Mean reaction time

The half-time (or half-life) of the reaction is independent of [A]o. The reciprocal of the rate constant, t = l/k, is referred to as the lifetime or the mean reaction time. In that time [A] falls to l/e of its initial value. The pharmaceutical industry refers to the shelf life or t90, the time at which [A]/[A]o reaches 0.90. Both t and t90 are also independent of [A]o. 

Most interest focuses on very fast reactions. This includes systems whose mean reaction times range from roughly 1 minute to 10 14 second. Reactions that involve bond making or breaking are not likely to occur in less than 10 13 second, since this is the scale of molecular vibrations. Some unimolecular electron transfer events may, however, occur more rapidly. 

The mean reaction time was around 5.5-6 s, which did not differ significantly from the control group. Hence it was inferred that these compounds did not show good analgesic activity. 

In this example, there are two reactors in which reactions 1, 2 and 3 can be performed. Equal mean reaction times for the different reactions in each of the reactors imply similar performances for the reactors. The overall process consists of four units, i.e. heater, reactor 1, reactor 2 and separator. In order to handle the usage of feed C in two distinct reactions, i.e. reactions 1 and 3, different states were assigned to each of the streams of feed C, i.e. states, v3 and s4, respectively. In this example, scheduling is performed over an 8-h time horizon. It should be noted that reactors 1 and 2 are suitable for performing reactions 1, 2 and 3, which implies that constraint (2.13) is crucial. Constraints that exhibit similar structure to those presented in example 1 are not repeated. 

The overall reaction rate r may be defined as the reciprocal of this mean reaction time. 

For the kinetics of a reaction, it is critical to know the rough time to reach equilibrium. Often the term "mean reaction time," or "reaction timescale," or "relaxation timescale" is used. These terms all mean the same, the time it takes for the reactant concentration to change from the initial value to 1/e toward the final (equilibrium) value. For unidirectional reactions, half-life is often used to characterize the time to reach the final state, and it means the time for the reactant concentration to decrease to half of the initial value. For some reactions or processes, these times are short, meaning that the equilibrium state is easy to reach. Examples of rapid reactions include H2O + OH (timescale < 67 /is at... 

The mean reaction time or reaction timescale (also called relaxation timescale relaxation denotes the return of a system to equilibrium) is another characteristic time for a reaction. Roughly, the mean reaction time is the time it takes for the concentration to change from the initial value to 1/e toward the final (equilibrium) value. The mean reaction time is often denoted as x (or Xr where subscript "r" stands for reaction). The rigorous definition of x is through the following equation (Scherer, 1986 Zhang, 1994) ... 

The above simple formula is one of the reasons why some authors prefer the use of the mean reaction time (or relaxation timescale) instead of the half-life. The mean reaction time is longer than the half-life. 

The mean reaction time during a reaction varies as the concentration varies if the reaction is not a first-order reaction. Expressions of mean reaction time of various types of reactions are listed in Table 1-2. In practice, half-lives are often used in treating radioactive decay reactions, and mean reaction times are often used in treating reversible chemical reactions. 

In general, for unidirectional elementary reactions, it is easy to handle non-isothermal reaction kinetics. The solutions listed in Table 1-2 for the concentration evolution of elementary reactions can be readily extended to nonisothermal reactions by replacing kt with a = j k df. The concepts of half-life and mean reaction time are not useful anymore for nonisothermal reactions. 

The concentration evolution curves of Figures 2-la and 2-lb may be used to estimate the half-life or mean reaction time. When Figures 2-la and 2-lb are compared, the mean reaction time is found to differ by four orders of magnitude Hence, for second-order reactions, the timescale to reach equilibrium in general depends on the initial conditions. This is in contrast to the case of first-order reactions, in which the timescale to reach equilibrium is independent of the initial conditions. 

The method to find the mean time to reach equilibrium without calculating the full concentration evolution curve is as follows. By comparing the definition of mean reaction time (Equation 1-60) d /dt=—( - ao)hr and Equation 2-17, the following is obtained ... 

