Editing Aerosol (section)

==Size distribution==
{{See also|Particulates#Size, shape, and solubility matter}}
[[File:Synthetic aerosol distribution in number area and volume space.png|thumb|upright=1.3|The same hypothetical log-normal (bi-modal) aerosol distribution plotted, from top to bottom, as a number vs. diameter distribution, a surface area vs. diameter distribution, and a volume vs. diameter distribution. Typical mode names are shown at the top. Each distribution is normalized so that the total area is 1000.|alt=graph showing the size distribution of aerosols over different variables]]
For a monodisperse aerosol, a single number—the particle diameter—suffices to describe the size of the particles. However, more complicated [[particle-size distribution]]s describe the sizes of the particles in a polydisperse aerosol. This distribution defines the relative amounts of particles, sorted according to size.<ref>{{cite journal| last1 = Jillavenkatesa | first1 = A | last2 = Dapkunas | first2 = SJ |last3 = Lin-Sien |first3 = Lum| title = Particle Size Characterization | journal = NIST Special Publication | volume = 960-1 | date = 2001}}</ref> One approach to defining the particle size distribution uses a list of the sizes of every particle in a sample. However, this approach proves tedious to ascertain in aerosols with millions of particles and awkward to use. Another approach splits the size range into intervals and finds the number (or proportion) of particles in each interval. These data can be presented in a [[histogram]] with the area of each bar representing the proportion of particles in that size bin, usually normalised by dividing the number of particles in a bin by the width of the interval so that the area of each bar is proportionate to the number of particles in the size range that it represents.{{sfn|Hinds|1999|pp=75-77}} If the width of the bins [[Limit (mathematics)|tends to zero]], the frequency function is:{{sfn|Hinds|1999|p=79}}

:<math> \mathrm{d}f = f(d_p) \,\mathrm{d}d_p</math>
where
:<math> d_p </math> is the diameter of the particles
:<math> \,\mathrm{d}f </math> is the fraction of particles having diameters between <math>d_p</math> and <math>d_p</math> + <math>\mathrm{d}d_p</math>
:<math>f(d_p)</math> is the frequency function

Therefore, the area under the frequency curve between two sizes a and ''b'' represents the total fraction of the particles in that size range:{{sfn|Hinds|1999|p=79}}
:<math> f_{ab}=\int_a^b f(d_p) \,\mathrm{d}d_p</math>

It can also be formulated in terms of the total number density ''N'':{{sfn|Hidy|1984|p=58}}
:<math> dN = N(d_p) \,\mathrm{d}d_p</math>

Assuming spherical aerosol particles, the aerosol surface area per unit volume (''S'') is given by the second [[Moment (mathematics)|moment]]:{{sfn|Hidy|1984|p=58}}
:<math> S= \pi/2 \int_0^\infty N(d_p)d_p^2 \,\mathrm{d}d_p</math>

And the third moment gives the total volume concentration (''V'') of the particles:{{sfn|Hidy|1984|p=58}}
:<math> V= \pi/6 \int_0^\infty N(d_p)d_p^3 \,\mathrm{d}d_p</math>

The particle size distribution can be approximated. The [[normal distribution]] usually does not suitably describe particle size distributions in aerosols because of the [[skewness]] associated with a [[long tail]] of larger particles. Also for a quantity that varies over a large range, as many aerosol sizes do, the width of the distribution implies negative particles sizes, which is not physically realistic. However, the normal distribution can be suitable for some aerosols, such as test aerosols, certain [[pollen]] grains and [[spore]]s.{{sfn|Hinds|1999|p=90}}

A more widely chosen [[log-normal distribution]] gives the number frequency as:{{sfn|Hinds|1999|p=90}}

:<math> \mathrm{d}f = \frac{1}{d_p \sigma\sqrt{2\pi}} e^{-\frac{(ln(d_p) - \bar{d_p})^2}{2 \sigma^2} }\mathrm{d}d_p</math>

where:

:<math> \sigma</math> is the [[standard deviation]] of the size distribution and
:<math> \bar{d_p}</math> is the [[arithmetic mean]] diameter.

The log-normal distribution has no negative values, can cover a wide range of values, and fits many observed size distributions reasonably well.{{sfn|Hinds|1999|p= 91}}<ref>There is also a practical advantage of modelling the aerosols size distributions with a log-normal distribution, as the n-th moment of a log-normally distributed variable X has a simple analytical expression using the two parameters <math>\sigma</math> and <math>\mu</math> which simplifies the model.</ref>

Other distributions sometimes used to characterise particle size include: the [[Weibull distribution|Rosin-Rammler distribution]], applied to coarsely dispersed dusts and sprays; the Nukiyama–Tanasawa distribution, for sprays of extremely broad size ranges; the [[Power law#Power-law probability distributions|power function distribution]], occasionally applied to atmospheric aerosols; the [[exponential distribution]], applied to powdered materials; and for cloud droplets, the Khrgian–Mazin distribution.{{sfn|Hinds|1999|pp=104-5}}