# Two versions of Diffusion coefficient

I found two versions of the Diffusion coefficient, first:

$$D=\frac{\pi \lambda }{8}\overline{c}$$

Where $$\overline{c}$$ ist the particles mean thermal velocity and $$\lambda$$ the particles mean free path. (Found in W. C. Hinds, Aerosol Technology. Wiley Interscience (1999). S.156)

and second the version from my lecture (also found on wikipedia):

$$D=\frac{ \lambda}{3} \langle v\rangle$$

Where $$\langle v\rangle$$ is the particles mean thermal verlocity.

Hinds gives two more equations (S.154):

$$c_{\text{rms}}=\sqrt{\frac{3kT}{m}}$$

$$\overline{c}=\sqrt{\frac{8kT}{\pi m}}$$

With $$c_{\text{rms}}$$ being the root mean square velocity of a particle, $$k$$ being the Boltzmann-constant, $$T$$ being the temperature and $$m$$ the particles mass. My first thought was that maybe there is a typo in the book or the lecture notes so I calculated

$$\frac{c_{\text{rms}}}{\overline{c}}=\sqrt{\frac{3\pi}{8}}$$

which led to

$$\overline{c}=\sqrt{\frac{8}{3\pi}}c_{\text{rms}}$$

so I replaced $$\overline{c}$$ in

$$D=\frac{\pi \lambda }{8}\overline{c}$$

just to come to

$$D=\frac{\pi \lambda }{8}\sqrt{\frac{8}{3\pi}}c_{\text{rms}}$$

Which (set $$\langle v\rangle=c_{\text{rms}}$$, due to the potential typo) because of the square root isn't

$$D=\frac{ \lambda}{3} \langle v\rangle$$

Where did I make a mistake?

In the two expressions you have, you see they'd cancel out were you to get rid of the square root. So, I believe $$\lambda$$ to be defined differently in your source material: In the one in Hinds, $$\lambda = \bar{c}\tau$$, whereas on the page on Wikipedia you found, I take it $$\lambda = c_\text{rms}\tau$$.