Reversible reaction rate equation

Many reactions show appreciable reversibility. This section introduces thermodynamic methods for estimating equilibrium compositions from free energies of reaction, and relates these methods to the kinetic approach where the equilibrium composition is found by equating the forward and reverse reaction rates. 

Reverse Reaction Rates. Suppose that the kinetic equilibrium constant is known both in terms of its numerical value and the exponents in Equation (7.28). If the solution is ideal and the reaction is elementary, then the exponents in the reaction rate—i.e., the exponents in Equation (1.14)—should be the stoichiometric coefficients for the reaction, and Ei mettc should be the ratio of... 

Rapid exchange between Xi and Xi is reported in reference (3). This means that the forward and reverse reaction rates of this step are mnch faster than all others, and hence this particnlar step can be treated as a qnasi-eqnihbrium. The two intermediates in that step are present at all times in concentrations related to one another by a thermodynamic eqnihbrium constant and can be Inmped into one pseudo-intermediate [Xs]. This approach is very useM in reducing the number of terms in the denominator of the rate equation, which is equal to the square of the number of intermediates in the cycle (7). 

If the forward and reverse reactions are nonelementary, perhaps involving the formation of chemical intermediates in multiple steps, then the form of the reaction rate equations can be more complex than Equations 5.33 to 5.36. 

Equation (38) applies the principle that the net reaction rate is the difference between the sum of all reverse reaction rates and the sum of all forward reaction rates. 

The reaction set was numerically modeled using the computer program CHEMK (9) written by G. Z. Whitten and J. P. Meyer and modified by A. Baldwin of SRI to run on a MINC laboratory computer. CHEMK numerically Integrates a defined set of chemical rate equations to reproduce chemical concentration as a function of time. Equilibria can be modeled by Including forward and reverse reaction steps. Forward and reverse reaction rate... 

The second motive of this chapter is concerned with evergreen topic of interplay of chemical kinetics and thermodynamics. We analyze the generalized form of the explicit reaction rate equation of the thermodynamic branch within the context of relationship between forward and reverse reaction rates (we term the corresponding problem as the Horiuti-Boreskov problem). We will compare our... 

Equation (3) is linear with respect to the reaction rate variable, R. In the further analysis of more complex, non-linear, mechanisms and corresponding kinetic models, we will present the polynomial as an equation, which generalizes Equation (3), and term it as the kinetic polynomial. We will demonstrate that the overall reaction rate, in the general non-linear case, cannot generally be presented as a difference between two terms representing the forward and reverse reaction rates. This presentation is valid only at the special conditions that will be described. 

Equation (10) can be presented as the difference of two terms, the forward and reverse reaction rates,... 

Sherwood and Pigford (S9) have discussed the problem of the absorption of a solute A by a solvent S upon solution, A may be converted into B according to the reaction A = B (k/ and krf being the forward and reverse reaction-rate constants, and K = k//k/). The concentration of A is maintained at cAo at the surface of the liquid S, and it is assumed that S is semiinfinite in extent. It is further assumed that B is nonvolatile that is, it cannot escape from solvent S. Equation (51) is then used to explain the diffusion of A and B, with DAg and DBs taken as concentration independent, and the term containing the molar average velocity w is neglected. Hence the mathematical statement of the problem is (for very dilute solutions of A and B)... 

Thermodynamic calculations reveal that there is a significant reverse reaction rate for the decomposition of HI (Brown, 2009). This reverse reaction rate requires an accurate consideration of several coupled reaction rate equations. These expressions are ... 

Equilibrium State reached by a reversible reaction when forward and reverse reaction rates are equal represented in chemical equations by =s instead of — . 

As Graven and Long noted, these expressions were based on conditions removed from equilibrium. If the forward and reverse reaction rates are equated to determine the equilibrium constant, there will be disagreement between the result and well-established correlations for the equilibrium constant. Therefore, we modified their model in the following manner to yield the correct equilibrium constant. 

Although more complex models have been proposed to describe the process [57, 85, 86], involving the Ps bubble state and its shrinking upon reaction, the equations based on a reversible reaction with a forward and reverse reaction rate constants as in scheme (X) enables the fitting of the data perfectly, as shown by the solid line in Figure 4.9. The kinetic equations corresponding to such a scheme are tedious to derive, particularly as concerns the intensities (still more when a magnetic field is applied). However, they do not present insuperable mathematical difficulties and should be used instead of the approximate expressions that have appeared casually (e.g., "steady state" treatment of the reversible reaction). From scheme (X), it is not expected that the variation of X3 with C be linear, but the departure from linearity may be rather small, so that the shape of the X3 vs C plots may not be taken as a criterion to ascribe the nature of the reaction. 

In equation (11), is the reverse reaction rate constant, and it is independent of s. The concentration of vacant states in the semiconductor and the concentration of filled states in the metal has been incorporated into the value of This rate is called the reverse rate, because it represents the formation of the species on the left-hand side of the reaction represented in equation (9). At equihbrium (when = so), these rates must be equal to each other (k = kn so)-... 

