Equilibrium constant, reverse reaction rates

We derived the relation between the equilibrium constant and the rate constant for a single-step reaction. However, suppose that a reaction has a complex mechanism in which the elementary reactions have rate constants ku k2, and the reverse elementary reactions have rate constants kf, k2, . .Then, by an argument similar to that for the single-step reaction, the overall equilibrium constant is related to the rate constants as follows ... 

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... 

In chemical equilibrium, the forward and reverse reaction rates are equal and there is no net production of intermediates. The equilibrium constant Keq is given as the ratio of reactants in equilibrium. For the elementary reaction shown in Eq. (21), we obtain... 

The reader should refer to the original tables for the reference material on which the thermochemical data are based. The reference state used in Chapter 1 was chosen as 298 K consequently, the thermochemical values at this temperature are identified from this listing. The logarithm of the equilibrium constant is to the base 10. The unit notation (J/K/mol) is equivalent to (JK mol ). Supplemental thermochemical data for species included in the reaction listing of Appendix C, and not given in Table A2, are listed in Table A3. These data, in combination with those of Table A2, may be used to calculate heats of reaction and reverse reaction rate constants as described in Chapter 2. References for the thermochemical data cited in Table A3 may be found in the respective references for the chemical mechanisms of Appendix C. 

A chemical relaxation technique that measures the magnitude and time dependence of fluctuations in the concentrations of reactants. If a system is at thermodynamic equilibrium, individual reactant and product molecules within a volume element will undergo excursions from the homogeneous concentration behavior expected on the basis of exactly matching forward and reverse reaction rates. The magnitudes of such excursions, their frequency of occurrence, and the rates of their dissipation are rich sources of dynamic information on the underlying chemical and physical processes. The experimental techniques and theory used in concentration correlation analysis provide rate constants, molecular transport coefficients, and equilibrium constants. Magde" has provided a particularly lucid description of concentration correlation analysis. See Correlation Function... 

Since the equilibrium constant, is a ratio of the forward to reverse reaction rate... 

The principle of detailed balance is a result of the microscopic reversibility of electron kinetics. A prerequisite for the establishment of thermal equihbrium requires that the forward and reverse rates are identical. For isothermal reactions, the equilibrium constant remains unchanged. The principle of detailed balance is of fundamental importance to estabhsh helpful relations between reaction and equilibrium constants because both are at the initial thermal equilibrium in addition, at the new equihbrium after the relaxation of the perturbation, the net forward and reverse reaction rates are zero. 

Both of these steps have the net stoichiometry A- B and therefore have the same free energy change AG° = AG (B) — AG (A). This, in turn, means that both reactions must have the same value for their equilibrium constant Ke. This latter point can also be seen from the equilibrium requirement that the forward and reverse reaction rate should be equal ... 

As an interesting fact, we can learn from Eq. 12-21 that the time to steady-state (or time to equilibrium) depends on the sum of the forward and reverse reaction rate constants. Thus, even if one rate constant is very small, time to equilibrium can be small, provided that the other rate constant is large. By using Eq. 4 in Box 12.1 (95% of equilibrium reached) we obtain ... 

Reversible chemical reactions. In any reversible process, we must consider rate constants for both the forward and the reverse reactions. At equilibrium a reaction proceeds in the forward direction at exactly the same velocity as in the reverse reaction so that no change occurs. For this reason there is always a relationship between the equilibrium constant and the rate constants. For Eq. 9-9, /c is the bimolecular rate constant... 

Wilson and Cannan (18) reported detailed observations on the equilibrium and velocity constants in the glutamic acid—pyrrolidone carboxylic acid system in dilute aqueous solution. They found that the conversion of glutamic acid to pyrrolidone carboxylic acid follows the equation for a reversible first-order reaction. The equilibrium constant and the rate at which the equilibrium is achieved depend on the pH of the solution and the temperature. In neutral solutions, the equilibrium favors almost complete conversion of glutamic acid to pyrrolidone carboxylic acid however, the rate of the reaction is very slow and thus only 1% conversion occurs after 2-3 hr at 100°. In weakly acid (pH 4) and alkaline (pH 10) solutions, the conversion of glutamic acid to pyrrolidone carboxylic acid is much faster and about 98% conversion occurs in less than 60 hr. In strong acid (2 N HC1) and base (0.5 N NaOH) the conversion of pyrrolidone carboxylic acid to glutamic acid proceeds rapidly and virtually to completion. Other studies have shown that the conversion of glutamic acid to pyrrolidone carboxylic acid can be carried out within 2 hr at 142° with little alteration of optical rotation (80). 

