Equilibrium open systems

In the case of interface equilibrium (open system conditions), the partition coefficient is valid only at the interface between solid and liquid (or at zero distance from the interface) and at time of crystallization (or melting) t ... 

Part II of this book represents the bulk of the material on the analysis and modeling of biochemical systems. Concepts covered include biochemical reaction kinetics and kinetics of enzyme-mediated reactions simulation and analysis of biochemical systems including non-equilibrium open systems, metabolic networks, and phosphorylation cascades transport processes including membrane transport and electrophysiological systems. Part III covers the specialized topics of spatially distributed transport modeling and blood-tissue solute exchange, constraint-based analysis of large-scale biochemical networks, protein-protein interactions, and stochastic systems. 

A system in which the dependent variables are constant in time is said to be in a steady or stationary state. In a chemical system, the dependent variables are typically densities or concentrations of the component species. Two fundamentally different types of stationary states occur, depending on whether the system is open or closed. There is only one stationary state in a closed system, the state of thermodynamic equilibrium. Open systems often exhibit only one stationary state as well however, multistability may occur in systems with appropriate elements of feedback if they are sufficiently far from equilibrium. This phenomenon of multistability, that is, the existence of multiple steady states in which more than one such state may be simultaneously stable, is our first example of the universal phenomena that arise in dissipative nonlinear systems. 

Some philosophic systems (e.g., Aristotle s) consider objects that retain their identity through time substances ) as fundamental, others (e.g., Whitehead s) deny that such coherences are so basic but stUl consider them important The Universe achieves its values by reason of its coordination into societies of societies, and societies of societies of societies (Whitehead 1967, 206). Recent progress in physical chemistry has identified new modes of dynamic coherence (which occur in far-from equilibrium open systems) that are critically important in many areas of science—and have shown how those integrations exemplify and extend current theory (Kondepudi and Prigogine 1998). This major advance is not yet appreciated by philosophers—in part at least because such coherences do not easily fit into prevailing categorial schemes. 

Ertl s research has demonstrated that a rather simple system (a chemical reaction occurring between two diatomic molecules on a well-defined single crystal surface with fixed external parameters and well-established mechanism) can be used to study (and model) a quite complex behavior. The conclusions which it allows us to draw about far-from-equilibrium open systems transcend catalysis and surface science and provide clues about laws believed to govern the whole of nature. 

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

Since the phase rule treats only the intensive state of a system, it apphes to both closed and open systems. Duhem s theorem, on the other hand, is a nJe relating to closed systems only For any closed system formed initially from given masses of preseribed ehemieal speeies, the equilibrium state is completely determined by any two propeities of the system, provided only that the two propeities are independently variable at the equilibrium state The meaning of eom-pletely determined is that both the intensive and extensive states of the system are fixed not only are T, P, and the phase compositions established, but so also are the masses of the phases. 

In Figure 2 the solubility and speciation of plutonium have been calculated, using stability data for the hydroxy and carbonate complexes in Table III and standard potentials from Table IV, for the waters indicted in Figure 2. Here, the various carbonate concentrations would correspond to an open system in equilibrium with air (b) and closed systems with a total carbonate concentration of 30 mg/liter (c,e) and 485 mg/liter (d,f), respectively. The two redox potentials would roughly correspond to water in equilibrium wit air (a-d cf 50) and systems buffered by an Fe(III)(s)/Fe(II)(s)-equilibrium (e,f), respectively. Thus, the natural span of carbonate concentrations and redox conditions is illustrated. 

Generally, in a system that is energetically and materially isolated from the environment without a change in volume (a closed system), the entropy of the system tends to take on a maximum value, so that any macroscopic structures, except for the arrangement of atoms, cannot survive. On the other hand, in a system exchanging energy and mass with the environment (an open system), it is possible to decrease the entropy more than in a closed system. That is, a macroscopic structure can be maintained. Usually such a system is far from thermodynamic equilibrium, so that it also has nonlinearity. 

Therefore, the development of an open system can be described by a set of nonlinear equations that usually have solutions in equilibrium at infinity. In some cases, the solutions change their states greatly before and after the specific values of physical parameters these phenomena are called bifurcations. Figure 1 shows a simple case of bifurcation. For example, the following nonlinear differential equation is considered,... 

Another feature of the equilibrium copolymerization, shown in the next equations, is specific for ring-opening systems ... 

The flow of matter and energy through an open system allows the system to self-organize, and to transfer entropy to the environment. This is the basis of the theory of dissipative structures, developed by Ilya Prigogine. He noted that self-organization can only occur far away from thermodynamic equilibrium [17]. 

Dissipative, open systems that allow for the flux of energy and matter may exhibit non-linear and complex behavior. Following the above argumentation, complex systems are usually far from thermodynamic equilibrium but, despite the flux, there may be a stable pattern, which may arise from small perturbations that cause a larger, non-proportional effect. These patterns can be stabilized by positive (amplifying)... 

