Steady-state approximation relaxation time

If a reaction system consists of more than one elementary reversible reaction, there will be more than one relaxation time in general, the number of relaxation times is equal to the number of states of the system minus one. (However, even for multistep reactions, only a single relaxation time will be observed if all intermediates are present at vanishingly low concentrations, that is, if the steady-state approximation is valid.) The relaxation times are coupled, in that each relaxation time includes contributions from all of the system rate constants. A system of more than... 

Molecular rotors are useful as reporters of their microenvironment, because their fluorescence emission allows to probe TICT formation and solvent interaction. Measurements are possible through steady-state spectroscopy and time-resolved spectroscopy. Three primary effects were identified in Sect. 2, namely, the solvent-dependent reorientation rate, the solvent-dependent quantum yield (which directly links to the reorientation rate), and the solvatochromic shift. Most commonly, molecular rotors exhibit a change in quantum yield as a consequence of nonradia-tive relaxation. Therefore, the fluorophore s quantum yield needs to be determined as accurately as possible. In steady-state spectroscopy, emission intensity can be calibrated with quantum yield standards. Alternatively, relative changes in emission intensity can be used, because the ratio of two intensities is identical to the ratio of the corresponding quantum yields if the fluid optical properties remain constant. For molecular rotors with nonradiative relaxation, the calibrated measurement of the quantum yield allows to approximately compute the rotational relaxation rate kor from the measured quantum yield [Pg.284]

These evaluations are made within the context of the two level model and the steady state approximation. The steady-state approximation is probably valid for this experiment. C is not (electronically, vibronically, or rotationally) a two level system. Other groups, particularly Daily (23) and Berg and Shackleford (18) have developed expressions which allow for the inclusion of more levels and provide for incomplete relaxation. Lucht and Laurendeau (28) have carefully considered the effect of rotational equilibration. There is not time here to discuss these models in detail. The theoretical models which include specifically more than two electronic levels require experimental measurements independently of the radiation coupling the various levels. We have not found a system experimentally tractible for testing the three electronic level model. 

The steady-state approximation applies only after a time t, the relaxation time. The relaxation time is the time required for the steady-state concentration of the reactive intermediates to be approached. Past the relaxation time, the steady-state approximation remains an approximation, but it is normally satisfactory. Below, a more quantitative description of the relaxation time is described. ... 

In the previous section the steady-state approximation was defined and illustrated. It was shown that this approximation is valid after a certain relaxation time that is a characteristic of the particular system under investigation. By perturbing the system and observing the recovery time, information concerning the kinetic parameters of the reaction sequence can be obtained. For example, with A > B ---> C, it was shown that the relaxation time when i 2 was Thus, relaxation methods can be very useful in determining the kinetic parameters of a particular sequence. 

As discussed in the text, the steady-state approximation applies only after a relaxation time associated with the reactive intermediates. Plot the time dependence of e (the deviation intermediate concentration from the steady-state value) for several values of, / 2 (0-5, 0.1, 0.05) and 1 2 = 0.1 What happens when (a) ki = k2 and (b) k > 2 ... 

Of course, the steady-state approximation applies only after a time t, called the relaxation time, necessary for the steady-state concentration of the... 

Even past the relaxation time, the steady-state approximation remains an approximation but it will normally be completely satisfactory. To make these remarks more quantitative, it is of interest to inquire further into the nature of the approximation by returning to the case of the simple sequence A B C. 

In general, as formulated in Rule I of Chapter 1, the rate of reaction decreases as the extent of reaction increases. In the case of more than one single reaction, exceptions to this statement can occur as a result of competition for active centers (see Section 5.2). But even when the reaction is stoichiometri-cally simple, such an acceleration of the rate is possible if the concentration of active centers builds up as reaction proceeds. Then the steady-state approximation apparently fails There does not seem to exist a short relaxation time beyond which the concentration of active centers ceases to depend explicitly on time (see p. 67). 

When condition (dai/dt)react (dai/dt)rei is not satisfied, the perturbation of equilibrium distribution is substantial. However, in this case realization of the quasi-steady-state condition is possible, provided the overall rate dai/dt is low compared to partial rates (daj/dt)rei and (dai/dt)i.eact Then, the microscopic kinetic equation can be solved by the quasi-steady-state approximation. The approximation implies that the non-equilibrium distribution functions depend on time implicitly via the total concentration of reactants rather than explicitly. This also means that the macroscopic reaction rates are low compared to microscopic reaction and relaxation rates. Since the distribution functions in this approximation depend on the total concentration only, the reaction rates, according to Eq. (8.50) also depend on the total concentration. Hence, we come to macroscopic kinetic equations that involve only the total concentration of reactants and certain combinations of microscopic rate constants that have the meaning of macroscopic constants. Note that these macroscopic equations need not be consistent with the macroscopic kinetic law as, besides elementary reactive processes, they involve unreactive processes. 

The quasi-steady-state approximation for non-equilibrium reactions is not valid from the very start of the reaction but after a time interval greater than the longest relaxation time of reactants. If only a small fraction of molecules has reacted during this interval, it can be disregarded and the macroscopic kinetic equations can be considered as valid from the reaction start. This yields the basic limitation for the validity of macroscopic equations the characteristic reaction time Tj-eact iTiust be longer than the relaxation time of the slowest relaxation process Trei- When this condition does not hold, the kinetic equations involving only total concentrations cannot be derived. This is just the case for the overlapping of the relaxation and reaction. Examples are provided by fast reactions in shock waves and plasmochemical reactions [87, 202, 369, 370, 472]. 

