P nlnug Jo V n CT

After evaluating the integral one finds that

We see that the mean diameter of a lognormal distribution depends on both Dpg and ag and increases as ag increases.

8.1.7 Plotting the Lognormal Distribution

The cumulative distribution function N(DP) for a lognormally distributed aerosol population is given by (8.39). Defining the normalized cumulative distribution

Mdp)

one obtains

The cumulative distribution function can be plotted on a log-probability graph using one of the scientific computer graphics programs. In these diagrams the x axis is logarithmic, log(x), and the v axis is scaled like the error function, erf(y). This scaling compresses the scale near the median (50% point) and expands the scale near the ends. In these graphs the cumulative distribution function of a log-normal distribution is a straight line (Figure 8.8). The point at N(DP) = 0.5 occurs when Dp = Dpg. Therefore the geometric mean, or median, of the distribution is the value of Dp where the straight line plot of N(Dp) crosses the 50th percentile.

The point at which N(Dp) = 0.84 occurs for In Dp+„ = In Dpg + In ag or Dp+a = Dpg<3g. The slope of the line is related to the geometric standard deviation of the distribution. Lognormal distributions with the same standard deviation when plotted on probability

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0.1 1 10 Diameter, nm

FIGURE 8.8 Cumulative lognormal aerosol number distributions plotted on log-probability paper. The distributions have mean diameter of 1 |im and r>K — 2 and 1.5, respectively.

0.1 1 10 Diameter, nm

FIGURE 8.8 Cumulative lognormal aerosol number distributions plotted on log-probability paper. The distributions have mean diameter of 1 |im and r>K — 2 and 1.5, respectively.

coordinates are parallel to each other. A small standard deviation corresponds to a narrow distribution and to a steep line on the log-probability graph (Figure 8.8). The geometric standard deviation can be calculated as the ratio of the diameter Dp, a for which N(Dp j a) = 0.84 to the median diameter

8.1.8 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) = nD2nN(Dp) and nv(Dp) = (it/6)DipnN(Dp), let us determine the forms of ns(Dp) and nv(Dp) when n(Dp) is lognormal. From (8.34) one gets ns(Dp)=-—p--exp ( — ---8.48

By letting Dp = 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 hn{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

or by letting Dp = exp(3 In Dp), expanding the exponential, and completing the square in the exponent, (8.51) becomes

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