Order, determination Integrated methods

Fixed-time integral methods are advantageous for systems in which the signal is a linear function of concentration. In this case it is not necessary to determine the concentration of the analyte or product at times ti or f2, because the relevant concentration terms can be replaced by the appropriate signal. For example, when a pseudo-first-order reaction is followed spectrophotometrically, when Beer s law... 

Equation 13.14 shows how [A]o is determined for a two-point fixed-time integral method in which the concentration of A for the pseudo-first-order reaction... 

Indices 1 and 2 denote the catalyzed and non-catalyzed reactions, respectively. If rate constant k2 and orders m2 and n2 have been determined Eq. (3) can be solved by an iterative integration method. 

ILLUSTRATION 3.2 USE OF A GRAPHICAL INTEGRAL METHOD FOR DETERMINING THE RATE CONSTANT FOR A CLASS II SECOND-ORDER REACTION... 

Use a graphical integral method to determine the order of the reaction and the reaction rate constant. 

Determine the reaction order and rate constant for the reaction by both differential and integral methods of analysis. For orders other than one, C0 will be needed. If so, incorporate this term into the rate constant. 

Use (a) the differential method and (b) the integral method to determine the reaction order, and the value of the rate constant. Comment on the results obtained by the two methods. 

Integration Method or Hit and Trial Method Here known quantities of standard solutions of reactants are mixed in a reaction vessel and the progress of the reaction is determined by determining the amount of reactant consumed after different intervals of time. These values are then substituted in the equations of first, second, third order and so on. The order of the reaction is the order corresponding to that equation which gives a constant value of K. 

We see that the half-life is always inversely proportional to k and that its dependence on [A]o depends on the reaction order. Thereby the method can be used to determine both the rate constant and the reaction order, even for reactions with noninteger reaction order. Similar to the integral method, the half-life method can be used if concentration data for the reactant are available as a function of time, preferably over several half-lives. Alternatively the half-life can be determined for different initial concentrations in several subsequent experiments. 

Olvera de la Cruz and Sanchez [76] were first to report theoretical calculations concerning the phase stability of graft and miktoarm AnBn star copolymers with equal numbers of A and B branches. The static structure factor S(q) was calculated for the disordered phase (melt) by expanding the free energy, in terms of the Fourier transform of the order parameter. They applied path integral methods which are equivalent to the random phase approximation method used by Leibler. For the copolymers considered S(q) had the functional form S(q) 1 = (Q(q)/N)-2% where N is the total number of units of the copolymer chain, % the Flory interaction parameter and Q a function that depends specifically on the copolymer type. S(q) has a maximum at q which is determined by the equation dQ/dQ=0. 

To determine the reaction order by the integral method, we guess the reaction order and integrate the differential equation used to model the batch system. If the order we assume is correct, the appropriate plot (detemtined from this integration) of the concentration-time data should be linear. The integral method is used most often when the reaction order is known and it is desired to evaluate the specific reaction rate constants at different temperatures to determine the activation energy. 

Determination of Integral Method When the concentrations of reactants and products in the cay order for phase change very slowly with time, the pseudo-steady-state forms of the order n balances can be used. For the irreversible nth-order reaction carried out... 

The decay of trapped radicals in polyethylene has been studied by many workers in recent years [226—242]. The usual pattern of step-like decay is observed below room temperature. The radicals studied, the experimental conditions and the measured reaction orders are given in Table 12. This clearly shows that disagreement exists between the results obtained by the various workers. The general problems of solid state reaction kinetics and the problems related to inter- and intraspur recombination have been discussed in section 3. It must be remembered, however, that usual kinetic analyses cannot be applied to step-like reactions and that the determination of the order of reaction by the integration method requires very accurate analytical methods. A strong dependence of the rate of radical decay on the crystallinity of the sample was nevertheless clearly demonstrated by Charlesby et al. [228] the specific rate... 

A kinetic study has been carried out to determine the order of the reaction and to establish a correlation between the structure and composition of the materials with their stability properties. Integration methods, lifetimes, and initial rate analysis were used to determine the order of the reaction [87]. A plot of ln(///0) versus time is lineal, confirming first-order kinetics for oxidation of SEBS. A pseudo-first order of the reaction is also confirmed, because half lifetime remains constant for different /<> and the double logarithmic plot of initial rates versus intensity maxima (which are proportional to the initial peroxides concentration) gave a straight fine. 

By simply knowing the phase equilibrium behavior and the composition within the beaker at the start of the experiment (x ), one can easily construct a residue curve by integrating Equation 2.8. Such integration is usually performed with the use of a numerical integration method (see later, Section 2.5.3), such as Runge Kutta type methods, remembering that at each function evaluation, a bubble point calculation must be performed in order to determine y(x). 

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