations for which AF 0, AG 0 or AH 0. For fluctuations in the equilibrium state, these conditions can be written more explicitly in terms of the second-order variations 6 V 0,5 G 0 and d H 0, which in turn can be expressed using the. second-order derivatives of these potentials. The conditions for stability obtained in this way are identical to those obtained in Chapter 12. 

A theory of stability that is based on the positivity of entropy production in spontaneous processes is more general than the classical Gibbs—Duhem theory of stability [1, 2], which is limited to the constraints expressed in (14.1.1) to (14.1.3) and the associated thermodynamic potentials. In addition, stability 

In our more general approach, the main task is to obtain an expression for the entropy production Ai5 associated with a fluctuation. A system is stable to fluctuations if the associated Ai 0. In Chapter 3 we have seen that the general form of entropy production due to irreversible processes takes the quadratic form 

In this section we shall present some simple cases for the calculation of A S and defer the more general theory to later chapters in which we consider the stability of nonequilibrium states. 

The Gibbs stability theory condition may be restrictive for nonequilibrium systems. For example, the differential form of Fourier s law together with the boundary conditions describe the evolution of heat conduction, and the stability theory at equilibrium refers to the asymptotic state reached after a sufficiently long time however, there exists no thermodynamic potential with a minimum at steady state. Therefore, a stability theory based on the entropy production is more general. 

The change of total entropy is dS d,.S + d S. The term dJS is the entropy exchange through the boundary, which can be positive, zero, or negative, while the term dtS is the rate of entropy production, which is always positive for irreversible processes and zero for reversible ones. The rate of entropy production is djS/dt = %JkXk. A near equilibrium system is stable to fluctuations if the change of entropy production is negative A S 0. For isolated systems, 

The second law for isolated systems shows that the excess entropy, A.V S SKI 0, increases monotonically in time, d(AS)/dt 0. Therefore, it plays the role of a Lyapunov function, and defines a global stability. So, dfi/dt is a Lyapunov function that guarantees the global stability of stationary states that are close to global equilibrium. 

Equations (12.26) and (12.27) do not depend on the boundary conditions imposed on the system. For a thermal fluctuation of Teq + j8, we have 

As before, a general thermodynamic theory of stability formulation is quadratic in the perturbations of 8T. 8 V, and 8Nk, because the forces and flows vanish at equilibrium 

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The concepts of destabilization of reactants and stabilization of products described for pyrophosphate also apply for ATP and other phosphoric anhydrides (Figure 3.11). ATP and ADP are destabilized relative to the hydrolysis products by electrostatic repulsion, competing resonance, and entropy. AMP, on the other hand, is a phosphate ester (not an anhydride) possessing only a single phosphoryl group and is not markedly different from the product inorganic phosphate in terms of electrostatic repulsion and resonance stabilization. Thus, the AG° for hydrolysis of AMP is much smaller than the corresponding values for ATP and ADP. 

TNC. 10. P. Glansdorff and I. Prigogine, Generalized entropy production and hydrodynamic stability, Phys.Lett. 7, 243-244 (1963). 

How could we have predicted which product would be favored The first step is to decide whether the prediction is to be based on (1) which of the two products is the more stable, or (2) which of the two products is formed more rapidly. If we make a decision on the basis of product stabilities, we take into account AH° values, entropy effects, and so on, to estimate the equilibrium constants Keq for the reactants and each product. When the ratio of the products is determined by the ratio of their equilibrium constants, we say the overall reaction is subject to equilibrium (or thermodynamic) control. Equilibrium control requires that the reaction be reversible. 

In order to extract the maximal energy out of the available foodstuff oxidative phosphorylation should operate at the state of optimal efficiency in vivo. Since a zero as well as an infinite load conductance both lead to a zero efficiency state, obviously there must be a finite value of the load conductance permitting the operation of the energy converter at optimal efficiency. For linear thermodynamic systems like the one given in equations (1) and (2) the theorem of minimal entropy production at steady state constitutes a general evolution criterion as well as a stability criterion.3 Therefore, the value of the load conductance permitting optimal efficiency of oxidative phosphorylation can be calculated by minimizing the entropy production of the system (oxidative phosphorylation with an attached load)... 

It is important to emphasize the role of the solid state in providing a medium for the formation of the molecular assembly 2(resorcinol) 2(4,4 -bpe). Indeed, that 2(resorcinol) 2(4,4 -bpe) is stabilized by weak forces (i.e. hydrogen bonds) comparable in strength to structure effects of solvent and entropy of the liquid phase [13] means that the components of 2(resorcinol)-2(4,4 -bpe) may assemble in solution to produce multiple equilibria involving individual molecules and undesirable (photostable) complexes. In effect, the solid state was used to sequester [23] 2(resorcinol) 2(4,4 -bpe) from the liquid phase, facilitating the formation of the desired photoactive complex and construction of the cyclobutane product. 

