Flory–Schulz distribution

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title: "Flory–Schulz distribution" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polymers", "continuous-distributions"] description: "Probability distribution in chemistry" topic_path: "science/chemistry" source: "https://en.wikipedia.org/wiki/Flory–Schulz_distribution" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Probability distribution in chemistry ::

| name = Flory–Schulz distribution | type = mass | pdf_image = Schulz-Flory-Verteilung Polydispersität.svg | cdf_image = | notation = | parameters = 0 | support = k ∈ { 1, 2, 3, ... } | pdf = a^2 k (1-a)^{k-1} | cdf = 1-(1-a)^k (1+ a k) | mean = \frac{2}{a}-1 | median = \frac{W\left(\frac{(1-a)^{\frac{1}{a}} \log (1-a)}{2 a}\right)}{\log (1-a)}-\frac{1}{a} | mode = -\frac{1}{\log (1-a)} | variance = \frac{2-2 a}{a^2} | skewness = \frac{2-a}{\sqrt{2-2 a}} | kurtosis = \frac{(a-6) a+6}{2-2 a} | entropy = | mgf = \frac{a^2 e^t}{\left((a-1) e^t+1\right)^2} | char = \frac{a^2 e^{i t}}{\left(1+(a-1) e^{i t}\right)^2} | pgf = \frac{a^2 z}{((a-1) z+1)^2}

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,{{cite journal|last=Flory|first=Paul J.|journal=Journal of the American Chemical Society|title=Molecular Size Distribution in Linear Condensation Polymers|date=October 1936|volume=58|issue=10|pages=1877–1885|issn=0002-7863|language=English|doi=10.1021/ja01301a016

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}{l} (a-1) (k+1) w_a(k)+k w_a(k+1)=0\text{,} \[10pt] w_a(0)=0\text{,} w_a(1)=a^2\text{.} \end{array}\right} As a probability distribution, one can note that if X and Y are two independent and geometrically distributed random variables with parameter a taking values in {0, 1, \cdots}, thenw_a(k) = \mathbb{P}\left(X + Y + 1 = k\right)This in turn means that the Flory-Schulz distribution is a shifted version of the negative binomial distribution, with parameters r = 2 and p = a.

References

References

1. "most probable distribution".

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