The Reaction Equilibrium Assumption

The quasi-steady-state assumption (QSSA) and the reaction equilibrium assumption allow us to generate reaction-rate expressions that capture the details of the reaction chemistry with a minimum... 

Since a is directly proportional to C/ q, a plot of E versus should produce a straight line if the reaction equilibrium assumption holds. Also, if oK 1, the intercept of this line is 2/Sh and it provides a measure of the external mass transfer resistance for the membrane. Figure 6 is a plot of E for the H S transport data from Figure 3. The data plot as a straight line with an intercept of 3.63 X 10". The correlation coefficient of the enhancement factor plot is 0.987. Therefore, the reaction equilibrium assumption is valid and external mass transfer resistances can be neglected. 

These predictions were made for 200 um membranes. Industrial application of this technology will require the use of membranes which are two orders of magnitude thinner. In order to use the model to predict facilitation factors for thinner membranes, it is necessary to determine whether the reaction equilibrium assumption still applies. The parameter (tanh )/ has a value of 0 if the system is diffusion limited and 1 if the facilitated transport system is reaction rate limited. At a thickness of Ipm, the value of (tanh X)/X is of the order 10 , which implies that the system is diffusion limited and that the simplified analytical model can be used to predict facilitation factors. If the solubility of HjS, the pressure and temperature dependence of the equilibrium constant and the diffusion coefficients are known, then F could be estimated at industrial conditions. 

Solution Example 11.5 treats a system that could operate indefinitely since the liquid phase serves only as a catalyst. The present example is more realistic since the liquid phase is depleted and the reaction eventually ends. The assumption that the gas-side resistance is negligible is equivalent to assuming that a = ag throughout the course of the reaction. Equilibrium at the interface then fixes a = ag/Ku at all times. Dropping the flow and accumulation terms in the balance for the liquid phase, i.e.. Equation (11.11), gives 0 = kiAiV(ag/KH - ai) - Vikafi... 

As will be discussed in the following chapter, most combustion systems entail oxidation mechanisms with numerous individual reaction steps. Under certain circumstances a group of reactions will proceed rapidly and reach a quasi-equilibrium state. Concurrently, one or more reactions may proceed slowly. If the rate or rate constant of this slow reaction is to be determined and if the reaction contains a species difficult to measure, it is possible through a partial equilibrium assumption to express the unknown concentrations in terms of other measurable quantities. Thus, the partial equilibrium assumption is very much like the steady-state approximation discussed earlier. The difference is that in the steady-state approximation one is concerned with a particular species and in the partial equilibrium assumption one is concerned with particular reactions. Essentially then, partial equilibrium comes about when forward and backward rates are very large and the contribution that a particular species makes to a given slow reaction of concern can be compensated for by very small differences in the forward and backward rates of those reactions in partial equilibrium. 

A specific example can illustrate the use of the partial equilibrium assumption. Consider, for instance, a complex reacting hydrocarbon in an oxidizing medium. By the measurement of the CO and C02 concentrations, one wants to obtain an estimate of the rate constant of the reaction... 

Thus, one observes that the rate expression can be written in terms of readily measurable stable species. One must, however, exercise care in applying this assumption. Equilibria do not always exist among the II2 02 reactions in a hydrocarbon combustion system—indeed, there is a question if equilibrium exists during CO oxidation in a hydrocarbon system. Nevertheless, it is interesting to note the availability of experimental evidence that shows the rate of formation of C02 to be (l/4)-order with respect to 02, (l/2)-order with respect to water, and first-order with respect to CO [17,18], The partial equilibrium assumption is more appropriately applied to NO formation in flames, as will be discussed in Chapter 8. 

If one invokes the steady-state approximation described in Chapter 2 for the N atom concentration and makes the partial equilibrium assumption also described in Chapter 2 for the reaction system... 

Readers might have noticed that the two chain reactions, (i) Reactions 2-121 and 2-122 and (ii) Reactions 2-116 and 2-117, are similar, but were treated differently. Reactions 2-116 and 2-117 were treated using the quasi-equilibrium assumption, but may also be treated using the steady-state concept. The result is a more complicated expression, which would reduce to the experimental reaction rate law if < ii6b- Readers can work this problem out as an exercise. Therefore, the... 

The chemical equilibrium assumption often results in modeling predictions similar to those obtained assuming infinitely fast reaction, at least for overall aspects of practical systems such as combustion. However, the increased computational complexity of the chemical equilibrium approach is often justified, since the restrictions that the equilibrium constraint places on the reaction system are accounted for. The fractional conversion of reactants to products at chemical equilibrium typically depends strongly on temperature. For an exothermic reaction system, complete conversion to products is favored thermodynamically at low temperatures, while at high temperatures the equilibrium may shift toward reactants. The restrictions that equilibrium place on the reaction system are obviously not accounted for by the fast chemistry approximation. 

