Forced chemical systems rate equations

The appropriate dimensionless rate equations for the surface coverages 6p and 6t are 

Of particular interest are the parameters p and r, which are the dimensionless partial pressures of the reactant gases flowing over the catalyst surface. If we wish to force this system, a conceptually simple way is to vary one of these concentrations periodically about some mean value. We will follow McKar-nin et al. (1988), who imposed a cosine variation on the partial pressure of R  

r0 is the mean value of the dimensionless partial pressure, rf is the amplitude of the forcing, so r varies between r0 + r, and a is the dimensionless forcing frequency (the forcing peiod if is given by In/co). 

We now have a total of six parameters four from the autonomous system (p, r0, and the desorption rate constants k, and k2) and two from the forcing (rf and co). The main point of interest here is the influence of the imposed forcing on the natural oscillations. Thus, we will take just one set of the autonomous parameters and then vary rf and co. Specifically, we take p = 0.019, r0 = 0.028, fq = 0.001, and k2 = 0.002. For these values the unforced model has a unique unstable stationary state surrounded by a stable limit cycle. The natural oscillation of the system has a period t0 = 911.98, corresponding to a natural frequency of co0 = 0.006 889 6. 

With a small forcing amplitude, rf = 0.002 in Fig. 13.9(b) and rf = 0.003 in 13.9(c), the oscillatory traces show only a slight deformation, with a shoulder 

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Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

Equation (1.104) is similar to Eq. (1.101) in relating the rate of entropy production to the product of the flows (here the rate of reaction) and the scalar thermodynamic force that is AIT. Equation (1.104) can be readily extended to several chemical reactions taking place inside the system... 

Within non-equilibrium thermodynamics, the driving force for relaxation is provided by deviations in the local chemical potential from it s equilibrium value. The rate at which such deviations relax is determined by the dominant kinetics in the physical system of interest. In addition, the thermal noise in the system randomly generates fluctuations. We thus describe the dynamics of a step edge by the equation. 

Although these arguments have been presented for reaction systems whose rates are forced by an external oscillator, they remain true for autonomous biochemical oscillations where ot and are nonlinear functions of metabolite concentrations. That is, the rate of removal of a labeled compound through a reaction step whose rate is oscillating due to nonlinear kinetics will be enhanced over an equivalent system that maintains the same mean chemical flux and mean concentrations of metabolites but does not oscillate. This has been demonstrated numerically ( 6) on the reaction system (1) from the previous section using the full kinetic equations... 

Reversibility — This concept is used in several ways. We may speak of chemical reversibility when the same reaction (e.g., -> cell reaction) can take place in both directions. Thermodynamic reversibility means that an infinitesimal reversal of a driving force causes the process to reverse its direction. The reaction proceeds through a series of equilibrium states, however, such a path would require an infinite length of time. The electrochemical reversibility is a practical concept. In short, it means that the -> Nernst equation can be applied also when the actual electrode potential (E) is higher (anodic reaction) or lower (cathodic reaction) than the - equilibrium potential (Ee), E > Ee. Therefore, such a process is called a reversible or nernstian reaction (reversible or nerns-tian system, behavior). It is the case when the - activation energy is small, consequently the -> standard rate constants (ks) and the -> exchange current density (jo) are high. 

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 relationship between the transient and stationary approaches to the relaxation times has been considered by Eigen and de Maeyer. For any chemical equilibrium a system of nonhomogeneous differential equations which represent the rates of concentration change may be set up. The complete solution of the system is the sum of two solutions. One of these depends on the initial conditions of the dependent variables and upon the forcing function (the transient solution), while the other depends on the differential equation system and on the forcing function (the forced solution). The latter does not depend on the initial conditions of concentration, etc. The step-function methods for studying chemical relaxation experimentally determine the transient behaviour, while the stationary methods determine the steady-state behaviour. 

The formulation for the gas-phase region is based on the mass, energy, and species transport for a multi-component chemically reacting system of N species, and accommodates finite-rate chemical kinetics and variable thermophysical properties. If body force, viscous dissipation, and kinetic energy are ignored, the conservation equations for an isobaric flow can be written as follows. 

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