Computer curves

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

Figure 7. Computed curves of dimensionless heat generation fg (0) and heat removal r ( ) functions showing stable behavior of dimensionless temperature for nonautonomous systems (3)...

The flatness of the computed curves in Fig. 6 shows that in the case of spheres in contact F is insensitive to rather large changes in V (or / ). Mason and Clark (M3) have measured the pendular bonding force as a function of the separation distance and bridge volume. Considering the experimental uncertainty involved, their results follow the expected trend quite satisfactorily. 

The residual plot of the difference between the best computed curve and the experimental data is shown in Figure 8. The largest variance is observed in the vicinity of the maximum isocyanate absorbance and probably arises because of the large changes in concentration which arise during the early stages of cure. 

Figure 8. The experimental data and computed curve of the active material utilization factor...

Using modern computer curve-fitting techniques,102 equations (17) and (27) can be fitted simultaneously with variables X, log CH+ and D, obtaining pATBH+. m, DB and DA as coefficients, which is a more precise technique than calculating log/ values first and using equation (17) and the error function61 discussed above. 

Dr. Spijkerman No, that will not do. The spectral lines should be resolved otherwise manual and computer curve fitting could give diflFer-ent results. 

Figure 25.7 Calculated peak values of Xno for p = 1 bar. Curves with symbols represent numerical computations curves without symbols represent the RRA analysis 1 — To = Tf = 300 K, 2 — 500 K... 

Fig. 12. Schematic curves (top, bottom) of the calculated free energy of hydration AGhydr of two hypothetical HT materials (top Si-rich/Ca-poor bottom Si-poor/Ca-rich) as a function of the pH. The shaded areas are the actual ranges of computed AGhydr of HT materials, categorized as a function of the HT process from which they originate. For comparison, the computed curve for the standard HLW glass SON68 is also shown (dotted curve).

In the first case, the multiplet skeleton is preserved. As the parameter a increases, the multiplet collapse proceeds. In Figures 3 and 4 there are shown examples of computed curves and the corresponding spectra... 

The compositions of the polymers were determined by simulation of their spectra using a computer curve simulator-plotting program developed by B. L. Bruner at the University of Kentucky. An example of the output of the program is shown in Fig. 4. The input data are the positions, amplitudes, and... 

Table 4 shows the effect of monomer concentration, coinitiator concentration, and conversion on the composition of poly(4-methyl-1-pentene) using EtAlCl2 coinitiator at — 50° C. The 1,2-, 1,3-, and 1,4-repeat unit concentrations in the polymer have been determined from polymer spectra by use of a computer curve simulator-plotting program and are rounded to the nearest percent. No limits of error are indicated since none could be determined analytically. A reasonable error is thought to be +15% of the measured value. 

Fig. 9.6 Computed curves of dimensionless flow rate versus dimensionless pressure gradient for isothermal flow of a power law model fluid in shallow screw channels with the power law exponent n as a parameter, for helix angles 6f as follows O, 30° A, 20° , 10° solid curves are for a helix angle 30°. Note that for n < 1, the reduced flow rate is less than 1, with the deviation diminishing with decreasing of the helix angle. [Reprinted with permission from R. M. Griffith, Fully Developed Flow in Screw Extruders, Ind. Eng. Chem. Fundam., 1, 180-187 (1962).]...

In order to achieve an amplification of chirality, it requires that/> 1. If P = 0 (no meso catalyst) or g = 1 (same reactivity of meso and homochiral catalysts), then/= 1. The condition/> 1 is achieved for 1 + p > 1 + g ), or g < 1. Thus the necessary condition for asymmetric amplification in the above model is for the heterochiral or meso catalyst to be less reactive than the homochiral catalyst. If the meso catalyst is more reactive, then/< 1, and hence a negative nonlinear effect is observed. The size of the asymmetric amplification is regulated by the value off, which increases as K does. The more meso catalyst (of the lowest possible reactivity) there is, the higher will be eeproduct. This is well illustrated by computed curves in Scheme 11. The variation of eeproduct with eeaux is represented for various values of g (the relative reactivity of the meso complex) with K = 4 (corresponding to a statistical distribution of ligands Scheme 11, top). The variation in the relative amounts of the three complexes with eeaux is also represented for a statistical distribution of ligands (Scheme 11, bottom). 

