First-order decay exponential

Possibly the most easily observable process is the phosphorescence decay.<17) While at 77°K the decay is exponential, at 1.6°K the observed decay for compounds like pyrazine is nonexponential and is composed of three first-order decays. Thus one can determine the values of kx, ky, and kz from this decay. Typical values are shown in Table 6.2. 

The parallel-replica method [5] is perhaps the least glamorous of the AMD methods, but is, in many cases, the most powerful. It is also the most accurate AMD method, assuming only first-order kinetics (exponential decay) i.e., for any trajectory that has been in a state long enough to have lost its memory of how it entered the state (longer than the correlation time icorr, the time after which the system is effectively sampling a stationary distribution restricted to the current state), the probability distribution function for the time of the next escape from that state is given by... 

Bakale et al. [397] pulse irradiated the hydrocarbons cyclopentane, cyclohexane and n-hexane with 0.9 MeV electrons of duration 10 or 100 ns. The transient conductivity decreased approximately exponentially with time for low doses of radiation. The first-order decay of the conductance is probably due to electrons reacting with impurities. With higher doses, the conductance decays approximately as inverse time, characteristic of a second-order recombination of free ions. No evidence for time-dependent geminate ion-pair recombination effects was observed. 

The ratio -ln[yp(r)]/T = 1 describes first-order decay that is unaffected by mass transport. When yp is calculated by Eq. 6 the ratio will not equal 1, and will express the deviation between the case of the measured first-order rate constant with flow and diffusion and the ideal case of no flow and diffusion. Figure 6 shows a plot of -ln[yp(r)]/T vs. z for the case when reaction zone at t = 0. The parameters are those from an investigation of the reaction flash photolysis of CF2ClBr in the presence of 02 and NO, where the reaction of CF2C102 radicals with N02 was studied [41]. For reference, rd = 0.1024 corresponds to a total pressure of 1 torr. Figure 6 clearly shows that at low pressures the deviation from exponential decay occurs at shorter times, z = kt, than at higher pressures. This is due to the pressure dependence of the diffusion coefficient. 

Among the methods utilizing the TST, the parallel replica method is the simplest and most accurate dynamics techniques, because the only assumption for the method is that of infrequent events obeying first-order kinetics (exponential decay). In practice, replicated MD simulations are started in a particular basin state in search for a transition to an adjacent basin state on M non-correlated processors. Whenever a transition is detected on any processor, the... 

The two random numbers, denoted r and r2, are randomly sampled from a uniform distribution over the interval [0,1], The time T, as dictated by any first-order decay process, is exponentially distributed ... 

Radioactive decay can be described as a first-order process. Thus, for any first-order decay process, the amount of material present declines in an exponential fashion with time. This is easy to see by integrating Equation (1.5.3) to give ... 

Such a chemical reaction, in which molecules are not colliding with other atoms or molecules, is called a first-order reaction because the rate at which chemical concentration changes at any instant in time is proportional to the concentration raised to the first power. Certain chemical processes, such as radioactive decay, are described by first-order kinetics. In the absence of any other sources of the chemical, first-order kinetics may lead to exponential decay or first-order decay of the chemical concentration (i.e., the concentration of the parent compound decreases exponentially with time) ... 

A decay transient for the 2-nitropropane radical anion, produced in the system described above, is shown in Fig. 19 together with the calculated transient. It can be seen that, for the low flow rate used, the transient is exponential, as predicted for the first-order decay, but at higher flow rates (or slower radical decay), one finds the transients are not perfectly exponential. The cause of this is convection of radical from the ESR cavity combining with the decay signal. This problem was shown to be easily overcome by a simple method of analysis [65] whereby the measured transients are treated as exponentials, as in eqn. (17), and an "apparent rate constant is deduced for the flow rates used. A plot of "apparent rate constant against u2/3 allowed extrapolation to zero flow rate and hence the deduction of the true value for the rate constant. Results obtained in this manner for the 2-nitropropane system were shown to be in agreement with the steady-state measurements. 