Data in Table 2 reveal that with a few exceptions, the stability of the complexes is enthalpy controlled and entropy destabilised. As previously stated, the variations observed in the thermodynamics of complexation of these systems as a result of the medium effect are the result of the solvation changes that the reactants (anion and receptor) and the product (complex anion) undergo in moving from one medium to another. This issue will be carefully addressed in the next section. 

Calix[4]pyrroles are versatile ligands to the extent that the composition of the anion receptor complex is solvent dependent. This chapter has been concerned with the affinity of calix[4]pyrrole for the fluoride anion. It was therefore considered of relevant to focus attention on the steps required for the derivation of reliable thermodynamic data. It is indeed the ratio between stability constants which defines quantitatively the selectivity factor. Thus, representative examples are given to demonstrate selectivity in terms of anion, receptor and solvent. The key role played by solvation in the complexation of these receptors with the fluoride anion is unambiguously demonstrated in the variations observed in the stability constants, enthalpies and entropies of complexation of these systems in the various solvents (Table 2). One convenient measure to assess solvation is through the thermodynamics of transfer of product and reactants from one... 

This chapter deals with formation of cyclic 2D and 3D structures in solution by self-assembly of two or more components using hydrogen bonds as the major interactions. Obviously, the formation of a defined supramolecular aggregate stabilized by non-covalent forces is a thermodynamically driven process which reflects a balance between enthalpy and entropy. Consequently, the product of a non-covalent macrocyclic synthesis must be evaluated and predicted in terms of thermodynamic minima in an equilibrium mixture. 

Unlike the reactants or products, a transition state is unstable and cannot be isolated. It is not an intermediate, because an intermediate is a species that exists for some finite length of time, even if it is very short. An intermediate has at least some stability, but the transition state is a transient on the path from one intermediate to another. The transition state is often symbolized by a superscript double dagger ( ), and the changes in variables such as free energy, enthalpy, and entropy involved in achieving the transition state are symbolized AG, A//, and AS. AG is similar to Ea, and the symbol AG is often used in speaking of the activation energy. 

The enthalpy change for the reaction is favorable because (1) electrostatic repulsion between the negative charges in ATP exceeds that in the reaction products, (2) the reaction products are resonance stabilized, and (3) the enthalpies of solvation of the products are larger than that for ATP. The entropy change for the reaction is favorable because of the release of a phosphate group. Note that this implies that ATP hydrolysis is strongly temperature-dependent [cf. Eq. 10.7)]. 

Minimum exergy loss or minimum entropy production at stationary state provides a general stability criterion. There are two important steady states identified in the cell static head (sh) and level flow (If). At the static head, where ATP production is zero since. /p = 0, the coupling between the respiratory chain and oxidative phosphorylation maintains a phosphate potentialXp, which can be obtained from Eq. (11.151) as (A p)sh = - qXJZ, and the static head force ratio xsh becomes xsh = q. The oxygen flow./, at the static head is obtained from Eqs. (11.151) and (11.152)... 

The term to the right of the equal sign in Eq. (12.32) is the excess entropy production. Equations (12.31) and (12.32) describe the stability of equilibrium and nonequilibrium stationary states. The term 82S is a Lyapunov functional for a stationary state. 

Since a definite function 82S leads to the stability condition, it operates as a Lyapunov function, and assures the stability of a stationary state. As the entropy production is the sum of the products of flows J and forces X, we have... 

For systems not far from equilibrium, the total entropy production reaches a minimum value and also assures the stability of the stationary state. However, for systems far from equilibrium, there is no such general criterion. Far from equilibrium, we may have order in time and space, such as, appearance of rhythms, oscillations, and morphological structurization. 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The Lyapunov function resembles the thermodynamic entropy production function and the asymptotic stability principle. If the eigenvalues of the coefficient matrix of the quadratic form of the entropy production are very large, then the convergence to equilibrium state will be rapid. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

Only the orbits infinitesimally close to the steady state may be considered stable, according to Liyapunov s theory of stability. However, at a finite distance from the steady state, two neighboring points belonging to two distinct cycles tend to be far apart from each other because of differences in the period. Such motions are called stable in the extended sense of orbital stability. The average concentrations of X and Y over an arbitrary cycle are equal to their steady-state values (Xs = 1 and Y.. A 1). Under these conditions, the average entropy production over one period remains equal to the steady-state entropy production. 

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