Kinetics for the return of the radical pair from Cp2Co+ Mn(CO)s" are measured by following the absorbance change of Mn(CO) at the monitoring wavelength of Amon = 800 nm, and the second-order kinetics for the disappearance of Mn(CO)5 is demonstrated by the excellent fit of the smooth computed curve to the experimental decay. The second-order rate constant k2 evaluated in this manner is insensitive to solvent variation. Disappearance of the radicals derived from such carbonylcobaltate salts as Cp2Co+ Co(CO)4 also follows the same second-order kinetics. It is noteworthy that the CT excitation of the quinolinium salt Q+ Co(CO)4- allows a pair of second-order rate constants to be extracted from the absorbance decays at 2 = 550 and 780 nm. Assignment of the latter to the 17-elec-... 

Figure 1. Computer curves for loss of folding endurance (washed Kraft, MD) with rate constants of Table IV. The curves pass through the data for (D) 70°C, (O) 80°C, and (+) 90°C. The data for the curve at 70°C has been displaced upward by 300 units for clarity.

A simple example of exchange occurs in N,N-dimethylformamide, as illustrated in Fig. 2.15. The C—N bond in an amide has partial double bond character hence rotation about this bond is highly restricted but not entirely precluded. At room temperature the rotation rate is sufficiently slow that separate sharp lines are observed for the two methyl groups, which are in different environments because of proximity to the carbonyl group (see Chapter 4). With increasing temperature, however, the barrier to rotation is gradually surmounted, and the observed spectra follow the theoretically computed curves of Fig. 2.14. 

Hence, if values for Dp and F are assumed, the curve of E vs. t can be computed from the model. Computed curves were fitted to the experimental curves by finding the values of Dp and F which minimized the sum of squares of errors in E at the experimental points these curves are shown as continuous lines in Figure 4. 

If two or more transitions overlap, the quantitative interpretation of the MCD spectrum becomes much more difficult and computational curve fitting that requires simple band shapes is needed. Often, the vibrational structure of the MCD spectrum, which will in general be different from the one in the absorption spectrum, will be so pronounced that this assumption is not fulfilled. Although uncommon, there are some cases known where some sections of the vibrational envelope associated with a single electronic transition are positive and some negative. Thus, the appearance of both positive and negative MCD peaks in a spectral region does not necessarily mean that... 

The quantitative comparisons with the available experimental data are less favorable in this case. The transition state and product in water appear to be shifted up in energy by about 15 kcal/mol. The computed curves are more in line with experimental data for ketones, where formation of hydrates is far less favorable than for formaldehyde. The discrepancy likely comes from an overly exothermic hydration energy for the charge-localized hydroxide ion, which lowers the reactant end of the pmfs. This results from the use of two-body potential functions, that is, the TIP4P water molecules are not polarized by the ion, so water-water repulsions between molecules in the first solvent shell are underestimated. Until the polarization can be explicitly treated, ions that have attractive interactions with single water molecules greater than about 18-20 kcal/mol should probably be avoided. For example, CN would be less problematic since its single molecule hydration enthalpy is only 14 kcal/mol, versus 25 for... 

Figure 1 Dynamic Polarizability, a(w) for H20 in the gas phase. Curves A and B are alternative interpretations of the experimental data (see text) C and D computed curves from Christiansen et al. and Kongsted et al45 filled points are from Poulsen et al46 open points from Jensen et al. All the theoretical values have been increased at all frequencies by 0.29 au, the estimated ZPVA correction...

The main purpose of this work was to reproduce the whole MWD and the objective function of the non-linear regression was to minimize the sum of relative errors. Determination of each basic model starts with one component and the number of components is increased until an acceptable fit is obtained between the computed curve and measured one. Agreement has to be reached also between the values of the computed and experimental molecular averages. 

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