The Thermodynamics of Irreversible Systems

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. 

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). 

Equilibrium thermodynamics was developed about 150 years ago. It is concerned only with the achievement of an equilibrium state, without taking into account the time which a system requires for the transition from an initial to a final state. Thus, only the thermodynamics of irreversible processes can be used to describe processes which lead to the formation of self-organising systems. Here, the time factor, and thus also the rate at which material reactions occur, is taken into account. Evolutionary processes are irreversibly coupled with temporal sequences, so that classical thermodynamics no longer suffices to describe them (Schuster and Sigmund, 1982). 

As already mentioned, a continual inflow of energy is necessary to maintain the stationary state of a living system. It is mostly chemical energy which is injected into the system, for example by activated amino acids in protein biosynthesis (see Sect. 5.3) or by nucleoside triphosphates in nucleic acid synthesis. Energy flow is always accompanied by entropy production (dS/dt), which is composed of two contributions  

Thc //ow.v these enter the system, and also leave it again (de.S7d/). 

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Eigen s theory describes the self-organisation of biological macromolecules on the basis of kinetic considerations and mathematical formulations, which are in turn based on the thermodynamics of irreversible systems. Evolutionary processes are irreversibly linked to the flow of time. Classical thermodynamics alone cannot describe them but must be extended to include irreversible processes, which take account of the arrow of time (see Sect. 9.2). Eigen s theory is based on two vital concepts ... 

The internal entropy production this represents the time-related entropy growth generated within the system (djS/df). The internal entropy production is the most important quantity in the thermodynamics of irreversible systems and reaches its maximum when the system is in a stationary state. The equation for the entropy production is then ... 

The theory of the thermodynamics of irreversible systems (Prigogine, 1979 Prigogine and Stengers, 1986) shows that the differential quotient of entropy with time (the change of entropy with time) can be expressed as the sum of products, the terms of which contain a force factor and a flow factor. In chemical systems, the... 

By taking into account the latest results on the behaviour of systems far away from equilibrium, Kondepudi and Nelson (1985) were able to show by calculation that L-amino acids are slightly favoured. There is a very tiny stabilisation effect due to the weak interaction amplification mechanisms cause this effect to reach 98% of the probability that L-enantiomers of amino acids are favoured for incorporation into polymers. The amplification mechanisms are explained by the thermodynamics of irreversible systems. 

A final remark should be made as to the validity of eq. (2.13). This equation suggests the existence of a set of independent relaxation mechanisms. A general proof for the existence of such mechanisms could be given for visco-elastic solids in terms of the thermodynamics of irreversible processes (52) at small deviation from equilibrium. For liquid systems, however, difficulties arise from the fact that in these systems displacements occur which are not related to the thermodynamic functions. 

Any dynamic system becomes stable eventually and comes to the rest point, i.e. attains its equilibrium or steady state. For closed systems, a detailed equilibrium is achieved at this point. This is not so simple as it would seem, as substantiated by a principle of the thermodynamics of irreversible processes. At a point of detailed equilibrium not only does the substance concentration remain unchanged (dcjdt = 0), but also the rate of each direct reaction is balanced by that of its associated reverse counterpart... 

According to the thermodynamics of irreversible processes, the mutual diffusion coefficient D may be a function of penetrant concentration ct, position x, and time t. In the present chapter we shall discuss sorption behavior of systems in which D varies with cx only, and shall use the notation D (cx) to indicate this condition. It is assumed that the sample film is so thin that diffusion takes place effectively in the direction of its thickness. At the beginning of an absorption or a desorption experiment the film is conditioned so that Cj is uniform everywhere in it. This initial concentration is denoted by cf. Then we have... 

Transference numbers are quantities which are treated in the thermodynamics of irreversible processes. In a continuous system, the average velocity Vi of a species i related to a reference velocity w, describes the diffusional motion of the species i. The diffusion current density Ji represents in moles/cm sec the flow of species i in unit time perpendicular to a surface of unit area which by itself is moving with velocity... 

The essential advantage of this formulation of the second principle is that the inequality (2) is valid whatever may be the exact conditions under which the system changes. The fundamental problem of the thermodynamics of irreversible phenomena is the explicit evaluation of the entropy production. 

