Deviations from First Order Appearance

Deviations from 1st order spectra occur when the high field condition D hv does not strictly apply  

The spectrum schematically shown in Fig. 4.10 is of the type first observed [24] for triplet state naphthalene molecules at low temperature. In this case the Ams = 2 

10 Schematic appetitance of the powder spectium of a triplet state. Axial symmetry is assumed so that the Ants = 1 perpendicular and parallel features are separated by D and 2- D, respectively. The = 1 - -1 Ants = 2 transition is a single, narrow line, unaffected by anisotropy of the zfs 

The half field line usually saturates less readily than the A.ms = 1 lines and can therefore be recorded at higher microwave power to enhance the intensity. The reason is that a weakly allowed transition is less efficient to equalize the populations of the magnetic levels than an allowed one so that saturation does not occur. Specially designed cavities employing parallel B and B fields make it even easier to observe the Ams = 2 transitions [25]. The observation of Arris = 2 lines is as mentioned of value experimentally particularly to verify the assignment of 5 = 1 species from powder spectra. 

Second order corrections can be relatively easily applied to systems with axial symmetry when 1st order treatment is insufficient, i.e. when D sc hv does not strictly apply. One can note from the equations for axial symmetry in Chapter 3 that, for the S = 1 case, 2nd order energies contribute equally to the = 1 0 and 

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The pressure-dependence was studied only at 200°C, between 0.12 and 16 mm Hg the rate appeared to be proportional to p0-75. If this deviation from first-order kinetics is neglected, their result can be rewritten as ... 

Any explanation of first-order kinetics should be general enough to account for the first-order kinetics (and deviations from first-order kinetics) on Ge (and perhaps C) surfaces. In particular, defect-mediated mechanisms that may appear feasible on Si surfaces would require modification since the defect density and bond energies on the Ge surface are very different. The prepairing model has already been shown to model the kinetics on Ge [44]. 

Petersen [85] used some of the data of Austin and Walker [107] to show how the first-order criterion would not be correct here the data are recalculated on the basis of Eq. 3.6.C-7. This reaction appears to be very strongly inhibited by adsorption of the product, carbon monoxide, which leads to large deviations from first-order beliavior. The rate equation, in concentration units, was of the standard adsorption type. 

The feasibility of designing polymer-chelator conjugates from which release of the chelator is triggered reversibly by the appearance of the target (toxic) metal in the circulatory system was tested by synthesis and measurement of the rates of metal-catalysis hydrolysis of the ester of poly(vinyl alcohol) and quinaldic acid at pH 7.5. Hydrolysis in 50% aqueous ethanol solution in the presence of the four m.etals, Cu(II), Ni(II), CodI) and Zn(II) at pH 7.5 was first order, with half lives of 67, 71, 172 and 476 min., respectively. The rate of hydrolysis in the absence of metals was not measurable. A double reciprocal plot of kobs vs. [M] for Ni(II) exhibited the expected linearity. For films of the ester, the deviation from first order kinetics was consistent with a contributing diffusion process. 

As the ratio of A 8v (the difference in chemical shift between two coupled nuclei) to J decreases, the relative intensities of the lines in a multiplet deviate further from first-order (e.g., Pascal triangle) ratios. Inner lines (those facing the coupled multiplet) increase in intensity, while outer lines lose intensity. This slanting of the multiplets is one type of second-order effect. At very small values of A Sv/J, not only may extra lines appear in the multiplets but also apparent line positions and spacings may not equate with true chemical shifts and coupling constants (e.g., deceptive simplicity and virtual coupling). 

K, the static disorder is certainly maintained. The results are presented as plots of formula in Fig. 7. The deviations from linearity of the plots is small enough to support such method of analysis. The slopes of the curves give the 5a values tabulated in Table 4. It follows that in the (1 x l)Co/Cu(lll) case the anisotropy of surface vibrations clearly appears in the measured values of 8a and 5aT There are two reasons for such anisotropy the first is a surface effect due to the reduced coordination in the perpendicular direction. cF is a mean-square relative displacement projected along the direction of the bond Enhanced perpendicular vibrational amplitude causes enhanced mean-square relative displacement along the S—B direction. The second effect is due to the chemical difference of the substrate (Fig. 8). S—B bonds are Co—Cu bonds and the bulk Co mean-square relative displacement, cr (Co), is smaller than the bulk value for Cu, aJ(Cu). Thus for individual cobalt-copper bonds, the following ordering is expected ... 

The magnitude of the correction to the simple (zero-order) autocatalator provided by the first-order equations, (14) and (15) for the time dependent behaviour and (17) for the stationary-states, can be seen quantitatively for a typical system from the first column of numerical results in Table 1. The corrections in ass and /3SS appear always to be an order of magnitude less than e and k, even when these parameters are of order unity. The second column, which gives the solutions of the full set of three equations (9a)-(9c), shows that the first-order equations actually overestimate the deviation from the e = k = 0 autocatalator. Returning to column I, we see that the stationary-state concentration of X is adequately given by ss = kol, even for e = k = 0.1, and is very accurately given by ss = Kass/3ss(l - e ) even for e = k = 1. Notice that for row (d) there are three stationary-state solutions at Tres = 225. 

The relative rates for the p-fluoro substituent deviate even more seriously. At first glance, the discrepancies appear far greater than could be tolerated in a simple first-order treatment (Fig. 45). Considerable scatter from a line defined by the average value of greater than for m-t-butyl (Fig. 42). Moreover, the absence of a relationship between log pf and the reaction constants is also certain. More careful inspection indicates that the data define a rather narrow band of reactivity. The behavior of this group is more clearly portrayed by another test equation which eliminates hydrogen as the abscissa (33). 

Hexanol In contrast, consider the 300 and 600 MHz spectra of 1-hexanol (Figure 3.52). At first glance at the 300 MHz spectrum, the three-proton triplet at 8 0.87 seems odd for the CH3 group since the peak intensities deviate from the first-order ratios of 1 2 1. Furthermore, the filled-in appearance of the expanded set is obvious despite a reasonable A vU value of 13 for the CH2—CH3 groups. 

In the crystallization of calcium sulfate dihydrate, Schierholtz (S7) found that the growth process appears to follow a first-order rate law for a large part of the reaction (see Fig. 20). The solid line shows the experimental results obtained during a typical run. The time interval preceding do is taken as the induction period. Deviations from linearity at the ends... 

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