Phenomenological equations chemical

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6-9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is N-n(t), and the concentration of product states is N-pU). We assume that we can define the reactants and products as distinct macrostates that are separated by a transition state dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1). 

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.)... 

States away from global equilibrium are called the thermodynamic branch (Figure 2.2). Systems not far from global equilibrium may be extrapolated around equilibrium state. For systems near equilibrium, linear phenomenological equations may represent the transport and rate processes. The linear nonequilibrium thermodynamics theory determines the dissipation function or the rate of entropy production to describe such systems in the vicinity of equilibrium. This theory is particularly useful to describe coupled phenomena, and quantify the level of coupling in physical, chemical, and biological systems without detailed process mechanisms. 

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. 

For situations not far removed from equilibrium (what this implies will be fully documented later), one postulates the usual linear relations between fluxes and forces. In the present context ( should not be confused with the generic chemical symbol A) we set up R phenomenological equations of the form... 

The relaxation of certain properties of the system can often be described by simple phenomenological equations called relaxation equations. In chemical kinetics, for example, the constrained state may be a mixture of gases in metastable equilibrium—for example, hydrogen and oxygen. A spark is then introduced and the gas mixture reacts. The concentration of the reactants and products change with time until a new equilibrium state is achieved. The relaxation equations are the familiar phenomenological equations of chemical kinetics and the relaxation times are related to the chemical rate constants. 

Notice first of all that equations (5.56) for the relative fluxes hold for the charged components as well, but for these components, the quantities should be understood as the electrochemical potentials depending on pressure, temperature, chemical composition and electric parameters. The last are, in turn, described by independent equations. As to phenomenological equations for concentrated mixtures, they are cumbersome and demand a lot of empirical parameters. 

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. 

The DSC can only measure a true total enthalpy change for a chemical or physical process when the specimen size and the scanning rate are such that the deviation of the sample from equilibrium remains in the range where the assumption of linear phenomenological equations is valid and when integration is carried out over the total range of temperature where the reaction or process may occur. 

Here, the reciprocal rules hold, and we have L12 = T2i- Introduction of the explicit form of chemical potential for a single component Afi = —SAT -I- VAP into the phenomenological equations yields... 

As pointed out hy Prigogine, the linear phenomenological equations as given in Eqn (8.16) are valid only when A/i r << 1, which requires that the chemical reaction to he quite close to equilibrium. In most reactions A RT. Consider the following cyclic reaction ... 

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