Steady-state conditions decay

If Xi and A,2are real numbers and both have negative values, the values of the exponential terms and hence the magnitudes of the perturbations away from the steady-state conditions, c, and T, will reduce to zero, with increasing time. The system response will therefore decay back to its original steady-state value, which is therefore a stable steady-state solution or stable node. 

When the transient effect is negligible, k can then be determined by measuring the luminescence intensities under steady state conditions and the lifetimes of the decay in both the absence and the presence of a scavenger at concentration c. Indicating the intensities by Ig and I, respectively, and the... 

Secular Equilibrium A condition that occurs when a chain of radionuclides has reached a steady state condition, in which the rate of decay of daughter nuclides is balanced by their rate of formation by decay of each parent. In this condition, the radioactivity (measured in disintigrations per minute) of each radionuclide in a chain is the same. 

Compare Eq. 3-229 with 3-224. The decay in monomer concentration depends on the orders of both initiator and activator initial concentrations with no dependence on deactivator concentration and varies with t2/3 under non-steady-state conditions. For steady-state conditions, there are first-order dependencies on initiator and activator and inverse first-order dependence on deactivator and the time dependence is linear. Note that Eq. 3-229 describes the non-steady-state polymerization rate in terms of initial concentrations of initiator and activator. Equation 3-224 describes the steady-state polymerization rate in terms of concentrations at any point in the reaction as long as only short reaction intervals are considered so that concentration changes are small. 

In some studies it is desirable to do constant infusion to achieve a steady state or equilibrium condition which is a function of input, extraction rate, tissue washout, and radioactive decay (23). Figure 6 shows the yield of Rb-82 at various elution rates to a steady-state condition. At the faster flow rate of 5.33 ml/min, there is 24% yield of Rb-82 and at the slower flow rate of 2.15 ml/min there is about 1% yield of Rb-82. The lower yield at the slower flow rate is mostly accounted for in decay during transit through the line to the patient. 

MAS NMR experiments characterizing catalysts in reaction environments in flow systems may be carried out under conditions close to those of industrial processes. The formation of catalytically active surface species and the cause of the deactivation of catalysts can be characterized best under flow conditions. When flow techniques are used for the investigation of reactions under steady-state conditions, a continuous formation and transformation of intermediates occurs. The lifetime of the species under study must be of the order of the length of the free-induction decay, which is ca. 100 ms for " C MAS NMR spectroscopy. 

The diffusion-controlled mechanism of triplet-triplet quenching (whether it be collisional or interaction at a distance) could still be applied to interpret these results, qualitatively at least, if one additional factor is taken into account. The additional factor is the extremely slow rate of diffusion in rigid medium which makes necessary the Consideration of nonsteady-state effects. Immediately after illumination is shut off, those triplet molecules situated close together will diffuse and interact more rapidly than others situated at greater distances. The more favorably situated pairs of triplets will thus be depleted more rapidly and the overall rate of interaction will be greater at shorter times than later when steady-state conditions will ultimately be approached. In fluid solvents at room temperature the steady state is reached after about 10-7 sec. In very highly viscous media, however, much longer times are required and this could explain the non-exponential decay observed with phenanthrene in EPA at 77°K. 

Suppose that one species (e.g., R) of a redox couple is subjected to a chemical reaction leading to nonelectroactive products. If this reaction occurs during a limiting steady-state condition (Fig. 3.36), i ism will decay until both R and O are completely exhausted. The rate constant for the coupled reaction can be calculated from the decay curve [59]. 

