Effusion

Effusion from an equilibrated container into outside vacuum can be calculated based on kinetic theory.[2] The number of atomic or molecular collisions with a wall of a container per unit area per unit time (impingement rate) is given by:

${\displaystyle {\begin{aligned}J_{\text{impingement}}&={\frac {P}{\sqrt {2\pi mk_{B}T}}}.\end{aligned}}}$

If a small area ${\displaystyle A}$ on the container is punched to become a small hole, the effusive flow rate will be

${\displaystyle {\begin{aligned}Q_{\text{effusion}}&=J_{\text{impingement}}\times A\\&={\frac {PA}{\sqrt {2\pi mk_{B}T}}}\\&={\frac {PAN_{A}}{\sqrt {2\pi MRT}}}\propto {\frac {1}{\sqrt {M}}}({\text{Graham's Law}}),\end{aligned}}}$

where ${\displaystyle M}$ is the molar mass. Note that the outside vacuum has zero pressure, so the pressure difference between two sides of the hole is simply ${\displaystyle P}$.

The average velocity of effused particles is

${\displaystyle {\begin{aligned}{\overline {v_{x}}}&={\overline {v_{y}}}=0\\{\overline {v_{z}}}&={\sqrt {\frac {\pi k_{B}T}{2m}}}.\end{aligned}}}$

Combined with the effusive flow rate, the recoil/thrust force on the system itself is

${\displaystyle F=m{\overline {v_{z}}}{\times }Q_{\text{effusion}}={\frac {PA}{2}}.}$

An example is the recoil force on a balloon with a small hole flying in vacuum.

According to the kinetic theory of gases, the kinetic energy for a gas at a temperature ${\displaystyle T}$ is

${\displaystyle {\frac {1}{2}}mv_{\rm {rms}}^{2}={\frac {3}{2}}k_{\rm {B}}T}$

where ${\displaystyle m}$ is the mass of one molecule, ${\displaystyle v_{\rm {rms}}}$ is the root-mean-square speed of the molecules, and ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant. The average molecular speed can be calculated from the Maxwell speed distribution as ${\displaystyle v_{\rm {avg}}={\sqrt {8/3\pi }}\ v_{\rm {rms}}\approx 0.921\ v_{\rm {rms}}}$(or, equivalently, ${\displaystyle v_{\rm {rms}}={\sqrt {3\pi /8}}\ v_{\rm {avg}}\approx 1.085\ v_{\rm {avg}}}$). The rate ${\displaystyle \Phi _{N}}$ at which a gas of molar mass ${\displaystyle M}$ effuses (typically expressed as the number of molecules passing through the hole per second) is then[3]

${\displaystyle \Phi _{N}={\frac {\Delta PAN_{A}}{\sqrt {2\pi MRT}}}.}$

Here ${\displaystyle \Delta P}$ is the gas pressure difference across the barrier, ${\displaystyle A}$ is the area of the hole, ${\displaystyle N_{A}}$ is the Avogadro constant, ${\displaystyle R}$ is the gas constant and ${\displaystyle T}$ is the absolute temperature. Assuming the pressure difference between the two sides of the barrier is much smaller than ${\displaystyle P_{\rm {avg}}}$, the average absolute pressure in the system (i.e. ${\displaystyle \Delta P\ll P_{\rm {avg}}}$), it is possible to express effusion flow as a volumetric flow rate as follows:

${\displaystyle \Phi _{V}={\frac {\Delta Pd^{2}}{P_{\rm {avg}}}}{\sqrt {\frac {\pi k_{B}T}{32m}}}}$

or

${\displaystyle \Phi _{V}={\frac {\Delta Pd^{2}}{P_{\rm {avg}}}}{\sqrt {\frac {\pi RT}{32M}}}}$

where ${\displaystyle \Phi _{V}}$ is the volumetric flow rate of the gas, ${\displaystyle P_{\rm {avg}}}$ is the average pressure on either side of the orifice, and ${\displaystyle d}$ is the hole diameter.

At constant pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that the number of lighter molecules passing through the hole per unit time is greater. This is why a balloon filled with low molecular weight helium (M = 4) deflates faster than an equivalent balloon full of higher molecular weight oxygen (M = 32). However the total mass of the escaping molecules is directly proportional to the square root of the molecular weight and is less for lighter molecules.

The Knudsen effusion cell is used to measure the vapor pressures of a solid with very low vapor pressure. Such a solid forms a vapor at low pressure by sublimation. The vapor slowly effuses through a pinhole, and the loss of mass is proportional to the vapor pressure and can be used to determine this pressure.[3] The heat of sublimation can also be determined by measuring the vapor pressure as a function of temperature, using the Clausius–Clapeyron relation.[5]