Diffusion

Consider an ideal gas of uniform temperature, $T$, that has a number density gradient along the $z$-axis, such that

$$n(z)= n_0 + \frac{\partial n}{\partial z}\,dz.$$ (5.230)

Let $F(v)$ be the distribution of molecular speeds. Repeating the analysis of Section 5.3.2, the number of molecules per unit area, per second, whose speeds lie between $v$ and $v+dv$, and whose directions of motion subtend an angle lying between $\theta$ and $\theta+d\theta$ with the $z$-axis, that cross the $x$-$y$ plane is

$$dJ_z = [n'\,F(v)\,dv]\,[g(\theta)\,d\theta]\,[v_z] = \left[n'\,F(v)\,dv\right]\left[\frac{1}{2}\,\sin\theta\,d\theta\right][v\,\cos\theta],$$ (5.231)

where $n'$ is the number density where the molecules last made a collision. [See Equation (5.167).] On average, the molecules move a distance $l$ (i.e., the mean free path) between collisions. Hence, $dz =- l\,\cos\theta$, and

$$n' = n_0 - \frac{\partial n}{\partial z}\,l\,\cos\theta.$$ (5.232)

Thus, the net flux of molecules across the $x$-$y$ plane is

$$J_z =\frac{1}{2}\int_0^\pi\left[n_0-\frac{\partial n}{\partial z}\,l\,\cos\theta\right]\cos\theta\,\sin\theta\,d\theta \int_0^\infty F(v)\,dv,$$ (5.233)

which gives

$$J_z = -D\,\frac{\partial n}{\partial z},$$ (5.234)

where

$$D = \frac{1}{3}\,l\,\langle v\rangle.$$ (5.235)

Here, $\langle v\rangle$ is the mean molecular speed. (See Section 5.5.9.) Thus, we conclude that the flux of molecules in the $z$-direction is proportional to minus the local number density gradient along the $z$-axis. This result is known as Fick's law, after Adolf Fick who discovered in experimentally in 1855.

Consider a slab of gas lying between $z$ and $z+dz$. The rate of change of the number of molecules contained in the slab is the difference between the flux of molecules into the slab and the flux of molecules out of the slab. In other words,

$$\frac{\partial (n\,A\,dz)}{\partial t}= \left[J_z(z,t)- J_z(z+dz,t)\right]A,$$ (5.236)

where $A$ is the cross-sectional area of the slab. The previous equation implies that

$$\frac{\partial n}{\partial t} = - \frac{\partial J_z}{\partial z}.$$ (5.237)

However, making use of Fick's law, (5.234), we obtain

$$\frac{\partial n}{\partial t} = D\,\frac{\partial^2 n}{\partial z^2}.$$ (5.238)

The previous equation is known as the diffusion equation, and the constant $D$ is known as the diffusivity.

It can be seen, by inspection, that one solution of the diffusion equation is

$$n(z,t) = n_0 + \frac{\delta n_0}{\sqrt{4\pi\,D\,t}}\,\exp\left(-\frac{z^2}{4\,D\,t}\right),$$ (5.239)

where $n_0$ and $\delta n_0$ are arbitrary constants. Note that at $t=0$,

$$n(z,0)= n_0+\delta n_0\,\delta(z),$$ (5.240)

where $\delta(z)$ is a delta function. (See Section 2.1.6.) Moreover, at large times,

$$n(z,t \rightarrow \infty) = n_0.$$ (5.241)

Thus, our solution describes an initially localized Gaussian (see Section 5.1.7) density perturbation that gradually spreads out, and eventually disperses entirely. It is easily demonstrated that the width (i.e., standard deviation) of the density perturbation, $\sigma_z$, grows in time as

$$\sigma_z= \sqrt{2\,D\,t}.$$ (5.242)

On the other hand, the maximum height of the perturbation decays in time as

$$h_z = \frac{\delta n_0}{\sqrt{4\pi\,D\,t}}.$$ (5.243)

Moreover, the area under the perturbation remains fixed as it evolves in time, which implies that the number of molecules associated with the density perturbation also remains fixed, as has to be the case (because we have not discussed any processes that create or destroy molecules). It is clear, from Sections 5.1.5 and 5.1.7, that the spreading of the density perturbation is due to a random walk of the excess molecules along the $z$-axis, under the action of molecular collisions.

Let us estimate the particle diffusivity in air at standard temperature ( $T=15^\circ$ C) and pressure ( $p=10^5\,{\rm N\,m}^{-2}$). The mean thermal speed of molecules of mass $m$ in an ideal gas of temperature $T$ is

$$\langle v\rangle= \sqrt{\frac{8}{\pi}\,\frac{k_B\,T}{m}}.$$ (5.244)

(See Section 5.5.9.) Hence, it follows from Equations (5.219), (5.220), and (5.235) that

$$D = \frac{2}{3\,\pi^{3/2}}\,\frac{1}{R^{\,2}\,p}\sqrt{\frac{(k_B\,T)^3}{m}},$$ (5.245)

where $R$ is the molecular diameter. Thus, the diffusivity scales as $1/p$ at constant temperature, as $T^{3/2}$ at constant pressure, and as $T^{1/2}$ at constant volume. Given that $m=29\,m_p$ for air, where $m_p$ is the proton mass, and $R=2\times 10^{-10}\,{\rm m}$, we deduce that

$$D = 3\times 10^{-5}\,{\rm m^2\,s^{-1}}.$$ (5.246)

This is a very small diffusivity. According to Equation (5.242), it takes about 4.6 hours for a molecule to diffuse a distance of 1 meter in air.