Time constant relaxation kinetics

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,... 

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. 

The first experimental data for a reaction involving proton transfer from a hydrogen-bonded acid to a series of bases which were chosen to give ApK-values each side of ApK=0 are given in Fig. 15 (Hibbert and Awwal, 1976, 1978 Hibbert, 1981). The results were obtained for proton transfer from 4-(3-nitrophenylazo)salicylate ion to a series of tertiary aliphatic amines in aqueous solution, as in (64) with R = 3-nitrophenylazo. Kinetic measurements were made using the temperature-jump technique with spectrophoto-metric detection to follow reactions with half-lives down to 5 x 10"6s. The reciprocal relaxation time (t ), which is the time constant of the exponential... 

The nature of the adsorbed species can be inferred from the usual chemical parameters, i.e. chemical shifts, linewidths and relaxation times. These latter allow the study of the mobility on the surfaces. As an analytical tool, C-NMR spectroscopy can also be used to determine the concentration of reactants or products as a function of time and hence kinetic constants can easily be determined. As a conclusion, a rather complete kinetic study can be carried out involving the nature of interaction between the admolecule and the surface and eventually the nature of the surface active centers. One can finally arrive at the proposition of a reaction mechanism. 

The temporal resolution of both methods is limited by the risetime of the IR detectors and preamplifiers, rather than the delay generators (for CS work) or transient recorders (SS) used to acquire the data, and is typically a few hundred nanoseconds. For experiments at low total pressure the time between gas-kinetic collisions is considerably longer, for example, approximately 8 /is for self-collisions of HF at lOmTorr. Nascent rotational and vibrational distributions of excited fragments following photodissociation can thus be obtained from spectra taken at several microseconds delay, subject to adequate SNR at the low pressures used. For products of chemical reactions, the risetime of the IR emission will depend upon the rate constant, and even for a reaction that proceeds at the gas-kinetic rate the intensity may not reach its maximum for tens of microseconds. Although the products may only have suffered one or two collisions, and the vibrational distribution is still the initial one, rotational distributions may be partially relaxed. 

The expression for the time course (equation 4.20) may be divided into various factors. There is first the exponential term with the rate constant, or in terms of relaxation kinetics, the reciprocal relaxation time 1/r given by... 

Consider a micellar solution at equilibrium that is subject to a sudden temperature change (T-jump). At the new temperature the equilibrium aggregate size distribution will be somewhat different and a redistribution of micellar sizes will occur. Aniansson and Wall now made the important observation that when scheme (5.1) represents the kinetic elementary step, and when there is a strong minimum in the micelle size distribution as in Fig. 2.23(a) the redistribution of micelle sizes is a two-step process. In the first and faster step relaxation occurs to a quasi-equilibrium state which is formed under the constraint that the total number of micelles remains constant. Thus the fast process involves reactions in scheme (5.1) for aggregates of sizes close to the maximum in the distribution. This process is characterized by an exponential relaxation with a time constant Tj equal to... 

Equation 4.35 shows that the concentration deviations based on a linearization analysis of the rate laws in Eqs. 1.54a and 1.54c will decay to zero exponentially ( relax ) as governed by the two time constants, r, and r2. These two parameters, in turn, are related to the rate coefficients for the coupled reactions whose kinetics the rate laws describe (Eqs. 4.36c-4.36e and 4.38). If the rate coefficients are known to fall into widely different time scales for each of the coupled reactions, their relation to the time constants can be simplified mathematically (Eq. 4.39 and Table 4.3). Thus an experimental determination of the time constants leads to a calculation of the rate coefficients.20 In the example of the metal complexation reaction in Eq. 1.50, with the assumptions that the outer-sphere complexation step is much faster than the inner-sphere complexation step and that dissociation of the inner-sphere complex is negligible (k b = 0 in Eq. 1.54c), the results for tx and r2 in the first row of Table 4.3 can be applied. The expression for tx indicates that measurements of this parameter as a function of differing equilibrium concentrations of the complexing metal and ligand will produce a straight line whose slope is kf and whose y-intercept is kb. The measured values of l/r2 at these same two equilibrium concentrations then lead to a calculation of kf. 

C. F. Bernasconi, Relaxation Kinetics, Academic Press, New York, 1976. This standard textbook discusses comprehensively the linearization of rate laws for a wide variety of reaction sequences as well as experimental methods of measuring time constants. ... 

In fact, caused by the short-range order relaxation, the residual-resistivity and heat-capacity relaxation kinetics at low temperatures cannot be represented by a single time constant and sometimes even by two time constants. According to the Khachaturyan s approach [1] (see also [7, 8]), in a general case, the short-range order... 

Mizutani and Kitagawa measured the time-dependent Stokes and anti-Stokes Raman intensities of the heme v4 band after photoexcitation and used the relative intensities to estimate its temperature and thermal relaxation dynamics (30). They found the population relaxation to occur biexponen-tially with 1.9 ps (93%) and 16 ps (7%) time constants. The dominant 1.9 ps population relaxation correlates with a 3.0 ps thermal relaxation, which is a factor of 2 faster than the ensemble averaged temperature relaxation deduced from the near-IR study of band III. The kinetic energy retained within a photoexcited heme need not be distributed uniformly among all the vibrational degrees of freedom, nor must the energy of all vibrational modes decay at the same rate. Consequently, a 6.2 ps ensemble-averaged estimate of the heme thermal relaxation is not necessarily inconsistent with a 3 ps relaxation of v4. 

The aimealing kinetics of the light-induced defects are shown in Fig. 6.29. Several hours at 130 °C are needed to anneal the defects completely, but only a few minutes at 200 C. The relaxation is nonexponential, and in the initial measurements of the decay the results were analyzed in terms of a distribution of time constants, Eq. (6.78) (Stutzmann, Jackson and Tsai 1986). The distribution is centered close to 1 eV with a width of about 0.2 eV. Subsequently it was found that the decay fits a stretched exponential, as is shown in Fig. 6.29. The parameters of the decay-the dispersion, p, and the temperature dependence of the decay time, t - are similar to those found for the thermal relaxation data and so are consistent with the same mechanism of hydrogen diffusion. The data are included in Fig. 6.23 which describes the general relation between x and D,. The annealing is therefore the process of relaxation to the equilibrium state with a low defect density. 

The overall time constant of the orientational relaxation process rfp and the kinetic process tJIP = [Ail + 2Aii(c — c/p)] is given by the relation... 

---