Third-order equations

With these kinetic data and a knowledge of the reactor configuration, the development of a computer simulation model of the esterification reaction is iavaluable for optimising esterification reaction operation (25—28). However, all esterification reactions do not necessarily permit straightforward mathematical treatment. In a study of the esterification of 2,3-butanediol and acetic acid usiag sulfuric acid catalyst, it was found that the reaction occurs through two pairs of consecutive reversible reactions of approximately equal speeds. These reactions do not conform to any simple first-, second-, or third-order equation, even ia the early stages (29). 

U 2 - JI density matrix is second-order and the initial state I) can evolve after the second scattering caused in both bra and ket states caused by the applied field. Further, three-photon absorptions with frequencies uj, 0J2, and can be described by p t) = ///dwidw2dw3/2(a i,u 2,W3)e= ( i+ 2+" ), which obeys a third-order equation of motion,... 

Po is the standard pressure and px is the partial pressure of component X. Rearrangement of Eq. (56) leads to a third-order equation in t, which can be solved iteratively. Notice that the pressures used in Eq. (55) should actually be replaced by activities, implying that they should be corrected by their respective fugacity coefficients, which are of importance when dealing with methanol and water. We leave it as an exercise for the reader to judge the influence of such effects, utUizing the relation between pressure and activity given in Eq. (39) of Chapter 2. 

In the polyesterification process p is directly calculated from the carboxyl group titer. Results for the polyesterification reaction between diethylene glycol and adipic acid at 166° and 202°C are plotted in Fig. 3 in accordance with the third-order equation (8). For comparison purposes, the course of the non-polymer-forming reaction of diethylene glycol with the monobasic acid, caproic, is also shown. Eq. (8) is not obeyed from zero to 80 percent esterification [l/(l—p) =l to 25], as is shown by the curvature over this region. From 80 to 93 percent esterification the reaction appears to be third order. The non-polymerforming esterification of diethylene glycol with caproic acid (and other... 

Higher order ODEs are reduced to a set of first order equations for solution by these softwares. Thus the third order equation,... 

AgC2H302 + NaCH02 = 2 Ag + C02 + HC2H302 + NaC2H302 The starting mixture has Aq =0.1 mol/liter and B0 = 0.05. The data are of t in minutes and change of concentration of the silver compound. Check second and third order equations. 

Combining the results of Kieffer s model and of laboratory experiments, Clayton and Kieffer (1991) obtained a set of third-order equations relating the reduced partition functions of various minerals to the inverse of the squared absolute temperature (table 11.9) according to... 

Consider the behavior for small angles, that is, nearly parallel plates for which a < 1. From Eq. 5.10 it is apparent that the system reduces to a particularly simple third-order equation... 

It may be noted that Eq. 5.45 could be solved as a third-order equation with a shooting technique. The equation could be also be transformed into a first-order and a second-order equation, which are solved as discussed in Section 5.2.2. In these cases the boundary conditions manifest themselves in somewhat different ways. 

The pressure-gradient terms can be eliminated by cross differentiating the momentum equations and subtracting. The following third-order equation emerges ... 

Generally speaking, we prefer to use the equations in the form of Eqs 6.40 and 6.41, rather than transform to the F form. For the numerical solutions used here, there is no advantage to the single third-order equation compared to the system of equations. Furthermore the F equation has lost any clear physical meaning. The physical form of the equations can accommodate variable densities or viscosities without difficulty, but the F form of the equations loses its appeal in this case. Finally, the overall objective is to include variable properties, as well as to consider the coupled effects of thermal energy and species transport. Therefore the discussion on the F form of the equations is included here mainly for historical perspective. 

Solution by Shooting Solution of the boundary value problem described by Eq. 6.59 is usually accomplished numerically by a shooting method. To implement a shooting method, the third-order equations is transformed to a system of three first-order equations as... 

In the computer model the simultaneous equations (5), (6) and (8) are combined into a single third-order equation for Ce, which is solved iterately for each integration step. For relevant values of the various parameters, the equation has a single positive solution. 

