Relaxation times steady-state parameters

In Table IV we list the temperature-dependent steady-state parameters static permittivity 8S density p Debye relaxation time td second-Debye relaxation time tD2 the contribution Aes to 8S due to fast vibrating HB molecules and the dipole-moment factor k involved in Eq. (3). 

Various theoretical and empirical models have been derived expressing either charge density or charging current in terms of flow characteristics such as pipe diameter d (m) and flow velocity v (m/s). Liquid dielectric and physical properties appear in more complex models. The application of theoretical models is often limited by the nonavailability or inaccuracy of parameters needed to solve the equations. Empirical models are adequate in most cases. For turbulent flow of nonconductive liquid through a given pipe under conditions where the residence time is long compared with the relaxation time, it is found that the volumetric charge density Qy attains a steady-state value which is directly proportional to flow velocity... 

Then let us examine the rate relaxation time constant x, defined as the time required for the rate increase Ar to reach 63% of its steady state value. It is comparable, and this is a general observation, with the parameter 2FNq/I, (Fig. 4.13). This is the time required to form a monolayer of oxygen on a surface with Nq sites when oxygen is supplied in the form of 02 This observation provided the first evidence that NEMCA is due to an electrochemically controlled migration of ionic species from the solid electrolyte onto the catalyst surface,1,4,49 as proven in detail in Chapter 5 (section 5.2), where the same transient is viewed through the use of surface sensitive techniques. 

A number of parameters have to be chosen when recording 2D NMR spectra (a) the pulse sequence to be used, which depends on the experiment required to be conducted, (b) the pulse lengths and the delays in the pulse sequence, (c) the spectral widths SW, and SW2 to be used for Fj and Fi, (d) the number of data points or time increments that define t, and t-i, (e) the number of transients for each value of t, (f) the relaxation delay between each set of pulses that allows an equilibrium state to be reached, and (g) the number of preparatory dummy transients (DS) per FID required for the establishment of the steady state for each FID. Table 3.1 summarizes some important acquisition parameters for 2D NMR experiments. 

The steady state is disturbed and the system exhibits transient behavior when at least one of its parameters is altered under an external stimulus (perturbation). Transitory processes that adjust the other parameters set in (response) and at the end produce a new steady state. The time of adjustment (transition time, relaxation time) is an important characteristic of the system. 

Hammes (4S6) has summarized some of the extensive studies from his laboratory on the interaction of a variety of nucleotides with RNase-A as seen by relaxation kinetic measurements. The bimolecular and isomerization steps that occur with each of the nucleotides are very much faster than the rate determining steps separating the different substances. Thus the kinetic parameters for the interaction of each nucleotide can be established separately and then combined with steady state kinetic data to provide a detailed kinetic picture. The bimolecular steps are recognized by the concentration dependence of the relaxation time and the isomerization steps by the lack of a concentration dependence. 

First, it is of common interest to unsteady processes and their models. Chemical unsteadiness must be taken into account in many cases. For example, studies with variations in catalyst activity, calculations of fluidized catalyst bed processes (when the catalyst grain "is shaking in a flow of the reaction mixture and has no time to attain its steady state), analyses of relaxational non-stationary processes and problems of control. Unsteady state technology is currently under development [14,15], i.e. the technology involving programmed variation of the process parameters (temperature, flow rate, concentration). The development of this technology is impossible without distinct interpretation of the unsteady reaction behaviour. 

In several experiments, in particular the study by Temkin and co-workers [224] of the kinetics in ethylene oxidation, slow relaxations, i.e. the extremely slow achievement of a steady-state reaction rate, were found. As a rule, the existence of such slow relaxations is ascribed to some "side reasons rather than to the purely kinetic ("proper ) factors. The terms "proper and "side were first introduced by Temkin [225], As usual, we classify as slow "side processes variations in the chemical or phase composition of the surface under the effect of reaction media, catalyst deactivation, substance diffusion into its bulk, etc. These processes are usually considered to require significantly longer times to achieve a steady state compared with those characterizing the performance of chemical reactions. The above numerical experiment, however, shows that, when the system parameters attain their bifurcation values, the time to achieve a steady state, tr, sharply increases. 

