Reversion rate

The usual situation, true for the first three cases, is that in which the reactant and product solids are mutually insoluble. Langmuir [146] pointed out that such reactions undoubtedly occur at the linear interface between the two solid phases. The rate of reaction will thus be small when either solid phase is practically absent. Moreover, since both forward and reverse rates will depend on the amount of this common solid-solid interface, its extent cancels out at equilibrium, in harmony with the thermodynamic conclusion that for the reactions such as Eqs. VII-24 to VII-27 the equilibrium constant is given simply by the gas pressure and does not involve the amounts of the two solid phases. 

A rather different method from the preceding is that based on the rate of dissolving of a soluble material. At any given temperature, one expects the initial dissolving rate to be proportional to the surface area, and an experimental verification of this expectation has been made in the case of rock salt (see Refs. 26,27). Here, both forward and reverse rates are important, and the rate expressions are... 

Because of their prevalence in physical adsorption studies on high-energy, powdered solids, type II isotherms are of considerable practical importance. Bmnauer, Emmett, and Teller (BET) [39] showed how to extent Langmuir s approach to multilayer adsorption, and their equation has come to be known as the BET equation. The derivation that follows is the traditional one, based on a detailed balancing of forward and reverse rates. 

In this equation, the primes on the imaginary parts indicate that the Lamior frequencies and coupling constants will be different. Also, if the equilibrium constant for the exchange is not 1, then the forward and reverse rates will not be equal. Note that the 1,2 block, in the top right, represents the rate from site 2 into site 1. 

The eonditions at whieh the reaetion N2 -1- 3H2 —> 2NH3 are performed are sueh that the reverse rate is negligible. 

If kj and kj denote tlie forward and reverse rate constants respectively, die general rate of disappearance (-r ) for any type of reaction is defined as ... 

The general principle that activation of para substitution is greater than of ortho substitution holds true also for an azinium moiety in the one instance studied. Thus, the activation energy for the 4-chloropyridine quaternary salt 280 (Table II, line 9) is 1 kcal lower than that for the 2-isomer (line 5). The rate relation (2- > 4-isomer) is controlled by the entropies of activation in this reaction due to electrostatic attraction in the transition state (281). The reverse rate relation (4- > 2-position) is predicted for aminations of such quaternary compounds due to electrostatic repulsion (282) plus the difference in E. A kinetic study of the 2- and 4-pyridine quaternary salts... 

This is the general expression for film growth under an electric field. The same basic relationship can be derived if the forward and reverse rate constants, k, are regarded as different, and the forward and reverse activation energies, AG are correspondingly different these parameters are equilibrium parameters, and are both incorporated into the constant A. The parameters A and B are constants for a particular oxide A has units of current density (Am" ) and B has units of reciprocal electric field (mV ). Equation 1.114 has two limiting approximations. 

Similar expressions may be written for the partial reverse rates / and /j, but under the conditions assumed here they may be neglected. Hence substituting equations 19.5 and 19.6 in equation 19.4. 

According to Eq. (3-7), a plot of In [A], - [AL will be linear. The plot has, as the negative of its slope, the sum k + k-. The implication that this data treatment yields a sum is at first surprising, because this rate constant characteristic of the equilibration is clearly larger than the forward rate constant alone. The net rate itself, on the other hand, is smaller than the forward rate, since the reverse rate is subtracted from it, as in Eq. (3-2). These statements are not contradictory, and they illustrate the need to distinguish between a rate and a rate constant. 

Kinetic data"6 for reaction (3-23) and the calculation of forward and reverse rate constants... 

Once the previously neglected reverse rate constant for the second step is included, the excess rate at Step 2 becomes... 

That step with the smallest excess rate is the one whose forward and reverse rates are closest to the net forward and reverse rates, v/ and vr. This feature can be taken as a characteristic of the RCS, it being the one whose forward and reverse rates exceed the net forward rate vf and the net reverse rate vr by the smallest amount. 

FORWARD AND REVERSE RATE CONSTANTS AND THE PRINCIPLE OF MICROSCOPIC REVERSIBILITY... 

We have noted previously that the forward and reverse rates are equal at equilibrium. It seems, then, that one could use this equality to deduce the form of the rate law for the reverse reactions (by which is meant the concentration dependences), seeing that the form of the equilibrium constant is defined by the condition for thermodynamic equilibrium. By and large, this method works, but it is not rigorously correct, since the coefficients in the equilibrium condition are only relative, whereas those in the rate law are absolute.19 Thus, if we have this net reaction and rate law for the forward direction,... 

