## Thursday, March 7, 2013

### Molecular Weight Distribution: The generalized exponential distribution

A generalized exponential (GEX) distribution described and plotted in a previous blog post, is often used to empirically approximate a wide variety of distribution. Unlike the lognormal distribution, which has two parameters, the GEX distribution has 3 parameters, which understandably permits greater flexibility.

The weight- and number- distributions are given by
$W(M) = \frac{m t^{\frac{k+1}{m}}}{\Gamma \left(\frac{k+1}{m}\right)} M^k e^{-t M^m}$
$N(M) = \frac{m t^{k/m}}{\Gamma \left(\frac{k}{m}\right)} M^{k-1}e^{-t M^m}$
The weight- and number-average molecular weights are:

$M_w = \int_0^\infty M W(M) dM = t^{-1/m} \frac{\Gamma\left( \frac{2 + k}{m} \right)}{\Gamma\left( \frac{1 + k}{m} \right)}$
$M_n = \int_0^\infty M N(M) dM = t^{-1/m} \frac{\Gamma\left( \frac{1 + k}{m} \right)}{\Gamma\left( \frac{k}{m} \right)}$