Flory–Schulz distribution

{{Probability distribution

| name = Flory–Schulz distribution

| type = mass

| pdf_image = Schulz-Flory-Verteilung Polydispersität.svg

| cdf_image =

| notation =

| parameters = 0 < a < 1 (real)

| support = k ∈ { 1, 2, 3, ... }

| pdf = $a^2\; k\; (1-a)^\{k-1\}$

| cdf = $1-(1-a)^k\; (1+\; a\; k)$

| mean = $\backslash frac\{2\}\{a\}-1$

| median = $\backslash frac\{W\backslash left(\backslash frac\{(1-a)^\{\backslash frac\{1\}\{a\}\}\; \backslash log\; (1-a)\}\{2\; a\}\backslash right)\}\{\backslash log$

(1-a)}-\frac{1}{a}

| mode = $-\backslash frac\{1\}\{\backslash log\; (1-a)\}$

| variance = $\backslash frac\{2-2\; a\}\{a^2\}$

| skewness = $\backslash frac\{2-a\}\{\backslash sqrt\{2-2\; a\}\}$

| kurtosis = $\backslash frac\{(a-6)\; a+6\}\{2-2\; a\}$

| entropy =

| mgf = $\backslash frac\{a^2\; e^t\}\{\backslash left((a-1)\; e^t+1\backslash right)^2\}$

| char = $\backslash frac\{a^2\; e^\{i\; t\}\}\{\backslash left(1+(a-1)\; e^\{i\; t\}\backslash right)^2\}$

| pgf = $\backslash frac\{a^2\; z\}\{((a-1)\; z+1)^2\}$

}}

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: $$w\_a(k)\; =\; a^2\; k\; (1-a)^\{k-1\}\backslash text\{.\}$$

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

}} and 0 is an empirically determined constant related to the fraction of unreacted monomer remaining.{{GoldBookRef | title = most probable distribution | file = M04035}}

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: $$\backslash left\backslash \{\backslash begin\{array\}\{l\}$$

(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\}