Linear phenomenological laws

When a system is close to equilibrium, a general theory based on linear relations between forces and flows could be formulated, bi the previous chapter we have seen that the entropy production per unit volume, a, can be written as 

In these equations, k is the heat conductivity, is the diffusion coefficient of compound k and Uk is the concentration of compound k. Ohm s law is usually stated as (16.1.5a) in which / is the electrical current, R is the resistance and V is the voltage. It can also be stated in terms of the electric current density I, the electric field E and the resistivity p (resistance per unit length per unit area of cross section). Other quantities in the above equations are as defined in Table 15.1 

As a specific example of the general relation (16.1.2), let us consider the thermoelectric phenomenon mentioned above (Fig. 16.1). The equations that 

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There exist a number of linear phenomenological laws describing irreversible processes in the form of proportionalities between the flow J, and the conjugate driving force Xk... 

For an overall reaction with / number of intermediate reactions, the linear phenomenological law is valid, if every elementary reaction satisfies the condition A/RT 1, and the intermediate reactions are fast and hence a steady state is reached. 

The linear phenomenological law of diffusion for a binary system is given by... 

We have thus demonstrated that Newton s law of viscosity, an inherently macroscopic result, can be obtained via linear response theory as the nonequilibrium average in the steady state. Furthermore, the distribution function for the steady state average is determined by microscopic equations of motion. Hence, the SLLOD equations, in the linear regime, reduce to the linear phenomenological law proposed by Newton. Moreover, all the quantities that are needed to compute the shear viscosity can be obtained from a molecular dynamics simulation. 

Here, T is the appropriate state variable conjugate to the flux J and X, and depends on the thermodynamic state of the system. These linear, phenomenological laws are fundamental to all processes involving the transfer of mass, momentum or energy but, in many practical circumstances encountered in industry, the fundamental transport mechanisms arise in parallel with other means of transport such as advection or natural convection. In those circumstances, the overall transport process is far from simple and linear. However, the description of such complex processes is often rendered tractable by the use of transfer equations, which are expressed in the form of linear laws such as... 

The topic of flow through an effectively infinite system of particles belongs to the more general domain of flow through porous media (C12, C13, PIO, R5, S3, S4). In the absence of physicochemical interaction between the particles and fluid, the slow quasi-static Newtonian flow of incompressible fluids through such media is governed by Darcy s law. This linear phenomenological law has the form... 

Linear phenomenological laws of nonequilibrium thermodynamics lead to a general relation between mobility Tk and the diffusion coefficient Dk- This relation can be obtained as follows. The general expression for the chemical potential in a field with potential x / is given by + xjtxl/, in which is the... 

Phenomenological laws and the cross effects between the flows were independently studied and, until the formalism presented here was developed in the 1930s, there was no unified theory of all the cross effects. Relating the entropy production to the phenomenological laws is the first step in developing a unified theory. For conditions under which the linear phenomenological laws (16.1.2) are valid, entropy production (16.1.1) takes the quadratic form... 

Onsager s theory begins with the assumption that, where linear phenomenological laws are valid, a deviation a, decays according to the linear law... 

The general linear phenomenological laws that follow from this are... 

The linear phenomenological law that relates the flux J i to the conjugate force in (16.4.8) is... 

In this section we shall look at the meaning of linear phenomenological laws in the context of chemical reactions. In a formalism in which the principle of detailed balance or microscopic reversibility is incorporated through the condition that forward rates of every elementary step balance the corresponding reverse rate, the Onsager reciprocity is implicit. No additional relations can be derived for the reaction rates if it is assumed that at equilibrium each elementary step is balanced by its reverse. Therefore, the main task in this section will be to relate the Onsager coefficients Ly and the experimentally measured reaction rates. In our formalism the Onsager reciprocal relations will be automatically valid. 

We have seen that the linear phenomenological laws are valid for chemical reactions with affinity A if the condition A/RT 1 is satisfied. However, if the overall chemical reaction... 

Though we considered a coupled set of unimolecular reactions (16.5.24) to obtain (16.5.31), the result is more generally valid. Thus, the linear phenomenological law is valid for an overall chemical reaction if A/RT 1... 

Thus nonequilibrium thermodynamics gives a unified theory of irreversible processes. Onsager reciprocal relations are general, valid for all systems in which linear phenomenological laws apply. 

For one of the reactions in Chapter 9, specify the conditions in which the linear phenomenological laws may be used. 

The entropy production in circuit elements (equations 17.1.24 to 17.1.26) is in the form of a product of a thermodynamic force and a flow. In each case we can write the following linear phenomenological law relating the flows and the forces ... 

Figure 17.3 Elementary circuit elements, such as a resistor R, a capacitor C and an inductance L, also dissipate energy and produce entropy. In the thermodynamic formalism there are no ideal circuit elements with no dissipation of energy. Linear phenomenological laws give expressions for the rate of entropy production and dissipation of energy...

Now, applying the linear phenomenological laws Vk = Lkk Ak/T) to this equation, we obtain... 

Using the linear phenomenological law Jq=Lqqd /T)/ bx, the above expression can be written as... 

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