Properties of the Lognormal Distribution

We have discussed the properties of the lognormal distribution for the number concentration. The next step is examination of the surface and volume distributions corresponding to a lognormal number distribution given by (8.34). Since ns(Dp) = nD nN(Dp) and nv(Dp) = (n/6)DpriN(Dp), let us determine the forms of ns(Dp) and nv(Dp) when n(Dp) is lognormal. From (8.34) one gets 

By letting E = exp(2 In Dp), expanding the exponential, and completing the square in the exponent, (8.48) becomes 

Thus we see that if the number distribution n (Dp) is lognormal, the surface distribution ns(Dp) is also lognormal with the same geometric standard deviation ag as the parent distribution and with the surface median diameter given by 

The calculations above can be repeated for the volume distribution, and one can show that 

Therefore if the number distribution is lognormal, the volume distribution ny(Dp) 

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