Third-order constants

The recombination of Br atoms in the presence of N2 at 1000°K has a third-order constant equal to 2 X 10 liters /mole -sec. losing the thermodynamic data for the dissociation of Br2, calculate at 1000 K the specific second-order rate constant for the rate of dissociation of Br2 by N2. What is the frequency factor for the rate constant How does it compare with collision freqticncies ... 

Actually, Baker s mechanism requires that at low alcohol concentrations at which fcs[A] <5C h the kinetics follow strictly third-order kinetics, second order with respect to alcohol concentration and first order with respect to the isocyanate concentration. Therefore, the proper way to handle Baker s data would be to calculate the third-order velocity constants for each initial alcohol concentration. A plot of the reciprocal of the third-order constant vs. the alcohol concentration could then be expected to give a straight line relationship. 

Diffusivity of oxygen in the liquid, 2) = 1.8 x I0 cm /s Third-order constant for the reaction... 

This shows directly that the nonlinearity of the dynamic resporrse is due both to geometrical nonlinearities (in conjimction with second-order material constants) and to material nonlinearities as expressed by the various third-order constants. 

Of course, if we speak of nonhnear behaviom this also includes effects of even higher order. The next step would be a further extension of the series expansion of the potentials to include fourth-order derivatives. Taking a short look at (6.10) (6.11), (6.12), (6.13), (6.14), (6.15), (6.16), and (6.17) and considering how the mixed derivatives would multiply, this becomes a tall order indeed. Considering the state of the art concerning third-order constants it is highly questionable if this further step would make much sense, at least for the time being. 

Adam W, Ticliy J, Kittinger E (1988) The different sets of eleetrical, mechanical, and electromechanical third-order constants for quartz. J Appl Phys 64 2556 Becker R, Sauter F (1973) Theorie der Elektrizitat (in German). Teubner, Stuttgart Boyd RW (1997) Nonlinear optics. Academic, San Diego, CA... 

The technique of Brillouin spectroscopy (Section 6.3.3) has been applied to determine the elastic constants of oriented polymer fibres. Early studies of this nature were undertaken by Kruger et al. [46,47] on oriented polycarbonate films, also determining the third-order constants, which define the elastic non-linear behaviour. Wang, Liu and Li [48,49] have described measurements on oriented polyvinylidene fluoride and polychlorotrifluoroethy-lene films. In the latter case the results were interpreted using an aggregate model differing in detail from that of Ward discussed in Section 8.6.2. 

The kinetics of the nitration of benzene, toluene and mesitylene in mixtures prepared from nitric acid and acetic anhydride have been studied by Hartshorn and Thompson. Under zeroth order conditions, the dependence of the rate of nitration of mesitylene on the stoichiometric concentrations of nitric acid, acetic acid and lithium nitrate were found to be as described in section 5.3.5. When the conditions were such that the rate depended upon the first power of the concentration of the aromatic substrate, the first order rate constant was found to vary with the stoichiometric concentration of nitric acid as shown on the graph below. An approximately third order dependence on this quantity was found with mesitylene and toluene, but with benzene, increasing the stoichiometric concentration of nitric acid caused a change to an approximately second order dependence. Relative reactivities, however, were found to be insensitive... 

The rate of a process is expressed by the derivative of a concentration (square brackets) with respect to time, d[ ]/dt. If the concentration of a reaction product is used, this quantity is positive if a reactant is used, it is negative and a minus sign must be included. Also, each derivative d[ ]/dt should be divided by the coefficient of that component in the chemical equation which describes the reaction so that a single rate is described, whichever component in the reaction is used to monitor it. A rate law describes the rate of a reaction as the product of a constant k, called the rate constant, and various concentrations, each raised to specific powers. The power of an individual concentration term in a rate law is called the order with respect to that component, and the sum of the exponents of all concentration terms gives the overall order of the reaction. Thus in the rate law Rate = k[X] [Y], the reaction is first order in X, second order in Y, and third order overall. 

To estimate the third-order derivatives of the function w with respect to y, we make use of the following fact (see Duvaut, Lions, 1972). Let O d E be a bounded domain with smooth boundary and let u be a distribution on O such that u, Du G Then u G L 0) and there is a constant c,... 

Thus, to third order in strain, the entropy along the Hugoniot is constant, and weak shock waves are nearly isentropic. For small strains, the Hugoniot can be replaced with the isentrope to a high degree of accuracy. At the initial state, the Hugoniot and isentrope have the same slope and curvature in the P-V plane. 

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. 

Nc = 0.0 gmol, Nq = 0.0 gmol, respectively. A mixture of A and B is charged into a 1-liter reactor. Determine the holding time required to achieve 90% fractional conversion of A (X = 0.9). The rate constant is k = 1.0 X 10 [(liter) /(gmoP min)] and the reaction is first order in A, second order in B and third order overall. 

Table 2.1. Third-order elastic constants determined from Hugoniot elastic limits (after Davison and Graham [79D01]).

A number of such phenomena or materials characteristics are listed in Table 2.5. The noted effects include mechanical, physical, and chemical processes. The positive third-order elastic constants were described in Sec. 2.2. 

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

What are the units of first-order, second-order, and third-order rate constants ... 

The isolation technique showed that the reaction is first-order with respect to cin-namoylimidazole, but treatment of the pseudo-first-order rate constants revealed that the reaction is not first-order in amine, because the ratio k Jc is not constant, as shown in Table 2-2. The last column in Table 2-2 indicates that a reasonable constant is obtained by dividing by the square of the amine concentration hence the reaction is second-order in amine. For the system described in Table 2-2, we therefore find that the reaction is overall third-order, with the rate equation... 

Bimolecular rate constants determined at temperatures giving conveniently measurable rates and calculated for the temperature given in parentheses, except for some of the catalyzed reactions (lines 1-4 and 14—19) which are third-order. 

Thermal initiation of styrene has been shown to be third order in monomer. The average rate constants for third order initiation determined by Hui and Hamielec is k = 105 34 e(,j8iaT) (M V).- "0 The rate constant for formation of the Mayo dimer determined in trapping experiments with nitroxidcs (Scheme 3.63) or acid (Scheme 3.64) as kn = 104 4 (M ls 1)j21 is substantially higher than is... 

In 75 % aqueous acetic acid, the bromination of fluorene at 25 °C obeys second-order kinetics in the presence of bromide ion and higher orders in its absence287, with Ea (17.85-44.85 °C) = 17.4, log A = 10.5 and AS = —12.4 however, these values were not corrected for the bromine-tribromide ion equilibrium, the constant for which is not known in this medium, and so they are not directly comparable with the proceeding values. In the absence of bromide ion the order with respect to bromine was 2.7-2.0, being lowest when [Br2]initial was least. Second- and third-order rate coefficients were determined for reaction in 90 and 75 wt. % aqueous acetic acid as 0.0026 and 1.61 (k3/k2 = 619), 0.115 and 12.2 (k3/k2 = 106) respectively, confirming the earlier observation that the second-order reaction becomes more important as the water content is increased. A value of 7.25 x 106 was determined for f3 6 (i.e. the 2 position of fluorene). 

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