A log-normal distribution in molecular weights often arises as a consequence of anionic polymer synthesis. You can see pictures of this asymmetric distribution on the linked wikipedia page.

Let us review some properties of this distribution. The number distribution of segments with molecular weight \(M\) has two parameters

\[ N(M;m, s) = \frac{1}{M s \sqrt{2 \pi}} \exp \left( {-\frac{(\ln M - m)^2}{2 s^2}} \right). \]

The quantity \(\ln M\) is normally distributed around \(m\) with variance \(s\). You can verify that:

\[ \int_0^\infty N(M) dM = 1 \]

\[M_n = \int_0^\infty M N(M) dM = \exp\left(m + \frac{s^2}{2} \right) \]

\[M_w = \int_0^\infty M W(M) dM =\exp\left(m + \frac{3 s^2}{2} \right), \]

where I tacitly used the relation for the weight distribution \(W(M) = N(M)/M_n\) in the last line. The polydispersity index \[p = M_w/M_n = \exp(s^2) \]

It is usual to report the \(p\) and \(M_w\) of an empirically obtained distribution. One can express the parameters of the log-normal distribution in terms of these numbers as:

\[ s^2 = \ln p \]

\[m = \ln M_w - \frac{3}{2} \ln p\]

One could use these to directly express the number distribution in the following somewhat ugly form:

\[N(M) = \frac{1}{\sqrt{2 \pi } M \sqrt{\ln (p)}} \exp \left[- \frac{\left( \ln M-\ln \left(\frac{2 M_n}{p} \right) \right)^2}{2 \ln (p)}\right]\]

Note: The cumulative distribution function is given by:

\[C(M; m,s) = \frac{1}{2} + \frac{1}{2} \text{erf} \left(\frac{\log M-\mu}{\sqrt{2}\sigma} \right)\]

Let us review some properties of this distribution. The number distribution of segments with molecular weight \(M\) has two parameters

\[ N(M;m, s) = \frac{1}{M s \sqrt{2 \pi}} \exp \left( {-\frac{(\ln M - m)^2}{2 s^2}} \right). \]

The quantity \(\ln M\) is normally distributed around \(m\) with variance \(s\). You can verify that:

\[ \int_0^\infty N(M) dM = 1 \]

\[M_n = \int_0^\infty M N(M) dM = \exp\left(m + \frac{s^2}{2} \right) \]

\[M_w = \int_0^\infty M W(M) dM =\exp\left(m + \frac{3 s^2}{2} \right), \]

where I tacitly used the relation for the weight distribution \(W(M) = N(M)/M_n\) in the last line. The polydispersity index \[p = M_w/M_n = \exp(s^2) \]

It is usual to report the \(p\) and \(M_w\) of an empirically obtained distribution. One can express the parameters of the log-normal distribution in terms of these numbers as:

\[ s^2 = \ln p \]

\[m = \ln M_w - \frac{3}{2} \ln p\]

One could use these to directly express the number distribution in the following somewhat ugly form:

\[N(M) = \frac{1}{\sqrt{2 \pi } M \sqrt{\ln (p)}} \exp \left[- \frac{\left( \ln M-\ln \left(\frac{2 M_n}{p} \right) \right)^2}{2 \ln (p)}\right]\]

Note: The cumulative distribution function is given by:

\[C(M; m,s) = \frac{1}{2} + \frac{1}{2} \text{erf} \left(\frac{\log M-\mu}{\sqrt{2}\sigma} \right)\]

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