Relaxation time constant

Figure C3.5.5. Vibronic relaxation time constants for B- and C-state emitting sites of XeF in solid Ar for different vibrational quantum numbers v, from [25]. Vibronic energy relaxation is complicated by electronic crossings caused by energy transfer between sites.

Ohmic charge decay processes obey a first order rate law from which the charge Q remaining at any time t can be expressed in terms of the initial charge Qq and relaxation time constant r. Using Eqs. (2-3.4) through (2-3.5) the time constant r can alternatively be expressed as... 

Figure 4.13. NEMCA Rate and catalyst potential response to step changes in applied current during C2H4 oxidation on Pt T=370°C, p02=4.6 kPa, Pc2h4=0.36 kPa. The experimental (t) and computed (2FNG/I) rate relaxation time constants are indicated on the figure. See text for discussion. ro=1.5-10 8 mol O/s, Ar=38.5-10 8 mol O/s, I/2F=5.2-10 12 mol O/s, pmax=26, Amax=74000, Ng=4.240 9 mol Pt.4 Reprinted with permission from Academic Press.

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. 

Figure 5.5. (a) Dependence of the NEMCA relaxation time constant x on 2FNc/I for C2H4 epoxidation on Agu and (b) for CO, C2H4 and CH3OH oxidation on Pt and Ag.12 Adapted from ref. 11 and reprinted from ref. 12 with permission from the American Chemical Society and from Elsevier Science respectively. 

It is worth emphasizing, however, that in both cases C2H4 oxidation exhibits electrophobic behaviour, that the relaxation time constants x can be estimated from similar formulae (equations (4.32) and (9.3)) and that the enhancement factors A can again be estimated from the same formula1 (equation 4.20). 

There is an additional important observation to be made in Fig. 9.25 regarding the magnitude of the relaxation time constant, x, upon current imposition Electrochemical promotion studies involving both solid electrolytes and aqueous alkaline solutions have shown that x (defined as the time required for the catalytic rate increase to reach 63% of its final steady-state value upon current application) can be estimated from ... 

The effects of anelasticity are seen clearly in the strain-time schematic plot of Fig. 7. The time independent strain, 0, occurs immediately as the stress is apphed. The additional strain, o, is associated with stress-induced anelasticity. The total strain, -i- o, is approached exponentially. When the stress is removed the time independent strain is recovered immediately, while the anelastic component relaxes exponentially with a relaxation time constant, k. 

The usefulness of NMR in such analysis is because the proton spin-relaxation time constants are different for different components, such as water, liquid fat and solid fat. For example, the signal from solid fat is found to decay rapidly while the liquid signals decay much slower. This phenomenon is the basis for an NMR technique to determine the solid fat content [20], However, as the relaxation time constant of water, for example, could depend on its local environment, such as protein concentration, it may overlap with that of oil and other components. As a result, it could be difficult to formulate a robust and universal relaxation analysis. It... 

However, T2 is sensitive to the molecular interactions of spins and dependent on the molecular environment [60]. Thus, T2 may overlap for different components in certain materials and this technique alone may not be sufficient to identify the components. The relaxation time distributions are often broad, e.g., in meat [21], thus making it more difficult to associate the relaxation time constants with the components. 

Another quite delicate step is the determination of the relaxation time constant r from the relaxation curve T(t). In this method, instead of the electrical heater, other heating sources, such as an optical source, can be used [13],... 

An example of Tc(t) is reported in Fig. 12.7. A single relaxation time constant r was always observed. 

The real power of MRI is the ability to exploit the inherent NMR properties of different tissues and tissue pathologies. The main MR parameters are known as longitudinal-relaxation time-constant (Tl), describing the time for the magnetization to recover to its equilibrium along the main-field axis after RF perturbation transverse-relaxation time-constant (T2), describing the... 

Within the accuracy of the experimental data the galvanostat-ic transient response of AV is identical to the transient rate response Ari andAr2, i.e. t = x where x is the relaxation time constant for the two rates (17. This s shown in figure 5 for two different reactors under similar operating conditions and also in figure 6 where the transient and the steady state Ar values from four reactors are plotted vs. the cell overvoltage AV. In view of the fact that r. is proportional to the surface area Q it follows from figure 6 tftat for constant gas phase composition... 

Figure 10. Effect of Ag catalyst-electrode surface area Q on the cell relaxation time constant. Conditions 400°C, PqJPet — 7.

Surface area effect on the relaxation time constant In a previous communication (17) we have developed a simple dynamic model which allows one to predict the change in the mole number of silver oxide S Ac, therefore AV, in terms of the imposed current, Pq and The dynamic equation of the model was... 

The relaxation time constants of the overvoltage (t ) and the rates (x ) are practically equal and of order 4FQ/i. They increase with increasing Po2 ET ratios. The new observations are in reasonably good agreement with a previously proposed model (17). 

The diode laser is scanned up and down in frequency by a triangle wave, so that the scan should be linear in time and have the same rate in both directions. In the thermal accommodation coefficient experiments, the external beam heats the microsphere to a few K above room temperature and is then turned off. The diode laser is kept at fairly low power ( 7 pW) so that it does not appreciably heat the microsphere. Displacement of a WGM s throughput dip from one scan trace to the next is analyzed to find the relaxation time constant as the microsphere returns to room temperature. Results from the two scan directions are averaged to reduce error due to residual scan nonlinearity. This is done over a wide range of pressures (about four orders of magnitude). The time constant provides the measured thermal conductivity of the surrounding air, and fitting the thermal conductivity vs. pressure curve determines the thermal accommodation coefficient, as described in Sect. 5.5.2. 

Relaxation is an inherent property of all nuclear spins. There are two predominant types of relaxation processes in NMR of liquids. These relaxation processes are denoted by the longitudinal (Ti) and transverse (T2) relaxation time constants. When a sample is excited from its thermal equihbrium with an RF pulse, its tendency is to relax back to its Boltzmann distribution. The amount of time to re-equilibrate is typically on the order of seconds to minutes. T, and T2 relaxation processes operate simultaneously. The recovery of magnetization to the equilibrium state along the z-axis is longitudinal or the 7 relaxation time. The loss of coherence of the ensemble of excited spins (uniform distribution) in the x-, y-plane following the completion of a pulse is transverse or T2... 

A pressure perturbation results in the shifting of the equilibrium the return of the system to the original equilibrium state (i.e., the relaxation) is related to the rates of all elementary reaction steps. The relaxation time constant associated with the relaxation can be used to evaluate the mechanism of the reaction. During the shift in equilibrium (due to pressure-jump and relaxation) the composition of the solution changes and this change can be monitored, for example by conductivity. A description of the pressure-jump apparatus with conductivity detection and the method of data evaluation is given by Hayes and Leckie (1986). 

The pH dependence of the inverse of the fast relaxation time constant, Xp, is shown in Figure 10 (error bars represent 95% confidence level) for pressure-jump magnitudes of 70-140 atmospheres. 

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