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4.2 Rectilinear Particle Motion 103

The diffusion coefficient for an aerosol particle is given by the Stokes-Einstein
equation [12].

kTC c
D p ¼ ð4:29Þ
3pld p
2
The particle diffusion coefficient has a unit of m /s. For large particles when
C c ¼ 1, the particle diffusion coefficient is inversely proportional to the particle
size. For smaller particles when the slip effect becomes significant, D p approaches a
2
value that is inversely proportional to d , which is very similar to the diffusion
p
coefficient for gas molecules. This is because the behavior of finer particles
approaches that of gas molecules.
Example 4.5: Particle diffusion coefficients
Calculate the diffusion coefficients for 100 nm and 10 μm particles at standard
conditions.

Solution
2
2
We know that k ¼ 1:3806 10 23 kg• m /(s K), T = 288 K; λ = 66 nm.
Case I For particles with d p ¼ 100 nm
Kn ¼ 2k=d p ¼ 2k=d e ¼ 2 66=100 ¼ 1:32

1:10
C c ¼ 1 þ Kn 1:257 þ 0:40 exp 0:001\Kn\100
Kn

1:10
C c ¼ 1 þ 1:32 1:257 þ 0:40 exp 2:66
1:32
kTC c
The particle diffusion coefficient D p ¼
3pld p
1:3806 10 23 288 2:66
¼
3p 1:81 10 5 100 10 9
2
¼ 6:21 10 10 m =s
Case II For particles with d p =10 μm, repeat the calculation and we get

C c ¼ 1:0166

kTC c 1:3806 10 23 288 1:0166 17 2
D p ¼ ¼ 5 6 ¼ 2:33 10 m =s
3pld p 3p 1:81 10 10 10