Thermodynamic equilibrium, stationary state

Example 6 Non-equilibrium stationary state as a characteristic of mechanochemical treatment Concurrent mechanochemical treatment of powder mixture of Bi203 and Xi02 in 2 3 molar ratio and pulverized Bi4Ti30i2 compound prepared by reactive sintering shows that after some milling time, a steady-state characterized by a very disordered, amorphous-like stmcture was reached. Thus, the systems evolves toward a non-equilibrium stationary state regardless of different initial thermodynamic states. 

In this book we offer a coherent presentation of thermodynamics far from, and near to, equilibrium. We establish a thermodynamics of irreversible processes far from and near to equilibrium, including chemical reactions, transport properties, energy transfer processes and electrochemical systems. The focus is on processes proceeding to, and in non-equilibrium stationary states in systems with multiple stationary states and in issues of relative stability of multiple stationary states. We seek and find state functions, dependent on the irreversible processes, with simple physical interpretations and present methods for their measurements that yield the work available from these processes. The emphasis is on the development of a theory based on variables that can be measured in experiments to test the theory. The state functions of the theory become identical to the well-known state functions of equilibrium thermodynamics when the processes approach the equilibrium state. The range of interest is put in the form of a series of questions at the end of this chapter. 

For the transient relaxation of X to the non-equilibrium stationary state AG is not a valid criterion of irreversibility or spontaneous reaction. We shall develop necessary and sufEcient thermodynamic criteria for such cases. 

What are the thermodynamic functions that describe the approach of such systems to a non-equilibrium stationary state, both the approach of each intermediate species and the reaction as a whole ... 

What are the thermodynamic forces, conjugate fluxes and applicable extremum conditions for processes proceeding to or from non-equilibrium stationary states What is the dissipation for these processes ... 

In Chap. 2 9 we presented a thermodynamic and stochastic theory of chemical reactions and transport processes in non-equilibrium stationary and transient states approaching non-equilibrium stationary states. We established a state function systems approaching equilibrimn reduces to AG. Since Gibbs free energy changes can be determined by macroscopic electrochemical measurements, we seek a parallel development for the determination of by macroscopic electrochemical and other measurements. 

Thermodynamics is one of the foundations of science. The subject has been developed for systems at equilibrium for the past 150 years. The story is different for systems not at equilibrium, either time-dependent systems or systems in non-equilibrium stationary states here much less has been done, even though the need for this subject has much wider applicability. We have been interested in, and studied, systems far from equilibrium for 40 years and present here some aspects of theory and experiments on three topics ... 

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. 

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... 

These have only one true stationary-state solution, ass = pss = 0 (as t - go), corresponding to complete conversion of the initial reactant to the final product C. This stationary or chemical equilibrium state is a stable node as required by thermodynamics, but of course that tells us nothing about how the system evolves in time. If e is sufficiently small, we may hope that the concentrations of the intermediates will follow pseudo-stationary-state histories which we can identify with the results of the previous sections. In particular we may obtain a guide to the kinetics simply by replacing p by p0e et wherever it occurs in the stationary-state and Hopf formulae. Thus at any time t the dimensionless concentrations a and p would be related to the initial precursor concentration by... 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

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. 

The equilibrium state we have just described has to be contrasted with the non-equilibrium conditions which prevail in practice. The general experimental condition is a stationary state in which the gaseous molecules, at a pressure P2, are not in equilibrium with the surface. The extent of adsorption remains constant for a given pressure, by reason of a mass balance between incident and desorbing material, but may be substantially greater than the thermodynamic coverage. 

Constant driving forces cause steady flows, which leads to a stationary state. For example, a constant temperature difference applied to a metal bar will induce a heat flow that will cause a change in all local temperatures. After a while, a constant distribution of temperature will be attained and the heat flow will become steady. The steady state flow and constant distribution of forces characterizing a system form the ultimate state of irreversible systems corresponding to the states of equilibrium in classical thermodynamics. 

Example 4.8 Chemical reactions and reacting flows The extension of the theory of linear nonequilibrium thermodynamics to nonlinear systems can describe systems far from equilibrium, such as open chemical reactions. Some chemical reactions may include multiple stationary states, periodic and nonperiodic oscillations, chemical waves, and spatial patterns. The determination of entropy of stationary states in a continuously stirred tank reactor may provide insight into the thermodynamics of open nonlinear systems and the optimum operating conditions of multiphase combustion. These conditions may be achieved by minimizing entropy production and the lost available work, which may lead to the maximum net energy output per unit mass of the flow at the reactor exit. 

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. 

There are two types of macroscopic structures equilibrium and dissipative ones. A perfect crystal, for example, represents an equilibrium structure, which is stable and does not exchange matter and energy with the environment. On the other hand, dissipative structures maintain their state by exchanging energy and matter constantly with environment. This continuous interaction enables the system to establish an ordered structure with lower entropy than that of equilibrium structure. For some time, it is believed that thermodynamics precludes the appearance of dissipative structures, such as spontaneous rhythms. However, thermodynamics can describe the possible state of a structure through the study of instabilities in nonequilibrium stationary states. 

The deduction of a criterion for the evolution of an open system to its stationary state resembles the classical thermodynamic problem of predict ing the direction of spontaneous irreversible evolution in an isolated system According to the Second Law of thermodynamics, in the latter case the changes go only toward the increase in entropy, the entropy being maximal at the final equilibrium state. 

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).

Inequalities (3.2) and (3.3) are generalizations of the principle of the minimal entropy production rate in the course of spontaneous evolution of its system to the stationary state. They are independent of any assump tions on the nature of interrelations of fluxes and forces under the condi tions of the local equilibrium. Expression (3.2), due to its very general nature, is referred to as the Qlansdorf-Prigogine universal criterion of evolution. The criterion implies that in any nonequilibrium system with the fixed boundary conditions, the spontaneous processes lead to a decrease in the rate of changes of the entropy production rate induced by spontaneous variations in thermodynamic forces due to processes inside the system (i.e., due to the changes in internal variables). The equals sign in expres sion (3.2) refers to the stationary state. 

THERMODYNAMIC CRITERIA OF THE STABILITY OF STATIONARY STATES THAT ARE FAR FROM EQUILIBRIUM... 

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. 

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