Molecular Flux

Suppose that the molecules in our gas are equally likely to be moving in any direction, and have a distribution of molecular speeds $F(v)$. (See Section 5.5.9.) In other words, the probability that a given molecule has a speed in the range $v$ to $v+dv$ is $F(v)\,dv$. Let $n$ be the total number of molecules per unit volume. Let us calculate how many molecules per unit area, per second, pass through the $x$-$y$ plane in the direction of increasing $z$. This quantity, ${\mit\Phi}_z$, is termed the molecular flux.

Let $\theta$ and $\phi$ be standard spherical polar angles. (See Section A.23.) We can write the Cartesian components of the velocity of a given molecule, whose molecular speed is $v$, as ${\bf v} = v\,(\sin\theta\,\cos\phi$, $\sin\theta\,\sin\phi$, $\cos\theta)$. If the molecules are equally likely to move in any direction then the number of molecules for which $\theta$ lies between $\theta$ and $\theta+d\theta$, and $\phi$ lies between $\phi$ and $\phi+d\phi$, is proportional to $\sin\theta\,d\theta\,d\phi$ (i.e., to the amount of solid angle contained in this range of angles). Thus, the number of molecules for which $\theta$ lies between $\theta$ and $\theta+d\theta$, and $\phi$ can take any value in the range 0 to $2\pi$, is proportional to $2\pi\,\sin\theta\,d\theta$. Hence, given that there are $4\pi$ steradians in a complete solid angle, the fraction of molecules for which $\theta$ lies between $\theta$ and $\theta+d\theta$ is $g(\theta)\,d\theta$, where

$$g(\theta) = \frac{1}{2}\,\sin\theta.$$ (5.167)

Consider molecules whose speeds lie between $v$ and $v+dv$. The number of such molecules per unit volume is $n\,F(v)\,dv$. The number of such molecules per unit volume whose directions of motion subtend an angle lying between $\theta$ and $\theta+d\theta$ with the $z$-axis is $[n\,F(v)\,dv]\,[g(\theta)\,d\theta]$. All such molecules for which $-v_z < z <0$ cross the $x$-$y$ plane in one second. Thus, the number of such molecules per unit area, per second, that cross the $x$-$y$ plane is

$$d{\mit\Phi}_z = [n\,F(v)\,dv]\,[g(\theta)\,d\theta]\,[v_z] = \left[n\,F(v)\,dv\right]\left[\frac{1}{2}\,\sin\theta\,d\theta\right][v\,\cos\theta].$$ (5.168)

Hence, the net flux of molecules across the $x$-$y$ plane in the direction of increasing $z$ (i.e., with $0\leq \theta\leq \pi/2$) is

$${\mit\Phi}_z =\frac{1}{2}\, n\int_0^{\pi/2}\sin\theta\,\cos\theta\,d\theta\,\int_0^\infty F(v)\,v\,dv,$$ (5.169)

which reduces to

$${\mit\Phi}_z = \frac{1}{4}\,n\,\langle v\rangle,$$ (5.170)

where

$$\langle v\rangle = \int_0^\infty F(v)\,v\,dv.$$ (5.171)

is the mean molecular speed. (See Sections 5.1.6 and 5.5.9.)

For example, if a low-pressure gas is held in a container, the wall of which contains a small hole of area $A$, then the number of escaping molecules per second is

$$\dot{N}=\frac{1}{4}\,n\,\langle v\rangle\,A.$$ (5.172)

This process of molecular escape is known as molecular effusion. (See Section 5.3.13.) It turns out that the previous formula is only accurate if the dimensions of the hole are small compared to the typical distance travelled by a molecule in the gas between collisions (this distance is know as the mean free path; see Section 5.3.8). In the opposite limit, the gas flows through the hole according to the laws of continuum fluid dynamics.