This reaction Fe2(/r-02CAr )4(4-CNPy)2 with water was investigated at —60°C with different excess amoimts of water. A plot of Inkobs versus hi[H20] was fitted to a linear equation, which had a slope of 0.99, indicating a first-order dependence on water. The pseudo-first-order rate constant is also linearly dependent on the concentration of water, with an intercept, which corresponds to the reverse reaction rate constant, near zero. This result indicates the reaction to be irreversible under these conditions. 

In acidic conditions the tetracyclines undergo epimerisation at carbon atom 4 to form an equilibrium mixture of tetracycline and the epimer, 4-epi-tetracycline (Scheme 4.7). The 4-epi-tetracycline is toxic and its content in medicines is restricted to not more than 3%. The epimerisation follows the kinetics of a first-order reversible reaction (see equation (4.24)). The degradation rate is pH-dependent (maximum epimerisation occurring... 

In this model, the first step is the dissociation of C02 at a carbon free active site (Cfas), releasing CO and forming an oxidized surface complex [C(O)]. In the second step, the carbon-oxygen complex subsequently produces CO and a new free active site. The reverse reaction is relatively slow compared with the forward reaction, so the second reaction can be treated as an irreversible reaction. In this model, desorption of the carbon-oxygen surface complex is the rate-limiting step. The rate for this mechanism can be described by the Lang-muir-Hinshelwood rate equation. Furthermore, the C/C02 reaction rate is dependent on the CO and C02 partial pressures and is inhibited by the presence of carbon monoxide. A widely utilized reaction rate equation based on this mechanism is... 

Whereas radioactive decay is never a reversible reaction, many first-order chemical reactions are reversible. In this case the characteristic life time is determined by the sum of the forward and reverse reaction rate constants (Table 9.5). The reason for this maybe understood by a simple thought experiment. Consider two reactions that have the same rate constant driving them to the right, but one is irreversible and one is reversible (e.g. k in first-order equation (a) of Table 9.5 and ki in first-order reversible equation (b) of the same table). The characteristic time to steady state tvill be shorter for the reversible reaction because the difference between the initial and final concentrations of the reactant has to be less if the reaction goes both ways. In the irreversible case all reactant will be consumed in the irreversible case the system tvill come to an equilibrium in which the reactant will be of some greater value. The difference in the characteristic life time between the two examples is determined by the magnitude of the reverse reaction rate constant, k. If k were zero the characteristic life times for the reversible and irreversible reactions would be the same. If k = k+ then the characteristic time for the reversible reaction is half that of the irreversible rate. 

If Q is greater than K, then the ratio of the concentration of products to the concentration of reactants, as given by the reaction quotient equation above, is greater than when at equilibrium, and the reaction increases reactants and decreases products. In other words, the reverse reaction rate will be greater than the forward rate. This is sometimes called a leftward shift in the equilibrium. Of course, the equilibrium constant does not change during this type of equilibrium shift. 

Here k and k-i are the forward and reverse reaction rate constants for substrate binding, while the process of conversion of the enzyme-substrate complex ES is assumed to be irreversible, thus having only one reaction rate constant, 2, for the forward reaction. The MM kinetic equation is derived assuming, among other things, that the reaction is operating away from the thermodynamic equilibrium and that the concentration of the ES complex is constant. Skipping the derivation, the final MM equation can be written in the familiar form... 

The quasi-equilibrium approximation relies on the assumption that there is a single rate-determining step, the forward and reverse rate constants of which are at least 100 times smaller than those of all other reaction steps in the kinetic scheme. It is then assumed that all steps other than the rds are always at equilibrium and hence the forward and reverse reaction rates of each non-rds step may be equated. This gives simple potential relations describing the varying activity of reaction intermediates in terms of the stable solution species (of known and potential-independent activity) that are the initial reactants or final products of the reaction. The variation of the activities of reaction intermediates is, however, restricted by either the hypothetical solubility limit of these species or, in the case of surface-confined reactions and adsorbed intermediates, the availability of surface sites. In both these cases, saturation or complete coverage conditions would result in a loss of the expected... 

It was concluded that the reverse reaction in Equation 7.35 involves a radical coupling. The rate constant for the forward homolytic reaction was estimated as kH = 197 s 1 in 40% acetonitrile the rapid homolysis, traced to the weak N-O bond in the peroxynitrate complex, leaves little time for it to engage in bimolecular reactions with added substrates. 

Note that equations need not be written for species Y, since its concentration does not affect the current or the potential. If reaction (12.2.2) were reversible, however, the concentration of species Y would appear in the equation for 5Cr(x, t) dt, and an equation for 5Cy(, t) dt and initial and boundary conditions for Y would have to be supplied (see entry 3 in Table 12.2.1). Generally, then, the equations for the theoretical treatment are deduced in a straightforward manner from the diffusion equation and the appropriate homogeneous reaction rate equations. In Table 12.2.1, equations for several different reaction schemes and the appropriate boundary conditions for potential-step, potential-sweep, and current-step techniques are given. 

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