Properties of the Log Normal Particle Size Distribution

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Properties of the Log-Normal Particle Size
Distribution
Jost Heintzenberg
To cite this article: Jost Heintzenberg (1994) Properties of the Log-Normal Particle Size
Distribution, Aerosol Science and Technology, 21:1, 46-48, DOI: 10.1080/02786829408959695
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Properties of the Log-Normal Particle Size 
Distribution 
Jost Heintzenberg 
Institute for Tropospheric Research, Permoserstrasse 15, 0-04303 Leipzig, Germany 
This note presents analytical relationships that con- ticle sensors. They also make pdssible calculating es- 
nect the parameters of any moment of the log-normal sential aerosol parameters that may not be amenable 
distribution to those of any other, allowing one to to direct measurements. 
harmonize experimental data derived with varying par- 
INTRODUCTION 
The log-normal distribution is a function 
distributing a dependent variable in a nor- 
mal or Gaussian fashion on a logarithmic 
scale of the independent variable. This 
function has been used for a long time to 
describe size distributions of particle 
properties in atmospheric aerosols. Foitzik 
(1964) used this functional relationship 
for the description of optical aerosol 
properties. Later, Whitby (1974) built a 
general concept for the multimodal na- 
ture of the atmospheric aerosol on this 
approach by fitting measured particle size 
distributions in a combination of three 
log-normal distributions. He also com- 
bined information from three different 
moments of the distribution in the log- 
normal modeling of the experimental data. 
Friedlander (1977) extended the use of 
moments of the size distribution up to the 
sixth moment. Since then, the log-normal 
function has gained wide acceptance in 
the modeling of aerosol-related physical 
and chemical processes. There are algo- 
rithms now available to automate the fit- 
ting of log-normal distributions to experi- 
Address for correspondence: Meteorology, Stock- 
holm University, S-106 Stockholm, Sweden. 
mental aerosol data from instruments of 
varying size channels (Whitby and Whitby, 
1989). 
The results of the log-normal approxi- 
mations are used in the modeling of 
aerosols, clouds, and precipitation as well 
as in the comparison of experimental data. 
In these applications, the need often arises 
for transformations between different mo- 
ments of the distributions. Here, some 
mathematically elegant features of this 
function come to bear: 
A distribution that is log-normal in one 
of its moments will be log-normal in 
any of its moments with the same 
geometric standard deviation, de- 
scribing the spread of the dependent 
variable. 
The median diameter of the distribu- 
tion of any moment is equal to the 
corresponding geometric mean di- 
ameter, and is related to the integral 
of that moment by a simple factor. 
The median size of any moment is con- 
nected to the median size of any 
other moment by an analytical re- 
lationship derived by Hatch and 
Choate (1929). 
When transforming log-normal distri- 
butions between different moments, we 
Aerosol Science and Technology 21:46-48 (1994) 
O 1994 Elsevier Science Inc. 
Log-Normal Particle Size Distribution 47 
frequently want to calculate integrals of ber concentration at the diameter d,,. 
higher moments of the distribution. This Higher moments of this distribution have 
is because most state-of-the-art aerosol the same form as Eq. 1 except for their 
instrumentation is based on particle- median diameters d,,, which are related 
counting sensors which yield number-size to that of the zeroth moment by the 
distributions, i.e., the zeroth moment of Hatch-Choate conversion equation: 
the size distribution. For those distribu- 
d,, 2 tions we often want to calculate total sur- 
- = exp[q(ln ug) ] (2) 
face or total volume (or mass). in order to d g ~ 
relate the measured size distribution to its 
effects or to complementary chemical in- 
formation derived on a particle mass ba- 
sis. I am not aware of any published solu- 
tion to this problem other than that by 
Heintzenberg and Baker (1976), relating 
number and volume integrals for two for- 
mulations of the log-normal distribution. 
Extensive discussions of the application 
of the log-normal function to the descrip- 
tion of aerosol size distributions can be 
found in Butcher and Charlson (1972) and 
Hinds (1982). Here I will collect the avail- 
able formulas relating properties of the 
log-normal distribution function and will 
complement them with useful relation- 
ships between the integrals of the differ- 
ent moments of the distribution. 
