Equilibrium in Open Systems

An open system is one which can undergo all the changes allowed for a closed system and in addition it can lose and gain matter across its boundaries. An open system might be one phase in an extraction system, or it might be a small-volume element in an electrophoretic channel. Such systems, which allow for the transport of matter both in and out, are key elements in the description of separation processes. 

In open systems, we must modify the expression describing dG at equilibrium in closed systems, namely 

The magnitude of the increment depends, as the above equation shows, on the rate of change of G with respect to nit providing the other factors are held constant. This magnitude is of such importance in equilibrium studies that the rate of change, or partial derivative, is given a special symbol 

Quantity is called the chemical potential. It is, essentially, the amount of G brought into a system per mole of added constituent i at constant T and p. Dimensionally, it is simply energy per mole. 

If T and p are now allowed to vary, we must add the increment that describes their contribution, which is identical to that for a closed system, Eq. 2.9. All of these contributions together give a general equilibrium expression for an open system subject to any combination of incremental changes 

---

Weill, D.F., and Fyfe, W.S., 1964, A discussion of the Korzhinskii and Thompson treatment of thermodynamic equilibrium in open systems Geochim. Cosmochim. Acta, v. 28, pp. 565-576. 

In the previous chapter we derived criteria for identifying equilibrium states for example, in a closed system at fixed T and P, the equilibrium state is the one that minimizes the Gibbs energy. That minimization is equivalent to satisfying the equality of component fugacities. More generally, we derived criteria for thermal, mechanical, and diffusional equilibrium in open systems. But although those criteria can be used to identify equilibrium states, they are not always sufficient to answer the question of observability. Observability requires stability. Thermodynamic states can be stable, metastable, or unstable a stable equilibrium state is always observable, a metastable state may sometimes be observed, and an unstable state is never observed. 

Regarding the electrode/electrolyte interface, it is important to distinguish between two types of electrochemical systems thermodynamically closed (and in equilibrium) and open systems. While the former can be understood by knowing the equilibrium atomic structure of the interface and the electrochemical potentials of all components, open systems require more information, since the electrochemical potentials within the interface are not necessarily constant. Variations could be caused by electrocatalytic reactions locally changing the concentration of the various species. In this chapter, we will focus on the former situation, i.e., interfaces in equilibrium with a bulk electrode and a multicomponent bulk electrolyte, which are both influenced by temperature and pressures/activities, and constrained by a finite voltage between electrode and electrolyte. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

The interesting feature of our representation is that many sub-terms of the "fourth", non-linear, term may contain the "potential term" (the cyclic characteristic C) as well. It means that even in the domain "far from equilibrium" the open system still may have a "memory" about the equilibrium. Particular forms of this general Equation (77), i.e. for the cases of step limiting and the vicinity of equilibrium, respectively, are presented. 

The great utility of the chemical potential in phase equilibrium problems arises in the following way. In open systems in which the number of moles of any component may increase or decrease, any change in the Gibbs free energy of the system as a whole may be expressed as the sum of the following contributions ... 

Note that the appearance of a generic time scale is a characteristic property of a dissipative system and T generates its time evolution in scaled time units. Such time operators are strictly speaking forbidden in standard Quantum Mechanics, see Ref. [24] for further aspects on the problem, however, in open systems far from equilibrium they do not only exist but might also be useful in many applications, see below and [4-10, 13-15], The form (15) has been investigated and obtained... 

Detailed equilibrium must occur in closed systems, whereas in open systems, particularly in those that are far from being in equilibrium due to their exchange with the environment, the situation is much more complicated. Primarily, steady-state solutions can be multiple, i.e. the rates of substance formation and consumption can be balanced on many points. 

A chemical system with conjugated chemical reactions is usually open, and a stationary state far from the chemical equilibrium is typical of it. Stationary proceeding of chemical reactions in open systems (stationary flow systems) is characterized by the equilibrated rates of the mass and energy transfer to the system from the environment and the inverse process. 

It is essential to note that nonlinear thermodynamics of nonequilib rium processes critically change the status of the Second Law of thermody namics. It appeared that in open systems that are far from equilibrium, this law determines not only the inevitability of the destruction of equilibrium... 

A fundamental corollary of the Glansdorf Prigogine criterion (3.2) is a potentiality of the formation of ordered structures at the occurrence of irreversible processes in the region of nonlinear thermodynamics in open systems that are far from their equilibrium. Prigogine created the term dissipative structures to describe the structures that arise when some controlling parameters exceed certain critical values and are classified as spatial, temporal, or spatial temporal. Some typical dissipative structures are discussed in Sections 3.5 and 4.6. 

