Vs. log time curves

The log (rate) vs. log (time) curves for the McLafferty rearrangement in the molecular ions of n-butyl acetate, 2-phenylethyl acetate, 2-phenyl-propyl acetate and 2-dimethylaminoethyl acetate have been found to be approximately parallel to each other and the rates at any time decreased significantly on going from n-butyl acetate to 2-phenylethyl acetate to... 

Figure 7.29 Examples of different formatted creep vs. log time curves (courtesy of Bayer)...

Figure 2.6 Isometric stress vs. log time curves derived from creep data...

The intensity of the crystalline bands was monitored simultaneously during the crystallization. To correct for changes in density or thickness in the different samples, the intensities were normalized by the reference band. These normalized intensities were plotted vs. log time for each of the blends at the different crystallization temperatures. The curves obtained are sigmoidal in nature and they level off when the final crystallinity is achieved. A typical curve for the normalized intensity of the 848-cm-1 band vs. log time is plotted in Figure 7 for PET. 

The first equation is much easier to manipulate than the second one. A simple check for separability of the variables is that curves of log strain vs. log time with stress as... 

Fig. 11. Stress Relaxation curves for NaPOa sample of hipest molecular weight at various temperatures. log ,(t) (dynes/cm ) vs. log time (sec)...

Application to a viscoelastic material involves the construction of a so-called master curve. Here experimental data such as log modulus are plotted vs. log time for a series of temperatures to form a family of curves. By horizontal shifts of the curves (the shifts corresponding to Aj), physical properties for very short or extended time ranges at a single temperature may be obtained. This is illustrated schematically in Figure 1.19 (Catsiff and Tobolsky, 1955, 1956 Nielsen, 1962, pp. 89-92). 

Figure 12.17 shows a master stress relaxation curve, presented as ((7 — (T.)/ (T — (7j) vs. log time. The value of a. is determined for t— oo by an approximation to the experimental stress relaxation master curve. This type of plot has been recommended by one of us [52-61]. It brings out the common features of stress relaxation curves for metals, polymers and other materials. This is true not only for experimental but also for computer generated stress relaxation curves [60-62]. For a discussion of these common features see also [62]. The type of plot recommended first in [52] is also being used successfully for instance by Wortmann and coworkers [63-66]. They investigated a variety of materials including wool fibers and also very stiff aramid fibers such as Kevlar. 

The similarity is also preserved when we plot the master curves as ((7 — — (7j) vs. log time (Figure 12.20). The initial stress <7 (0)... 

The stress-strain-time data can be plotted as creep curves of strain vs. log time (Fig. 3.10 top view). Different methods are also used to meet specific design requirements. Examples of methods include creep curves at constant times to yield isochronous stress versus strain curves or at a constant strain, giving isometric stress versus log-time curves, as shown in the bottom views in Fig. 3.10. 

Fig. 6.14 The master time-humidity stress relaxation curve for ASl/3501-6. Tensile stress relaxation modulus vs. Log time. (Crossman and Warren 1985)...

Figure 38 The departure from equilibrium S vs. log time following steps in temperature to 35 °C from 40°C ( ) and 30°C (O) for poly(vinyl acetate) The curves are computed from the RSC model (after ref. 225, with permission)...

For a given extent of reaction, Eq. (3-33) is an equation with the two unknowns r and d. The procedure, in essence, is to measure F at two times and to solve the two simultaneous equations. In practice the problem is more difficult than this because an analytical solution cannot be obtained moreover d is itself dependent upon time. Swain " constructed tables of d (and of log d) as a function of r for three different extents of reaction. Curves of log d vs. log r are plotted. The curve... 

Plot a log/linear curve of activity vs time and analyse it into its components. Deduce the half -lives and... 

The voltammetric response depends on the equilibrium constant K and the dimensionless chemical kinetic parameter e. Figure 2.30 illustrates variation of A f, with these two parameters. The dependence AWp vs. log( ), can be divided into three distinct regions. The first one corresponds to the very low observed kinetics of the chemical reaction, i.e., log( ) < —2, which is represented by the first plateau of curves in Fig. 2.30. Under such conditions, the voltammetric response is independent of K, since the loss of the electroactive material on the time scale of the experiment is insignificant. The second region, —2 < log( ) < 4, is represented by a parabolic dependence characterized by a pronounced minimum. The descending part of the parabola arises from the conversion of the electroactive material to the final inactive product, which is predominantly controlled by the rate of the forward chemical reaction. However, after reaching a minimum value, the peak current starts to increase by an increase of . In the ascending part of the parabola, the effect of... 