PROPERTIES OF THE LOG-NORMAL 
SIZE DISTRIBUTION 
The zeroth moment of the particle size 
distribution, expressed as the number of 
particles per logarithmic size interval, can 
be formulated with the log-normal func- 
tion as 
dn 
- 
-- 
F m o 
dlnd, G l n u , 
with d, being the particle diameter, ug 
being the geometric standard deviation of 
the distribution, d,, being the number- 
median diameter, and Fmo being the num- 
with q being the moment of the distribu- 
tion. The integral of an exponential func- 
tion over all values of the independent 
variable between zero and infinity can be 
calculated (cf. Bronstein and Semendja- 
jew, 1962). Thus, the integral I, of any 
moment of the log-normal distribution is 
related to the median value of that mo- 
ment Fmq by 
We can relate the integral of any higher 
moment to the integral of the zeroth mo- 
ment by expressing it as a function of the 
zeroth moment of the distribution at the 
median diameter of that particular mo- 
ment. If we write the general transforma- 
tion to the higher moment q as A(d,)q, 
we connect the median value of that mo- 
ment to the zeroth moment by employing 
Eq. 1 as 
(In dgq - In d,,)' 
2011 u,12 1 
According to Eq. 3, the corresponding 
expression for the integral is 
(In d,, - In dgo) 
I, = ~ ( d , , ) ~ ~ ~ , e x ~ 
2(1n 1 
Frequently, we need integral transforma- 
tions from total particle number N to 
surface and volume. Assuming spherical 
J. Heintzenberg 
particles, the total particle surface S, can 
be written as a function of parameters of 
number-size distribution only by combin- 
ing Eqs. 4, 3, and 2: 
(5) 
The corresponding expression for the to- 
tal particle volume V, has the form 
(6) 
A caveat concerning the use of these for- 
mulas should be added. When calculating 
parameters of higher (or lower) moments 
of a size distribution, the deduced data 
may be at the size limits of available parti- 
cle information (or even beyond). Thus, 
the predicted moments may have little or 
. no validity. Any error in dgN will propa- 
gate with its second power to S and with 
its third power to V. The propagation of 
errors in o,, on the other hand, is not as 
straightforward to predict. However, as 
with the median diameter, their influence 
on V will be higher than on S. 
SUMMARY AND CONCLUSIONS 
With the analytical relationships summa- 
rized or derived above, the parameters of 
any moment of the log-normal distribu- 
tion can be connected to each other, al- 
lowing one to harmonize the experimental 
data derived with varying particle sensors. 
These formulas also allow the calculation 
of essential aerosol parameters which may 
not be amenable to direct measurements.As an example, small ice crystals in cirrus 
clouds are suspected to have a number 
and mass mode of their size distribution 
in the aerodynamic diameter range 10-50 
,urn. The most sensitive sensors in this 
size range measure total number and mass 
concentrations ( N , M ) see Strom and 
Heintzenberg (1993.) Assuming a log-nor- 
ma1 size distribution we can bound the 
range of possible distribution parameters 
(d,,, o,) with the help of the above equa- 
tions and measured values of N and M. 
REFERENCES 
Bronstein, I. N., and Semendjajew, K. A. (1962). Hand- 
book of Mathematics. (in German). Teubner, Leip- 
zig, 584 pp. 
Butcher, S. S., and Charlson, R. J. (1972). An Introduc- 
tion to Air Chemistly. Academic Press, New York, 
241 pp. 
Foitzik, L. (1964). Gerlands Beitr. Geophys. 73:199-206. 
Friedlander, S. K. (1977). Smoke, Dust and Haze, Fun- 
damentals of Aerosol Behavior. John Wiley & Sons, 
New York, 317 pp. 
Hatch, T., and Choate, S. P. 1929. J. Franklin Inst. 
2O7:36Y. 
Heintzenberg, J., and M. Baker (1976). Appl. Opt. 
15:1178-1181. 
Hinds, W. C. (1982). Aerosol Technology. John Wiley & 
Sons, New York, 424 pp. 
Strom, J., and Heintzenberg, J. (1993). J. Atmos. Sci., 
in print. 
Whitby, K. T. (1974). Modeling of multimodal aerosol 
distributions. Paper delivered at the GAF meeting, 
Bad Soden, Germany, Oct. 17, 1974. 
Whitby, E., and Whitby, N. L. (1989). DISTFIT Com- 
puter Program. Distributed by TSI, Inc., St. Paul, 
MN. 
Received July 26, 1993; accepted November 3, 1993