Linear phenomenological coefficients

A very important property of the linear phenomenological coefficients is On-sager s reciprocity relation... 

Now, the condition dL/dt 0 as required for the Liapunov-stability of the equilibrium is reduced to the condition that the matrix of the linear phenomenological coefficients is positive definite. This latter property, however, is a direct consequence of the second law of thermodynamics as we have shown in (3.82). With this conclusion we have reconfirmed our preliminary result of the preceding section in a formally precise way a sufficient condition for the stability of the equilibrium is a) a positive-definite capacitance matrix such that L 0 and b) the second law of thermodynamics such that dL/dt 0. Let us emphasize once more the significance of the equivalence between dL/dt < 0 and the second law in the form of (3.82). This equivalence, however, is valid only in the range of validity of the linear relations in (3.81). If the fluxes I were some nonlinear functions of the forces F as will be the case in situations far from the thermodynamic equilibrium, dL/dt 0 is no longer guaranteed by the second law and possibly may no longer be valid at all. 

The constant of proportionality, Lk, is the linear phenomenological coefficient for diffusional flow. We have seen earlier that in an ideal fluid mixture the chemical potential can be written as p(p, T,Xk) = ii(p, T) + RThiXk, in which Xk is the mole fraction per unit volume of k, generally a function of position. If Utot is the total mole number density and n is the mole number density of... 

Having identified all the linear phenomenological coefficient in terms of the experimentally measured quantities, we can now turn to the reciprocal relations, according to which one must find... 

From the above considerations it is clear that the entropy production can be written in terms of A2 and A3 instead of A1 and A2. There is no unique way of writing the entropy production. In whatever way the independent affinities and velocities are chosen, the corresponding linear phenomenological coefficients can be obtained. The entropy production <7 can be written in terms of different sets of independent reaction velocities and affinities ... 

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]. 

A method is described for fitting the Cole-Cole phenomenological equation to isochronal mechanical relaxation scans. The basic parameters in the equation are the unrelaxed and relaxed moduli, a width parameter and the central relaxation time. The first three are given linear temperature coefficients and the latter can have WLF or Arrhenius behavior. A set of these parameters is determined for each relaxation in the specimen by means of nonlinear least squares optimization of the fit of the equation to the data. An interactive front-end is present in the fitting routine to aid in initial parameter estimation for the iterative fitting process. The use of the determined parameters in assisting in the interpretation of relaxation processes is discussed. 

Spiegler (164) followed another way. Instead of the Lik s, he introduces the i2iJt s, which have the character of frictional coefficients and are defined by the phenomenological coefficients in the equations, which now express the forces as linear functions of the fluxes. It can be demonstrated that equations (1) maybe written as ... 

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 /... 

Irreversible processes are driven by generalized forces, X, and are characterized by transport (or Onsager) phenomenological coefficients, L [21,22], where these transport coefficients, Lip are defined by linear relations between the generalized flux densities,./, which are the rates of change with time of state variables, and the corresponding generalized forces X . 

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. 

The linear phenomenological equations in terms of the resistance coefficients are... 

In a two-flow system, there are two degrees of freedom in choosing the phenomenological coefficients. With the linear relations of flows and forces, there is one degree of freedom that is I.]2 = hi, and L22 is proportional to V... 

We can compare these linear phenomenological equations with Eq. (3.277) to obtain the phenomenological coefficients... 

There is no definite sign for Eq. (3.317). When the generalized flows are expressed by linear phenomenological equations with constant coefficients obeying to the Onsager relations... 

From these linear relations, we can define the following relations between the phenomenological coefficients ... 

Equations (8.178) and (8.180) show that JT is a complicated function of P, T, and composition, and it cannot be expressed in the form of linear phenomenological equations with constant coefficients. 

The linear phenomenological reaction flows with vanishing cross-coefficients are... 

To estimate the flow of borate, and assuming that the phenomenological coefficients are dependent on the average concentration (CBI + CBII)/2 linearly, we have... 

If we consider a membrane having the same solute concentration on both sides, we have All 0 However, a hydrostatic pressure difference AP exists between the two sides, and we have a flow Jv that is a linear function of AP. The term Lp is called the mechanical filtration coefficient, which represents the velocity of the fluid per unit pressure difference between the two sides of the membrane. The cross-phenomenological coefficient Ldp is called the ultrafiltration coefficient, which is related to the coupled diffusion induced by a mechanical pressure of the solute with respect to the solvent. Osmotic pressure difference produces a diffusion flow characterized by the permeability coefficient, which indicates the movement of the solute with respect to the solvent due to the inequality of concentrations on both sides of the membrane. 

This equation shows that a stationary state imposes a relation between the diffusion and chemical reactions, and is of special interest in isotropic membranes where the coupling coefficients vanish. For a homogeneous and isotropic medium the linear phenomenological equations are... 

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... 

The phenomenological coefficient L is dependent on the partial rate of reaction at equilibrium. If, however, A/(RT) -> co then (c c Jc ) co 0r cHI ->0. This shows a sort of saturation effect with respect to the affinity. Under this condition, entropy production becomes a linear function of the affinity. 

They have shown that the first term on the right is negative definite even for cases for which the linear phenomenological equations do not hold. By introducing the linear phenomenological equations J, = LikXk with constant coefficients we get... 

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. 

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