Droplet suspensions (gas-liquid, two-component system) Since the inertia of a liquid suspended in the gas phase is higher than the inertia of the gas, the time for the displacement of liquid under the pressure waves should be considered. Temkin (1966) proposed a model to account for the response of suspension with pressure and temperature changes by considering the suspensions to move with the pressure waves according to the Stokes s law. The oscillatory state equation is thereby approximated by a steady-state equation with the oscillatory terms neglected, which is valid if the ratio of the relaxation time to the wave period is small, or... 

After the electrode reaction starts at a potential close to E°, the concentrations of both O and R in a thin layer of solution next to the electrode become different from those in the bulk, cQ and cR. This layer is known as the diffusion layer. Beyond the diffusion layer, the solution is maintained uniform by natural or forced convection. When the reaction continues, the diffusion layer s thickness, /, increases with time until it reaches a steady-state value. This behaviour is also known as the relaxation process and accounts for many features of a voltammogram. Besides the electrode potential, equations (A.3) and (A.4) show that the electrode current output is proportional to the concentration gradient dcourfa /dx or dcRrface/dx. If the concentration distribution in the diffusion layer is almost linear, which is true under a steady state, these gradients can be qualitatively approximated by equation (A.5). 

The iFi terms are the fluorescence lifetimes of fractional contributions a, and the xRJ indicate decay constants due to solvent relaxation (or other excited-state processes) of fractional contribution Pj. The negative sign is indicative of a relaxation process (red shift). Usually, the relaxation process is approximated to a single relaxation time x R by assuming an initial excited state and a final fully relaxed state (see, e.g., Ref. 128). A steady-state fluo-... 

The steady-state reaction rate and relaxation time are determined by these two constants. In that case their effects are coupled. For the steady state we get in first-order approximation instead of Equation (13) ... 

The experiments discussed above were all carried out with total pressures below 10-4 Torr. However, Hori and Schmidt (187) have also reported non-stationary state experiments for total pressures of approximately 1 Torr in which the temperature of a Pt wire immersed in a CO—02 mixture was suddenly increased to a new value within a second. The rate of C02 production relaxed to a steady-state value characteristic of the higher temperature with three different characteristic relaxation times that are temperature dependent and vary between 3 and 100 seconds between 600 and 1500 K. The extremely long relaxation time compared with the inverse gas phase collision rate rule out an explanation based on changes within the chemisorption layer since this would require unreasonably small sticking coefficients or reaction probabilities of less than 10-6. The authors attribute the relaxation times to characteristic changes of surface multilayers composed of Pt, CO, and O. The effects are due to phases that are only formed at high pressures and, therefore, cannot be compared to the other experiments described here. 

In Eq. (5) r defines the dielectric relaxation time (r = e/cr) according to which obviously a charge perturbation decays exponentially in a conductor. This defines a parallel R-C circuit as a good approximation of a homogeneous conductor (see Section III). In the following part of this section we consider the steady state, in which the conduction current represents the total current and capacitive contributions have vanished. 

Two other paradigms demonstrate that the band width of relaxation times is extremely broad the time required to achieve a new steady state in a chemostat culture is approximately determined by (pmax-H) 1. The time required by a culture to consume a considerable fraction of small amounts of residual... 

Problem 3,9 Integrate the Doi-Edwards equation (3-71) using the Currie expression for the Q tensor, Eq. (3-75), for steady-state shearing, for yr = 0.1, 0.3, 1.0, 3.0, and 10.0, using only one relaxation time in the spectrum. Plot the values of dimensionless shear stress OnjG versus yr on the same plot as in Problem 3.8. How close is the prediction of the approximate differential model to that of the exact integral model ... 

J. Relaxed steady-state or sliding regime (T rj. When the input varies rapidly relative to the characteristic response time, the state oscillates with a very small amplitude. The quasi-steady-stale approximation can be applied to the state using the time-averaged value of the control. The performance of the system can be predicted using the performance in comparable steady-state operation. 

The rates kj and apply at times sufliciently short that the equilibrium population in the vicinity of Rj is maintained. At long times this is depleted because the configurational relaxation of the end segments is slow. A steady-state rate is eventually attained, with a rate constant approximated by... 

For nonlinear elliptic partial differential equations, successive relaxation or finite difference approximations can be used in both the coordinates.[7] [12] [13] (Constantinides Mostoufi, 1999 Davis, 1984, Finlayson, 1980) As illustrated by Schiesser (1991),[2] a method of lines was used for 2D and 3D steady state problems by adding a pseudo time derivative, applying finite differences in all the... 

The fluctuating variables aie thereby projected onto pair-density fluctuations, whose time-dependence follows from that of the transient density correlators q(,)(z), defined in (12). Tliese describe the relaxation (caused by shear, interactions and Brownian motion) of density fluctuations with equilibrium amplitudes. Higher order density averages are factorized into products of these correlators, and the reduced dynamics containing the projector Q is replaced by the full dynamics. The entire procedure is written in terms of equilibrium averages, which can then be used to compute nonequilibrium steady states via the ITT procedure. The normalization in (10a) is given by the equilibrium structure factors such that the pair density correlator with reduced dynamics, which does not couple linearly to density fluctuations, becomes approximated to ... 

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