Irreversible thermodynamics difficulties

It is well known that a flow-equilibrium must be treated by the methods of irreversible thermodynamics. In the case of the PDC-column, principally three flows have to be considered within the transport zone (1) the mass flow of the transported P-mer from the sol into the gel (2) the mass flow of this P-mer from the gel into the sol and (3) the flow of free energy from the column liquid into the gel layer required for the maintenance of the flow-equilibrium. If these flows and the corresponding potentials could be expressed analytically by means of molecular parameters, the flow-equilibrium 18) could be calculated by the usual methods 19). However, such a direct way would doubtless be very cumbersome because the system is very complicated (cf. above). These difficulties can be avoided in a purely phenomenological theory, based on perturbation calculus applied to the integrated transport Eq. (3 b) of the PDC-column in a reversible-thermodynamic equilibrium. 

While the formalism of irreversible thermodynamics provides an elegant framework for describing molecular displacements, it provides too little substance and too much conceptual difficulty to justify its development here. For instance, it provides no values, not even estimates, for various transport coefficients such as the diffusion coefficient. Cussler has noted the disappointment of scientists in several disciplines with the subject [7]. It is the author s opinion that a clearer understanding of the transport processes and interrelationships that underlie separations can be obtained from a mechanical-statistical approach. This is developed in the subsequent sections. 

From a physical point of view, it seems that measurable quantities are mixture invariant (cf. end of Sect. 4.4). Such are the properties of mixture like y, T (see (4.94), (4.236), (4.240), (4.225)) but also the chemical potentials ga. Note that also heat flux is transformed as (4.118) (with functions (4.223)) and therefore heat flux is mixture invariant in a non-diffusing mixture (all = o) in accord with its measurability. But heat flux is mixture non-invariant in a diffusing mixture, consistently with our expectation of difficulties in surface exchange (of masses) of different constituents with different velocities together with heat. We note that all formulations of heat flux used in linear irreversible thermodynamics [1, 120] (cf. Rems. 11 in this chapter, 14 in Chap. 2) are contained (by arbitrariness of rjp) in expression (4.118) for heat flux in a diffusing mixture. 

Concentrating initially on time dependence for infinitesimal strains, the correspondence principle relating viscoelastic and elastic behaviour, well established for isotropic systems, may be simply extended to apply to the anisotropic case. There is, however, a difficulty in showing that the compliance matrix Sy will necessarily have the same symmetry properties in the viscoelastic case as in the classically elastic case. This difficulty arises from the thermodynamic nature of part of the argument used in proving symmetry. In the viscoelastic case the proof would depend upon the less well established principles of irreversible thermodynamics. No discussion on this point will be attempted the symmetry properties of Sij as determined in elastic theory will be accepted and its validity examined in the light of the experimental data available. This data shows that there may be systematic deviations from the assumptions in work at finite strains and further work is needed in this area. However, the manner in which these deviations occur does not detract significantly from the utility of the simple formalism in many cases. 

In spite of the above difficulties, a simple theory [26-28] based on Helmholtz model yields a microscopic picture which is useful in understanding the role of pore size and channel length along with the electrical characteristics of this interface in electro-kinetic phenomena. Whereas the macroscopic theory based on irreversible thermodynamics does not depend on any model, the theory discussed below would be valid provided the situation conforms to the model. Both approaches are complementary in understanding the phenomena. 

A common difficulty in all the aforementioned cases is the irregular geometry of the porous medium. In addition, a precise analysis will have to consider that the diffusion in the pores is modified by surface diffusion and surface reactions. A completely phenomenological approach based on irreversible thermodynamics would give theoretically consistent transport expressions but would be too complicated for experimental determination of the transport coefficients and for the solution of the conservation equations. 

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. 

In practice, this requires a marked increase in the amount of experimental data necessary to characterize an enzyme kinetically (see above). There also may be increased difficulty in interpreting the kinetic analysis (see above). While many reactions may operate far from thermodynamic equilibrium in vivo, there also are examples of reactions that operate near equilibrium and actually reverse direction under physiological conditions. Thus, one generally cannot assume rate laws for irreversible reactions. 

The transition from cellulose I to cellulose II is irreversible cellulose II is the thermodynamically more stable arrangement. In contrast to cellulose I the chains do not point in the same direction as seen in Figure 3.4, bottom. The corner chain points upwards whereas the centre chain points downwards. This poses some difficulties in explaining the solid-state phase transition from natural cellulose to cellulose II after alkali treatment and thus an interdiffusion model was proposed for chains from adjacent natural cellulose crystallites of opposite orientation [19]. 

The extensive properties of the overall system that is not in equilibrium, such as volume or energy, are simply the sums of the (almost) equilibrium properties of the subsystems. This simple division of a sample into its subsystems is the type of treatment needed for the description of irreversible processes, as are discussed in Sect. 2.4. Furthermore, there is a natural limit to the subdivision of a system. It is reached when the subsystems are so small that the inhomogeneity caused by the molecular structure becomes of concern. Naturally, for such small subsystems any macroscopic description breaks down, and one must turn to a microscopic description as is used, for example, in the molecular dynamics simulations. For macromolecules, particularly of the flexible class, one frequently finds that a single macromolecule may be part of more than one subsystem. Partially crystalhzed, linear macromolecules often traverse several crystals and surrounding liquid regions, causing difficulties in the description of the macromolecular properties, as is discussed in Sect. 2.5 when nanophases are described. The phases become interdependent in this case, and care must be taken so that a thermodynamic description based on separate subsystems is still valid. 

However, the ptractical apphcation of the second law in the analysis of equilibrium irreversible trajectories faced great difficulties. Clausius and then Helmholtz, Boltzmann, J. Thomson, Planck and other researchers tried to harmonize the second law of thermodynamics with the principle of the least action and derive the equation that meets this painciple similar to the equations (3) or (4) for dissipative macroscopic systems (in which the organized energy forms turn into a non-organized form, i.e. heat, due to friction). As is known their attempts were unsuccessful and resulted in understanding the necessity to statistically substantiate thermodynamics (Polak, 2010). 

The most serious obstacle to the study of models of the uptake of metal cations, in spite of the lively experimental activity, is probably the theoretical difficulties in energetics and molecular structure at the cellular barrier membranes, the chemistry of which seems to be unexpectedly complex. In principle, the uptake of metal cations is divided into the passive, physicochemical phase and an active phase that depends on metabolic sources of energy. The active uptake has been adequately demonstrated and the sites of input for different groups of cations have been revealed by competitive experiments. In spite of many attempts, however, the active carriers of cations have not been found. It is an appropriate time for new experimental and/or theoretical ideas. For example, the new, developing thermodynamics of irreversible processes has energetically outlined some very interesting possibilities of the stationary state for the coupling of the processes which are worthy of closer experimentation. 

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