Relaxation amplitudes

It has been stated above that the difference of partial molar volumes of the LS and HS isomers AK° can be obtained from the relaxation amplitude A of ultrasonic absorption. An independent method for the determination of AE° is based on the pressure dependence of the equilibrium constant. The pressure derivative of being determined by ... 

Measurement of an ultrasonic relaxation curve enables evaluation of both the relaxation time, t, and the relaxation amplitude, A. Interpretation of the relaxation time requires knowledge of the equilibrium constant. For a intramolecular isomerization such as a high-spin low-spin equilibrium, the forward and reverse rate constants, kl and respectively, can be evaluated from the relaxation time and the equilibrium constant from Eq. (8) (17). 

Figure 7. Relaxation amplitude analysis variation of AH with [Fe(CN)64 ]. AH = normal enthalpy of the relaxation mode observed (14). (a) At 23.5°C, plot according to linearized form of Equation 2. Total azurin 2-7 X 10 5 M. (b) At 6.5°C, plot according to Equation 2. Total azurin 2-11 X 10 5 M. The line drawn is the best fit to the experimental points with Kt = 1.0 X 102 M 1 and AH/ = —5.5 kcal/mole.

Figure 8, Kapp is plotted against [Fe(II)], and it shows a behavior corresponding to Equation 4. The association constant Kx (57M"1 at 25°), obtained from the ratio of negative slope to intercept, is in good agreement with the value derived from the relaxation amplitude )54 M 1 at the same temperature). Equation 5 relates relaxation time, equilibrium con-...

Temperature-jump 31) measurements (with a time resolution in the microsecond range) only yielded a lower limiting value for the rate of the monactin-Na+ system. The relaxational amplitudes of these measurements could be evaluated in order to obtain the stability constant (Kgtab) and heat of reaction (/IH) 18). The time constant for the reaction of monactin with Na+ was obtained by the sound absorption technique 32). The rate constant of recombination was found to be ( form) 2—3 X 10 M i sec i. The complex formation of trinactin with Na+ (at high ionic strength) was just detectable with the T-jump technique since the time constant for this system was sufficiently larger than the time constant of heating 33). 

In general, upon applying a perturbation to a chemical equilibrium, the larger the shift in the equilibrium (relaxation amplitude), the more similar the equilibrium population of the species involved. Therefore, systems with very small or very large equilibrium constants are relatively insensitive to perturbation. Thus, it is not surprising that no relaxations were also observed in SiO2 (K < 10) and 7-AI2O3 (K = suspensions. The... 

Fig. 6. Relaxation amplitude (rb) as a function of the logarithm of concentration ratio = ( stand/ sample...

The amplitudes of chemical relaxation processes are determined by the equilibrium concentrations (and strictly speaking, associated activity coefficients) and by thermodynamic variables appropriate for the particular perturbation method used. Thus, for example, an analysis of the amplitudes of relaxation processes associated with temperature jump measurements can lead to determination of the equilibrium constants and enthalpies associated with the mechanism under study. As might be anticipated from our previous discussion, the relaxation amplitudes are determined by normal mode thermodynamic variables which are linear combinations of the thermodynamic variables associated with the individual steps in the mechanism. The formal analysis of relaxation amplitudes has been developed in considerable detail [2, 5,7],... 

These authors base their assignment on the observations, first, that the relaxation time is independent of concentration and, secondly, that a change in the concentration of Ac" influences the relaxation amplitude much more than does a change in the concentration of Ni + (indicating that this reaction involves more Ac than Ni + ) On this basis, the values of the rate constants at 10 °C are kn = (3.2 1.6) x 10 lmol" s" and Am = (3.2 0.4) x 10 s. Clearly, if this interpretation is the correct one, we must somehow find a way of rationalizing the apparent discrepancy between these results and those of Puentes et al. ... 

For a single equilibrium present, the thermodynamic analysis (9) of the process state-2 — state-3 results in an expression for the relaxation amplitude A (i.e. the change of the extent of reaction during relaxation) as a function of the dilution ratio n, the equilibrium constants and in the initial and final... 

We normally use a measuring cell of 10 x 10 x 2 mm made from quartz-infrasil glass. The laser beam should have a cross-section of 8 mm and is reflected at the back of the measuring cell after passing a layer of 2 mm. This minimizes temperature inhomogeneities inside the heated water layer, which generate locally different relaxation amplitudes as well as unwanted shock waves. Shock waves may disturb the detection channel and locally diffe-... 

In pressure (-release ) jump techniques the perturbation of the chemical system is achieved by raising the pressure from the ambient to some higher value and a subsequent sudden drop back to the ambient value. To which extent the actual concentrations can assume their equilibrium values at high static pressure before the jump depends on the ratio of the relaxation time of the system and the pressure duration time 0, and also slightly on the time to needed for the pressure increase (see fig. 2). Here y and y refer to generalized concentration shifts where y is the equilibrium and y the actual shift, y is the equilibrium shift at high static pressure. Obviously the choice of the pressure duration time 6 is critical for the relaxation amplitude which is measured after the jump, because a jump before t 6i would lead to a reduction as compared to yQ. 

For the effect of integration which changes the correct relaxation amplitudes reference (3) should be consulted.)... 

Case II. One isomer is thermodynamically strongly favored over the other but - orms at a comparable rate. In such a situation there co id not be any pmr evidence but there should be an additional relaxation time which essentially refers to the reaction PS2(cis)PS2(trans) via PS as steady state intermediate. This additional relaxation time would however only be detectable if the extinction coefficients of PS2(cis) and PS2(trans) were sufficiently different from each other as to produce a significant relaxation amplitude (12b). In view of the expected similarity of the spectra of the two isomers this may not be the case. 

RELAXATION AMPLITUDES IN CONCENTRATION JUMP EXPERIMENTS. APPLICATION TO ANTHOCYANINS. 

The calculation of relaxation amplitudes by temperature jump is complex, except for very simple systems of the type A B or A + B C. This is unlike their calculation by concentration jump, since, in this case, the perturbation occurs at constant T and P, and the equilibrium constants remain unchanged. Moreover, we shov in this paper, that by proper use of the concentration jump method, one is able to measure the equilibrium constants of complicated as well as simple chemical systems. Whenever possible, the perturbation should be performed by rapidly modifying the concentration of a species that is characteristic of the type of reaction studied H or 0H for proton transfer, nucleophile for nucleophilic addition or substitution, etc... Thus, one can always write Z [X ] (t) = C, where represents the different... 

From an experimental standpoint, the observation of relaxation amplitudes by concentration jump requires only a classical spectrophotometer fitted with a stirred thermostatted cell. With... 

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