Rate constant cross relation

The macroscopic rate constant is related to the relative speed of the reactants and the reaction cross-section, and the expression contains a weighted average over all possible quantum states and velocities of the reactants, and sums and integrals over all possible quantum states and velocities of the products. 

The most important point about this equation is that both sides are to be evaluated at the same total energy E. To insure a sharp value of the translational energy, i and /are indices of quantum levels and g is the degeneracy of the level. At a given total energy, the equilibrium condition is that all quantum states are equally probable. So the equihbrium concentrations are proportional to the number of quantum states. There are gip- E- ), Ej = E - Ei, states of the reactants and similarly for the products. The reaction rate constants are related to the cross-section as k(i J) = ViCf (i j), where v is the relative velocity of the reactants and similarly for the products. Since d T = Vid/ i we have that v, /Ot( t) = A A /(27r) where, as usual, p = hk. For our immediate purpose, it is more convenient to write detailed balance in its chemical form, Eq. (6.36). At a given total energy we have from (6.37)... 

In evaluating the kinetics of copolymerization according to the chemical control model, it is assumed that the termination rate constants k,AA and A,Br are known from studies on homopolymerization. The only unknown in the above expression is the rate constant for cross termination (AtAB)- The rate constant for this reaction in relation to klAA and kmB is given by the parameter . 

These equations do not necessarily show the actual charges the important point is that all three are single-electron events. The asterisks can be thought of as an isotopic label, but need not be anything that concrete, since certain line-broadening techniques (Section 11.5) provide EE rate constants without them. The Marcus cross relation is an expression for kA% as a function of kAA, bb> and A, the equilibrium constant for Eq. (10-67). It reads,... 

Data are given in Table 10-7 to illustrate certain facets of the Marcus cross relation. They refer to six reactions in which the cage complex Mn(sar)3+ is reduced or Mn(sar)2+ oxidized.34 These data were used to calculate the EE rate constant for this pair. The results of the calculation, also tabulated, show that there is a reasonably self-consistent value of fcEE for Mn(sar)3+/Mn(sar)2+ lying in the range 3-51 L mol-1 s-1. When values34 for an additional 13 reactions were included the authors found an average value of kEE = 17 L mol 1 s l. 

Equation (49) applies to both the forward and reverse rate constant, /clh and Ichl- Consequently, the thermodynamic parameters for the intersystem crossing process are related according to ... 

If we let and t2 represent the times corresponding to reaction progress variables and <5J, respectively, the time ratio t2/tl for fixed values of <5 and <5 will depend only on the ratio of rate constants k. One may readily prepare a table or graph of <5 versus k t for fixed k and then cross-plot or cross-tabulate the data to obtain the relation between k and ktt at a fixed value of <5. Table 5.1 is of this type. At specified values of <5 and S one may compute the difference log(fe1t)2 — log f) which is identical with log t2 — log tj. One then enters the table using experimental values of t2 and tx and reads off the value of k = k2/kv One application of this time-ratio method is given in Illustration 5.5. 

The permeability of a porous medium (K) is defined as the proportionality constant that relates the flow rate through the medium to the pressure drop, the cross-sectional area, the fluid viscosity, and net flow length through the medium ... 

According to equations 6.4-16 and -16a, EA and A are somewhat dependent on T. The calculated values for A, usually agree with measured values within an order of magnitude, which, considering the approximations made regarding the cross-sections, is satisfactory support for the general concepts of the theory. SCT provides a basis for the estimation of rate constants, especially where experimental values exist for related reactions. Then, values of p and E can be estimated by comparison with the known system. 

Also, the observed rates probably refer to outer-sphere pathways, and the rate constants for the corresponding homogeneous self-exchange reactions are available or can be estimated from rate data for closely related cross reactions (15). These h ex... 

Quantitative data on rates of reaction have been obtained for some of the triplet reactions. Assuming triplet quenching to be approximately diffusion controlled, the rate constants for the reactions between excited species and nucleophile are 10 -10 1 mole s . The data show that in comparing and interpreting quantum yields—even in the case of related systems—one should proceed to determine separately rate constants as well as intersystem crossing efficiencies and lifetimes of the reacting excited species. 

