Thermodynamic phenomenological coefficient

This give us a relation between the thermodynamic phenomenological coefficient and the diffusion coefficient. In Chapter 16 we will consider diffusion in detail and see how the diffusion of one species affects the diffusion of another species, using the modem theory of nonequilibriUm thermodynamics. 

Irreversible thermodynamics has also been used sometimes to explain reverse osmosis [14,15]. If it can be assumed that the thermodynamic forces responsible for reverse osmosis are sufficiently small, then a linear relationship will exist between the forces and the fluxes in the system, with the coefficients of proportionality then referred to as the phenomenological coefficients. These coefficients are generally notoriously difficult to obtain, although some progress has been made recently using approaches such as cell models [15]. 

Some of the elements of thermodynamics of irreversible processes were described in Sections 2.1 and 2.3. Consider the system represented by n fluxes of thermodynamic quantities and n driving forces it follows from Eqs (2.1.3) and (2.1.4) that n(n +1) independent experiments are needed for determination of all phenomenological coefficients (e.g. by gradual elimination of all the driving forces except one, by gradual elimination of all the fluxes except one, etc.). Suitable selection of the driving forces restricted by relationship (2.3.4) leads to considerable simplification in the determination of the phenomenological coefficients and thus to a complete description of the transport process. 

The transport of both solute and solvent can be described by an alternative approach that is based on the laws of irreversible thermodynamics. The fundamental concepts and equations for biological systems were described by Kedem and Katchalsky [6] and those for artificial membranes by Ginsburg and Katchal-sky [7], In this approach the transport process is defined in terms of three phenomenological coefficients, namely, the filtration coefficient LP, the reflection coefficient o, and the solute permeability coefficient to. 

R D - research and development AG, AH, AS, q, w - classical thermodynamic significance J, X, L - fluxes, forces and phenomenological coefficients of irreversible thermodynamics ... 

In comparison with the qualitative description of diffusion in a binary system as embodied by Eqs. (11), (12) or (14), the thermodynamic factors are now represented by the quantities a, b, c, and d and the dynamic factors by the phenomenological coefficients which are complex functions of the binary frictional coefficients. Experimental measurements of Dy in a ternary system, made on the basis of the knowledge of the concentration gradients of each component and by use of Eqs. (21) and (22), have been reviewed 35). Another method, which has been used recently36), requires the evaluation of py from thermodynamic measurements such as osmotic pressure and evaluation of all fy from diffusion measurements and substitution of these terms into Eqs. (23)—(26). 

The model active transport system described by Dr. Thomas is based on an asymmetric arrangement of two enzymes. A model active transport system was also described by Blumenthal et al. several years ago based on a single enzyme immobilized between asymmetric boundaries [Blumenthal, Caplan, and Kedem, Biophys. J., 7, 735 (1967)]. In the latter case the phenomenological coefficients were measured, and it was possible to demonstrate Onsager symmetry and the correlation between the thermodynamic coefficients and the kinetic constants. 

The forces Fk involve gradients of intensive properties (temperature, electrochemical potential). The Ljk are called phenomenological coefficients and the fundamental theorem of the thermodynamics of irreversible processes, due originally to Onsager (1931a, b), is that when the fluxes and forces are chosen to satisfy the equation... 

Although irreversible thermodynamics neatly defines the driving forces behind associated flows, so far it has not told us about the relationship between these two properties. Such relations have been obtained from experiment, and famous empirical laws have been established like those of Fourier for heat conduction, Fick for simple binary material diffusion, and Ohm for electrical conductance. These laws are linear relations between force and associated flow rates that, close to equilibrium, seem to be valid. The heat conductivity, diffusion coefficient, and electrical conductivity, or reciprocal resistance, are well-known proportionality constants and as they have been obtained from experiment, they are called phenomenological coefficients Li /... 

The transport of solutes through a membrane can be described by using the principles of irreversible thermodynamics (IT) to correlate the fluxes with the forces through phenomenological coefficients. For a two-components system, consisting of water and a solute, the IT approach leads to two basic equations [83],... 

The phenomenological coefficients are important in defining the coupled phenomena. For example, the coupled processes of heat and mass transport give rise to the Soret effect (which is the mass diffusion due to heat transfer), and the Dufour effect (which is the heat transport due to mass diffusion). We can identify the cross coefficients of the coupling between the mass diffusion (vectorial process) and chemical reaction (scalar process) in an anisotropic membrane wall. Therefore, the linear nonequilibrium thermodynamics theory provides a unifying approach to examining various processes usually studied under separate disciplines. 

