Chemical reactions phenomenological equations

The discussion of Kapral s kinetic theory analysis of chemical reaction has been considered in some detail because it provides an alternative and intrinsically more satisfactory route by which to describe molecular scale reactions in solution than using phenomenological Brownian motion equations. Detailed though this analysis is, there are still many other factors which should be incorporated. Some of the more notable are to consider the case of a reversible reaction, geminate pair recombination [286], inter-reactant pair potential [454], soft forces between solvent molecules and with the reactants, and the effect of hydrodynamic repulsion [456b, 544]. Kapral and co-workers have considered some of the points and these are discussed very briefly below [37, 285, 286, 454, 538]. 

All local concentrations C of particles entering the non-linear functions F in equation (2.1.40) are taken at the same space points, in other words, the chemical reaction is treated as a local one. Taking into account that for extended systems we shouldn t consider distances greater than the distinctive microscopic scale Ao, the choice of equation (2.1.40) means that inside infinitesimal volumes vo particles are well mixed and their reaction could be described by the phenomenological reaction rates earlier used for systems with complete reactant mixing. This means that Ao value must exceed such distinctive scales of the reaction as contact recombination radius, effective radius of a dynamical interaction and the particle hop length, which imposes quite natural limits on the choice of volumes v0 used for averaging. 

The equation suggested by Boreskov accounts for the presence in the catalytic system of two time scales, namely a "fast one due to the surface chemical transformations and a "slow one due to the effect of the reaction mixture on the catalyst. (It shoud be noted that, in general, one can hardly discriminate between the constituents in the way it has been done in this phenomenological equation.)... 

A few years ago the concept considered was introduced also in the low-temperature chemistry of the solid.31 Benderskii et al. have employed the idea of self-activation of a matrix due to the feedback between the chemical reaction and the state of stress in the frozen sample to explain the so called explosion during cooling observed by them in the photolyzed MCH + Cl2 system. The model proposed in refs. 31,48,49 is unfortunately not quite concrete, because it includes an abstract quantity called by the authors the excess free energy of internal stresses. No means of measuring this quantity or estimating its numerical values are proposed. Neither do the authors discuss the connection between this characteristic and the imperfections of a solid matrix. Moreover, they have to introduce into the model a heat-balance equation to specify the feedback, although they proceed from the nonthermal mechanisms of selfactivation of reactants at low temperatures. Nevertheless, the essence of their concept is clear and can be formulated phenomenologically as follows the... 

On the other hand, we have the following linear phenomenological equation for chemical reaction i... 

For an elementary chemical reaction, the local entropy production and the linear phenomenological equation are... 

Here, Jq is the total heat flow, J, the mass flow of component i, and Jrj the reaction rate (flow) of reaction j. For chemical reactions, linear phenomenological equations are... 

Using a dissipation function or entropy production equation, the conjugate flows and forces are identified and used in the phenomenological equations for simultaneous heat and mass transfer. Consider the heat and diffusion flows in a fluid at mechanical equilibrium not undergoing a chemical reaction. The dissipation function for such a system is... 

In chemical kinetics, the reaction rates are proportional to concentrations or to some power of the concentrations. Phenomenological equations, however, require that the reaction velocities are proportional to the thermodynamic force or affinity. Affinity, in turn, is proportional to the logarithms of concentrations. Consider a monomolecular... 

Despite its limitations for chemical reactions, the linear net theory has a useful conceptual base. Consider the linear phenomenological equations for two chemical reactions with flows ofand Jl2... 

Nonisothermal reaction-diffusion systems represent open, nonequilibrium systems with thermodynamic forces of temperature gradient, chemical potential gradient, and affinity. The dissipation function or the rate of entropy production can be used to identify the conjugate forces and flows to establish linear phenomenological equations. For a multicomponent fluid system under mechanical equilibrium with n species and A r number of chemical reactions, the dissipation function 1 is... 

The linear phenomenological equations help determine the degree of coupling between a pair of flows the degree of coupling between heat and mass flows qSq and between the chemical reaction and the transport process of heat... 

Example 9.10 Chemical reaction velocity coupled to heat flow In this case, LSl and LlS vanish. Still, heat and mass flows are coupled. The new phenomenological equations are... 

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... 

Example 10.8 Coupled system of flows and a chemical reaction For a specific membrane, the phenomenological equations relating the flows and forces of either vectorial or scalar character may be written. Such flows and forces must be derived from an appropriate dissipation function. Consider the following dissipation function ... 

Some processes may have forces operating far away from equilibrium where the linear phenomenological equations are no longer applicable. Such a domain of irreversible phenomena, such as some chemical reactions, periodic oscillations, and bifurcation, is examined by extended nonequilibrium thermodynamics. Extending the methods of thermodynamics to treat the linear and nonlinear phenomena, and such dissipative structures are attracting scientists from various disciplines. 

This example illustrates the fundamental principle that if one describes coupled reactions in terms of a set of linearly independent steps, then sufficiently close to equilibrium the reaction rates may be described in terms of phenomenological equations involving the chemical affinities as driving forces. 

Earlier [26,27,43,46] a phenomenological approach, based on the premise that the thermodynamics of irreversible processes [29] joined with Nemst-Planck equations for ion fluxes, would be useful was applied to the solution of intraparticle diffusion controlled ion exchange (IE) of fast chemical reactions between B and A counterions and the fixed R groups of the ion exchanger. In the model, diffusion within the resin particle, was considered the slow and sole controlling step. 

The equation is a simple case of a mechanistic model. Models such as this may give better predictions but may not always apply because of the complexity of the reactions. Phenomenological models arc expressed by simple rate equations which ignore the details of the reaction. Phenomenological models are typically used to follow cure rates in polymeric systems which are difficult to follow by chemical analysis. This is because reaction products become insoluble during the course of the reaction and, consequently, are not detected in an analysis of the solution. 

In case of the anisotropic media, for example, anisotropic crystals, the phenomenological equations in the absence of chemical reactions are similar to (5.206) and (5.207), but the quantities Lqq, Lqi, Liq and L are tensors. In particular the tensor Lqq is proportional to heat conductivity tensor. 

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. 

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