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Milds Flory–Schulz distribution

Parameters 0 < a < 1 (real) k ∈ { 1, 2, 3, ... } $a^{2}k(1-a)^{k-1}$ $1-(1-a)^{k}(1+ak)$ ${\frac {2}{a}}-1$ ${\frac {W\left({\frac {(1-a)^{\frac {1}{a}}\log(1-a)}{2a}}\right)}{\log(1-a)}}-{\frac {1}{a}}$ $-{\frac {1}{\log(1-a)}}$ ${\frac {2-2a}{a^{2}}}$ ${\frac {2-a}{\sqrt {2-2a}}}$ ${\frac {(a-6)a+6}{2-2a}}$ ${\frac {a^{2}e^{t}}{\left((a-1)e^{t}+1\right)^{2}}}$ ${\frac {a^{2}e^{it}}{\left(1+(a-1)e^{it}\right)^{2}}}$ ${\frac {a^{2}z}{((a-1)z+1)^{2}}}$ The Flory–Schulz distribution is a 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) has form:

$w_{a}(k)=a^{2}k(1-a)^{k-1}$ and gives the mass fraction (chemistry) of chains of length k.

In this equation, k is a variable characterizing the chain length (e.g. number average molecular weight, degree of polymerization), 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. Apart from polymerization processes, this distribution is also relevant to the Fischer–Tropsch process that is conceptually related, 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:

$\left\{{\begin{array}{l}(a-1)(k+1)w_{a}(k)+kw_{a}(k+1)=0,\\[10pt]w_{a}(0)=0,w_{a}(1)=a^{2}\end{array}}\right\}$ Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.