The successful development of the thermodynamics of irreversible phenomena depends on the possibility of an explicit evaluation of the production of entropy, and for this it is necessary to assume that the thermodynamic definition of entropy can be applied equally to systems which are not in equilibrium, that is to states whose mean lifetime is limited. We are thus confronted immediately with the problem of the domain of validity of the thermodynamic treatment of irreversible phenomena, which can be determined only by a comparison of the results of the thermodynamic treatment with those obtained by the use of statistical mechanics. This problem wall be dealt with in more detail in the third volume of this work meanwhile the main conclusions can be summarized as follows. 

Haase, R., The Thermodynamics of Irreversible Processes, Addison-Wesley, London, England, 1969. Haase, R. and Siry, M., Diffusion im kritischen Entmischungsgebiet binarer fliissiger Systeme, ... 

The thermodynamics of irreversible processes should be set up from the scratch as a continuum theory, treating the state parameters of the theory as field variables [32]. This is also the way in which classical fluid mechanic theory is formulated. Therefore, in the computational fluid dynamics literature, the transport phenomena and the extensions of the classical thermodynamic relations are both interpreted as closures of the fluid dynamic theory. The validity of the thermodynamic relations for fluid dynamic systems has been approached from the viewpoint of the kinetic theory of gases [13]. However, any Arm distinction between irreversible thermodynamics and fluid mechanics... 

The last term is characteristic of the thermodynamics of irreversible processes. Its magnitude becomes positive if the system s processes are irreversible. Typical irreversible processes are the adsorption or desorption of surfactants at liquid interfaces. The derivative of the second term of Eq. (2C.2), as local entropy production is... 

Onsager relations - An important set of equations in the thermodynamics of irreversible processes. They express the symmetry between the transport coefficients describing reciprocal processes in systems with a linear dependence of flux on driving forces. 

In contrast to equilibrium thermodynamics, the thermodynamics of irreversible processes portray the application of thermodynamic methods as dynamic and therefore time-dependent procedures. The name Prigo-gine must be mentioned in relationship to this—he received for his work in this area the Nobel Prize in the year 1977. A new, very complex thermodynamics originated from his examination method for chemical reactions, and was developed by us, to come to a successful description of heterogenous multiphase polymer systems. This theory interprets crazing fracture energy dissipation and fracture mechanism in a totally new way on the basis of dissipative structures in polymer blends and their dynamics, For a list of abbreviations used in this section sec page 610,... 

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. 

Transport of a substance caused by a thermodynamic force which arises from the gradient of its chemical potential is termed diffusion. It can be formulated phenomenologically by application of the thermodynamics of irreversible processes [19]. In what follows we confine ourselves to binary systems consisting... 

It is the fundamental requirement of the thermodynamics of irreversible processes that for a two-component system / evaluated from D and II agrees with that from 6 within experimental error. An excellent check of this requirement, described by Nystrom and Roots [31], is reproduced in Figure 7-9. However,... 

Open system 1. a system in dynamic equilibrium (see Steady state) with its surroundings, i.e. there is a continual exchange of material, energy and information with the environment. Application of the theory of O. s. to living systems (by Bertalanffy) involves the thermodynamics of irreversible processes. 

This law (1931) was the first attempt to use the thermodynamics of irreversible processes as a useful instrument for the rationalisation of physical-chemical systems. An example of the application of the above is the Seebeck effect in the thermoelectric field. If the extremities of a bimetallic couple reach different temperatures, there will be a flow of electrical current I caused by the potential difference. Considering Table 4.1 and the discontinuous system gives ... 

In the form of eq. (5-30), Pick s second law applies only to one-dimensional problems in an isotropic medium. The index i on the diffusion coefficient has been removed in order to make it clear that this is no longer the component diffusion coefficient D,-, but rather, it is the chemical interdiffusion coefficient. Normally, the chemical interdiffusion coefficient will be a function of the individual component diffusion coefficients Di because of the coupling of the fluxes in the lattice system. When local thermodynamic equilibrium prevails, the coefficients Di are, in turn, unique functions of the composition. From the thermodynamics of irreversible processes it can be shown [6] that in binary systems there is only one independent transport coefficient, and in general, in n-component systems there can only be (n - 1) /2 independent transport coefficients. 

Let us also examine Eq. (2.82) for a convection-free system at isothermal conditions, where D = 0, and VT = 0. For the system to have no entropy production (that is, to be at equilibrium state), V/i, == —g. Since in a gravity field gx —gv = 0 and g = g, then djijdz = —g or dpii = —M gdz, which is the same as Eq. (2.13). With the above background, we now switch to the expression for the total diffusion flux, which can be derived from the thermodynamics of irreversible processes. 

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