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The secondary reactions are series with respect to the chloromethane but parallel with respect to chlorine. A very large excess of methane (mole ratio of methane to chlorine on the order of 10 1) is used to suppress selectivity losses. The excess of methane has two effects. First, because it is only involved in the primary reaction, it encourages the primary reaction. Second, by diluting the product, chloromethane, it discourages the secondary reactions, which prefer a high concentration of chloromethane. 

Polyethylbenzenes (diethylbenzene, triethylbenzene, etc.) are also formed as unwanted byproducts through reversible reactions in series with respect to ethylbenzene but parallel with respect to ethylene. For example,... 

The secondary reactions are parallel with respect to ethylene oxide but series with respect to monoethanolamine. Monoethanolamine is more valuable than both the di- and triethanolamine. As a first step in the flowsheet synthesis, make an initial choice of reactor which will maximize the production of monoethanolamine relative to di- and triethanolamine. 

Correspondingly, Equation (3.9) can be solved, expanding the solution in a series with respect to this parameter ... 

Referring back to the rate equation for a first-order reaction (Equation A1.2), we have a differential equation for which the derivative of the variable ([S]) is proportional to the variable itself. Such a system can be described by an infinite series with respect to time ... 

These reactions may be regarded as parallel with respect to species A and series with respect to species B, C, and D. Only species A and B are present at time zero. 

Although applications of perturbation theory vary widely, the main idea remains the same. One starts with an initial problem, called the unperturbed or reference problem. It is often required that this problem be sufficiently simple to be solved exactly. Then, the problem of interest, called the target problem, is represented in terms of a perturbation to the reference problem. The effect of the perturbation is expressed as an expansion in a series with respect to a small quantity, called the perturbation parameter. It is expected that the series converges quickly, and, therefore, can be truncated after the first few terms. It is further expected that these terms are markedly easier to evaluate than the exact solution. 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

This integral of the electronic dipole moment operator is a function of a nuclear coordinate Q. The integral may be expanded in a Taylor series with respect to Q (equation 4) and... 

An example of a reacting system with a network involving reactions in series is the decomposition of acetone (series with respect to ketene) (C)... 

This network is series with respect to HCHO and parallel with respect to CH4 and 02.)... 

A kinetics scheme for a set of (irreversible) reactions occurring in series with respect to species A, B, and C may be represented by... 

Consider the following mechanism for step-change polymerization of monomer M (Px) to P2, P3,..., Pr,. The mechanism corresponds to a complex series-parallel scheme series with respect to the growing polymer, and parallel with respect to M. Each step is a second-order elementary reaction, and the rate constant k (defined for each step)1 is the same for all steps. 

This network is a series-parallel network series with respect to HCHO in steps (1) and (2), parallel with respect to CH4 in steps (1) and (3), and parallel with respect to 02 and H20 in all steps. The rate constants kl, k2, and are step rate constants (like k in equation... 

Following the approach of Runge and Gross [13], Ghosh and Dhara [14] proved that if the two potentials (r, t) and A(r, t) can be expanded into Taylor series with respect to time around t = t0, both the potentials are uniquely (apart from only an additive TD function) determined by the current density j(r, t) of the system. 

As mentioned earlier, to solve explicitly for the temperature T2 and the product composition, one must consider a, mass balance equations, (/j, a) nonlinear equilibrium equations, and an energy equation in which one of the unknowns T2 is not even explicitly present. Since numerical procedures are used to solve the problem on computers, the thermodynamic functions are represented in terms of power series with respect to temperature. 

For a range of potential in which the interfacial charge is relatively small, the reciprocal of the interfacial electric capacity, C, of metal electrodes has conventionally been represented by a Laurent series with respect to the Debye length L-o of aqueous solution as shown in Eqn. 5-25 [Schmickler, 1993] ... 

In this Section we discuss the ROPM integral equations for the recursive scheme introduced in Sec. 2.2, restricting the analysis to the order s. An extension to higher order can proceed along the same lines. Expanding the energy and the potential into a power series with respect to A-,... 

This form has the advantage of not containing concentration value m and thus permits the explicit expression of x. from the basic equation. The expansion of the general function given by "Equation 17 into the MacLaurin series with respects to molalities m yields the equation ... 

Here, reaction proceeds in parallel with respect to B, but in series with respect to A, R, and S. 

K being a constant which is usually determined experimentally during cell calibration. Lj is the heat of evaporation of the solvent, the density of the solution, and c the polymer concentration. Finally, because the given deviation is valid only for ideal solutions but only real solutions can be studied in practice, the above equation is developed in a power law series with respect to c ... 

This situation is encountered when one or more chiral units are created and the diastereo-dif-ferentiating reaction is performed in the nonracemic series. With respect to configurational assignment, three cases may be specified. 

The determination of gas-solid virial coefficients can be a useful technique to explain the interaction between an adsorbed gas and a solid surface. The terms are defined so that the number of adsorbate molecules interacting can be readily ascertained. For example, the second order gas-solid interaction involves one adsorbate molecule and the solid surface the third order gas-solid interaction involves two adsorbate molecules and the surface, and so on. The number of adsorbed molecules under consideration is expanded in a power series with respect to the density of the adsorbed phase. 

The initial system can be constructed as a series with respect to powers of [39], A zero approximation here is a solution of the degenerated system. This approach is, however, very rarely used since the increase of accuracy results in a significant complication of calculations. 

Equations (4.212) are solved sequentially beginning from Eq. (4.213) by expanding in a power series with respect tox. The right-hand sides of Eqs. (4.212) and (4.213) are proportional to an exponentially small parameter A,. For this reason alone, we did not take into account the corrections of the order ul n l when deriving Eqs. (4.167). However, the quantities... 

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