The error function erf z is defined as dD*

and erf(O) = 0, erf(oo) = 1. If we divide the integral in (8.37) into one from — oo to 0 and the second from 0 to (In Dp — In Dpg)/\/2 In og, then the first integral is seen to be equal to \/n/2 and the second to (v/7t/2)erf[(ln Dp — In Dpg)/\/2 In cta,]. Thus for the lognormal distribution

and we see that Dpg = Dmcd is the median diameter, that is, the diameter for which exactly one-half of the particles are smaller and one-half are larger. To understand the role of og let us consider the diameter Dpa for which ag = Dpo/Dpg. At that diameter, using (8.39)

Thus ag is the ratio of the diameter below which 84.1% of the particles lie to the median diameter and is termed the geometric standard deviation. A monodisperse aerosol population has ag = 1. For any distribution, 67% of all particles lie in the

JPg! ug "J ^pg^g range from Dpg/r>g to Dpgog and 95% of all particles lie in the range from Dpg/o2

Let us calculate the mean diameter Dp of a lognormally distributed aerosol. By definition, the mean diameter is found from

1 f00

Mf Jo which we wish to evaluate in the case of n^(Dp) given by (8.34). Therefore

1 f°° ( (in D„ — lnD„„)2N\ Dp = - / exp - ^--dDp 8.43

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