Relaxation time , reciprocal

Fig. 19 shows the temperature dependence of the rate constant (i.e. of the relaxation time reciprocal) for the process... 

The relaxation time (reciprocal of the dispersion frequency) is given by... 

Of the adjustable parameters in the Eyring viscosity equation, kj is the most important. In Sec. 2.4 we discussed the desirability of having some sort of natural rate compared to which rates of shear could be described as large or small. This natural standard is provided by kj. The parameter kj entered our theory as the factor which described the frequency with which molecules passed from one equilibrium position to another in a flowing liquid. At this point we will find it more convenient to talk in terms of the period of this vibration rather than its frequency. We shall use r to symbolize this period and define it as the reciprocal of kj. In addition, we shall refer to this characteristic period as the relaxation time for the polymer. As its name implies, r measures the time over which the system relieves the applied stress by the relative slippage of the molecules past one another. In summary. 

Not surprisingly, we find that the relaxation is a first-order process with rate constant A , + A i. It is conventional in relaxation kinetics to speak of the relaxation time T, which is the time required for the concentration to decay to Me its initial value. In Chapter 2 we found that the lifetime defined in this way is the reciprocal of a first-order rate constant. In the present instance, therefore,... 

We have next to consider the measurement of the relaxation times. Each t is the reciprocal of an apparent first-order rate constant, so the problem is identical with problems considered in Chapters 2 and 3. If the system possesses a single relaxation time, a semilogarithmic first-order plot suffices to estimate t. The analytical response is often solution absorbance, or an electrical signal proportional to absorbance or to another physical property. As shown in Section 2.3 (Treatment of Instrument Response Data), the appropriate plotting function is In (A, - Aa=), where A, is the... 

Turning from chemical exchange to nuclear relaxation time measurements, the field of NMR offers many good examples of chemical information from T, measurements. Recall from Fig. 4-7 that Ti is reciprocally related to Tc, the correlation time, for high-frequency relaxation modes. For small- to medium-size molecules in the liquid phase, T, lies to the left side of the minimum in Fig. 4-7. A larger value of T, is, therefore, associated with a smaller Tc, hence, with a more rapid rate of molecular motion. It is possible to measure Ti for individual carbon atoms in a molecule, and such results provide detailed information on the local motion of atoms or groups of atoms. Levy and Nelson " have reviewed these observations. A few examples are shown here. T, values (in seconds) are noted for individual carbon atoms. 

If the equilibrium is suddenly displaced, the results obtained in Chapter 3 show that the re-equilibration process will follow first-order kinetics. It is customary in this field to refer to r, the relaxation time, which is defined as reciprocal of the first-order rate constant for re-equilibration. In this case, we have... 

The roots of Eq. (11-26) are the eigenvalues of the secular equation they are the reciprocal of the relaxation times (of which there are two since Eq. (11-26) is a quadratic). 

Relaxation kinetics in single-stage reactions. Derive expressions for the reciprocal relaxation times of the following single-stage reactions ... 

The time constant r, appearing in the simplest frequency equation for the velocity and absorption of sound, is related to the transition probabilities for vibrational exchanges by 1/r = Pe — Pd, where Pe is the probability of collisional excitation, and Pd is the probability of collisional de-excitation per molecule per second. Dividing Pd by the number of collisions which one molecule undergoes per second gives the transition probability per collision P, given by Equation 4 or 5. The reciprocal of this quantity is the number of collisions Z required to de-excite a quantum of vibrational energy e = hv. This number can be explicitly calculated from Equation 4 since Z = 1/P, and it can be experimentally derived from the measured relaxation times. 

When [El] has been reduced to the point that it is equal in concentration to [EI Je, the natural logarithm of the ratio /f/, /[ 7]0 would be ln(l/< ) = -1.0. The length of time required for [EI to decay by this amount can be obtained from the semilog plot shown in Figure A1.4B. This time interval is referred to as the relaxation time or the time constant and is given the symbol x. The relaxation time is the reciprocal of the rate constant ... 

Debye and Falkenhagen predicted that the ionic atmosphere would not be able to adopt an asymmetric configuration corresponding to a moving central ion if the ion were oscillating in response to an applied electrical field and if the frequency of the applied field were comparable to the reciprocal of the relaxation time of the ionic atmosphere. This was found to be the case at frequencies over 5 MHz where the molar conductivity approaches a value somewhat higher than A0. This increase of conductivity is caused by the disappearance of the time-of-relaxation effect, while the electrophoretic effect remains in full force. 

The term T, which is known as the spin-lattice relaxation time, is essentially the reciprocal of a first-order rate constant. At steady state... 

