Effusion

Into a vacuum

Effusion from an equilibrated container into outside vacuum can be calculated based on kinetic theory. The number of atomic or molecular collisions with a wall of a container per unit area per unit time (impingement rate) is given by: $${\displaystyle J_{\text{impingement}}=\frac{P}{\sqrt{2\pi m{k}_{\text{B}}T}}.}$$ assuming mean free path is much greater than pinhole diameter and the gas can be treated as an ideal gas.

If a small area ${\displaystyle A}$ on the container is punched to become a small hole, the effusive flow rate will be $${\displaystyle \begin{array}{cc}{Q}_{\text{effusion}}& =J_{\text{impingement}}\times A\\ & =\frac{PA}{\sqrt{2\pi m{k}_{\text{B}}T}}\\ & =\frac{PA{N}_{\text{A}}}{\sqrt{2\pi MRT}}\end{array}}$$ where ${\displaystyle M}$ is the molar mass, ${\displaystyle N_{\text{A}}}$ is the Avogadro constant, and ${\displaystyle R=N_{\text{A}}{k}_{\text{B}}}$ is the molar gas constant.

The average velocity of effused particles is $${\displaystyle \begin{array}{cc}\overline{v_x}& =\overline{v_y}=0\\ \overline{v_z}& =\sqrt{\frac{\pi k_{\text{B}}T}{2m}}.\end{array}}$$

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.

Measures of flow rate

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

${\displaystyle \frac{1}{2}m{v}_{\mathrm{r}\mathrm{m}\mathrm{s}}^2=\frac{3}{2}{k}_{\mathrm{B}}T}$

where ${\displaystyle m}$ is the mass of one molecule, ${\displaystyle v_{\mathrm{r}\mathrm{m}\mathrm{s}}}$ is the root-mean-square speed of the molecules, and ${\displaystyle k_{\mathrm{B}}}$ is the Boltzmann constant. The average molecular speed can be calculated from the Maxwell speed distribution as $v_{\mathrm{a}\mathrm{v}\mathrm{g}}=\sqrt{8/3\pi }{v}_{\mathrm{r}\mathrm{m}\mathrm{s}}\approx 0.921v_{\mathrm{r}\mathrm{m}\mathrm{s}}$ (or, equivalently, $v_{\mathrm{r}\mathrm{m}\mathrm{s}}=\sqrt{3\pi /8}{v}_{\mathrm{a}\mathrm{v}\mathrm{g}}\approx 1.085v_{\mathrm{a}\mathrm{v}\mathrm{g}}$). The rate ${\displaystyle {\mathrm{\Phi }}_N}$ 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

${\displaystyle {\mathrm{\Phi }}_N=\frac{\mathrm{\Delta }PA{N}_A}{\sqrt{2\pi MRT}}.}$

Here ${\displaystyle \mathrm{\Delta }P}$ is the gas pressure difference across the barrier, ${\displaystyle A}$ is the area of the hole, ${\displaystyle N_{\text{A}}}$ 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 ${\displaystyle P_{\mathrm{a}\mathrm{v}\mathrm{g}}}$, the average absolute pressure in the system (i.e. ${\displaystyle \mathrm{\Delta }P\ll P_{\mathrm{a}\mathrm{v}\mathrm{g}}}$), it is possible to express effusion flow as a volumetric flow rate as follows:

${\displaystyle {\mathrm{\Phi }}_V=\frac{\mathrm{\Delta }P{d}^2}{P_{\mathrm{a}\mathrm{v}\mathrm{g}}}\sqrt{\frac{\pi k_{\text{B}}T}{32m}}}$

or

${\displaystyle {\mathrm{\Phi }}_V=\frac{\mathrm{\Delta }P{d}^2}{P_{\mathrm{a}\mathrm{v}\mathrm{g}}}\sqrt{\frac{\pi RT}{32M}}}$

where ${\displaystyle {\mathrm{\Phi }}_V}$ is the volumetric flow rate of the gas, ${\displaystyle P_{\mathrm{a}\mathrm{v}\mathrm{g}}}$ is the average pressure on either side of the orifice, and ${\displaystyle d}$ is the hole diameter.

Effect of molecular weight

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.

Graham's law

Scottish chemist Thomas Graham (1805–1869) found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles.

${\displaystyle \frac{{\text{Rate of effusion of gas}}_1}{{\text{Rate of effusion of gas}}_2}=\sqrt{\frac{M_2}{M_1}}}$

where ${\displaystyle M_1}$ and ${\displaystyle M_2}$ represent the molar masses of the gases. This equation is known as Graham's law of effusion.

The effusion rate for a gas depends directly on the average velocity of its particles. Thus, the faster the gas particles are moving, the more likely they are to pass through the effusion orifice.

Knudsen cell

The Knudsen 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. The heat of sublimation can also be determined by measuring the vapor pressure as a function of temperature, using the Clausius–Clapeyron relation.