Second-order linear coefficients

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

Although individual runs for the first set of experiments follow the second-order rate law, the observed second-order rate coefficients, k, are strongly dependent on the initial amine concentrations, with the rate increasing regularly as the amine concentration increases. Nevertheless, for all of the measurements, a plot of k versus the initial amine concentration is linear, and the data can be fitted withegn.(4), with k equal to 1.87 x 10-4 l.mole-1. sec-1 and k" equal to 5.63 x 10-412. mole-2. sec-1. 

In conforming to an expected linear free energy relationship, the Ce(lV) oxidation of various 1,10-phenanthroline and bipyridyl complexes of Ru(II) in 0.5 M sulphuric acid are consistent with the requirements of the Marcus treatment . The results for the oxidation of the 3- and 5-sulphonic-substituted ferroin complexes by Ce(IV) suggest that the ligand does not function as an electron mediator, and that the mechanism is outer-sphere in type. Second-order rate coefficients for the oxidation of Ru(phen)j, Ru(bipy)3, and Ru(terpy)3 are 5.8x10, 8,8 X 10, and 7.0 x 10 l.mole . sec, respectively, in 0.5 M H2SO4 at 25 °C a rapid-mixing device was employed. 

Some problems with diffusion or dispersion give rise to the second order linear equation with constant coefficients,... 

Eq (9) is non-homogeneous second order linear with constant coefficients and can be solved readily. Boundary conditions are,... 

Second-Order Linear ODEs With Constant Coefficients... 

Kramer, M. A., Calo, J. M Rabitz, H., and Kee, R. J., AIM The Analytically Integrated Magnus Method for Linear and Second Order Sensitivity Coefficients. SAND82-8231, Sandia National Laboratories, Livermore, August, 1982b. 

It was found in transesterification of ethyl acrylate in the liquid phase over a non-porous KU-2 catalyst [464], that the structure of the alcohol influenced the value of the limiting sorption of alcohol by the ion exchanger, the logarithm of this value being a linear function of the dielectric constant of the alcohol. As the second-order rate coefficients yielded the same sequence as the limiting sorption values, viz. allyl alcohol > 1-butanol > 3-methyl-l-butanol, Filippov et al. [464] assumed a relation between the dielectric constant and the reactivity of the alcohols. 

As discussed in Vol. 2, pp. 354 and 372, logarithms of second-order rate coefficients for acid or base catalysis are linearly related to the pK values of the acids or bases, respectively (Bronsted relation). 

For the A-Sg2 mechanism, the entropy changes on activation are expected to be negative as translational and rotational freedom is also lost in the transition state of this mechanism. The experimental AS values for A-Sfc 2 reactions compiled by Matesich [52] (and supplemented by the writer, Table 4 [24, 44, 68—82]) are between —1 and —37 eu as far as they are concerned with reactions of H30+ with electrically neutral substrates. According to Matesich s findings [52], there is a rough linear correlation between AS and log k2 at 25 °C. The lower the second-order rate coefficient, k2, the more negative is AS. ... 

Following a flash, with C2N2 first-order decay was observed at fixed C2N2 pressure. The time constant of the decay increased linearly with C2N2 pressure. Over the temperature interval 301-447 °K the resulting second-order rate coefficient for the disappearance of CN was... 

The second-order perturbed coefficients in the original third derivative equation (19), and those introduced by the Handy-Schaefer device, appear in linear combinations like... 

Note that the viscosity // will appear linearly in each of the second-order tensor coefficients in (7 20) and (7 21) and thus has been factored out in (7 22) that is, A = // A,... 

The mass balance for a first-order reaction in a tubular reactor with a flow velocity of v and the concentration of reactant in feed, Cp undergoing a reaction with the rate constant of k is governed by the following second-order linear equation with constant coefficients ... 

Methods for solving mass and heat transfer problems. The convective diffusion equation (3.1.1) is a second-order linear partial differential equation with variable coefficients (in the general case, the fluid velocity depends on the coordinates and time). Exact closed-form solutions of the corresponding problems can be found only in exceptional cases with simple geometry [79,197, 270, 370, 516]. This is especially true of the nonlinear equation (3.1.17). Exact solutions are important for adequate understanding of the physical background of various phenomena and processes. They can serve as test solutions to verify whether the problem is well-posed or to estimate the accuracy of the corresponding numerical, asymptotic, and approximate methods. 

For neutral nucleophiles, we have utilized a series of ring-substituted N,N-dimethylanilines. The second-order rate coefficients should now be independent of nucleophile concentration, and this was confirmed by showing that log (k/k0 obtained from the product ratios, was independent of the amine concentration for 0.008 to 0.08 M N,N-dimethyl-p-toluidine. The log (k/k0) values could also be conveniently determined for m-CH3-, H-, p-Br-, and m-Cl-substituted derivatives (equation 13). For the m-N02 derivative, even at 0.32 M, the dominant reaction is solvolysis and only an approximate value for log (k/k0) could be obtained. A Hammett plot against the tabulated a values (43) (omitting the approximate m-N02 data) led to a linear plot and a slope (p value) of —2.77 0.15 (r = —0.996). This value is similar to values for reaction with other ethyl derivatives, derived from kinetically determined k values —3.60 for reaction with ethyl iodide in nitrobenzene at... 

Substituting the coefficients from (126) into (127), we obtain the system of two second-order linear differential equations... 

A plot of Atpbs versus [L] thus enables both k.y and ky to be determined and thus the binding equilibrium constant. For tightly binding ligands Ac, is often too small to measure by this method, i.e. the line appears to pass through the origin and only A , the second-order rate coefficient is obtainable. But, as Fig. 4 shows, when the affinity is relatively low both values may be determined. The initial (approximately) linear portion of the plot yields ky = 22/M/sec, k y = 1.2/sec. 

Y. Sibuya, Global Theory of a Second-Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North-Holland, Amsterdam, 1975. 

First-Order Linear Ordinary Differential Equation / 2.3.2 Second-Order Linear ODEs with Constant Coefficients / 2.3.3 Nth-Order Linear ODEs with Constant Coefficients... 

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