If enzyme inactivation is described by a one-step first-order mechanism, exponential decay of enzyme activity ensues, so that a straight line should be obtained in a semilog plot of residual activity versus time, as suggested by Eq. 3.126. Despite its limitation, this model has been used (and sometimes abused by forcing the data) to describe enzyme inactivation. It is quite frequent to observe behaviors that clearly depart from that simple model as revealed by semilog residual activity versus time profiles of the following type ... 

Figure 3.4 shows the first order decay of nd the exponential curve is typical of any simple radioactive decay process. A characteristic feature is that the time taken for the number of nuclides present at time t, A, to decrease to... 

Fig. 2.3. Upper panel First-order decay of the LIF signal from C2(a n.u) in the presence of 1.6 X 10 molecules cm of NO at 145 K in N2, fitted to a single-exponential decay, with residual shown below. The abscissa corresponds to the deiay time between the photolysis and probe laser pulses. Lower panel First-order decay constants for C2(a II ) at 145K in N2 plotted against the concentration of NO.

The exponential decay form is derived from assuming first order decay (ie. n=l) and integrating equation (1.5a) with =1 at t=0. The power law given by equation (1.7) is the integrated form of equation (1.5a) with the unrealistic limits of equal to infinity at zero contact time. A more adequate form of the general power law is given by Wojciechowski (1974) which is the integrated expression of equation (1.5a) with =1 at t=0 ... 

A second order reaction does not follow exponential decay and one cannot talk about time constants, except when first order conditions are imitated (see below for solution under condition (b)). We have to revert to the term half life, which differs by a factor of ln2 from the time constant of a first order reaction. For a first order decay the half life (note its relation to the time constant, see section 2.1), which is defined as the time taken for half completion of a reaction, is independent of the starting point. For a second order decay the half life is inversely proportional to the reactant concentration under the condition Ca(0) = Cb(0). 

In the case of many dynamical processes, the rate of change of a quantity is linearly proportional to the same quantity. Such processes include first-order decays in chemistry or radioactive decays in physics. The change of the quantity can then be described by an exponential function as shown in Eq. (6.1), and therefore the rate of change can be characterised by the time period needed to decrease the original quantity by e, where e is the basis of the natural logarithm having an approximate value of 2.71828. 

G(t) is often, but not necessarily, a simple function decaying exponentially to zero on either side of t = 0 with a time constant x, the correlation time for the motion under consideration. For example, random jumps of otherwise fixed interactions will give G(t) the same first-order decay as occurs with random radioactive fission of nuclei. Random angular diffusion will achieve the same. In the exponential case, for any frequency co,... 

Figure 20 contains a typical ESR transient for the 2-nitropropane system already discussed. It may be seen that the decay is approximately exponential, as might be predicted for first-order decay. In fact, the curve is not a perfect exponential because there is a contribution to the decay from convection of the radical out of the cavity. Obviously, this is most noticeable at fast flow rates and for slowly decaying radicals. Theory has been presented which describes the transient shape under any conditions.Conveniently, however, it was found that for all but the most stable radicals (fcj <5s ), a simple method of analysis was applicable. This involves treating the measured transients as if they were exponentials and deducing the effective first-order rate constants at the flow rates employed. These may then be extrapolated... 

Figure Al.5.3 shows that, as in interactions between other species, the first-order energy for Fte-Fle decays exponentially with interatomic distance. It can be fitted [70] within 0.6% by a fimction of the fonn... 

The conditions chosen make the reaction appear to be first-order overall, although the reaction is really not first-order overall, unlessjy and happen to be 2ero. If a simple exponential is actually observed over a reasonable extent (at least 90—95%) of decay the assumptions are considered vaUdated and is obtained with good precision. The pseudo-first-order rate constant is related to the k in the originally postulated rate law by... 

This equation demonstrates the exponential decay of the rate of formation of products in a first-order reaction widr time. When... 

Evidently simple first-order behavior is predicted, the reactant concentration decaying exponentially with time toward its equilibrium value. In this case a complicated differential rate equation leads to a simple integrated form. The experi-... 

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