The kinetics of the reduction of perruthenate(VII) by [FefCbOe]" and [W(CN)g]" and the oxidation of ruthenate(VI) by [Mo(CN)g] and [Ru(Cb06] have been studied in aqueous alkaline solutions. The cross-reaction data have been treated according to the Marcus relations and yield a self-exchange rate constant of 10 s at 25.0 °C and 1.0 M ionic strength for the... 

A model has been considered for Sn2 reactions, based on two interacting states. Relevant bond energies, standard electrode potentials, solvent contribntions (nonequi-librinm polarization), and steric effects are included. Applications of the theory are made to the cross-relation between rate constants of cross- and identity reactions, experimental entropies and energies of activation, the relative rates of Sn2 and ET reactions, and the possible expediting of an outer sphere ET reaction by an incipient SN2-type interaction (Marcus, 1997). 

Fig. 14. Schematic representation of energy levels and transitions for fluorescence and related processes kic, rate constant for interval conversion fcF, rate constant for fluorescence fcISC, rate constant for intersystems crossing fc[cp> rate constant for internal conversion from triplet state kp, rate constant for phosphorescence S, energy level for the first excited singlet state after solvent rearrangement for a polarity probe in a polar solvent.

At specified mass flow rate and inlet conditions Py and V), Eq. (6.68) predicts a relation between the area ratio A2IAX and the pressure ratio P-JPy when isentropic flow prevails. It turns out that, as the pressure falls, the cross section at first narrows, reaches a minimum at which the velocity becomes sonic then the cross section increases and the velocity becomes supersonic. In a duct of constant cross section, the velocity remains sonic at and below a critical pressure ratio given by... 

The value of ket > 6 x 108 cm8mol ls - - in the usual units for a bimolecular rate constant for homogeneous solution is > 6 x 105 M-ls l. The ferrocene self-exchange constant is 5 x 106 M-Ts-1 (29). Various cross reactions of substituted ferrocenes and ferricenium derivatives have bimolecular rate constants that exceed 10 M - -s l where the equilibrium constant exceeds unity (30). Further, in the cross reactions, the rate constants varied by almost two orders of magnitude for a change in driving force of -0.25 V (30). Thus, the data in Table II relating to the ferrocene-like molecules is reasonable. 

The free energies in (18) are illustrated in Fig. 10. It can be seen that GA is that part of AG ° available for driving the actual reaction. The importance of this relation is that it allows AGXX Y to be calculated from the properties of the X and Y systems. In thermodynamics, from a list of n standard electrode potentials for half cells, one can calculate j (m — 1) different equilibrium constants. Equation (18) allows one to do the same for the %n(n— 1) rate constants for the cross reactions, providing that the thermodynamics and the free energies of activation for the symmetrical reactions are known. Using the... 

In this connection kinetic models can also be separated into microscopic and macroscopic models. The relations between these models are established through statistical physics equations. Microscopic models utilize the concepts of reaction cross-sections (differential and complete) and microscopic rate constants. An accurate calculation of reaction cross-sections is a problem of statistical mechanics. Macroscopic models utilize macroscopic rates. 

In this chapter, we define the cross-section and derive its relation to the rate constant. We show the following. 

We begin by establishing the relation between the so-called reaction cross-section bimolecular rate constant. Let us consider an elementary gas-phase reaction,... 

The relation between the key quantities (the rate constant k(T), the cross-section... 

A theoretical determination of the rate constant for a chemical reaction requires a calculation of the reaction cross-section based on the dynamics of the collision process between the reactant molecules. We shall develop a general relation, based on classical dynamics, between reaction probabilities that can be extracted from the dynamics of the collision process and the phenomenological reaction cross-section introduced in Chapter 2. That is, we give a recipe for how to calculate the reaction cross-section in accord with the general definition in Eq. (2.7). 

In the previous chapter, we have discussed the reaction dynamics of bimolecular collisions and its relation to the most detailed experimental quantities, the cross-sections obtained in molecular-beam experiments, as well as the relation to the well-known rate constants, measured in traditional bulk experiments. Indeed, in most chemical applications one needs only the rate constant—which represents a tremendous reduction in the detailed state-to-state information. 

Fig. 8.1.2 An illustration of the relations between the rate constant k(T), the reaction cross-section o, and the reaction probability P as obtained from either quantum mechanics, quasi-classical mechanics, or various assumptions (approximations) for the reaction dynamics.

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