General Formulation for a Single Index

It is better to define the continuous mixture in terms of an index variable (Aris and Gavalas, 1966 Astarita and Ocone, 1988) rather than to use the dimensional rate constant k. Let x be an index variable, which without loss of generality we can take to be in the interval [0, °°). The initial concentration of material characterized by indices in the interval (x, x + dx) is 

The distribution /(x) will have one or more shaping parameters and we shall have occasion to use the gamma distribution 

If we approach the problem from the other direction and ask what distribution will produce a given alias, we have first to solve Eq. 9 and then find the distribution/of which it is the Laplace transform. Now (formally) 

We also see immediately that there are certain kinetic laws that cannot be generated. For example the so-called zero-order reaction that just takes place at a constant rate so long as there is any reactant present. Thus F is a constant and by Eq. 1 must be 1, giving 

Inversion of this transform gives the gamma distribution 

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