Given each scheme, it is easy to write down the expression for the reverse rates,... 

Note, first, that our prediction of the form of the reverse rate is ambiguous does or does not the reverse rate depend on [A2] Then, examine what results when both of the forward and reverse rates are set equal to one another at equilibrium. 

Both formulations give the correct equilibrium condition. Clearly, however, this is a special case. In nearly all real examples the reverse rate law and rate constant can be deduced correctly from the forward rate constant and the equilibria condition. To illustrate this characteristic, consider a two-step reaction and the expressions for the rates ... 

Ignoring for the moment the principle under consideration, we obtain the following expression for the equilibrium constant by equating, without further provision, the total forward and reverse rates at equilibrium ... 

Relaxation experiments. Use the relaxation times for the equilibrium shown to calculate the forward and reverse rate constants. The values are expressed in terms of the total concentration of chromium(VI), or [Cr(VI)]i = [HCrOj] + 2[Cr202 ] ... 

In the latter case one would like to know the length Apb of the metal-solid electrolyte-gas three-phase-boundaries (tpb) (in m or in metal mols, for which we use the symbol Ntpb throughout this book) and the value of the exchange current I0, where (W2F) expresses the value of the (equal and opposite under open-circuit conditions) forward and reverse rates of the charge-transfer reaction 4.1. 

The equilibrium constant for an elementary reaction is equal to the ratio of the forward and reverse rate constants of the reaction. 

FIGURE 13.21 The equilibrium constant for a reaction is equal to the ratio of the rate constants for the forward and reverse reactions, (a) A forward rate constant (A) that is relatively large compared with the reverse rate constant means that the forward rate matches the reverse rate when the reaction has neared completion and the concentration of reactants is low. (b) Conversely, if the reverse rate constant (A ) is larger than the forward rate constant, then the forward and reverse rates are equal when little reaction has taken place and the concentration of products is low. 

The model predicts the behavior of the active state LRG to be analogous to cell activation itself. LRG rises in seconds, disappears in minutes as binding equilibrates, and, when binding is interrupted, disappears in a few seconds as this state disappears, transduction also "collapses" and cell responses decay. The model should not be viewed as complete, however. For example, amplification steps, which permit the activation of multiple G proteins by a single receptor, would be built into the model by adding a reverse rate from LR to LRG. Such amplification would have to be verified experimentally. 

More complicated rate expressions are possible. For example, the denominator may be squared or square roots can be inserted here and there based on theoretical considerations. The denominator may include a term k/[I] to account for compounds that are nominally inert and do not appear in Equation (7.1) but that occupy active sites on the catalyst and thus retard the rate. The forward and reverse rate constants will be functions of temperature and are usually modeled using an Arrhenius form. The more complex kinetic models have enough adjustable parameters to fit a stampede of elephants. Careful analysis is needed to avoid being crushed underfoot. 

Example 7.11 showed how reaction rates can be adjusted to account for reversibility. The method uses a single constant, Kkinetic or Kthemo and is rigorous for both the forward and reverse rates when the reactions are elementary. For complex reactions with fitted rate equations, the method should produce good results provided the reaction always starts on the same side of equilibrium. 

Equation (7.28) may not provide a good fit for the equilibrium data if the equilibrium mixture is nonideal. Suppose that the proper form for Kkmetic is determined through extensive experimentation or by using thermodynamic correlations. It could be a version of Equation (7.28) with exponents different from the stoichiometric coefficients, or it may be a different functional form. Whatever the form, it is possible to force the reverse rate to be consistent with the equilibrium constant, and this is recommended whenever the reaction shows appreciable reversibility. 

Reverse rate constant for reversible adsorption step Exam. 10.2... 

Forward rate constant for reversible surface reaction Exam. 10.2 Reverse rate constant for reversible surface reaction Exam. 10.2 Mass transfer coefficient for a catalyst particle 10.2... 

Within the deterministic approach, this can also be expressed in terms of the forward and reverse rate constants, for and rev and equivalently in our discrete CA model the forward and reverse transition probabilities, Pt(A B) and Pj(B A), respectively... 

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