Rate equation near equilibrium

Before considering the special forms of rate equations near equilibrium, we consider how equilibrium perturbations can be introduced. Since we know from thermodynamics that... 

The plot of ( versus q which results from Eqn. 9-9 is a polarization curve this polarization crirve is usually divided into two ranges of polarization as shown in Fig. 9-3 one is a range of polarization where a linear rate equation holds near the equilibrium potential (t) - 0) the other is a range of polarization (the Tafel range) where an exponential rate equation applies at potentials away from the equilibrium potential (ti 0). 

The solution found when the rate equations are pul equal to zero corresponds to equilibrium in the case of a uniform reaction environment, but also characterizes the steady state if it is assumed that the linear lattice separates two two-dimensional spaces such that on the one side the reaction is all 0 —> 1 according to ku k2, and k3 and on the other all 1 —> 0 according to k2 k2 and k3. As the k s can include functions of the environment within them such as the concentrations of a transported substance with which the lattice reacts, this model can be used to discuss transport through membranes with reactions governed by near neighbor effects. It will be clear that the reactivity of the linear lattice must be defined in an asymmetric fashion in order to obtain transport. 

When the HSS solution of the chemical rate equations (la)—(lc) first becomes unstable as the distance from equilibrium is increased (by decreasing P, for example), the simplest oscillatory instability which can occur corresponds mathematically to a Hopf bifurcation. In Fig. 1 the line DCE is defined by such points of bifurcation, which separate regions of stability (I,IV) of the HSS from regions of instability (II,III). Along section a--a, for example, the HSS becomes unstable at point a. Beyond this bifurcation point, nearly sinusoidal bulk oscillations (QHO, Fig. 3a) increase continuously from zero amplitude, eventually becoming nonlin-... 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

Morse (30) carried out an examination of the near-equilibrium dissolution kinetics of calcium carbonate-rich deep sea sediments. His results are summarized in Figure 14. The sediment samples from different ocean basins have distinctly different reaction orders and empirical rate constants. The dissolution rate equations for the different sediment samples are ... 

At equilibrium Q = K and R = 0.) Eq. (2.107) perhaps applies to the rate of feldspar dissolution in a fresh, unconfined groundwater, whereas Eq. (2.108) is more consistent with the near-equilibrium conditions we might expect in older confined groundwaters. Far from equilibrium at pH values above 7 to 8, feldspar dissolution rates again increase, and the rate equation has the general form... 

Pseudo-Order Reactions As mentioned above, complex reactions can often be expressed by the simple equations of zeroth-, first-, or second-order elementary reactions under certain conditions. For example, the dissolution of many minerals at conditions close to equilibrium is a strong function of the free energy of the reaction (Lasaga, 1998, 7.10), but far from equilibrium the rate becomes nearly independent of the free energy of reaction. In other words, the rate of dissolution will be virtually constant under these conditions, or pseudo-first-order. 

Since A is very small near equilibrium, we can expand the exponential in series to obtain Q/K = 1 — AfRT + . This brings the rate equation to the form... 

In the preceding examples the isomers A and B were in equilibrium, that is, k2 and kj were assumed to be much greater than and k. However, it is also possible that the various rates are more nearly equal. In that case the coupled differential rate equations must be solved (Baer et al., 1975, 1983). The formation rate of P and P2 is... 

Most mechanisms are described by a series of coupled reactions. The number of independent concentration variables is simply the number of concentration variables minus the number of mass conservation equations. If there are n independent concentration variables in the mechanism, n independent rate equations are obtained furthermore, near equilibrium all these are linear first-order differential equations of the form... 

The advantages of having the fiiU rate equation in hand, rather than just an initial velocity equation, are numerous. The entire equation describes the steady-state Idnetic behavior of the system at any time and at any point, whether near or far from equilibrium. It easily yields simplified equations for situations where one or more reactants are at zero concentration, and thus saves the considerable time and effort necessary to derive separately a series of initial velocity and product inhibition equations. Since the product inhibition experiments are often the best means of distinguishing different mechanisms, this is one of the most important uses for the fuU equations (Fromm, 1975 Rudolph, 1979 Cooper Rudolph, 1995). 

