Concentration dependence of relaxation

Concentration dependence of relaxations is shown in Fig. 7.11 for samarium sulphate. 

During a long period, Richards and co-workers [51 52 245 246 262j262] have studied alkali and halide ion relaxation in solutions of alkali halides and interpreted their data rather differently from Hertz. In their first paper, studies of NaBr and CsBr solutions were performed at different concentrations and the concentration dependence of relaxation compared with that of viscosity. For NaBr the line width was found to increase linearly with viscosity although direct proportionality between the two quantities is not displayed by the data. For CsBr solutions, where there is a minimum in the plot of viscosity versus concentration, the bromine line width divided by viscosity was found to increase approximately linearly with salt concentration. The interpretation of these findings was that in the case of NaBr, ion-solvent interactions dominate, whereas with CsBr also ion-ion interactions are of importance. The basis for comparing relaxation rate... 

Stengle et al, [335] presented Cl relaxation data over the whole solvent composition range for water-acetonitrile mixtures. However, since no concentration dependence of relaxation was given and the data were obtained with tetramethylammonium chloride, which displays particular relaxation effects in aqueous systems, these results are difficult to compare quantitatively with theory. Gentzler et al. [337] obtained a value for Cl" relaxation in nitromethane but also here no concentration dependence was presented. 

The analysis of the observed concentration dependence of relaxation times Tjp(c) of the ion-pair relaxation region, reveals two superimposed modes, a diffusional rotation mode of the ion-pair dipoles and a kinetic mode due to the re-establishment of the undisturbed equilibrium = Cd2+, M +)... 

Here p is the solution density, v the sound velocity, ctp the coefficient of thermal expansion, Cp the specific heat, and F the concentration dependence of the equilibrium, r = [LS] -f- [HS] . The measurement of ultrasonic relaxation thus enables the determination of both the relaxation time x and the... 

The dipole interaction depends on the distance between the ions (6.4). Therefore, the transition probability increases with increasing concentration of magnetic ions. Studies of the concentration dependence of the relaxation can be conveniently performed on samples of amorphous frozen solutions with a uniform distribution... 

The measurement of viscosity is important for many food products as the flow properties of the material relate directly to how the product will perform or be perceived by the consumer. Measurements of fluid viscosity were based on a correlation between relaxation times and fluid viscosity. The dependence of relaxation times on fluid viscosity was predicted and demonstrated in the late 1940 s [29]. This type of correlation has been found to hold for a large number of simple fluid foods including molten hard candies, concentrated coffee and concentrated milk. Shown in Figure 4.7.6 are the relaxation times measured at 10 MHz for solutions of rehydrated instant coffee compared with measured Newtonian viscosities of the solution. The correlations and the measurement provide an accurate estimate of viscosity at a specific shear rate. 

When (DEB), is much smaller than unity, the polymer relaxation is relatively rapid compared to diffusion. In this case, conformational changes take place instantaneously and equilibrium is attained after each diffusional jump. This is the type of diffusion encountered ordinarily and is called viscous diffusion. Therefore, the transport will obey classical theories of diffusion. When (DEB), is much larger than unity, the molecular relaxation is very slow compared to diffusion and there are no conformational changes of the medium within the diffusion time scale. In this case, Fick s law is generally valid, but no concentration dependence of the diffusion coefficient is expected. This is termed elastic diffusion. When (DEB), is in the neighborhood of unity, molecular rearrangment... 

The equilibrium constant and dissociation rate constant were determined simultaneously by non-linear least-squares fitting, unless the absorption signal was too low157 or no dependence of relaxation frequency on concentration was observed.159,161,162 The association rate constant was then calculated from the definition of the equilibrium constant. The equilibrium constants determined from the dynamics in this manner agree fairly well with equilibrium constants determined independently. 

When counterion binding reaction II is extremely rapid, the concentration dependence of the relaxation time is given by (11)... 

For reactions VII and VIII, the concentration dependences of the relaxation times are given by... 

The concentration dependence of polymer or solvent motion has been studied only rarely over a wide range in concentration. Typically, polymer carbon-13 relaxation is not concentration dependent up to 20-30 percent polymer. Little is known concerning the concentration dependence of the solvent motion. 

Figure 5. Temperature and concentration dependence of the ester methylene C-13 relaxation (9), 10% 20%> (O), 30%.

Figure 9. Temperature and concentration dependence of the solvent proton-polymer proton relaxation rate (0), 10% polymer (X), 20% polymer.