Therefore, given the initial conditions, first find z, then the mean reaction time as a function of can be calculated. According to Vieta s formulas, the two roots of Equation 2-18 satisfy... 

Hence, as long as the equilibrium concentration is known, the mean reaction time can be calculated at any given species concentration. Although k does not appear in the above equation, it does not indicate that the mean reaction time is independent of k because the equilibrium concentration depends on the equilibrium constant that equals kf/k. ... 

In summary, because the forward reaction rate constant is similar to the back reaction rate constant, the concentration evolution of a second-order isotopic exchange reaction often reduces to that of a first-order reaction (exponential evolution) but the rate "constants" and mean reaction time for the reduced reaction depend on total concentrations. 

The mean reaction time for path 2 is hence 1/kgf x 23 s. The total rate for HC03 production is the combination of the two paths. By comparing the rates of two paths, the dominant path can be inferred. 

Given measured species concentrations for a homogeneous reaction in a rock, cooling rate at Tae can be found as follows if the equilibrium constant K and the forward reaction rate coefficient k as a function of temperature are known. First, the apparent equilibrium temperature is calculated from the species concentrations. Then kf and kb at Tae are calculated. Then the mean reaction time Xr at Tae is calculated using expressions in Table 2-1. From x, the cooling rate q at Tae can be obtained using Equation 5-125. Two examples are given below. 

Solution From Table 2-1, the reaction timescale for reversible first-order reaction is Xr= l/(kf+kb). Hence, knowing Tae = 857.93 K, we find the mean reaction time Xr at Tae as follows ... 

March et al. found that, as a result of the participation of reaction (11), the apparent value of klb decreased with reaction time, and they made an assumed linear extrapolation of a log klb-r plot (r is the mean reaction time) to r = 0 to estimate an absolute value for k7b 4 x 10 liter... 

The mean reaction time is very short, therefore only very small a- and //-crystallites (about 20 nm) and amorphous Si3N4 are formed. By an additional heat treatment fine a- or /1-rich powders can be produced, which allow the production of Si3N4 ceramics with mean grain sizes of about 100 nm [219],... 

Figure 3 Mean PVT reaction times (msec) and false starts (errors of commission) during 88 hr of total sleep deprivation and 88 hours of sleep deprivation with two 2-hour nap opportunities each day. Subjects in the total sleep deprivation (TSD) group (n = 13) are represented by the open circles. Subjects in the 88-hr sleep deprivation plus two 2-hr nap opportunities (NAP) group (n = 15) are represented by the closed squares. Nap opportunity periods were at 02 45-04 45 and 14 45-16 45 each day. The top panel illustrates mean reaction times ( s.e.m.) for each test bout across the experimental protocol. Subjects in the NAP group demonstrated little variation in reaction times across the experimental period, while subjects in the TSD group experienced significant impairment in performance, reflected in the increasing reaction times as time awake increased, with circadian variation in performance capability evident. The bottom panel illustrates mean number of errors ( s.e.m.) per test bout across the experimental protocol. A similar pattern of performance degradation in this variable was evident for both the NAP and TSD groups. (From Ref. 44.)...

The principle behind such terminology lies in the following method of analysis. In a system of consecutive reaction steps where reactant passes through a number of intermediate stages, the total time to produce a molecule of product is simply the sums of the discrete times necessary to pass through each consecutive stage of the reaction. The mean reaction time tp is thus ... 

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

Quite interestingly, the above argument gives also the rate of the reaction kAB in terms of Zab and the mean reaction time tAB defined as... 

Equivalently, we could use q- x) instead of g+(x) in this expression.) Since the derivation of (43) from (42) involves a few technical steps we postpone it till the end of this section. To the best of our knowledge, this expression was first given in [14]. It provides an exact alternative to the expression for the reaction rate given by TST and the reactive flux formalism (see Sect. 5). Finally, notice that (11) and (43) can be combined with (16) to give the following expression for the mean reaction time tAB defined in (14) ... 

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