Nth order

EMPIRICAL RATE EQUATIONS OF THE nth ORDER IRREVERSIBLE REACTIONS... 

The results of the types of reaetion being eonsidered show that the treatment of kinetie data heeomes rapidly more eomplex as the reaetion order inereases. In eases where the reaetion eonditions are sueh dial the eoneentrations of one or more of the speeies oeeurring in the rate equation remain eonstant, these terms may he ineluded in the rate eonstant k. The reaetions ean he attributed to lower order reaetions. These types of reaetions ean be defined as pseudo-nth order, where n is the sum of the exponents of those eoneentrations that ehange during the reaetion. An example of this type of reaetion is in eatalytie reaetion, where the eatalyst eoneentration remains eonstant during the reaetions. 

The half-life equation for the general nth-order reaetion is... 

Considering an nth order iiTeversible reaetion represented by A —> bB with (-r ) = kC, the design equation is... 

Equation 5-60 is the generalized equation, whieh ean be used to determine the reaetion time of an nth order reaetion with b, the stoiehiometrie eoeffieient of the produet eomponent B. 

Equation 5-247 is a polynomial, and the roots (C ) are determined using a numerical method such as the Newton-Raphson as illustrated in Appendix D. For second order kinetics, the positive sign (-r) of the quadratic Equation 5-245 is chosen. Otherwise, the other root would give a negative concentration, which is physically impossible. This would also be the case for the nth order kinetics in an isothermal reactor. Therefore, for the nth order reaction in an isothermal CFSTR, there is only one physically significant root (0 < C < C g) for a given residence time f. 

Assuming an nth order reaetion at steady state eondition, (dC/dQ) = 0, Equation 8-122 beeomes... 

For an nth order reaction of a macro fluid. Equation 9-16 is substituted into Equation 9-6 to give... 

From this it would appear that the ( - l)th-order wave function is required for calculating the th-order energy. However, by using the turnover rule and the nth and lower-order perturbation equations (4.32), it can be shown that knowledge of the nth-order wave function actually allows a calculation of the (2n-i-l)th-order energy. 

The nth-order property is the nth-order derivative of the energy, d EjdX" (the factor 1 /n may or may not be included in the property). Note that the perturbation is usually a vector, and the first derivative is therefore also a vector, the second derivative a matrix, the third derivative a (third-order) tensor etc. 

Derivative techniques consider the energy in the presence of the perturbation, perform an analytical differentiation of the energy n times to derive a formula for the nth-order property, and let the perturbation strength go to zero. 

Without the Lagrange teehnique for non-variational wave functions (Cl, MP and CC), the nth-order CPHF is needed for the nth-derivative. Consider for example the MP2 energy correction, eq. (4.45). 

An algorithm for an assesment of chromatographic peak purity was proposed. In this study ethyl 8-methyl-4-oxo-4/7-pyrido[l, 2-u]pyrimidine-3-carboxylate was also used (97MI13). Ethyl 7-methyl-4-oxo-4//-pyrido[l,2-u]pyrimidine-3-carboxylate, among other compounds, was applied to show practical mathematical tools for the creation of several figures of merit of nth order instrumentation, namely selectivity, net analyte signal and sensitivity (96ANC1572). 

The influence of an impurity (Y) on the absorption spectrum of a substance (X) can often be eliminated by considering derivative curves as shown in Fig. 17.15 the second-order plot of the mixture is identical with that of pure X. When the interference spectrum can be described by an nth-order polynomial, the interference is eliminated in the (n+ 1) derivative. 

If an order of n with respect to total add is obtained experimentally, it is out of the question to consider it as an nth order relative to free add. Unfortunately, this assumption has been made and it can be said that relative studies are a priori erroneous. 

Writing the rate equation in the logarithmic form for an nth order of reaction, we have... 

A review of epoxy-novolac reaction mechanisms and kinetics is provided by Biernath et al.85 Depending on the structures of the novolac and the epoxy, reactions have been reported to proceed through an nth-order mechanism or an autocatalytic mechanism.88-92... 

The reaction rate per unit volume of particle 5H, for an nth-order reaction is then given by ... 

The basic differential equation for mass transfer accompanied by an nth order chemical reaction in a spherical particle is obtained by taking a material balance over a spherical shell of inner radius r and outer radius r + Sr, as shown in Figure 10.12. 

Relatively few kinetic schemes admit analytical solutions. This section is concerned with those special cases that do, and also with some cases where preliminary analytical work will ease the subsequent numerical studies. We begin with the nth-order reaction. 

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