Irreversible thermodynamics direct coefficients

We will introduce basic kinetic concepts that are frequently used and illustrate them with pertinent examples. One of those concepts is the idea of dynamic equilibrium, as opposed to static (mechanical) equilibrium. Dynamic equilibrium at a phase boundary, for example, means that equal fluxes of particles are continuously crossing the boundary in both directions so that the (macroscopic) net flux is always zero. This concept enables us to understand the non-equilibrium state of a system as a monotonic deviation from the equilibrium state. Driven by the deviations from equilibrium of certain functions of state, a change in time for such a system can then be understood as the return to equilibrium. We can select these functions of state according to the imposed constraints. If the deviations from equilibrium are sufficiently small, the result falls within a linear theory of process rates. As long as the kinetic coefficients can be explained in terms of the dynamic equilibrium properties, the reaction rates are directly proportional to the deviations. The thermodynamic equilibrium state is chosen as the reference state in which the driving forces X, vanish, but not the random thermal motions of structure elements i. Therefore, systems which we wish to study kinetically must first be understood at equilibrium, where the SE fluxes vanish individually both in the interior of all phases and across phase boundaries. This concept will be worked out in Section 4.2.1 after fluxes of matter, charge, etc. have been introduced through the formalism of irreversible thermodynamics. 

When multicomponent diffusion is significant, it is best described with a generalized form of Pick s law containing (n - 1) diffusion coefficients in an -component system. This form of diffusion equation can be rationalized using irreversible thermodynamics. Concentration profiles in these multicomponent cases can be directly inferred from the binary results. However, multicomponent diffusion coefficients are difficult to estimate, and experimental values are fragmentary. As a result, you should make very sure that you need the more complicated theory before you attempt to use it. 

Development of the "flow" MEIS with the form reminding the models of nonequilibrium thermodynamics seems to be a very promising direction in equilibrium modeling of physical and chemical systems. Application of these models opens prospects for simpler analysis and solution of many complex problems related to the calculations of processes considered to be irreversible in principle. Certainly the flows in MEIS are interpreted statically as the coordinates of states. Thermodynamic interpretations are naturally extended to the kinetic coefficients that relate these flows with forces. Correctness of such interpretations is confirmed by the application of MP, being the theory of equilibrium states, as the terms for MEIS description. 

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... 

In this article we shall not utilize the generalized Fokker-Planck equations 6 which have been successfully used to calculate coefficients of viscosity and thermal conductivity.13 14 Rather, we shall find it more convenient to proceed directly from the Liouville equation. To obtain an expression for the contribution of the intermolecular forces to the heat flux, we shall postulate a plausible generalization of the usual phenomenological equations of the thermodynamics of irreversible processes to the space of molecular pairs. Although we shall not prove it here, it may be shown that the same results can also be obtained (with greater labor) from the Fokker-Planck equations ... 

For a simple order, the rate expression can be integrated and special plots utilized to determine the rate coefficient. A plot of MCa versus t or xjj -x versus t is used similarly for a second-order irreversible reaction. For reversible reactions with first order in both directions a plot of ln(Ci - Qeq)/(Qo - Qeq) or ln(l - XaIxa versus f yields ( i + ki) from the slope of the straight line. Using the thermodynamic equilibrium constant K = kxlkz, both k and kz are obtained. Certain more complicated reaction rate forms can be rearranged into such linear forms. These plots are useful for an estimate of the "quality" of the fit to the experimental data and can also provide initial estimates to formal regression techniques that will be systematically discussed in Chapter 2. 

---