## It Nt

Therefore if the number distribution n,\/{Dp) is lognormal, the volume distribution nv{Dp) is also lognormal with the same geometric standard deviation <Jg as the parent distribution and with the volume median diameter given by

The constant standard deviation for the number, surface, and volume distributions for any lognormal distribution is one of the great advantages of this mathematical representation.

Plotting the surface and volume distributions of a log-normal aerosol distribution on a log-probability graph would also result in straight lines parallel to each other (same standard deviation). For the distribution shown in Figure 8.8 with Dps = 1.0 (Am and cr^ = 2,0, the resulting surface area and volume median diameters are approximately 2.6 ¡.im and 4.2 |im, respectively.

The Power-Law Distribution A variety of other mathematical functions have been proposed for the description of atmospheric aerosol distributions. The power law, or Junge, distribution was one of the first used in atmospheric science (Prup-pacher and Klett 1980), where C and a are positive constants. Values of a from 2 to 5 are used for ambient aerosol distributions.

t. What is the shape of the power law distribution when plotted in log-log coordinates? What is the meaning of a and C based on this plot?

2. Calculate the volume distribution function nJ for the power-law distribution.

3. Figure 8.9 shows a typical urban aerosol size distribution and its fit by a power-law distribution using C = 91.658 and a = 3.746. Using these graphs and also Figure 8.6, discuss the advantages and disadvantages of the use of the power law distribution for atmospheric aerosols.

1. Taking the logarithm of both sides of the power-law equation, we find that log naN — log C — st logDp, and therefore this distribution will he a straight line on a log-log plot with slope equal to —a. For Dp = I pm we find that C = n°N and, as a result, the parameter C is the value of the distribution function at 1 jim.

2. Using (8.10), the volume distribution function will be

This is once more a straight line on a log-log plot.

3. The main advantage of the power-law distribution is its mathematical simplicity compared to other distribution functions. It is much easier to perform calculations with the simple power-law expression compared to the lognormal distribution given by (8,33). Figure 8.9 indicates that the power-law distribution can provide a reasonable approximation to atmospheric aerosol number distributions IXJUL

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