Reduced time plots

Characteristic features of a—time curves for reactions of solids are discussed with reference to Fig. 1, a generalized reduced-time plot in which time values have been scaled to t0.s = 1.00 when a = 0.5. A is an initial reaction, sometimes associated with the decomposition of impurities or unstable superficial material. B is the induction period, usually regarded as being terminated by the development of stable nuclei (often completed at a low value of a). C is the acceleratory period of growth of such nuclei, perhaps accompanied by further nucleation, and which extends to the... 

Fig. 2. Reduced time plots for the Avrami—Erofe ev equation [eqn. (6)] with n = 2, 3 and 4 and tT = (t/ta.g) the Prout—Tompkins expression [eqn. (9)] is included as the broken line.

Fig. 3. Reduced time plots, tr = (t/t0.9), for the contracting area and contracting volume equations [eqn. (7), n = 2 and 3], diffusion-controlled reactions proceedings in one [eqn. (10)], two [eqn. (13)] and three [eqn. (14)] dimensions, the Ginstling— Brounshtein equation [eqn. (11)] and first-, second- and third-order reactions [eqns. (15)—(17)]. Diffusion control is shown as a full line, interface advance as a broken line and reaction orders are dotted. Rate processes become more strongly deceleratory as the number of dimensions in which interface advance occurs is increased. The numbers on the curves indicate the equation numbers.

The influence of the nature of the anion on the intercalation process was also studied. The intercalation of 5 M solutions of liX (with X = Br, NO3, OH and ISO4) were followed at 120 °C. The extent of reaction plots vary greatly between the different salts (Fig. 9). The plots shown in Fig. 8 are reduced time plots, in which the time is divided by the half-life of the reaction. 

An alternative is to use reduced time plots. In this a computer was used to generate standard sets of data points for an imaginary reaction with some suitable value of rate constant k, arbitrarily chosen. This was repeated for all of the 18 model mechanisms, producing 18 tables of data. The time scales of these tables and the experimental data were completely different. However, time scales could be standardised by finding the time when a standard fraction of the reaction was complete. The obvious point... 

Figure 12 Kinetics of decomposition of calcium oxalate. (A) Reduced time plot and (B) Arrhenius plot...

Figure 2 Reduced-time plots for a Green River oil shale sample...

In addition to the elimination rate constant, the half-life (T/i) another important parameter that characterizes the time-course of chemical compounds in the body. The elimination half-life (t-1/2) is the time to reduce the concentration of a chemical in plasma to half of its original level. The relationship of half-life to the elimination rate constant is ti/2 = 0.693/ki,i and, therefore, the half-life of a chemical compound can be determined after the determination of k j from the slope of the line. The half-life can also be determined through visual inspection from the log C versus time plot (Fig. 5.40). For compounds that are eliminated through first-order kinetics, the time required for the plasma concentration to be decreased by one half is constant. It is impottant to understand that the half-life of chemicals that are eliminated by first-order kinetics is independent of dose. ... 

Mcllvried and Massoth [484] applied essentially the same approach as Hutchinson et al. [483] to both the contracting volume and diffusion-controlled models with normal and log—normal particle size distributions. They produced generalized plots of a against reduced time r (defined by t = kt/p) for various values of the standard deviation of the distribution, a (log—normal distribution) or the dispersion ratio, a/p (normal distribution with mean particle radius, p). 

A disadvantage inherent in the reduced time method of analysis, as discussed by Sharp et al. [70] Geiss [488] and others [30,33] is that it involves the comparison of curves. An alternative, and widely used, method of preliminary identification of the rate law providing the most satisfactory fit to a set of data is through a plot of the form... 

Jones et al. [73] have provided an alternative approach to the linearization of data using the tabulated reduced time values given by Sharp et al. [70]. The experimental data are expressed in the form ae as a function of (t/t0.s)ei where the subscript e refers to the experimental data. Three broadly equivalent methods of plotting can be used. 

The magnitude of t0 can be measured from the intercept of a f(a)—time plot. The existence of the induction period can introduce uncertainty into a reduced time analysis if the temperature coefficient of t0 differs from that later applicable, and it is necessary to plot (t — t0)/(tb — t0) against a where tb is the time at which the selected common value of a is attained. The occurrence of a slow initial process can be reflected in deviations from linearity in the f(a) time plot, though in favourable systems the contribution may be subtracted before analysis [40]. 

There have been few discussions of the specific problems inherent in the application of methods of curve matching to solid state reactions. It is probable that a degree of subjectivity frequently enters many decisions concerning identification of a best fit . It is not known, for example, (i) the accuracy with which data must be measured to enable a clear distinction to be made between obedience to alternative rate equations, (ii) the range of a within which results provide the most sensitive tests of possible equations, (iii) the form of test, i.e. f(a)—time, reduced time, etc. plots, which is most appropriate for confirmation of probable kinetic obediences and (iv) the minimum time intervals at which measurements must be made for use in kinetic analyses, the number of (a, t) values required. It is also important to know the influence of experimental errors in oto, t0, particle size distributions, temperature variations, etc., on kinetic analyses and distinguishability. A critical survey of quantitative aspects of curve fitting, concerned particularly with the reactions of solids, has not yet been provided [490]. 