In open systems, which are characterized by an exchange of matter with the surrounding medium, the evolution towards stable thermodynamic equihbrium may appear to be impossible in principle. However, the spon taneous evolution of such systems leads also to some state with its proper ties being dependent on the boundary conditions for the system. We shall consider, in general, that the system exists in a dynamic equilibrium if the imposed boundary conditions are compatible with such equilibrium. The latter means that, for example, the system may achieve a stationary state implying no change in the matter concentration and/or temperature field distribution in time. The typical and limit example of the dynamic equihbrium is indeed the stable thermodynamic equilibrium. 

In real situations surface and volume changes are often made with systems that are at equilibrium with their environment, characterized by a set of chemical potentials p, rather than keeping In ] fixed, as in [2.2.7 and 8j. In other words, area changes in open systems are considered. In statistical thermodynamics the conversion from closed to open implies the transition from the canonical to the grand canonical ensemble. The characteristic function of the latter is nothing other than the sum of the bulk and surface mechemical work terms (see [1.3.3.12] and [I.A6.23D which are the quantities of interest ... 

Nicholas, J., K.R. Cameron R.W. Honess. 1992. Herpesvirus saimiri encodes homologues of G protein-coupled receptors and cyclins. Nature 355 362-5. Nicolis, G. 1971. Dissipative structures in open systems far from equilibrium. Adv. Chem. Phys. 19 209-324. 

According to Prigogine [77a,b], dissipative structures are to be expected, if in open systems the distance from thermodynamic equilibrium exceeds some critical value, this means d, > fifj.cnr. In that region, the relations between flows (fluxes) and forces are non-linear. So the entropy change of the (irreversible) dispersion process can be calculated according to irreversible thermodynamics of non-linear processes. In principle, diffusion and dispersion can be treated in a similar way, they apparently lead to formally similar structures. Both processes are irreversible and non-linear. 

By considering the thermodynamical data of this carbothermal reduction it becomes obvious that the temperatures used in industrial scale applications are significantly higher than the one needed to get a negative Gibbs energy for the reaction. (ArG becomes zero at about 1800 K.) This is not to mention that in the case of such an equilibrium where one of the products may escape in open systems, the reaction would take place also already at lower temperatures. Thus, the need for such high temperatures is predominantly due to kinetic reasons. 

A second procedure, using the methods of thermodynamics applied to Irreversible processes, offers another new approach for understanding the failure of materials. For example, the equilibrium thermodynamics of closed systems predicts that a system will evolve In a manner that minimizes Its energy (or maximizes Its entropy). The thermodynamics of Irreversible processes In open systems predicts that the system will evolve In a manner that minimizes the dissipation of energy under the constraint that a balance of power Is maintained between the system and Its environment. Application of these principles of nonlinear Irreversible thermodynamics has made possible a formal relationship between thermodynamics, molecular and morphological structural parameters. 

Control in open systems (those that have inputs and outputs from their environment) implies the need for communication. Bertalanffy distinguished between closed systems, in which unchanging components settle into a state of equihbrium, and open systems, which can be thrown out of equilibrium by exchanges with their environment. 

The chapter divides in two in early sections we describe the behavior of nomeact-ing systems, while in later sections we deal with systems in which reactions occur. In 7.1 we combine the first and second laws to obtain criteria for identifying limitations on the directions of processes and for identifying equilibrium in closed multiphase systems. Then in 7.2 we develop the analogous relations for heat, work, and material transfers in open systems. With the material in 7.2 as a basis, we then present in 7.3 the thermodynamic criteria for equilibrium among phases. 

This section contains several models whose spatiotemporal behavior we analyze later. Nontrivial dynamical behavior requires nonequilibrium conditions. Such conditions can only be sustained in open systems. Experimental studies of nonequilibrium chemical reactions typically use so-called continuous-flow stirred tank reactors (CSTRs). As the name implies, a CSTR consists of a vessel into which fresh reactants are pumped at a constant rate and material is removed at the same rate to maintain a constant volume. The reactor is stirred to achieve a spatially homogeneous system. Most chemical models account for the flow in a simplified way, using the so-called pool chemical assumption. This idealization assumes that the concentrations of the reactants do not change. Strict time independence of the reactant concentrations cannot be achieved in practice, but the pool chemical assumption is a convenient modeling tool. It captures the essential fact that the system is open and maintained at a fixed distance from equilibrium. We will discuss one model that uses CSTR equations. All other models rely on the pool chemical assumption. We will denote pool chemicals using capital letters from the start of the alphabet. A, B, etc. Species whose concentration is allowed to vary are denoted by capital letters... 

---