Thermodynamic equilibrium, stationary

The general theoretical treatment of ion-selective membranes assumes a homogeneous membrane phase and thermodynamic equilibrium at the phase boundaries. Obvious deviations from a Nemstian behavior are explained by an additional diffusion potential inside the membrane. However, allowing stationary state conditions in which the thermodynamic equilibrium is not established some hitherto difficult to explain facts (e.g., super-Nemstian slope, dependence of the selectivity of ion-transport upon the availability of co-ions, etc.) can be understood more easily. 

Here Jta(x) denotes the a-th component of the stationary vector x of the Markov chain with transition matrix Q whose elements depend on the monomer mixture composition in microreactor x according to formula (8). To have the set of Eq. (24) closed it is necessary to determine the dependence of x on X in the thermodynamic equilibrium, i.e. to solve the problem of equilibrium partitioning of monomers between microreactors and their environment. This thermodynamic problem has been solved within the framework of the mean-field Flory approximation [48] for copolymerization of any number of monomers and solvents. The dependencies xa=Fa(X)(a=l,...,m) found there in combination with Eqs. (24) constitute a closed set of dynamic equations whose solution permits the determination of the evolution of the composition of macroradical X(Z) with the growth of its length Z, as well as the corresponding change in the monomer mixture composition in the microreactor. 

Equation [19] ensures that the thermodynamic equilibrium distribution of Eq. [20] is the stationary (long-time) limit of the Markov chain generated by Eq. [18]. It does not specify the transition rates uniquely, however. Let us write them in the following way ... 

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 their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

We can design a reactor to separate the products and achieve complete conversion by admitting pure A into the center of the tube with the sohd moving countercurrent to the carrier fluid. We adjust the flows such that A remains nearly stationary, product B flows backward, and product C flows forward. Thus we feed pure A into the reactor, withdraw pure B at one end, and withdraw pure C at the other end. We have thus (1) beat both thermodynamic equilibrium and (2) separated the two products from each other. 

Fig. 14. Schematic illustration of a drop ofliquid spreading in contact with a solid surface, showing the relations between the relevant parameters the contact angle, 0 the solid/vapor interfacial free energy, Ysv the liquid/vapor interfacial free energy, yLV and the solid/liquid interfacial free energy, ySL. Young s equation describes the relationship between these parameters for a stationary drop at thermodynamic equilibrium [175]...

Throughout we use the superscript e for the thermodynamic equilibrium, and s for any stationary, i.e., time-independent solution of the master equation. 

This equation has the same form as (3.7) for an isolated system the stationary solution of the master equation ps is identical with the thermodynamic equilibrium pe. 

In the case of a photoconductor p is increased by a constant y proportional to the incident light intensity. The system is no longer closed and the new P is no longer connected with a by detailed balance. The stationary solution (9.2) is no longer identical with the thermodynamic equilibrium. Another remark is, that it is possible to represent the effect of the incident photons in this simple way of adding y to the generation probability only if the arrival times of the photons are uncorrelated (shot noise). When they are correlated the number n is no longer a Markov process, and a more sophisticated description is needed, see XV.3. 

Thus the response of a spatially uniform system in thermodynamic equilibrium is always characterized by translationally invariant and temporaly stationary after-effect functions. This article is restricted to a discussion of systems which prior to an application of an external perturbation are uniform and in equilibrium. The condition expressed by Eq. (7) must be satisfied. Caution must be exercised in applying linear response theory to problems in double resonance spectroscopy where non-equilibrium initial states are prepared. Having dispensed with this caveat, we adopt Eq. (7) in the remainder of this review article. 

This paper shows that the conditions of thermodynamic equilibrium in a mix-tine of chemically reacting ideal gases always have a solution for the concentrations of the mixture components and that this solution is unique. The paper has acquired special significance in the last few years in connection with the intensive study of systems in which this uniqueness does not occur. Such anomalies may be related either to nonideal components, or to treatment of stationary states, rather than truly equilibrium ones, in which the system exchanges matter or energy with the surrounding medium. 

For systems that have not reached their stationary state (steady state or thermodynamic equilibrium), the behavior with regards to time cannot be determined without knowing the initial conditions, or the values of the state variables at the start, i.e., at time = 0. When the initial conditions are known, the behavior of the system is uniquely defined. Note that for chaotic systems, the system behavior has infinite sensitivity to the initial conditions however, it is still uniquely defined. Moreover, the feed conditions of a distributed system can act as initial conditions for the variations along the length. 

By using external reservoirs, some of these parameters can be kept at values different from those of thermodynamic equilibrium, / /, j = 1, , m < n. As a result, a non-equilibrium state arises, which is characterized by nonvanishing values of some fluxes /,-, i = 1, s < r and of the corresponding forces Xj. Examples of such processes are diffusion and related effects, Peltier effect, etc.45,46. Such a state can either be stationary or time-dependent, stable or unstable. 