One of the most crucial influencing factors in planar chromatography is the vapor space and the interactions involved. The fact that the gas phase is present, in addition to stationary and mobile phases, makes planar chromatography different from other chromatographic techniques. Owing to the characteristic of an open system the stationary, mobile, and vapor phases interact with each other until they all are in equihbrium. This equilibrium is much faster obtained if chamber saturation is employed. This is the reason for differences in separation quality when saturated and unsaturated chambers are used. However, the humidity of the ambient air can also influence the activity of the layer and, thus, separation. Especially during sample application, the equihbrium between layer activity and relative humidity of the... 

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. 

Figure 4. Evolution of the (N2/N1) ratio in a reservoir in the two cases of closed system evolution (as a function of t/T2, where t is the time since fractionation), or in an open-system, steady-state reservoir (the steady-state (N2/N1) ratio is plotted as a function of x/ T2, where x is the residence time of the magma in the reservoir). Initial fractionation results in an arbitrarily chosen ratio of 2, which is kept constant for the iirfluent magma in the continuously replenished reservoir. The diagram shows that radioactive equilibrium is reached sooner in a closed system evolution. It also illustrates the fact that the radioactive parent-daughter pair should be chosen such as T2 is commensmate with the residence time of the magma in the reservoir (e.g., x/ T2 between 0.1 and 10). If T2 is much longer than the residence time x, then the (N2/N1) ratio will remain close to the initial value (here 2). If T2 is much shorter than x, equilibrium will be nearly established in the reservoir.

In an early work by Mertz and Pettitt, an open system was devised, in which an extended variable, representing the extent of protonation, was used to couple the system to a chemical potential reservoir [67], This method was demonstrated in the simulation of the acid-base reaction of acetic acid with water [67], Recently, PHMD methods based on continuous protonation states have been developed, in which a set of continuous titration coordinates, A, bound between 0 and 1, is propagated simultaneously with the conformational degrees of freedom in explicit or continuum solvent MD simulations. In the acidostat method developed by Borjesson and Hiinenberger for explicit solvent simulations [13], A. is relaxed towards the equilibrium value via a first-order coupling scheme in analogy to Berendsen s thermostat [10]. However, the theoretical basis for the equilibrium condition used in the derivation seems unclear [3], A test using the pKa calculation for several small amines did not yield HH titration behavior [13],... 

Only in the last decades has the thermodynamics of open systems been treated intensively and successfully. The thermodynamics of irreversible systems was studied initially by Lars Onsager, and in particular by Ilya Progogine and his Brussels school both studied systems at conditions far from equilibrium. Certain systems have the capacity to remain in a dynamic state far from equilibrium by taking up free energy as a result, the entropy of the environment increases (see Sect. 9.1). 

Systems with dissipative self-organisation are important for processes which lead to biogenesis. These are open systems, the internal state of which is dominated by a disequilibrium far away from the equilibrium state. 

A final observation is in order the quantitative application of the equilibrium thermodynamical formalism to living systems and especially to ecosystems is generally inadequate since they are complex in their organisation, involving many interactions and feedback loops, several hierarchical levels may have to be considered, and the sources and types of energy involved can be multiple. Furthermore, they are out-of-equilibrium open flow systems and need to be maintained in such condition since equilibrium is death. Leaving aside very simple cases, in the present state of the art we are, therefore, limited to general semiquantitative statements or descriptions (e.g. ecosystem narratives ). 

Allowing an open system to close and come to equilibrium implies an extent of reaction - a measure of how close the system has come towards the equilibrium condition. For a single species the extent of the reaction towards an equilibrium concentration is given by ... 

The simplest open-system model involves a reactant which, if it is a mineral, is undersaturated in an initial fluid. The reactant is gradually added into the equilibrium system over the course of the reaction path (Fig. 2.3). The reactant dissolves irreversibly. The process may cause minerals to become saturated and precipitate or... 

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 ,... 

Recall that the equilibrium condition for a closed system at constant T and p was given by eq. (1.41). For an open system the corresponding equation is... 

Put in ordinary terms, the more successful we are in causing a separation, the more propensities there are for a re-mixing of the components. There are many ways this can occur but there are a fewer number of important routes to mixing. It seems reasonable that we examine these before we consider all the possible ways in which thermodynamics can be controlled in general terms. In almost all equilibrium separation systems, the separation can occur either in a packed bed of particles or fibers or in an open channel or tube. The stationary phase is either coated on the walls of the channel or on the particles/fibers of the packed bed. If there were no mixing mechanisms an infinitely narrow packet containing the components would become a series of infinitely narrow packets of pure components moving at different velocities toward the end of the packed bed or tube. 

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