Flory–Schulz distribution

The Flory–Schulz distribution is a discrete probability distribution named after Paul Flory and Günter Victor Schulz that describes the relative ratios of polymers of different length that occur in an ideal step-growth polymerization process. The probability mass function (pmf) for the mass fraction of chains of length $k$ is: $$w_a(k)=a^{2}k(1-a{)}^{k-1}\text{.}$$

In this equation, k is the number of monomers in the chain, and 0<a<1 is an empirically determined constant related to the fraction of unreacted monomer remaining.

The form of this distribution implies is that shorter polymers are favored over longer ones — the chain length is geometrically distributed. Apart from polymerization processes, this distribution is also relevant to the Fischer–Tropsch process that is conceptually related, where it is known as Anderson-Schulz-Flory (ASF) distribution, in that lighter hydrocarbons are converted to heavier hydrocarbons that are desirable as a liquid fuel.

The pmf of this distribution is a solution of the following equation: $$\left\{\begin{array}{c}(a-1)(k+1)w_a(k)+k{w}_a(k+1)=0\text{,}\\ w_a(0)=0\text{,}{w}_a(1)=a^2\text{.}\end{array}\right\}$$