Relaxation time in general

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... 

Very nice, you might say, but where does the viscosity come in Well the time required for a chain to diffuse out of its tube, is something we will refer to as a relaxation time. In general, relaxation times in liquids measure the period necessary for a system to relieve an applied stress by the molecules slipping past one another. It is thus directly proportional to the viscosity and we can write Equation 13-70 ... 

Rocks tend to have a very broad distribution of pore sizes, and there is therefore a distribution of observed relaxation times. In general, the magnetization signal, M(t), from a spin-echo experiment (used to determine the Ti distribution) is expected to obey (based on Eq. (9))... 

As can be seen, the Maxwell-Weichert model possesses many relaxation times. For real materials we postulate the existence of a continuous spectrum of relaxation times (A,). A spectrum-skewed toward lower times would be characteristic of a viscoelastic fluid, whereas a spectrum skewed toward longer times would be characteristic of a viscoelastic solid. For a real system containing crosslinks the spectrum would be skewed heavily toward very long or infinite relaxation times. In generalizing, A may thus he allowed to range from zero to infinity. The concept that a continuous distribution of relaxation times should be required to represent the behavior of real systems would seem to follow naturally from the fact that real polymeric systems also exhibit distrihutions in conformational size, molecular weight, and distance between crosslinks. 

The term relaxometry is normally used in context with techniques for the measurement of spin-lattice relaxation times in general. Transverse relaxation and effects due to residual dipolar couplings will be considered in the next section. 

Whether a viscoelastic material behaves as a viscous Hquid or an elastic soHd depends on the relation between the time scale of the experiment and the time required for the system to respond to stress or deformation. Although the concept of a single relaxation time is generally inappHcable to real materials, a mean characteristic time can be defined as the time required for a stress to decay to 1/ of its elastic response to a step change in strain. The... 

Carbon-13 NMR was utilized to study different aspects of the reactivity of the metal complexes as a function of certain structural features in the selected oxocyano complexes of Mo(IV), W(IV), Tc(V), Re(V), and Os(VI) as depicted in Scheme 1 and illustrated in Figs. 1-4. The NMR spectral properties were similar to those obtained from 13C NMR in general, i.e., very sharp lines indicative of fairly long relaxation times in the order of a few seconds. The large quadrupolar moment ofTc-99 (7 = 9/2, 100% abundance) led to a very broad bound 13C signal (Fig. 5), thus excluding the quantitative study of the cyanide exchange by 13C NMR. However, 16N NMR was successfully used instead. 

From this comparison it follows that the observation of the structural relaxation by standard relaxation techniques in general might be hampered by contributions of other dynamic processes. It is also noteworthy that the structural relaxation time at a given temperature is slower than the characteristic time determined for the a-relaxation by spectroscopic techniques [105]. An isolation of the structural relaxation and its direct microscopic study is only possible through investigation of the dynamic structure factor at the interchain peak - and NSE is essential for this purpose. 

In the previous discussion, the electron-nucleus spin system was assumed to be rigidly held within a molecule isotropically rotating in solution. If the molecule cannot be treated as a rigid sphere, its motion is in general anisotropic, and three or five different reorientational correlation times have to be considered 79). Furthermore, it was calculated that free rotation of water protons about the metal ion-oxygen bond decreases the proton relaxation time in aqua ions of about 20% 79). A general treatment for considering the presence of internal motions faster than the reorientational correlation time of the whole molecule is the Lipari Szabo model free treatment 80). Relaxation is calculated as the sum of two terms 8J), of the type... 

NMR spectroscopy provides spin-lattice (ri) and spin-spin (Tj) relaxation times. Making appropriate assumptions with regard to the magnetic interactions responsible for the relaxation process, these relaxation times can be related to molecular motions. Since nuclear spin relaxation results from all processes which cause a fluctuation in the magnetic field at the nucleus, the correlation function will generally correspond to more than one kind of motion involving all possible interactions. The equations for the relaxation times are generally of the form... 

In general, the dielectric relaxation in a polar polymer exhibits a wide distribution of relaxation time. In this case, the real and imaginary parts of the dielectric constant are written in the form ... 

This is the kinetic equation for a simple A/AX interface model and illustrates the general approach. The critical quantity which will be discussed later in more detail is the disorder relaxation time, rR. Generally, the A/AX interface behaves under steady state conditions similar to electrodes which are studied in electrochemistry. However, in contrast to fluid electrolytes, the reaction steps in solids comprise inhomogeneous distributions of point defects, which build up stresses at the boundary on a small scale. Plastic deformation or even cracking may result, which in turn will influence drastically the further course of any interface reaction. 

However, a single Tt is generally observed in heterogeneous systems partly because it is difficult to resolve T2 components unless they differ by at least a factor of two. Deuterium T2 relaxation times in crosslinked resins, also exhibit only a singleexponential relaxation process, indicating that different 2H in the same molecule have essentially the same spin-lattice relaxation times44). 

These A and E states have long electron relaxation times which generally result in broad, unobservable NMR spectra. Hence it would... 

The importance of hydrogen bonding is apparent from Table 1.1, which indicates that solvents with weak hydrogen-bonding capabilities have much shorter relaxation times. In addition, their lower viscosities reduce the interaction of molecules in the rotational process, which is induced by the microwave coupling and therefore will generally not heat as effectively in a microwave cavity. 

Cobalt(II) is a d7 ion which can be high spin or low spin. Generally, it is low spin in planar or some square pyramidal compounds. The effective electron relaxation times in the low spin state are long enough (10-9-10-1° s) so that EPR spectra can be recorded at room temperature [78] and the proton NMR lines are broad. This is due to the high energy of the first excited state (Fig. 5.29). In... 

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