For reactions which do not proceed virtually to completion, it is necessary to include the kinetics of the reverse reaction, or the equilibrium constant, in the rate equation. 

The exchange current density is the electrode reaction rate at the equilibrium potential (identical forward and reverse reaction rates) and depends on the electrode properties and operation. The typical expression for determining the exchange current density is the Arrhenius law (3.23), where the constant A depends on the gas concentration. Costamagna et al. [40] provide the following expressions for the anodic and cathodic exchange current density, respectively ... 

At equilibrium, depending on the temperature of the reaction, almost any concentration of the substances present can exist. If, at equilibrium in the reaction between sodium thiosulfate and silver ions, mostly silver ions and thiosulfate ions are present, we would not be successful in removing the silver ions to preserve a photo image. We need a system that shows us which substances are in excess at equilibrium, the reactants or products. It is possible to have equal concentrations of reactants and products at equilibrium, but this is usually not the case. At equilibrium, forward and reverse reaction rates are equal, not the amounts of reactants and products. At equilibrium, the rate of reactants making products equals the rate of products making reactants. This results in constant product and reactant concentrations. 

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. 

It should be mentioned that Arrhenius s paper [iv] was preceded by van t Hoff s book [v] (-> Hoff, Jacobus Hendricus van t), in which an equation compatible with that described above, was proposed on the basis of the relationships between equilibrium constants and the rate constants for the forward and reverse reactions. Nevertheless, the equation was named Arrhenius equation [vi-vii]. 

The equilibrium constant for reaction (2), 3.9 x 10"5 M, was obtained from the rates of the forward and reverse reactions. (Note that in many discussions the concentration of water is included in the equilibrium constant.) Thus AfG° for e"q is 55 kJ/mol greater than for H. Reaction (3) was estimated to have AG° = 0. The other data are available in standard... 

The forward reaction rate follows the Arrhenius law Kf j = Afj exp where Afj and are the pre-exponential factor and the activation energy. The reverse reaction rate is deduced from the equilibrium relation Krj = Kfj/Keq where Keq is the equilibrium constant. 

Note that the equilibrium constant for reactions 3 and 4, namely Ks 4, may be calculable from spectroscopic and thermal data, so that hi = kz/Ks 4 may be calculated. This permits us to calculate kb = kA/kt and also the rate constant for reaction 6 (the reverse of h) from kb == kb/Kb e, where 7v is the cahiulable equilibrium constant. All of these data are given in Table XII.5. 

Whereas the alkoxysilane hydrolysis is already frequently investigated, there is only little information on the reverse silanol alcoholysis. Thus, we investigated the reaction of silanols XMe2SiOH (for X see Table 1) with methanol and other alcohols with respect to equilibrium constants K and rate constants k and of the acid and base-catalyzed reaction and their substituent dependence. 

Figure 6 An energy diagram of the charge-transfer process at an n-type semiconductor/metal interface when an external potential (F) is applied across the semiconductor electrode. The applied potential changes the electric potential difference between the semiconductor surface and the bulk region. This perturbs the concentration of electrons at the surface of the semiconductor (ns), and a net current flows through the semiconductor/metal interface. The forward reaction represents the transfer of electrons from the semiconductor to the metal and the reverse reaction represents the injection of electrons into the semiconductor from the metal. The width of the arrows indicates schematically the relative magnitude of the current, (a) The reverse bias condition for an n-type semiconductor (V > 0). The forward reaction rate is reduced relative to its equilibrium value, while the reverse reaction rate remains constant. A net positive current exists at the electrode surface, (b) The forward bias condition (V < 0), the forward reaction rate increases compared to its equilibrium value, while the reverse reaction rate remains unaffected. A net negative current exists at the electrode surface...

The reverse reaction is determined by the nature of R. The generating [Pt(PR3)2] moiety is a rather stable intermediate. A comparison of the results obtained with the NMR data (22) of substituted phenylacetylenes led to the conclusion that there is a relation between equilibrium constant, first-order rate constants, and chemical shifts of the acetylenic proton they all depend on substituent effects on the electron density in the triple bond. 

The characteristic time for approach to equilibrium is given by the reciprocal of the sum of the rate constants l/(fci + /C2). This result has important consequences. In Figure 2.10 a case is considered where the equilibrium condition favors product over reactant. Note that the ratio of the rate constants gives the position of equilibrium, while their reciprocal sum gives the characteristic time for approach to equilibrium. The reader is referred to Stone and Morgan (1990) for further examples of reversible reaction rates. 

---