Time-dependent rate constant, changing

The physical meaning of the individual symbols is as follows k t) is the time-dependent rate constant of the radiative depletion of the excited state, p v,t) is the normalized time-dependent emission spectmm, i.e., the denominator k f)p v, t))) describes the total fluorescence, S t), P2 is the Legendre polynomial of the second order which correlates with the mutual angular orientations of /energy states. He also employed the master equation, which describes the time change of the conditional probabilityp(/, i2 tj(j, J2o t = 0) that the fluorophore is in energy state i and... 

NAH - I would like to say that the term "time dependent rate constant" is a dangerous term. The time dependence arises from a change in local concentration and is not due to a change in the "true" rate constant. 

To get the solution of Eq. 21-4 for a time-dependent rate k(t) you could use Eq. 9 of Box 12.1. Since in the above examples, k(t) is constant except for an abrupt change at t = 30 d, you can also use the solution for time-independent coefficients J and k (Box 12.1, Eq. 6) for the first 30 days and then insert the final concentration as the initial value for the second 30 days. The three concentration curves are shown in Fig. 2. Their behavior can be understood qualitatively with the following considerations ... 

Thus the approach to equilibrium always follows a first-order rate law, Equation (14), with the pH-dependent rate constant kobs = kE + kK. Figure 1 shows the concentration changes in time starting from a 1m solution of pure enol (full line) and of pure ketone (dashed line). The individual, unidirectional rate constants kE and kK can be determined as follows For most ketones the equilibrium enol concentration is quite small, i.e., tE = cE(oo)/ ck(oo)<<1. Hence kE kK [Equation (1)], so that enol ketonization is practically irreversible and kE may be neglected, kobskK. The rate constant of enolization kE, on the other hand, is equal to the observed rate constant of reactions for which enolization is rate-determining, such as ketone bromination (Scheme 2). 

It is not easy to visualize directly how the composition depends on any particular rate constant for such a complex expression as Eq. (274). Consequently, numerical calculations must be made to determine the effect of changes in the values of particular rate constants just as one does in the method given in this article. Using Eq. (274), the composition at various reaction times must be calculated for a given set of rate constants, the values of particular rate constants changed, the calculations repeated, and the results compared. This is not a small task. 

The reactor point effectiveness can now be readily determined from the overall pellet effectiveness developed earlier together with the known time dependence of the change of active surface area. Under the assumptions of negligible external mass transfer resistance and an isothermal pellet, the reactor point effectiveness is simply the pellet effectiveness multiplied by (ks/ki,), where ks and kj, are the rate constants evaluated at the pellet surface and bulk-fluid temperatures ... 

Eracture mechanics concepts can also be appHed to fatigue crack growth under a constant static load, but in this case the material behavior is nonlinear and time-dependent (29,30). Slow, stable crack growth data can be presented in terms of the crack growth rate per unit of time against the appHed R or J, if the nonlinearity is not too great. Eor extensive nonlinearity a viscoelastic analysis can become very complex (11) and a number of schemes based on the time rate of change of/have been proposed (31,32). 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Time-dependent fluids are those for which structural rearrangements occur during deformation at a rate too slow to maintain equilibrium configurations. As a result, shear stress changes with duration of shear. Thixotropic fluids, such as mayonnaise, clay suspensions used as drilling muds, and some paints and inks, show decreasing shear stress with time at constant shear rate. A detailed description of thixotropic behavior and a list of thixotropic systems is found in Bauer and Colhns (ibid.). 

This is an old, familiar analysis that applies to any continuous culture with a single growth-limiting nutrient that meets the assumptions of perfect mixing and constant volume. The fundamental mass balance equations are used with the Monod equation, which has no time dependency and should be apphed with caution to transient states where there may be a time lag as [L responds to changing S. At steady state, the rates of change become zero, and [L = D. Substituting ... 

In systems such as the 2- and 6-hydroxypteridines, sudden addition of an alkaline solution to a neutral buffer, or of a neutral solution to an alkaline buffer, displaces the equilibrium between hydrated and anhydrous species (because the anions are less hydrated than the neutral molecules). By measuring the time-dependent change of optical density at a selected wavelength, a first-order rate constant, obs5 can be obtained. This constant is a composite one, and to see its relationship to other quantities some discussion is necessary. 

Whereas heat capacity is a measure of energy, thermal diffusivity is a measure of the rate at which energy is transmitted through a given plastic. It relates directly to processability. In contrast, metals have values hundreds of times larger than those of plastics. Thermal diffusivity determines plastics rate of change with time. Although this function depends on thermal conductivity, specific heat at constant pressure, and density, all of which vary with temperature, thermal diffusivity is relatively constant. 

FIGURE 9.4. The autocorrelation function of the time-dependent energy gap Q(t) = (e3(t) — 2(0) for the nucleophilic attack step in the catalytic reaction of subtilisin (heavy line) and for the corresponding reference reaction in solution (dotted line). These autocorrelation functions contain the dynamic effects on the rate constant. The similarity of the curves indicates that dynamic effects are not responsible for the large observed change in rate constant. The autocorrelation times, tq, obtained from this figure are 0.05 ps and 0.07ps, respectively, for the reaction in subtilisin and in water. 

A constant current flowing across the electrolyte solution/eleetrode interface causes a potential shift because of the changing concentrations of educts and produets, which arc consumed and generated respee-tively. The change of the electrode potential as a funetion of time is recorded in a ehronopotentiometrie experiment. Depending on the rate of the electrode reaetion various mathematical treatments are possible providing access to rate constants for details see e.g. [OlBar]. (Data obtained with this method are labelled CH.)... 

This equation is known as the rate law for the reaction. The concentration of a reactant is described by A cL4/df is the rate of change of A. The units of the rate constant, represented by k, depend on the units of the concentrations and on the values of m, n, and p. The parameters m, n, and p represent the order of the reaction with respect to A, B, and C, respectively. The exponents do not have to be integers in an empirical rate law. The order of the overall reaction is the sum of the exponents (m, n, and p) in the rate law. For non-reversible first-order reactions the scale time, tau, which was introduced in Chapter 4, is simply 1 /k. The scale time for second-and third-order reactions is a bit more difficult to assess in general terms because, among other reasons, it depends on what reactant is considered. 

For SCVCP in general, DB strongly depends on the comonomer ratio (y=[monomer]o/[inimer]o) [73,74]. In the ideal case,when all rate constants are equal, for y>>l, the final value of DB decreases with y as DB=2/(y+l) which is four times higher than the value expected from dilution of inimer molecules by monomers. For low values of (yreactivity ratios, the structure of polymer obtained can change from macroinimers when the monomer M is much more reactive than the vinyl groups of inimer or polymer molecules to hyperstars in the opposite limiting case. 

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