Second-order derivative

The above difference in reactivity of benzene and toluene is close to that observed in a less viscous system so that the results for compounds of the reactivity of benzene (and perhaps toluene) or less in these media are meaningful, but not for compounds which are normally more reactive. This conclusion was confirmed for nitration in nitric acid-perchloric acid of a range of compounds which normally are fairly reactive. The derived second-order rate coefficients (10k2) for nitration... 

Dewar and Mole236 derived second-order rate coefficients for chlorination at 25 °C of benzene (6xl0-7), diphenyl (6.9 xlO-4), naphthalene (6.3 xlO-2), phenanthrene (2.9xl0 1) and triphenylene (2.2xlO-2) in Analar acetic acid and of diphenyl (9 x 10-7), naphthalene (1.9 x 10-4), phenanthrene (1.3 x 10-3),... 

The rate-acidity profile for pyrimidin-2-one indicated reaction on the free base but since the derived second-order rate coefficient is 104 times greater than that for 2-pyridone, and the acidity dependence in the H0 region was also greater, the slope of log kt versus —H0 plot being 0.45, cf. 0.15 for 2-pyridone reaction was, therefore, postulated as occurring via a covalent hydrate, hydration taking place at the 4 position. Methyl substitution increased the rate as expected and N-methyl substitution produced a larger effect than 4,6-dimethyl substitution and this may be due to alteration of the amount of covalent hydration at equilibrium. The data... 

Finally, the relative effectiveness of metal ions in catalysing this reaction has been measured628 and the first-order coefficients together with the derived second-order coefficients are given in Table 203 for the protodeboration of 2,6-dimethoxybenzeneboronic acid at 90 °C, p = 0.14 and pH = 6.70. 

Quantitative measurements of simple and enzyme-catalyzed reaction rates were under way by the 1850s. In that year Wilhelmy derived first order equations for acid-catalyzed hydrolysis of sucrose which he could follow by the inversion of rotation of plane polarized light. Berthellot (1862) derived second-order equations for the rates of ester formation and, shortly after, Harcourt observed that rates of reaction doubled for each 10 °C rise in temperature. Guldberg and Waage (1864-67) demonstrated that the equilibrium of the reaction was affected by the concentration ) of the reacting substance(s). By 1877 Arrhenius had derived the definition of the equilbrium constant for a reaction from the rate constants of the forward and backward reactions. Ostwald in 1884 showed that sucrose and ester hydrolyses were affected by H+ concentration (pH). 

Table XIV contains data for computationally derived second-order hyperpolarizabilities, principally by Marder, Ratner, and their co-workers, and more recently by Humphrey et al. The focus of these studies has been similar to that of experimental investigations, namely ferrocenyl, metal carbonyl, and metal acetylide complexes, and very recently carboranes. Calculated values were all obtained using ZINDO, the methodology for which is described in Section II,E.

The false-transient method can be applied to convective diffusion equations in a manner similar to that used for velocity profiles. Finite-difference approximations are written for the spatial derivatives. Second-order approximations can be used for first derivatives since they involve only the same five points needed for the second derivatives. The result is a set of simultaneous ODEs with (false) time as the independent variable. The computational template of Figure 16.3 is unchanged. The next two examples illustrate its application to problems where axial diffusion is negligible. Such problems are also readily solved by the method of lines as described in Chapter 8. Cases with significant axial diffusion are troublesome for the method of lines and require special boundary conditions for the method of false transients. They are treated in Section 16.2.4. 

A kinetic study and mechanistic investigation by UV spectrophotometry has been carried out on the reactions between triphenylphosphine, dialkyl acetylenedicarboxylates and NH-acids (such as Harman acids). A Bronsted-type plot of the derived second-order rate constants for the reactions of a series of primary amines (in water) with benzoyl methyl... 

A stopped-flow radiolysisstudy of the reaction of superoxide with ferricyto-chrome c has been reported. The derived second-order rate constant of 2.6 x 10 1 mol S" at pH 9.0 is higher than previously reported values under the same conditions and refers to the configuration below pH 7.4, which undergoes a slow equilibration if stored above pH 9. 

In order to evaluate the derivatives, second-order response theory is employed within either a relativistic or a nonrelativistic formulation. 

From (1) it is clear that the phase contrast can be interpreted simply in terms of tbe variation (second order derivative) of the projected image density, and increases with improving resolution of the system, in agreement with the findings of [3]. 

If the desorption rate is second-order, as is often the case for hydrogen on a metal surface, so that appears in Eq. XVIII-1, an equation analogous to Eq. XVIII-3 can be derived by the Redhead procedure. Derive this equation. In a particular case, H2 on Cu3Pt(III) surface, A was taken to be 1 x 10 cm /atom, the maximum desorption rate was at 225 K, 6 at the maximum was 0.5. Monolayer coverage was 4.2 x 10 atoms/cm, and = 5.5 K/sec. Calculate the desorption enthalpy (from Ref. 110). 

Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. 

A very simple procedure for time evolving the wavepacket is the second order differencing method. Here we illustrate how this method is used in conjunction with a fast Fourier transfonn method for evaluating the spatial coordinate derivatives in the Hamiltonian. 

Following the derivation of the linear susceptibility, we may now readily deduce the second-order... 

To calculate the matrix elements of second-order derivatives, we have... 

The last kind of second-order derivative considered is of the following fomi ... 

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

For the combined scheme (21), (23), second-order error bounds are derived in [14], These bounds hold independently of the size of the eigenvalues of T, and without assumptions about the smoothness of the solution, which in general is highly oscillatory. 

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