Since the total concentration a + r + s follows the time evolution d(a + r + s)/ dt = F - k(a + r + s), it approaches the steady state value F/k with a relaxation time 1 /k. This is a consequence of unbiased outflow (Eq. 47) of all reactants with the same rate k. Consequently, even though we are dealing with an open system under a flow, the analysis is similar to the closed system by replacing the total concentration c with the steady state value F/k. Instead of recycling, therefore, constant supply of the substrate allows the system to reach a certain fixed point with a definite value of the order parameter 0i, independent of the initial condition. 

Here, w = m, n, and S. V represents the membrane potential, n is the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. Ic and Ik are the calcium and potassium currents, gca = 3.6 and gx = 10.0 are the associated conductances, and Vca = 25 mV and Vk = -75 mV are the respective Nernst (or reversal) potentials. The ratio r/r s defines the relation between the fast (V and n) and the slow (S) time scales. The time constant for the membrane potential is determined by the capacitance and typical conductance of the cell membrane. With r = 0.02 s and ts = 35 s, the ratio ks = r/r s is quite small, and the cell model is numerically stiff. The calcium current Ica is assumed to adjust immediately to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards the voltage-dependent steady-state values noo (V) and S00 (V). Together with the ratio ks of the fast to the slow time constant, Vs is used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady-state value for the gating variable S attains one-half of its maximum value. The other parameters are assumed to take the following values gs = 4.0, Vm = -20 mV, Vn = -16 mV, 9m = 12 mV, 9n = 5.6 mV, 9s = 10 mV, and a = 0.85. These values are all adjusted to fit experimentally observed relationships. In accordance with the formulation used by Sherman et al. [53], there is no capacitance in Eq. (6), and all the conductances are dimensionless. To eliminate any dependence on the cell size, all conductances are scaled with the typical conductance. Hence, we may consider the model to represent a cluster of closely coupled / -cells that share the combined capacity and conductance of the entire membrane area. 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

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. 

Another important parameter in electrochemical promotion studies is the characteristic rate relaxation time, t, needed for the catalytic rate to reach steady state upon imposition of a constant current (galvanostatic transient). As one would expect and as experiment has clearly shown [13,14], t is always of the order of 2FNg/I (Figure 2) ... 

The parameter pco characterizes a short-time, high frequency viscosity and models viscous processes which require no structural relaxation, like in the general case (15). Together with F, it is tire only model parameter affected by Hl. Steady state shear stress under constant shearing, and viscosity then follow via integrating up the generalized modulus ... 

Fig. 21 Steady state incoherent intermediate scattering functions d> (r) as functions of accumulated strain yt for various shear rates y the data were obtained in a col loidal hard sphere dispersion at packing fraction

Another powerful contrast parameter is spin-lattice, or Tj, relaxation. Spin-lattice relaxation contrast can again be used to differentiate different states of mobility within a sample. It can be encoded in several ways. The simplest is via the repetition time, between the different measurements used to collect the image data. If the repetition time is sufficiently long such that Tj )) Tj for all nuclei in the sample, then all nuclei will recover to thermal equilibrium between measurements and will contribute equally to the image intensity. However, if the repetition time is reduced, then those nuclei for which Tr < Tj will not recover between measurements and so will not contribute to the subsequent measurement. A steady state rapidly builds up in which only those nuclei with Tj contribute in any significant manner. As with -contrast, single images recorded with a carefully selected may be used to select cmdely a short component of a sample. 

If the relaxation time, e.g. time to reach a steady-state state is longer than the duration of a catalytic experiment, than the reaction occurs under nonstationary conditions. Sometimes it could be even beneficial to perform a reaction under such conditions, periodically changing initial parameters of the reaction system., e.g. temperature, pressure, concentrations, or flow velocity. 

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