Above investigation for the behaviour of the solution at the free stream can be extended to other equations of (2.4.15) to check their usefulness in obtaining the eigen function. At the free stream the characteristic roots for Eqn. (2.4.15a) are —a, —Q. Equation (2.4.15b) being a third order equation has three roots given by [—a, —Q, a- -Q ]. Thus, this equation is also violently unstable, even for low wave numbers and Reynolds numbers. Equation (2.4.15c) has the asymptotic behaviour for large y s as dictated by the characteristic roots given by [—a, —Q, Finally, the characteristic... 

A number of analytical PVT relationships are referred to as cubic eqaations of state because, when expanded, they yield third-order equations for the specific volume. The van der Waals equation of stale is the earliest of these expressions, and it remains useful for discussing deviations from ideal behavior. 

The third order equation provides the "z-average molecular weight" and is calculated from the expression ... 

These third-order equations have been used in many applications in which molecular EAs have been computed for a wide variety of species as illustrated in Ref. [16]. Clearly, all the quantities needed to form the second- or third-order EOM matrix elements Hj. are ultimately expressed in terms of the orbital energies sj and two-electron integrals j, k l, h) evaluated in the basis of the neutral molecule s Hartree-Eock orbitals that form the starting point of the Mpller-Plesset theory. However, as with most electronic stmcture theories, much effort has been devoted to recasting the working EOM equations in a manner that involves the atomic orbital (AO) two-electron integrals rather than the molecular orbital based integrals. Because such technical matters of direct AO-driven calculations are outside the scope of this work, we will not delve into them further. 

The full third-order equation is relatively complex. It is normally written in the form (for 3 components) ... 

If we move to a third-order equation, we obtain three solutions, two of which are imaginary as shown in what follows ... 

The oxygen-oxygen interaction parameter y enters the TB solution only when two components of the k vector differ from zero. The energy levels in M can be obtained by solving the third order equation... 

A similar analysis can be made for any other type of inhibition. An interesting situation occurs in the case of uncompetitive inhibition by high substrate concentration. In this case, a steady-state analysis renders a third-order equation in ps that for certain values of the kinetic and diffusion parameters may give three positive values of Ps for one value of Po and a stability analysis should be made to assess the right value. The intermediate value is always unstable but the upper or lower... 

Using the data of Table 1.2, develop an equation relating the molarity as the dependent variable and molality as the independent variable. Does a second or third order equation significantly improve the fit ... 

In the absence of a solvent and excess of chloral, the reaction rate for this reaction may be expressed by a third-order equation-second order with respect to dimethyl H-phosphonate and first order with respect to the chloral [177]. In dioxane solution and excess dimethyl H-phosphonate, the dependence of the reaction rate on the chloral concentration is the same. In addition to the chloro-containing a-hydroxyalkyl phosphonate, which is the main product under the above conditions, a side product formed as a result of dehydrochlorination of the main product, followed by phosphonate-phosphate rearrangement, has been also isolated. The presence of a base such as triethylamine, alkali metal alkoxides and hydroxides, or sodium carbonate accelerates the dehydrochlorination process. An example of this side reaction is the transformation of dialkyl-2,2,2-trichloro-l-hydroxyethyl phosphonates into dialkyl-2,2-dichlQrovinyl phosphates in the presence of sodium hydroxide [181,182]. 

The methanolysis of p-nitrobenzoyl chloride in acetonitrile is a mixed second-and third-order reaction in methanol (equation A). When a chloride salt is present, the kinetics are strictly third-order (equation B) ... 

This is a third-order equation for x , and it might occur that it has three positive roots. For the special case P = 0 (5.162) yields... 

Jl, J2 and J3 are invariant coefficients of stress and, therefore, independent of the choice of coordinate system. liquation (1.23) is a third-order equation for stress, a, with three roots, ttmt. Iso termed principal stresses . Such solutions may be inserted into Eq. (1.22c) by utilizing the relation ... 

The solvability condition (Fredholm alternative) for the third-order equation in the J-hierarchy originated from the nonlinear eq. (97) implies the following relationship ... 

The corresponding solution for the asymptotic forms (9.254) is obtained by eliminating (t) in (9.254a) by the application of (9.2546). The result is a third-order equation involving a delayed term. 

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