Irreversible phenomenon

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

J. C., Irreversibility phenomena associated with heat transfer and fluid friction in laminar flows through singly connected ducts, Int. J. Heat Mass Transfer 40 (1997) 905-914. 

In addition to the general problem of the kinetics of the approach towards equilibrium, the statistical mechanics of irreversible phenomena concern in particular the study of transport phenomena. The latter are calculated in a stationary or quasi-stationary form (the distribution functions do not vary or vary in hydrodynamic fashion). Therefore, let us consider (see, for... 

LS.4. I. Prigogine, Etude Thermodynamique des Phenomenes Irreversibles, (Thermodynamic Study of Irreversible Phenomena) (These d agregation de I Enseignement superieur, Universite Libre de Bruxelles), Dunod, Paris et Desoer, Liege. 

THL.7. I. Prigogine et J. M. Wiame, Biologic et thermodynamique des phenomenes irreversibles (Biology and thermodynamics of irreversible phenomena), Experientia 2, 451 53 (1946). 

THL.13. I. Prigogine, Quelques aspects de la thermodynamique des phenomenes irreversibles (Some aspects of thermodynamics of irreversible phenomena). Coll. Thermodynamique, Un. Intemat. Phys., Bruxelles, 1948. 

THL.29. I. Prigogine et P. Mazur, Sur 1 extension de la thermodynamique aux phenomenes irreversibles lies aux degres de hbeerte internes (On the extension of thermodynamics to the irreversible phenomena related to internal degrees of freedom), Physica 19, 241-254 (1953). 

THL.30. I. Prigogine, Sur la theorie variationnelle des phenomenes irreversibles (On the variational theory of irreversible phenomena), Bull. Cl. Sci. Acad. Roy. Belg. 40, 471 83 (1954). 

MSN.5. G. Klein et I. Prigogine, Sur la mecanique statistique des phenomenes irreversibles, I (On statistical mechanics of irreversible phenomena, I), Physica 19, 74-00 (1953). 

The irreversible phenomena represent entropy gain through irrecoverable heat losses as follows, where X is the thermal conductivity and lis the length ... 

Reversibility. Apparent irreversibility phenomena of ion exchange in NaX were studied with zinc and cobalt ions using a temperature-variation method described in the experimental section. In view of the high selectivity of NaX for bivalent cations at low zeolite loading, the concentration of bivalent ions in the equilibrium solution is quite sensitive to small changes in the surface composition. In fact, the adsorption removal of bivalent cations at low loading, below 0.2, is quantitative or nearly so (99.5% or better). Consequently the value of the equilibrium concentration is an ideal criterion for assessing either reversibility or equilibrium conditions. 

The cell reaction for cells without liquid junction can be written as the sum of an oxidation reaction and a reduction reaction, the so-called half-cell reactions. If there are C oxidation reactions, and therefore C reduction reactions, there are C C — 1) possible cells. Not all such cells could be studied because of irreversible phenomena that would take place within the cell. Still, a large number of cells are possible. It is therefore convenient to consider half-cell reactions and to associate a potential with each such reaction or electrode. Because of Equation (12.88), there would be (C - 1) independent potentials. We can thus assign an arbitrary value to the potential associated with one half-cell reaction or electrode. By convention, and for aqueous solutions, the value of zero has been assigned to the hydrogen half-cell when the hydrogen gas and the hydrogen ion are in their standard states, independent both of the temperature and of the pressure on the solution. 

S. L. Ginzburg, Irreversible Phenomena in Spin Glasses, Nauka, Moscow, 1989. 

However, the presented interpretation of equilibrium processes turns out to be unsatisfactory for the analysis of possibility to use equilibrium descriptions for irreversible phenomena. The interpretation of interrelations between equilibrium and reversibility that was given by... 

It will apparently be possible to provide coordination between the capabilities of equilibrium models in (1) the analysis of perfection of the energy and substance transformation processes and (2) the analysis of different irreversible phenomena on the basis of dual interpretation of equilibrium processes as being both reversible and irreversible at a time. In the first case they are convenient for interpretation as reversible in terms of the system interaction with the environment and in the second case—as irreversible in terms of their inner content according to Gorban. It is clear that to explain the dual interpretation it is necessary to extend the analysis by Gorban to the nonisolated thermodynamic systems with other characteristic functions to be used along with entropy. 

L. Boltzmann came to the following conclusion The assumptions used by Clausius and Maxwell are sufficient to give a unified interpretation of irreversible phenomena. In particular, they give a kinetic meaning to the monotonic increase of entropy with time.6... 

So the stationary state is maintained through the decrease in entropy exchanged between the system and its surrounding. Entropy change inside an elementary volume by irreversible phenomena is the local value of the sum of entropy increments. By the second law of thermodynamics, the entropy production c/,5 is always positive for irreversible changes and zero for reversible changes. 

Classical thermodynamics states that the change of entropy production as a result of the irreversible phenomena inside a closed adiabatic system is always positive. This principle allows for the entropy to decrease at some place in the systems as long as a larger increase in the entropy at another place compensates for this loss. 

If we disregard the higher order terms, these expansions become linear relations, and we have the general type of linear phenomenological equations for irreversible phenomena... 

Some processes may have forces operating far away from equilibrium where the linear phenomenological equations are no longer applicable. Such a domain of irreversible phenomena, such as some chemical reactions, periodic oscillations, and bifurcation, is examined by extended nonequilibrium thermodynamics. Extending the methods of thermodynamics to treat the linear and nonlinear phenomena, and such dissipative structures are attracting scientists from various disciplines. 

Since a flux may be considered to be driven by a corresponding force, no flux can occur without a force field, in which case all irreversible phenomena cease l now vanishes, and Eq. (6.2.16) becomes a conservation condition for entropy. 

Steady—state conditions are characterized by the requirement that the fluxes and forces characterizing irreversible phenomena in a system be independent of time. Some of these forces and/or fluxes may vanish in the extreme case where all JL and X are zero the system is necessarily in an equilibrium state. 

By arguments now familiar we have introduced a new function of state, G = E — TS + PV, the Gibbs (free) energy, involving the indicated combination of other functions of state. As is evident, changes in this quantity are tracked by the reversible performance of work at constant T and P other than mechanical. Moreover, if irreversible phenomena occur under isothermal-isobaric conditions in the absence of any work performance then... 

Strictly speaking, the differential d (in dX) should only be used to indicate an infinitesimal variation of X along a thermodynamically describable path, and should not be applied to extrathermodynamic, irreversible phenomena. In our case, however, we demand only that the initial and final points of the process be representable as equilibrium states. The use of yet another symbol for a nonequilibrium differential process seems excessive. 

This shows that in the absence of external driving forces TodS and PodV any ongoing process will take place internally, and will not be subject to experimental control. Thus, with d9 > 0, the energy of the system spontaneously diminishes until these irreversible phenomena have run their course, at which point E assumes a minimum value consistent with the imposed constraints. Eq. (l.B.le) coincides with Eq. (1.12.13b) that holds when no work is performed. By contrast, if the conditions of the surroundings and of the system are in near balance, so that T To and P Pq, then any process becomes essentially reversible, and the differential energy function (1.13.Id) assumes the standard form... 

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