A linear concentration dependence of the reciprocal relaxation time upon total ion-pair concentration would therefore point to a main recombination process between a simple ion and a triple ion. 

Considering these different limiting forms of the recombination term an Important tentative conclusion emerges the concentration dependence of the reciprocal relaxation time is a direct measure of the main ionic recombination process and yields therefore information on the ionic species present in solution. A linear dependence on total ion-pair concentration would therefore indicate unilateral triple ion formation or, if both kinds of triple ions are present as indicated by conductance, a sufficient difference in their stability. At this point it should be noted that the usual method of Fuoss and Draus... 

Williams has derived the molecular weight and concentration dependence of a viscoelastic time constant t0 (actually the characteristic time governing the onset of shear rate dependence in the viscosity) from his theory (217-219). Employing a dimensional argument, he equates the parameters which control the shear rate dependence of chain configuration and the intermolecular correlation function. The result agrees with the observed form of characteristic relaxation time in concentrated systems [Eq.(6.62)] ... 

To measure the effect of the presence of ions on the hydrogen-bond structure of bulk water, we studied the concentration dependence of the orientational relaxation of bulk water in aqueous solutions of Mg(C104)2. We chose Mg(C104)2 because Mg2+ is considered to be a strong structure-making ion the viscosity of 1 M Mg(C104)2 is about 30 %... 

Two practical points should be noted. The kinetic mechanisms in equations 4.71 and 4.74 may be distinguished by the concentration dependence of l/r2. For 4.71 this increases with increasing [S] for 4.74 it decreases. But there are situations that are difficult to resolve. For example, in equation 4.74, if [E ] > [E] there will be a burst of formation of ES with relaxation time r(, followed by a small increase at relaxation time r2 as E converts to E. The concentration dependence of r2 will be small, since kl > k-y for [E ] > [E], This can be mistaken for the scheme in equation 4.71, where only a little ES is formed. In this case also, the concentration dependence of 1/t2 is small, because k-2 > k2. In both cases the amplitudes of the changes will often be small and the rate constants difficult to measure accurately. 

Any change in the medium, i.e. the solvent, the concentration, the pH value, and in the temperature will affect the mobility of the molecules and hence also the spin-lattice relaxation. However, few systematic studies have so far been performed on the concentration dependence or on the precise influence of the macroscopic viscosity, the main reason, in the case of 13C, lying in the need for highly protracted measurements in concentration studies. Moreover, little is known about the pH dependence of 13C relaxation [188], Nevertheless, the concentration dependence of 13C relaxation is apparent in the case of saccharose (Table 3.20) [166], and intramolecular hydrogen bonds can be detected by measuring the concentration dependence of Tx [189]. 

Relaxation times Tt and T2 have been determined as a function of temperature and surface coverage in various zeolites, particularly of the faujasite type. The early experiments have been troubled by the very strong dependence of relaxation rates on the concentration of paramagnetic impurities. In order for the relaxation values to be meaningful, such impurities expressed as Fe content must be below ca. 6 ppm. Figure 38 shows the variation of Tt and T2 for water adsorbed in a particularly pure sample of zeolite Na-X (248). The authors (248) account for the experimental results using a model of the intracrystalline fluid, which is about 30 times as viscous as bulk water at room temperature. It shows a broad distribution of molecular mobilities (the ratio T,/T2 at the minimum in Tt is much larger... 

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. 

Kakizaki, M. and Hideshima, T., Effect of distribution of free volume on concentration dependence of dielectric relaxation in water mixtures with poly(ethylene glycol) and glucose, Jpn. J. Appl. Phys., Part 1, 1998, 37, 900. 

The concentration dependence of ionic mobility at high ion concentrations and also in the melt is still an unsolved problem. A mode coupling theory of ionic mobility has recently been derived which is applicable only to low concentrations [18]. In this latter theory, the solvent was replaced by a dielectric continuum and only the ions were explicitly considered. It was shown that one can describe ion atmosphere relaxation in terms of charge density relaxation and the elctrophoretic effect in terms of charge current density relaxation. This theory could explain not only the concentration dependence of ionic conductivity but also the frequency dependence of conductivity, such as the well-known Debye-Falkenhagen effect [18]. However, because the theory does not treat the solvent molecules explicitly, the detailed coupling between the ion and solvent molecules have not been taken into account. The limitation of this approach is most evident in the calculation of the viscosity. The MCT theory is found to be valid only to very low values of the concentration. 

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