Thermodynamic equilibrium closed system

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... 

In a strict sense, the term Rayleigh fractionation should only be used for chemically open systems where the isotopic species removed at every instant are in thermodynamic and isotopic equilibrium with those remaining in the system at the moment of removal. Furthermore, such an ideal Rayleigh distillation is one where the reactant reservoir is finite and well mixed, and does not re-react with the product (Clark and Fritz, 1997). However, the term Rayleigh fractionation is commonly applied to equilibrium closed systems and kinetic fractionations as well because the situations may be computationally identical. Isotopic fractionations are strongly affected by whether a system is open or closed. 

Thermodynamics. Consider an equilibrium closed system of a liquid, L 2, in contact with a plane surface of an isotropic solid, S (which is not soluble in L 2), and saturated vapor, Vj, of liquid L2. The Young-Dupr equation is... 

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. 

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. 

From all possible statistical systems the equilibrium closed systems are allocated. For such systems the physical theory, referred to as equilibrium thermodynamics or simply thermodynamics, is well developed. Thermodynamics is the phenomenological doctrine of heat. Classical thermodynamics asserts that the isolated thermodynamic system cannot spontaneously change it s state . This statement is sometimes referred to as the zeroth law of thermodynamics, another assertion of which is that If two thermodynamic systems are in thermodynamic equilibrium with some third body they are in thermodynamic equilibrium with each other . 

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. 

As we all know from thermodynamics, closed systems in equilibrium have minimum free energy and maximum entropy. If such a system were brought out of equilibrium, i.e. to a state with lower entropy and higher free energy, it would automatically decay to the state of equilibrium, and it would lose all information about its previous states. A system s tendency to return to equilibrium is given by its free energy. An example is a batch reaction that is run to completion. 

A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

In order to obtain a qualitative view of how the transition regime differs from the continuum flow or the slip flow regime, it is instructive to consider a system close to thermodynamic equilibrium. In such a system, small deviations from the equilibrium state, described by thermodynamic forces X, cause thermodynamic fluxes J- which are linear functions of the (see, e.g., [15]) ... 

In the present case, thermodynamic constraints on the rate indicate that the first stage effluent should correspond to a conversion in the vicinity of 0.87 and an effluent temperature near 824.4 °K. Beyond this point, the system is so close to thermodynamic equilibrium that substantial increments in the reactor length do not produce noticeable increments in the conversion. 

Some of the terms used in classical thermodynamics which refer to equilibrium states and closed systems have become important outside the boundaries of physics one example is the term adapted state in Darwinian evolution theory, which represents a type of equilibrium state between the organism and its environment. 

The microcanonical ensemble in quantum statistics describes a macroscopi-cally closed system in a state of thermodynamic equilibrium. It is assumed that the energy, number of particles and the extensive parameters are known. The Hamiltonian may be defined as... 

In the above sense, the system may be considered as a thermodynamically closed system that will attain equilibrium if a non-equilibrium fluctuation were produced by some external means. 

The equations that describe equilibrium conditions between two phases of the same substance are derivable from the two laws of thermodynamics with the aid of the functions that we defined in the preceding chapter. Let us represent the equilibrium in a closed system at any given temperature and pressure by the equation... 

The exchange in the alkoxyamine-based polymer occurs in a radical process that is tolerant of many functional groups. The exchange process is therefore applicable to polymers with various functional groups. TEMPO-based polyester 43 and polyurethane 44 were synthesized for studies of the scrambling of disparate polymers imder thermodynamic control (Fig. 8.11) [37], Two kinds of TEMPO-based polymers were mixed and heated in a closed system. After 24 hours when the crossover reaction achieved equilibrium, GPC and NMR analyses revealed that they were totally scrambled through bond recombination on the TEMPO units. 

An example of such a system is one in which the internal energy, U, and the volume, V, are constant. If these are the only two constraints on the system then, at thermodynamic equilibrium, the entropy, S, is at a maximum. On the other hand, if entropy and volume are constant for the isolated system then, at thermodynamic equilibrium, the internal energy is at a minimum. See also Closed System Open System... 

One particular pattern of behaviour which can be shown by systems far from equilibrium and with which we will be much concerned is that of oscillations. Some preliminary comments about the thermodynamics of oscillatory processes can be made and are particularly important. In closed systems, the only concentrations which vary in an oscillatory way are those of the intermediates there is generally a monotonic decrease in reactant concentrations and a monotonic, but not necessarily smooth, increase in those of the products. The free energy even of oscillatory systems decreases continuously during the course of the reaction AG does not oscillate. Nor are there specific individual reactions which proceed forwards at some stages and backwards at others in fact our simplest models will comprise reactions in which the reverse reactions are neglected completely. 

We are thus, in many instances, more interested in the transient behaviour early in a reaction than we are in the more easily studied final or equilibrium state. With this in mind, we shall be concerned in our early chapters with simple models of chemical reaction that can satisfy all thermodynamic requirements and yet still show oscillatory behaviour of the kind described above in a well-stirred closed system under isothermal or non-isothermal conditions. 

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