The form of the expressions for the rate of entropy production does not uniquely determine the thermodynamic forces or generalized flows. For an open system, for example, we may define the energy flow in various ways. We may also define the diffusion in several alternative ways depending on the choice of reference average velocity. Thus, we may describe the flows and the forces in various ways. If such forces and flows, which are related by the phenomenological coefficients obeying the Onsager relations, are subjected to a linear transformation, then the dissipation function is not affected by that transformation. 

As shown by Prigogine, for diffusion in mechanical equilibrium, any other average velocity may replace the center-of-mass velocity, and the dissipation function does not change. When diffusion flows are considered relative to various velocities, the thermodynamic forces remain the same and only the values of the phenomenological coefficients change. 

The phenomenological coefficients are not a function of the thermodynamic forces and flows on the other hand, they can be functions of the parameters of the local state as well as the nature of a substance. The values of Lik must satisfy the conditions... 

If the dissipation function T 7 is used to identify the thermodynamic forces, then the phenomenological coefficient is... 

We can express the phenomenological coefficients in terms of the frictional forces assuming that for a steady-state flow, the thermodynamic forces Xare counterbalanced by a sum of suitable frictional forces F. Thus, for a solute in an aqueous solution, we have... 

Stucki (1980, 1984) applied the linear nonequilibrium thermodynamics theory to oxidative phosphorylation within the practical range of phosphate potentials. The nonvanishing cross-phenomenological coefficients Ly(i v /) reflect the coupling effect. This approach enables one to assess the oxidative phosphorylation with H+pumps as a process driven by respiration by assuming the steady-state transport of ions. A set of representative linear phenomenological relations are given by... 

In the linear region of the thermodynamic branch and with constant phenomenological coefficients, we have... 

Linear nonequilibrium thermodynamics has some fundamental limitations (i) it does not incorporate mechanisms into its formulation, nor does it provide values for the phenomenological coefficients, and (ii) it is based on the local equilibrium hypothesis, and therefore it is confined to systems in the vicinity of equilibrium. Also, properties not needed or defined in equilibrium may influence the thermodynamic relations in nonequilibrium situations. For example, the density may depend on the shearing rate in addition to temperature and pressure. The local equilibrium hypothesis holds only for linear phenomenological relations, low frequencies, and long wavelengths, which makes the application of the linear nonequilibrium thermodynamics theory limited for chemical reactions. In the following sections, some of the attempts that have been made to overcome these limitations are summarized. 

Figure 3-18. Forces, flux terms, and phenomenological coefficients in irreversible thermodynamics (see Eq.

Here Ca is the concentration of isotopically labelled species at a point where the concentration of unlabelled species is Ca. La a and La a are the straight and cross phenomenological coefficients of the irreversible thermodynamic formulation of diffusion. The original relation, Equation 1, assumes a zero cross coefficient, which in dense intracrystalline fluids certainly is not likely to be true. 

Previously introduced, the thermodynamic surface tension 7 represents the elastic resistance to surface dilation. Furthermore, two types of viscosities are defined within the interface, a dilational viscosity and a shear viscosity. For a surfactant monolayer, the surface shear viscosity rjS is analogous to the three-dimensional shear viscosity the rate of yielding of a layer of fluid due to an applied shear stress. The phenomenological coefficient s represents the surface dilational viscosity, and expresses the magnitude of the viscous forces during a rate expansion of a surface element. Figures 10a and 10b illustrate the difference between the two surface viscosities. 

In this framework, the intensity of an entropy source is represented by a quadratic form of thermodynamic forces. The corresponding phenomenological coefficients form a matrix with remarkable properties. These properties, formulated as the Onsager reciprocity theorem, allow to reduce the number of independent quantities and to find relations between various physical effects. 

Expressions (4.514), (4.515) are known as phenomenological equations of linear irreversible or non-equilibrium thermodynamics [1-5, 120, 130, 185-187], in this case for diffusion and heat fluxes, which represent the linearity postulate of this theory flows (ja, q) are proportional to driving forces (yp,T g) (irreversible thermodynamics studied also other phenomena, like chemical reactions, see, e.g. below (4.489)). Terms with phenomenological coefficients Lgp, Lgq, Lqg, Lqq, correspond to the transport phenomena of diffusion, Soret effect or thermodiffusion, Dtifour effect, heat conduction respectively, discussed more thoroughly below. 

Kinetic theory, non-equilibrium statistical mechanics and non-equilibrium molecular dynamics (NEMD) have proved to be useful in estimating both straight and cross-coefficients such as thermal conductivity, viscosity and electrical conductivity. In a typical case, cross-coefficient in case of electro-osmosis has also been estimated by NEMD. Experimental data on thermo-electric power has been analysed in terms of free electron gas theory and non-equilibrium thermodynamic theory [9]. It is found that phenomenological coefficients are temperature dependent. Free electron gas theory has been used for estimating the coefficients in homogeneous conductors and thermo-couples. 

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