Fig. 9 Reduced time (t/to.5) plot for the intercalation of LiX into gibbsite. X = NO3 (o), Br (A), Cl (0), OH ( ), SO4 (V). Reproduced with permission from Chem Mater (1999) 11 1771-1775...

Figure 5.19 shows the evolution of the gel volume fraction as a function of reduced time K(t — to). The solid line represents the theoretical curve obtained using Eq. (5.22). For (p values between 20% and 80%, the theoretical curve is roughly linear with a slope of 1 (see dashed line). Equivalently, within the same (p range, the volume fraction should vary linearly with time with a slope equal to K. The experimental data (Fig. 5.17) were recalculated in order to be plotted in reduced coordinates. For each initial volume fraction, K is deduced from the initial slope of the curve (p = f t) (for cp between roughly 20% and 60%). All the data lie within a unique curve that is in reasonable agreement with the theoretical one. 

One of the simplest criteria specific to the internal port cracking failure mode is based on the uniaxial strain capability in simple tension. Since the material properties are known to be strain rate- and temperature-dependent, tests are conducted under various conditions, and a failure strain boundary is generated. Strain at rupture is plotted against a variable such as reduced time, and any strain requirement which falls outside of the boundary will lead to rupture, and any condition inside will be considered safe. Ad hoc criteria have been proposed, such as that of Landel (55) in which the failure strain eL is defined as the ratio of the maximum true stress to the initial modulus, where the true stress is defined as the product of the extension ratio and the engineering stress —i.e., breaks down at low strain rates and higher temperatures. Milloway and Wiegand (68) suggested that motor strain should be less than half of the uniaxial tensile strain at failure at 0.74 min.-1. This criterion was based on 41 small motor tests. 

We will now summarize the conclusion of the Kirkwood-Bethe theory. Fig 15 shows the computed peak pressure and computed reduced tunc constant for TNT plotted VS the inverse reduced distance. The dotted lines are a power function fit thru the computed peak pressures. The x s are drawn in by the writer to compare computed and measured reduced time constants (taken from Fig 7.9, p 240 of Ref 1). Comparison of other computed and measured shock parameters on the basis of the power functions shown below (in Cole s notation and in English units) is made in Table 11 (from p 242 of Ref 1)... 

Figures 2-5 show typical rate curves computed by means of the closure assumption. Here n l is plotted as a function of reduced time kxt, assuming throughout values of the rate constants consistent with the...

Fig. 4. Weight average length of runs of 0 s, <0>w. Plot of 0>w — 1) as a function of reduced time, kLt. Conditions and numbering as in Fig. 2.

From these relations and the tabulated data, D/L() and R are plotted against reduced time tft. 

As the basis for further comparison, let us return to the case of a fixed volume V which may be a single tank (Fig. 2.7a) or two tanks (Fig. 2.7b) or three tanks and so on. In the general case there will be i tanks and the concentration of the tracer leaving the last tank will be C,. If we now plot C,/C0, where C0 = n0/V, against the reduced time (vt/V), the family of curves shown in Fig. 2.9a is obtained. The curves are reduced C (i.e. outlet concentration) curves, as already indicated in Section 2.1.2. 

Figure 17. Normalized velocity autocorrelation function of a two-dimensional fluid plotted against reduced time. The plot shows the presence of the r 1 tail in Cv(t) at long time. The plot is at p = 0.6 and T = 0.7. The time is scaled by rsc = y/ma2/e. This figure has been taken from Ref. 188.

Figure 21. The long-time tails of Cv(t) obtained from simulations plotted against tjtf at various densities at T = 1.0. t is the reduced time and tc is the time at which the long-time tail of C (r) started approaching zero. The different symbols from top to bottom represent the Cv(t) at reduced densities 0.3,0.4,0.6, and 0.82, respectively. The figure shows the dominance of 1 /r3 decay in C (f) at low and intermediate densities. This figure has been taken from Ref. 186.

Figure 22. Gaussian decay of the dynamic structure factor F(q, t) in the relevant time range. In this figure, the ratio F(q, t)/S(q) has been plotted against the reduced time at a small value of the wavenumber, ql = 0.1206. This figure has been taken from Ref. 186.

Once again Fit) can be calculated from Eq. 9.2-36 in conjunction with Eq. 9.2-28. Figure 9.11 plots the RTD function F(t) versus reduced time t/t and compares it to the RTD function of Newtonian laminar flow in a pipe and that in a well-stirred vessel. The RTD function in the melt extruder is quite narrow, approaching plug-type flow. Only about 5% of the flow rate stays more than twice the mean residence time in the extruder. 

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