Increasing the solvent polarity results in a red shift in the -t -amine exciplex fluorescence and a decrease in its lifetime and intensity (113), no fluorescence being detected in solvents more polar than tetrahydrofuran (e = 7.6). The decrease in fluorescence intensity is accompanied by ionic dissociation to yield the t-17 and the R3N" free radical ions (116) and proton transfer leading to product formation (see Section IV-B). The formation and decay of t-17 have been investigated by means of time resolved resonance Raman (TR ) spectroscopy (116). Both the TR spectrum and its excitation spectrum are similar to those obtained under steady state conditions. The initial yield of t-1 is dependent upon the amine structure due to competition between ionic dissociation and other radical ion pair processes (proton transfer, intersystem crossing, and quenching by ground state amine), which are dependent upon amine structure. However, the second order decay of t-1" is independent of amine structure... 

Half-life is useful because it indicates the time required to attain 50% of steady state—or to decay 50% from steady-state conditions—after a change in the rate of drug administration. Figure 3-3... 

Assuming that the transport (e.g., physical mixing and bioturbation), depositional inputs (e.g., sources/quality and amounts of POM), mass sediment accumulation rate, temperature, and decomposition are in steady state, as well as lateral homogeneity of the deposit, the concentration of POM will not change over time. Although these conditions are almost never met in estuarine systems (Berner, 1980), steady-state conditions will be used here for the general purposes of this discussion. Thus, assuming steady-state conditions, the GDE for POM decay is as follows (Rice and Rhoads, 1989) ... 

Here rm is the lifetime for monomolecular decay and g the efficiency of generation of the photoinduced species. The bimolecular decay constant (3 determines the intensity of the PIA signal via (1.13), where a = n/uTb and Tb — (gI/3) 05, the bimolecular lifetime under steady-state conditions. It is important to note that the bimolecular lifetime T > depends on experimental conditions such as concentration and pump beam intensity. 

Actually, the apparent rate constant of the photoisomerisation in a viscous media, like biological membranes, was found to be dependent upon the medium relaxation rate. Hence, it is possible to study the dynamics of proteins and biological membranes in the vicinity of the incorporated stilbene probe by monitoring the steady-state fluorescence decay of the stilbene probe with the conventional constant-illumination spectrofluorimeter. The experimental values of a (j>n Icx can be measured independently or can be omitted by comparison with photoisomerisation kinetics of the same probe and similar conditions in a medium with known macro- and microviscosity. A combined analysis of the trans-cis photoisomerisation kinetics of a stilbene probe and its polarization allows the establishment of the mechanism and the estimation of the... 

For an idealized case (perfectly homogeneous magnet, low transverse fields, steady-state conditions, no saturation effects), an NMR absorption line in the frequency domain or the Fourier transform of a free induction decay has a Lorentzian profile ... 

From these rate equations, a number of experimental observables can be simulated. Figure 6 shows the time-dependent upconversion decay curves calculated for a set of hypothetical three-level systems involving (a, b) pure ESA, (c, d) pure ETU, and (e, f) mixed ESA ETU (40 60) rates under steady-state conditions, calculated to simulate the two experimental observables of decay following short pulses (a, c, e), as described above, and decay following long square-wave pulses during which the system has reached steady state (b, d, f). 

The technique of microwave-recovery provides crucial information about the substates involved in the ODMR transitions. For this experiment, Pd(2-thpy)2 is optically excited by a c. w. source. This leads to specific populations of the three triplet substates. At low temperature, they are thermally decoupled and thus emit according to their specific populations and their individual decay constants (e. g. see Sect. 3.1.3 and Table 2). In the microwave recovery experiment, the steady state conditions are perturbed by a microwave pulse being in resonance with the zero-field transition at 2886 MHz. Due to the microwave pulse, the populations of the two states involved are changed. Subsequently, one monitors the recovery of the emission intensity in time until the steady state situation is reached again. The microwave pulses have, for example, a duration of 20 ps and are applied repeatedly to enable a detection with signal averaging [61]. 

At steady-state conditions, CO can be replaced by CO while maintaining all other process parameters (e.g., temperature, flow rate) constant. The outlet from the reactor can be continuously monitored by mass spectroscopy. The decay of the concentration of CFLj and the increase in the concentration of CH4 can provide... 

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