When Ts does not correspond to any of the characteristic cases described above, the evolution of sequences coupled with conformations can be still defined in the same way. One has only to remember that if T Ts and both T and Ts are finite, there is a flow of heat between conformational space and sequence space, so that full thermodynamic equilibrium is impossible. Still, we can be in a stationary regime corresponding to the sequences tending to a certain fixed point, and possessing (or not possessing) information complexity. 

A solute undergoing chromatographic migration partitions between the stationary and mobile phases, a process driven by thermodynamic equilibrium. At equilibrium (established fully only at the zone center), the concentration in the stationary phase (c,) relative to that in the mobile phase (cm) is given by the thermodynamic distribution constant K, as shown by comparing Eqs. 2.18 and 2.19. Thus... 

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 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. 

It follows from the Prigogine theorem that in cases where the system exists near its thermodynamic equilibrium, any deviation from the system stationary state due to a disturbance of some internal parameters causes an increase in the rate of entropy production. Simultaneously, the spontaneous evolution of the system will make the entropy production rate decreasing again to its minimal value. Hence, the stationary state of an open system nearly its equilibrium is stable. It is obvious here that the stability condi tion of the stationary state is inequality 8P > 0 at the appearance of any disturbance (fluctuation) of those internal parameters whose values are determined by the condition of the system stationarity. 

The preceding conclusions about the stabiHty of stationary states near stable thermodynamic equilibrium are graphically interpreted in Figure 2.5. Indeed, if an incidental fluctuation of thermodynamic force X, around its stationary magnitude X, results in a minor deviation of the system from the stationary state near thermodynamic equilibrium, the internal trans formations must happen according to inequality (2.31), which wiU affect the value of X, and return the system again to its initial stationary state (see Figure 2.5A). Thus, if the system is near thermodynamic equilibrium in the stationary state, it cannot escape this state spontaneously due to... 

Figure 2.5 The rate of energy dissipation (entropy production) near the stationary point in a system close to thermodynamic equilibrium dependence of P = Td S/dt on thermodynamic driving forces nearby stationary point Xj (A) time dependence of P(7, 3) and dP/dt 2, 4) on approaching the stationary state (B). The vertical dashed line stands for the moment of approaching the stationary state by the system, and wavy line for escaping the stationary state caused by an internal perturbation (fluctuation).

A typical problem in thermodynamics of systems that are far from their equilibrium is the analysis of the stability of stationary states of the system. Thermodynamic criteria of the stability of stationary states are found the same way as for systems that are far from and close to thermodynamic equilibrium (see Section 2.4) by analyzing signs of thermodynamic fluxes and forces arising upon infinitesimal deviation of the system from the inspected stationary state. If the system is in the stable stationary state, then any infinitesimal deviation from this state must induce the forces that push it to return to the initial position. 

We saw in Section 2.4 that in cases where the stationary state occurs near thermodynamic equilibrium and, therefore, is stable, an increase in the energy dissipation rate caused by the fluctuation of internal para meters is positive and equal at the first approximation to the product... 

When a reactive system is far from its thermodynamic equilibrium, corol laries of the Prigogine theorem, which were derived for the case of the linear nonequilibrium thermodynamics, cannot be applied to analysis directly. Nevertheless, tools of thermodynamics of nonequilibrium pro cesses allow the deduction of some important conclusions on properties of the system, even though strongly nonequilibrium, including in some cases on the stability of stationary states of complex stepwise processes. For several particular cases, theorems similar to the Prigogine theorem can be proved, too. 

If the positively defined Lyapunov function exists for a particular kinetic scheme, this scheme has the stationary state that is stable in respect to the concentrations of the intermediates, whether they are close to or far from thermodynamic equilibrium. 

As earlier, R and P are the starting reactant and final product, respectively, of the stepwise transformation (see also Section 3.4.1). External parameter R can be taken here as the controlling parameter. The kinetic irreversibility of the second step means that this step is a priori far from thermodynamic equilibrium. This is the necessary condition of the instability of stationary states. Let us check this. 

Indeed, one can analyze In the same manner the evolution of the system under consideration under conditions of reversibility of all of the elementary reactions in scheme (3.30). Unfortunately, in this situation the analytic solution of the eigenvalue equation in respect to parameter X appears unreasonably awkward. However, if the kinetic irreversibility of both nonlinear steps are a priori assumed, it is easy to find stationary valued (Y, Z ), and we come to the preceding oscillating solution. At the same time, near thermodynamic equilibrium (i.e., at R aa P), there exits only a sole and stable stationary state of the system with (Y Z R). 

When any complex catalytic system, even those that include inter mediate nonlinear steps, is close to its thermodynamic equilibrium, the stationary rate of the catalytic stepwise process is necessarily proportional to the affinity of the conjugating—that is, catalyzed—reaction as it was shown for non catalytic reaction in examples of Section 1.3.2. 

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