The variation of the peak current with the electrode kinetic parameter k and chemical kinetic parameter e is shown in Fig. 2.31. When the quasireversible electrode reaction is fast (curves 1 and 2 in Fig. 2.31) the dependence is similar as for the reversible case and characterized by a pronounced minimum If the electrode reaction is rather slow (curves 3-5), the dependence A fJ, vs. log( ) transforms into a sigmoidal curve. Although the backward chemical reaction is sufficiently fast to re-supply the electroactive material on the time scale of the reverse (reduction) potential pulses, the reuse of the electroactive form is prevented due to the very low kinetics of the electrode reaction. This situation corresponds to the lower plateau of curves 3-5 in Fig. 2.31. 

From the general formulae describing the kinetics of electron tunneling reactions it follows that with the constant value of the parameter ae the kinetic curves represent the function of the dimensionless parameter. t. Therefore, kinetic curves obtained at different temperatures with the coordinates concentration vs. log observation time must be just shifted with respect to the time axis. Consequently, the lack of temperature dependence... 

The stability of enzyme electrodes is difficult to define because an enzyme can lose some of its activity. Deterioration of immobilized enzyme in the potentiometric electrodes can be seen by three changes in the response characteristics (a) with age the upper limit will decrease (e.g., from 10-2 to 10 3 moll-1), (b) the slope of the analytical (calibration) curve of potential vs. log [analyte] decrease from 59.2 mV per decade (Nernstian response) to lower value, and (c) the response time of the biosensor will become longer as the enzyme ages [59]. The overall lifetime of the biosensor depends on the frequency with which the biosensor is used and the stability depends on the type of entrapment used, the concentration of enzyme in the tissue or crude extract, the optimum conditions of enzyme, the leaching out of loosely bound cofactor from the active site, a cofactor that is needed for the enzymatic activity and the stability of the base sensor. 

An essential difference between the data of Ref. [212] and those of the earlier works was that the kinetics of P700+ decay at T < 240 K was found not to follow a first-order law, but to show linear or near-linear dependences in the coordinates n(t) vs log t (see Fig. 28). The curves for the ranges 220-160,160-80 and 80-5 K are seen to have different characters. Actually, as seen from Fig. 28, below 80 K the curves show linear dependencies in the coordinates n(t) vs log t over the whole time range investigated. At 160 K > T > 80 K the kinetic curves demonstrate linear dependencies only in the initial section while a decrease in the slope of the curves is observed at longer times. In the temperature range 220-160 K notable deviations from linearity are observed throughout the whole time interval. The slope of the initial sections of the kinetic curves in the coordinates n(t) vs log t remains practically constant at 80 K < T < 160 K and becomes temperature-dependent at T < 80 K and at T > 160 K. 

Below 500 K heating of the solid salt results primarily in the vaporization of the covalent molecule as a monomer. In this temperature range the only thermal decomposition, into NOz and 02, is exhibited by the solid. The vapor is more stable. The vapor pressure of Cu(N03)2 was determined by Addison and Hathaway48 by extrapolating pressure-time curves to zero time in order to subtract the pressures of N02 and 02. These vapor pressures increased from 0.32 torr at 430 K to 3.59 torr at 405 K. A plot of log P vs. 1/T is linear and yields a sublimation enthalpy of 67.0 kJ. Above 500 K both the solid and the vapor phase decompose to N02 + 02. 

The rate constant k and the order of reaction can be determined from the plot of log (-d[A] / dt) vs. log [A], Figure 5.7 illustrates these procedures. First, the rate is determined from the concentration vs. time curve (Figure 5.7a). Second, the rate vs. concentration is plotted in a log-log plot (Figure 5.7b). The slope and the y-intercept furnish the order of reaction and the rate constant, respectively. 

Another technique of the differentiation method is the initial rate measurement. A series of experiments are carried out for different initial concentrations over a short time period (5 to 10% or less conversion). This approach is different from the experimental run discussed in Figure 5.7. Each rate measurement requires a new experiment with a different initial concentration. The initial rate of the reaction is determined from the curve of the concentration vs. time, as shown Figure 5.9a. The log of the initial rate is then plotted against the log of the initial concentration (Figure 5.9b). If the order of the reaction calculated from the concentration-time curve is different from the one determined by initial rate experiments, interference by the reaction products is expected, leading to complex reaction kinetics. 

Figure 14. Time evolution of the composition of the percolating water in the downstream part of the alteration profile of a pyrite-rich sandstone. = 10. The concentration of the dissolved species are given in mol/kg and the quantities of neoformed minerals are given in mol as a function of the parameter of advancement of the reaction t All data are represented as the logarithm of the molality (log m) vs. log. XU(a) corresponds to [U/Fe] = 5 X 10 (molar ratio) leached within the sandstone. U(l)) corresponds to the maximum possible dissolved uranium concentration. All the curves are direct Benson plots from the computer.

More complicated dielectric properties may be described by superimposing such terms with different relaxation times and the corresponding values of Aco- This always results in a flattening of the dispersion curves (i.e. e and e plotted vs. log to) in comparison with the basic Debye functions of (10). 

---