The first experimental data for a reaction involving proton transfer from a hydrogen-bonded acid to a series of bases which were chosen to give ApK-values each side of ApK=0 are given in Fig. 15 (Hibbert and Awwal, 1976, 1978 Hibbert, 1981). The results were obtained for proton transfer from 4-(3-nitrophenylazo)salicylate ion to a series of tertiary aliphatic amines in aqueous solution, as in (64) with R = 3-nitrophenylazo. Kinetic measurements were made using the temperature-jump technique with spectrophoto-metric detection to follow reactions with half-lives down to 5 x 10"6s. The reciprocal relaxation time (t ), which is the time constant of the exponential... 

For a reaction of the type shown in (74) with hydroxide ion in excess, the expected variation of the time constant (t-1) for the first-order approach to equilibrium after a temperature perturbation is given by (75). Thus a plot of reciprocal relaxation time (t 1) against hydroxide ion concentration is... 

Fig. 19 Reciprocal relaxation time for the equilibration of 4-(4-sulphonatophenyl-azo)resorcinol with hydroxide ion in H20 (O) and D20 ( )...

To provide information which may help the choice between the mechanisms in Schemes 9 and 10, the reciprocal relaxation time for the equilibration between the bisphenylazoresorcinol monoanions [69], [70], and [71] and the corresponding dianions was measured over a range of hydroxide-ion concentrations (Hibbert and Simpson, 1983, 1985). In each case a minimum... 

Fig. 20 Reciprocal relaxation time for the equilibration of the monoanions and dianions of 2,4-bis(phenylazo)resorcinol ( ) and 4,6-bis(phenylazo)resorcinol (O) with hydroxide ion...

The tautomerisation of the N(9)H- and N(7)H-isomers of adenine (93) has been studied in aqueous solution using the temperature-jump technique with Joule heating (Dreyfus et al., 1975). The equilibrium constant K = [N(7)H]/[N(9)H] has a value of 0.28 at 20°C. The reciprocal relaxation time... 

Fig. 22 Variation with pH of the reciprocal relaxation time for the equilibration between N(7)H and N(9)H isomers of cytosine...

At low frequencies the loss modulus is linear in frequency and the storage modulus is quadratic for both models. As the frequency exceeds the reciprocal of the relaxation time ii the Rouse model approaches a square root dependence on frequency. The Zimm model varies as the 2/3rd power in frequency. At high frequencies there is some experimental evidence that suggests the storage modulus reaches a plateau value. The loss modulus has a linear dependence on frequency with a slope controlled by the solvent viscosity. Hearst and Tschoegl32 have both illustrated how a parameter h can be introduced into a bead spring... 

When a constant electric field is suddenly applied to an ensemble of polar molecules, the orientation polarization increases exponentially with a time constant td called the dielectric relaxation time or Debye relaxation time. The reciprocal of td characterizes the rate at which the dipole moments of molecules orient themselves with respect to the electric field. 

In recent years, evidence has been found that both mechanisms of proton transfer can occur for certain intramolecularly hydrogen-bonded acids. Also, new kinetic behaviour has been obtained which allows a much more detailed examination of the reaction steps in (22). Kinetic data for the second ionization of substituted phenylazoresorcinols in the presence of hydroxide ions (25) were some of the first to be obtained for an intramolecularly hydrogen-bonded acid. The reciprocal relaxation time (t ) for the approach to equilibrium in a temperature-jump experiment was measured at different hydroxide-ion concentrations. A linear dependence of x on [OH] was obtained of the form of (26) (Eigen and Kruse, 1963 Inskeep et al., 1968 Rose and Stuehr, 1971). However, careful measurements at lower hydroxide-ion concentrations (Perlmutter-Hayman and Shinar, 1975 Perl-mutter-Hayman et al., 1976 Yoshida and Fujimoto, 1977) revealed that the... 

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by... 

Until 1984, all of the stopped-flow and temperature-jump kinetic studies of alpha cyclodextrin inclusion-complex formation were explainable in terms of a single-step, binding mechanism. According to this mechanism, the observed rate constant, kobs, (for stopped-flow) and the reciprocal relaxation time, 1/t, (for temperature-jump) should show a linear dependence on the edpha cyclodextrin concentration. Sano and coworkers, however, in the case of the iodide-alpha cyclodextrin interaction, and Hersey and Robinson,in the case of various azo dye-alpha cyclodextrin interactions (see Fig. 7), found that certain guest species exhibit a limiting value of kobs and 1/t at high concentrations of alpha cyclodextrin. This behavior can most simply be explained in terms of a mechanism of the type,... 

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