Kinetics relaxation

With the availability of perturbation techniques for measuring the rates of rapid reactions (Sec. 3.4), the subject of relaxation kinetics — rates of reaction near to chemical equilibrium — has become important in the study of chemical reactions.Briefly, a chemical system at equilibrium is perturbed, for example, by a change in the temperature of the solution. The rate at which the new equilibrium position is attained is a measure of the values of the rate constants linking the equilibrium (or equilibria in a multistep process) and is controlled by these values. 

The system is initially at equilibrium with concentrations Ca and c. Now we rapidly perturb the system so as to alter the magnitude of the equilibrium constant. Let the new equilibrium concentrations, toward which the actual concentrations will relax, be Ca and c. (Clearly one of these will be greater than and one will be less than the initial concentrations.) The concentrations at any time t are Ca and c.  

Define a reaction variable jc as the concentration by which the actual concentrations differ from the new equilibrium values thus. 

However, the equilibrium constant at the new conditions (after the perturbation) is K = kjk-, = cz/ca, so we obtain 

Not surprisingly, we find that the relaxation is a first-order process with rate constant A , + A i. It is conventional in relaxation kinetics to speak of the relaxation time T, which is the time required for the concentration to decay to Me its initial value. In Chapter 2 we found that the lifetime defined in this way is the reciprocal of a first-order rate constant. In the present instance, therefore, 

Equation (4-13) is nonlinear in the concentration jc. We, therefore, impose the linearization condition, namely, that the perturbation is sufficiently small that the term in is negligible. The result is 

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The individual reactions need not be unimolecular. It can be shown that the relaxation kinetics after small perturbations of the equilibrium can always be reduced to the fomi of (A3.4.138t in temis of extension variables from equilibrium, even if the underlying reaction system is not of first order [51, fil, fiL, 58]. 

Perturbation or relaxation techniques are applied to chemical reaction systems with a well-defined equilibrium. An instantaneous change of one or several state fiinctions causes the system to relax into its new equilibrium [29]. In gas-phase kmetics, the perturbations typically exploit the temperature (r-jump) and pressure (P-jump) dependence of chemical equilibria [6]. The relaxation kinetics are monitored by spectroscopic methods. 

Figure B2.5.2. Schematic relaxation kinetics in a J-jump experiment, c measures the progress of the reaction, for example the concentration of a reaction product as a fiinction of time t (abscissa with a logaritlnnic time scale). The reaction starts at (q. (a) Simple relaxation kinetics with a single relaxation time, (b) Complex reaction mechanism with several relaxation times x.. The different relaxation times x. are given by the turning points of e as a fiinction of ln((). Adapted from [110].

Relaxation kinetics may be monitored in transient studies tlirough a variety of metliods, usually involving some fonn of spectroscopy. Transient teclmiques and spectrophotometry are combined in time resolved spectroscopy to provide botli tire stmctural infonnation from spectral measurements and tire dynamical infonnation from kinetic measurements that are generally needed to characterize tire mechanisms of relaxation processes. The presence and nature of kinetic intennediates, metastable chemical or physical states not present at equilibrium, may be directly examined in tliis way. 

In contrast to statics, the relaxational kinetics of living polymers and of giant wormlike micelles is unique (and different in both cases). It is entirely determined by the processes of scission/recombination and results in a nonlinear approach to equilibrium. A comparison of simulational results and laboratory observations in this respect is still missing and would be highly desirable. 

If the step is initially prepared to be straight, it relaxes to its fluctuating shape in the due course of time. This time evolution of step width depends on the relaxation kinetics, and can be used to determine the values of various kinetic coefficients [3,16-18,64-66], For example, if the attachment and detachment kinetics of adsorbed atoms at a step is rate limiting, the step width increases as [65]... 

This device of A, the displacement from equilibrium, is used in the study of very fast reversible reactions by relaxation kinetics. We will see, in Chapter 4, that if A is very small, all reactions follow first-order kinetics, thus simplifying the interpretation of the kinetics. This approach might be extended to slow reversible reactions. 

This treatment illustrates several important aspects of relaxation kinetics. One of these is that the method is applicable to equilibrium systems. Another is that we can always generate a first-order relaxation process by adopting the linearization approximation. This condition usually requires that the perturbation be small (in the sense that higher-order terms be negligible relative to the first-order term). The relaxation time is a function of rate constants and, often, concentrations. 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

Concentration-jump methods, such as the pH-jump technique cited earlier, can be used in relaxation kinetics, but this approach is described later (Section 4.4). 

Bemasconi, C.F. "Relaxation Kinetics Academic Press New York, 1976. 

One useful means of studying reversible reactions is to effect a sudden change that perturbs a previously attained equilibrium. One might do this in several ways by injecting one component, by suddenly diluting with solvent (if An = 0), or by rapidly changing the temperature. The perturbation must be made rapidly compared with the rate of re-equilibration. This procedure is referred to as relaxation kinetics. 

Relaxation kinetics in single-stage reactions. Derive expressions for the reciprocal relaxation times of the following single-stage reactions ... 

Relaxation kinetics with a reaction intermediate. Show that the kinetic scheme with a steady-state intermediate I corresponds to the single relaxation time shown ... 

Relaxation kinetics. It is sometimes necessary to cause a larger perturbation, such that the terms in S2 [Eq. (11-8)] cannot be dropped. Can one then ignore the early part of the re-equilibration curve, and fit the later stages as a monoexponential Can one use King s method, as presented in Ref. 3, Chapter 3 ... 

Relaxation kinetics. In the course of a study of hemiacetal formation with a phenol nucleophile, an equation was given for the relaxation time in the system ... 

Reaction scheme, defined, 9 Reactions back, 26 branching, 189 chain, 181-182, 187-189 competition, 105. 106 concurrent, 58-64 consecutive, 70, 130 diffusion-controlled, 199-202 elementary, 2, 4, 5, 12, 55 exchange, kinetics of, 55-58, 176 induced, 102 opposing, 49-55 oscillating, 190-192 parallel, 58-64, 129 product-catalyzed, 36-37 reversible, 46-55 termination, 182 trapping, 2, 102, 126 Reactivity, 112 Reactivity pattern, 106 Reactivity-selectivity principle, 238 Relaxation kinetics, 52, 257 -260 Relaxation time, 257 Reorganization energy, 241 Reversible reactions, 46-55 concentration-jump technique for, 52-55... 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

Let us compare in detail the differential theory results with those obtained for rotational relaxation kinetics from the memory function formalism (integral theory). Using R(t) from Eq. (1.107) as a kernel of Eq. (1.71) we can see that in the low-density limit... 

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