To produce geochemical rate models, rates determined by reactor experiments must be converted into rate equations that summarize how the rate varies with solution composition, temperature, and other rate-determining variables. If the rates are determined at near-equilibrium conditions, the rate data must be fit to an equation that takes into account both the forward and reverse rate. Most geochemical rate experiments are designed to measure rates for far-from-equilibrium conditions where the reverse reaction rate is effectively zero. These experimental rates can be fit to a simple equation that relates the rate to the product of the concentration (w, molal) of each reacting species raised to a power (n). 

The nucleation rate, growth rate, and transformation rate equations that we developed in the preceding sections are sufficient to provide a general, semiquantitative understanding of nucleation- and growth-based phase transformations. However, it is important to understand that the kinetic models developed in this introductory text are generally not sufficient to provide a microstructurally predictive description of phase transformation for a specific materials system. It is also important to understand that real phase transformation processes often do not reach completion or do not attain complete equilibrium. In fact, extended defects such as grain boundaries or pores should not exist in a true equilibrium solid, so nearly all materials exist in some sort of metastable condition. Many phase transformation processes produce microstructures that depart wildly from our equilibrium expectation. The limited atomic mobilities associated with solid-state diffusion can frequently cause (and preserve) such nonequilibrium structures. In this section, we will focus more deeply on solidification (a liquid-solid phase transformation) as a way to discuss some of these issues. In particular, we will examine a few kinetic concepts/models... 

This is quite a complex integrated rate equation. However, if we study the kinetics of the reaction at points in time near the establishment of equilibrium, we make the assumption that the forward and reverse rates are becoming equal (as when equilibrium is really established). At equilibrium we define [x] as [x]e, where the extent of reaction is as far as it is going to go, which leads to W[ A]o - [x]c) = fcr([B]o + [x]e). Solving this equality for fcf[ A] - A r[B] , and substituting the result into Eq. 7.41, leads to Eq. 7.42. This tells us that as one approaches equilibrium, the rate appears first order with an effective rate constant that is the sum of the forward and reverse rate constants. This is an approximation because we defined [.v] as [. ]e to obtain this answer, but it is a very common way to analyze equilibrium kinetics. Chemists qualitatively estimate that the rate to equilibrium is the sum of the rates of the forward and reverse reactions. 

Electron transfer reactions at the electrode may not be rapid enough to maintain equilibrium concentrations of the redox couple species near the electrode surface. It is therefore necessary to consider the kinetics of the electron transfer process. The rate equation for heterogeneous electron transfer (Equation 8) expresses the flux of electrons at the electrode surface (Figure 1-9) ... 

There are three important points to note. First, for purely catalystic eflFectors (i.e., when the eflFector does not participate in the reaction catalyzed) the value of r is zero and = r. Second, the above treatment has been applied to a very simple reaction to illustrate the approach but it can be applied to any reaction or process no matter how complex the rate equation. Third, r is always calculated by assuming the reaction to be nonequilibrium if the reaction is near-equilibrium in vivo, the eflFects of nearness to equilibrium (i.e., reversibility) are incorporated into the intrinsic sensitivity at a later stage (see below). 

The situation becomes a little more complex if one of the reactions of the cycle is near-equilibrium. [For example, the proposed cycle between acetoacetate and acetoacetyl-CoA in the liver may contain such a nearequilibrium reaction (35).] Nonetheless the approach used here can simplify matters since any metabolic communication can be broken down into the elementary communications described in Table I and the intrinsic sensitivity can be obtained by applying the addition and product rules to the relevant elementary sensitivities. The intrinsic sensitivity (which will contain terms involving concentrations, fluxes, and kinetic parameters from the rate equations) then describes the response to the effector concentration produced by the combined effects of all the component mechanisms. By including the effects of near-equilibrium and cycling in the intrinsic sensitivity, the approach effectively reduces a step to a single nonequilibrium reaction. For example, the complex step. 

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