To eliminate the second unknown variable, the oxygen atom concentration, we introduce another simplifying assumption, the partial-equilibrium assumption. Reactions that are fast in both the forward and reverse direction may be assumed in partial equilibrium. Usually this assumption can only be applied to reactions that involve radicals both as reactants and as products, such as the reaction N + OH NO + H (R11). However, here we will assume that the oxygen atom is in partial equilibrium with molecular oxygen,... 

Thermal dissociation of 02 has a high activation energy and is usually quite slow. However, at the high temperatures in combustion systems where thermal NO is important, both O and 02 are involved in a number of reactions that are fast compared to the thermal NO formation. Due to the fast exchange between O and 02 in these reactions, the partial equilibrium assumption for (R12) is a reasonable approximation. Assuming partial equilibrium, we can relate the oxygen atom concentration to the 02 concentration and the equilibrium constant for the reaction,... 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

For multisubstrate enzymatic reactions, the rate equation can be expressed with respect to each substrate as an m function, where n and m are the highest order of the substrate for the numerator and denominator terms respectively (Bardsley and Childs, 1975). Thus the forward rate equation for the random bi bi derived according to the quasi-equilibrium assumption is a 1 1 function in both A and B (i.e., first order in both A and B). However, the rate equation for the random bi bi based on the steady-state assumption yields a 2 2 function (i.e., second order in both A and B). The 2 2 function rate equation results in nonlinear kinetics that should be differentiated from other nonlinear kinetics such as allosteric/cooperative kinetics (Chapter 6, Bardsley and Waight, 1978) and formation of the abortive substrate complex (Dalziel and Dickinson, 1966 Tsai, 1978). 

Recently, however, experimental studies of reaction processes have cast doubt on the local equilibrium assumption. When that assumption is not valid, understanding of reaction processes requires the smdy of global aspects of the phase space in multidimensional chaotic dynamics [1]. 

Subsurface solute transport is affected by hydrodynamic dispersion and by chemical reactions with soil and rocks. The effects of hydrodynamic dispersion have been extensively studied 2y 3, ). Chemical reactions involving the solid phase affect subsurface solute transport in a way that depends on the reaction rates relative to the water flux. If the reaction rate is fast and the flow rate slow, then the local equilibrium assumption may be applicable. If the reaction rate is slow and the flux relatively high, then reaction kinetics controls the chemistry and one cannot assume local equilibrium. Theoretical treatments for transport of many kinds of reactive solutes are available for both situations (5-10). 

They found the heat release rate to be proportional to the product [H] [O21 [H2O], and the dependence of H on pressure and mass flow to be also consistent with the removal of H by reaction (iv). Similar conclusions about the recombination were reached by Getzinger and Schott [181] from shock tube experiments, in which OH concentrations were measured and used to calculate total radical concentrations by means of the partial equilibrium assumption. 

Several key questions must be answered initially in a study of reaction chemistry. First, is the reaction sufficiently fast and reversible so that it can be regarded as chemical-equilibrium controlled Second, is the reaction homogeneous (occurring wholly within a gas or liquid phase) or heterogeneous (involving reactants or products in a gas and a liquid, or liquid and a solid phase) Slow reversible, irreversible, and heterogeneous (often slow) reactions are those most likely to require interpretation using kinetic models. Third, is there a useful volume of the water-rock system in which chemical equilibrium can be assumed to have been attained for many possible reactions This may be called the local equilibrium assumption. 

Ishikawa, H., Maeda, T., Hikita, H and Miyatake, K. (1988) The computerised derivation of rate equations for enzyme reactions on the basis of the pseudo-steady-state assumption and the rapid-equilibrium assumption. Biochem. J. 251, 175-181. 

Most applications of coupled models use the local equilibrium assumption. It is well known that most heterogeneous and redox reactions in the low temperature, near-surface systems are kinetically controlled (Hunter et al., 1998). However, we lack both theory and data to model kinetic reactions satisfactorily. In addition, inclusion of kinetic reactions makes the results even more difficult to comprehend completely and hence more costly. 

The Michaelis-Menten kinetics, represented by Eq. (18), may be extended to more complicated reactions by looking at the structure of the adsorption term. This procedure shown below is valid as long as the rapid equilibrium assumption is made. This is not valid in all cases. 

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