Fractional-order reaction Characteristics

The relationship between the characteristic rate coefficient k and the time tx required to reduce the distance from equilibrium to the fraction x is analogous to that for irreversible first-order reactions (see eqn 3.23) ... 

The Effect of Heat on the Active Constituents of a Solution. The thermal stability of components of a solution may determine the type of evaporator to be used and the conditions of its operation. If a simple solution contains a hydrolyzable material and the rate of its degradation during evaporation depends on its concentration at any time, an exponential relation between the remaining fraction, F, and the time, t, characteristic of a first-order reaction, is obtained, as shown in Eq. (2). 

For cellulase, typically 0.15 < r < 0.7. This fractional reaction order is characteristic of many effects by cellulase, not just production of glucose. The fractional reaction order indicates a diminishing return on increasing enzyme dosage. The relationship between extent of hydrolysis and reaction time is also expressed by a fractional exponent in time, which indicates a loss of enzyme effectiveness over time. 

Remember the important characteristics of a first-order reaction, which is evident from Eqs. (3.2) and (3.3). If a certain fraction of the reactant is converted in a given time, doubling the time will convert the same fraction of the remaining reactant. Thus for 90% conversion in t seconds, 99% would be converted in 2t seconds. 

Rate Laws Experimental measurement of the rate leads to the rate law for the reaction, which expresses the rate in terms of the rate constant and the concentrations of the reactants. The dependence of rate on concentrations gives the order of a reaction. A reaction can be described as zero order if the rate does not depend on the concentration of the reactant, or first order if it depends on the reactant raised to the first power. Higher orders and fractional orders are also known. An important characteristic of reaction rates is the time required for the concentration of a reactant to decrease to half of its initial concentration, called the half-life. For first-order reactions, the half-hfe is independent of the initial concentration. 

The time required to convert a given fraction of the limiting reagent is a characteristic of the rate equation. A comparison of successive half-times, or any other convenient fractional time, reveals whether a reaction follows any simple-order rate law. Thus, the ratio of the time to reach 75 percent completion to that for 50 percent is characteristic of the reaction order. Values of this ratio for different orders are as follows ... 

The rate constants in the reactions (29) may be conveniently envisaged as elements of symmetric matrix k. In order to calculate the statistical characteristics of a particular polycondensation process along with matrix k parameters should be specified which characterize the functionality of monomers and their stoichiometry. To this end it is necessary to indicate the matrix f whose element fia equals the number of groups A in an a-th type monomer as well as the vector v with components Vj,... va,..., v which are equal to molar fractions of monomers M1,...,Ma,...,M in the initial mixture. The general theory of polycondensation described by the ideal model was developed more than twenty years ago [2]. Below the key results of this theory are presented. 

It is unfortunate that many workers have not appreciated how essential a clue to the kinetics can be provided by the kinetic order of the whole reaction curve. The use of initial rates was carried over from the practice of radical polymerisation, and it can be very misleading. This was in fact shown by Gwyn Williams in the first kinetic study of a cationic polymerization, in which he found the reaction orders deduced from initial rates and from analysis of the whole reaction curves to be signfficantly different [111]. Since then several other instances have been recorded. The reason for such discrepancies may be that the initiation is neither much faster, nor much slower than the propagation, but of such a rate that it is virtually complete by the time that a small, but appreciable fraction of the monomer, say 5 to 20%, has been consumed. Under such conditions the overall order of the reaction will fall from the initial value determined by the consumption of monomer by simultaneous initiation and propagation, and of catalyst by initiation, to a lower value characteristic of the reaction when the initiation reaction has ceased. 

Models may also be tested by utilizing the time required for a given fraction of a reactant to disappear, since this varies with the initial concentration in a fashion characteristic of the reaction order. For example, if the half-life of a reaction is defined as the time required for one-half of the initial amount of reactant to be consumed, then Eq. (4) may be written... 

The most detailed modelling approach summarized in Figure 1 is found at the mechanistic level. These models are explicit accounts of the chemistry of elementary steps. Thus the hierarchy of the levels, i.e., reaction models in Figure 1, now becomes quite clear. Mechanistic models, which provide the temporal and many times spatial variation of the composition of each component and reaction intermediate, are based at the lowest modelling level. Their output, however, is typically phrased in terms of ensembles of stable molecular constituents which is more characteristic of the intermediate level molecular models. The molecular models, in turn, require subsequent organization in order to connect to the global reaction models and relevant product fractions at the top or global level. 

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