Time constant apparent

The time constant R /D, and hence the diffusivity, may thus be found dkecdy from the uptake curve. However, it is important to confirm by experiment that the basic assumptions of the model are fulfilled, since intmsions of thermal effects or extraparticle resistance to mass transfer may easily occur, leading to erroneously low apparent diffusivity values. 

For example, a temperature-measuring device, having its sensor placed in a protecting rube, is a system of second order. For such a system no single rime constant exists in the same way as a first-order system. The behavior of such a system is often given by a response time. Another concept is to give the apparent time constant t, which can be constructed by placing a line in the inflection point of the step response curve see Fig. 12.14. 

Figure 4 shows one of the better results obtained through the use of the control scheme illustrated in Figure 2. It is apparent that the large fluctuations in S shown in Figure 3 have been effectively eliminated, although the total transient time has increased. The time constant introduced at A for this case was 83 minutes.

As already indicated, a physical description in terms of a process with a single time constant is fair but an assessment on the basis of two time-scales gives even improved results. Therefore, more research has to be carried out to determine the characteristics of the most important additional phenomena. As an example a two time-scale model is applied to the previously reported measurements of Fig. 8.3 and displayed in Fig. 8.4. Clearly, the release is governed by two rates, typically a smaller and a larger time scale appear compared with the single rate case. However, the single rate results are still very valuable because they describe the apparent rate very well and this would be the only thing that can be described in coarse scale models of devolatilization, e.g., in CFD of biomass conversion. 

The case of several populations of fluorophores having their own fluorescence decay i (t) and time constants characterizing r (t) deserves particular attention. In Section 5.3, it was concluded that an apparent or a technical emission anisotropy r(t) can be obtained by considering that the measured polarized components, I(t) and I (t), are the sums of the individual components (i.e. of each population) and by using Eq. (6.43). Hence... 

The binding kinetics were characterized in terms of the apparent time constant (K pp = kf C + k ) where C = analyte concentration kf = association rate constant and k = dissociation rate constant. In closed loop experiments, a plateau value for K pp of 0.0024/s was reached at a linear flow rate of 2.67 mL/min. K ppWas foxmd to decrease with decreasing antigen concentration (C), with equilibrium achieved only at the highest level (1 pg/mL). The association rate constant Kf was calculated at 3.6 x 10 M/s for IgG binding. 

It has been suggested that P BChl (where BChl is one of the two monomeric or "accessory BChls that are not part of P) is a transient state prior to P "I (14,16,19), although the evidence supporting this view has been criticized (23, 24) Recent subpicosecond studies find no evidence for P "BChl (8,9) These new results do not preclude some involvement of a monomeric BChl in the early photochemistry, only that P BChl apparently is not a kinetically resolved transient state Perhaps P itself contains some charge-transfer character between its component BChls, or between P and one or both of the monomeric BChls (8,9,25-27) One of the two monomeric BChls apparently can be removed by treatment of the reaction center with sodium borohydride (28) and subsequent chromatography, with no impairment of the primary electron transfer reactions (29) Thus, at present it appears that P I is the first resolved radical-pair state, and it forms with a time constant of about 4 ps in Rps sphaeroides ... 

The effect of the detector time constant on the apparent efficiency depends only on the time width of the bands. It has been shown by Sch-mauch 41) and by Me William and Bolton 42) that the profile recorded with a detector having a time constant r is wider than the actual profile by a factor (1 -f r/ert), where is the time standard deviation of the profile, provided this factor is less than about 1.2. Moreover, the peak heigh becomes smaller although the peak area remains unchanged. 1 he (list mu ment (retention time) of a peak increases by r and the retention time of the... 

The optical rotation of the mixture approaches zero (a racemic mixture) over time, with apparent first-order kinetics. This observation was supported by the semi-log plot [ln(a°D/ aD) vs time], which is linear (Figure 1). It has been shown that this racemization process does in fact follow a true pseudo-first-order rate equation, the details of which have been described by Eliel.t30 Therefore, these processes can be described by the first-order rate constant associated with them, which reflects precisely the intrinsic rate of racemization. Comparison of the half-lives for racemization under conditions of varying amino acid side chain, base, and solvent is the basis for this new general method. 

Extrapolation to zero time of apparent rate constant 26... 

Where T0 is the initial temperature of the depolarization scan. It is assumed that the relaxation time constant r is related to the barrier height or apparent activation energy Ea in the Arrhenius equation. 

The detector time constant is the response time of the detector to the signal passing through it. A slower time constant will result in less apparent noise, but it will compromise signal and also resolution for closely eluting peaks. For closely eluting peaks, therefore, it is important to use a faster time constant. If there is plenty of resolution, a slower time constant will provide a smoother baseline. The effect of the time constant is illustrated in Figure 8.6. 

These lags are cumulative as the liquid passes each tray on its way down the column. Thus, a 30-tray column could be approximated by 30 first-order exponential lags in series having approximately the same time constant. The effect of increasing the number of lags in series is to increase the apparent dead time and increase the response curve slope. Thus, the liquid traffic within the distillation process is often approximated by a second-order lag plus dead time (right side of Figure 2.82). 

The general equations for chemical reaction in a turbulent medium are easy to write if not to solve (2). In addition to those for velocities (U = U + uJ and concentrations (Cj = Cj + Cj), balance equations for q = A u, the segregation ( , and the dissipations e and eg can be written (3). Whatever the shape of the reactor under consideration (usually a tube or a stirred tank), the solution of these equations poses difficult problems of closure, as u S, 5 cj, cj, and also c cj, c Cj in the reaction terms have to be evaluated. The situation is even more complicated when the temperature and the density of the reacting mixture are also fluctuating. Partial solutions to this problem have been proposed. In the case of instantaneous reactions (t << Tg) the "e-quilibrium assumption" applies the mixed reactants are immediately converted and the apparent rate of reaction is simply that of the decrease of segregation, with Corrsin s time constant xs. For instance, with a stoichiometric proportion of reactants, the extent of reaction X is given by 1 - /T ( 2), a simple result which can also be found by application of the IEM model (see (33)). 

Figure 7. Time-resolved mass spectrometry. AU-trcms-(2, 4, 6, 8) decatetraene was excited to its 5 2 electronic origin with a femtosecond pulse at A-pump — 287 nm. The excited-state evolution was probed via single-photon ionization using a femtosecond pulse at ApIObe = 235 nm. The time resolution in these experiments was 290 fs (0.3 ps). The parent ion CioH signal rises with the pump laser, but then seems to stay almost constant with time. The modest decay observed can be fit with a single exponential time constant of 1 ps. Note that this result is in apparent disagreement with the same experiment performed at Xprobe — 352 nm, which yields a lifetime of 0.4 ps for the S2 state. The disagreement between these two results can be only reconciled by analyzing the time-resolved photoelectron spectrum.

A time dependent apparent decay constant can be calculated to be... 

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